Computer Modeling for Injection Molding: Simulation, Optimization, and Control 9780470602997, 9781118444887

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COMPUTER MODELING FOR INJECTION MOLDING

COMPUTER MODELING FOR INJECTION MOLDING Simulation, Optimization, and Control

Edited by HUAMIN ZHOU Huazhong University of Science and Technology Wuhan, Hubei, China

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright  2013 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750–8400, fax (978) 750–4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748–6011, fax (201) 748–6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762–2974, outside the United States at (317) 572–3993 or fax (317) 572–4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Computer modeling for injection molding : simulation, optimization, and control / edited by Huamin Zhou. p. cm. Includes bibliographical references and index. ISBN 978-0-470-60299-7 (cloth) 1. Injection molding of plastics–Computer simulation. 2. Injection molding of plastics– Computer-aided design. 3. Plastics–Molding. I. Zhou, Huamin. TP1150.C665 2013 668.4’1200113–dc23 2012016535 Printed in Singapore 10 9 8 7 6 5 4 3 2 1

CONTENTS

PREFACE

xiii

CONTRIBUTORS

PART I 1

BACKGROUND

Introduction

xv

1 3

Huamin Zhou

1.1

1.2

1.3

1.4

2

Introduction of Injection Molding, 3 1.1.1 The injection molding process, 3 1.1.2 Importance of molding quality, 3 Factors Influencing Quality, 5 1.2.1 Molding polymer, 5 1.2.2 Plastic product, 6 1.2.3 Injection mold, 7 1.2.4 Process conditions, 7 1.2.5 Injection molding machine, 8 1.2.6 Interrelationship, 9 Computer Modeling, 10 1.3.1 Review of computer applications, 11 1.3.2 Computer modeling in quality enhancement, 11 1.3.3 Numerical simulation, 13 1.3.4 Optimization, 14 1.3.5 Process control, 15 Objective of This Book, 17 References, 18

Background

25

Huamin Zhou

2.1

Molding Materials, 25 2.1.1 Rheology, 25 2.1.2 Thermal properties, 27 v

vi

CONTENTS

2.2

2.3

2.4

2.5

PART II

2.1.3 PVT behavior, 29 2.1.4 Morphology, 30 Product Design, 31 2.2.1 Wall thickness, 31 2.2.2 Draft, 32 2.2.3 Parting plane, 32 2.2.4 Sharp corners, 33 2.2.5 Undercuts, 33 2.2.6 Bosses and cored holes, 33 2.2.7 Ribs, 33 Mold Design, 34 2.3.1 Mold cavity, 34 2.3.2 Parting plane, 35 2.3.3 Runner system, 36 2.3.4 Cooling system, 37 Molding Process, 37 2.4.1 The molding cycle, 38 2.4.2 Flow in the cavity, 40 2.4.3 Orientation, 41 2.4.4 Residual stresses, shrinkage, and warpage, 41 Process Control, 43 2.5.1 Characteristics of injection molding as a batch process, 45 2.5.2 Typical control problems in injection molding, 45 References, 47

SIMULATION

3 Mathematical Models for the Filling and Packing Simulation Huamin Zhou, Zixiang Hu, and Dequn Li

3.1

3.2

3.3 3.4

3.5

3.6

Material Constitutive Relationships and Viscosity Models, 51 3.1.1 Newtonian fluids, 51 3.1.2 Generalized Newtonian fluids, 52 3.1.3 Viscoelastic fluids, 54 Thermodynamic Relationships, 56 3.2.1 Constant specific volume, 57 3.2.2 Spencer–Gilmore model, 57 3.2.3 Tait model, 57 Thermal Properties Model, 58 Governing Equations for Fluid Flow, 59 3.4.1 Mass conservation equation, 59 3.4.2 Momentum conservation equation, 60 3.4.3 Energy conservation equation, 62 3.4.4 General transport equation, 64 Boundary Conditions, 65 3.5.1 Pressure boundary conditions, 66 3.5.2 Temperature boundary conditions, 66 3.5.3 Slip boundary condition, 66 Model Simplifications, 67 3.6.1 Hele–shaw model, 67 3.6.2 Governing equations for the filling phase, 68 3.6.3 Governing equations for the packing phase, 69 References, 69

49 51

CONTENTS

4

Numerical Implementation for the Filling and Packing Simulation

71

Huamin Zhou, Zixiang Hu, Yun Zhang, and Dequn Li

4.1

4.2

4.3

5

Numerical Methods, 71 4.1.1 Finite difference method, 72 4.1.2 Finite volume method, 76 4.1.3 Finite element method, 85 4.1.4 Mesh-less methods, 95 Tracking of Moving Melt Fronts, 101 4.2.1 Overview, 101 4.2.2 FAN, 104 4.2.3 VOF, 105 4.2.4 Level set methods, 110 Methods for Solving Algebraic Equations, 113 4.3.1 Overview, 113 4.3.2 Direct methods, 114 4.3.3 Iterative methods, 116 4.3.4 Parallel computing, 121 References, 125

Cooling Simulation

129

Yun Zhang and Huamin Zhou

5.1 5.2

5.3

5.4

5.5

6

Introduction, 129 Modeling, 131 5.2.1 Cycle-averaged temperature field, 131 5.2.2 Cycle-averaged boundary conditions, 132 5.2.3 Coupling calculation procedure, 134 5.2.4 Calculating cooling time, 135 Numerical Implementation Based on Boundary Element Method, 136 5.3.1 Boundary integral equation, 136 5.3.2 Numerical implementation, 138 Acceleration Method, 143 5.4.1 Analysis of the coefficient matrix, 143 5.4.2 The approximated sparsification method, 144 5.4.3 The splitting method, 145 5.4.4 The fast multipole boundary element method, 146 5.4.5 Results and discussion, 148 Simulation for Transient Mold Temperature Field, 150 References, 154

Residual Stress and Warpage Simulation Fen Liu, Lin Deng, and Huamin Zhou

6.1

6.2

Residual Stress Analysis, 157 6.1.1 Development of residual stress, 157 6.1.2 Model prediction, 159 6.1.3 Numerical simulation, 163 6.1.4 Case study, 165 Warpage Simulation, 170 6.2.1 Development of warpage, 172 6.2.2 Model prediction, 173 6.2.3 Implementation with surface model, 182 6.2.4 Case study, 186 References, 190

157

vii

viii

CONTENTS

7 Microstructure and Morphology Simulation

195

Huamin Zhou, Fen Liu, and Peng Zhao

7.1

7.2

7.3

7.4

7.5

7.6

7.7

Types of Polymeric Systems, 195 7.1.1 Thermoplastics and thermosets, 195 7.1.2 Amorphous and crystalline polymers, 196 7.1.3 Blends and composites, 196 Crystallization, 196 7.2.1 Fundamentals, 196 7.2.2 Modeling, 197 7.2.3 Case study, 202 Phase Morphological Evolution in Polymer Blends, 203 7.3.1 Fundamentals, 205 7.3.2 Modeling, 207 7.3.3 Case study, 213 Orientation, 214 7.4.1 Molecular orientation, 215 7.4.2 Fiber orientation, 216 7.4.3 Case study, 218 Numerical Implementation, 220 7.5.1 Coupled procedure, 220 7.5.2 Stable scheme of the FEM, 221 7.5.3 Formulations of the velocity and pressure equations, 222 7.5.4 Formulations of temperature and microstructure equations, 223 Microstructure-Property Relationships, 224 7.6.1 Effect of crystallinity on property, 224 7.6.2 Effect of phase morphology on property, 225 7.6.3 Effect of orientation on property, 226 Multiscale Modeling and Simulation, 228 7.7.1 Molecular scale methods, 229 7.7.2 Microscale methods, 229 7.7.3 Meso/macroscale methods, 230 7.7.4 Multiscale strategies, 231 References, 231

8 Development and Application of Simulation Software Zhigao Huang, Zixiang Hu, and Huamin Zhou

8.1

8.2 8.3

8.4

Development History of Injection Molding Simulation Models, 237 8.1.1 One-dimensional models, 238 8.1.2 2.5D models, 238 8.1.3 Three-dimensional models, 240 Development History of Injection Molding Simulation Software, 240 The Process of Performing Simulation Software, 243 8.3.1 Geometry modeling, 244 8.3.2 Selection of material, 245 8.3.3 Setting processing parameters, 246 Application of Simulation Results, 246 8.4.1 Dynamic display of melt flow front, 246 8.4.2 Cavity pressure, 246 8.4.3 Pressure at injection location, 247 8.4.4 Polymer temperature, 247 8.4.5 Shear rate, 247 8.4.6 Shear stress, 247

237

CONTENTS

8.4.7 Weld lines, 247 8.4.8 Air traps, 248 8.4.9 Shrinkage index, 250 8.4.10 Cooling evaluation, 250 8.4.11 Warpage prediction, 251 References, 251

PART III OPTIMIZATION 9

Noniterative Optimization Methods

255 257

Peng Zhao, Yuehua Gao, Huamin Zhou, and Lih-Sheng Turng

9.1

9.2

9.3

9.4

9.5

9.6

10

Taguchi Method, 258 9.1.1 Orthogonal arrays, 258 9.1.2 Analysis of the S/N ratio, 259 9.1.3 Analysis of variance, 259 9.1.4 Taguchi technology, 259 Gray Relational Analysis, 260 9.2.1 Data preprocessing, 260 9.2.2 Gray relational coefficient and gray relational grade, 260 Expert Systems, 261 9.3.1 Knowledge base, 262 9.3.2 Inference engine, 263 Case-Based Reasoning, 266 9.4.1 Case representation, 266 9.4.2 Case retrieval, 267 9.4.3 Case adaptation, 267 Fuzzy Systems, 268 9.5.1 Fuzzy theory, 269 9.5.2 Fuzzy inference, 272 9.5.3 A fuzzy system for part defect correction, 274 Injection Molding Applications, 274 9.6.1 Review of noniteration optimization methods, 274 9.6.2 Application of the taguchi method, 276 9.6.3 Application of case-based reasoning and fuzzy systems, 278 References, 281

Intelligent Optimization Algorithms Yuehua Gao, Peng Zhao, Lih-Sheng Turng, and Huamin Zhou

10.1

10.2

10.3

10.4 10.5

Genetic Algorithms, 283 10.1.1 Chromosome representation, 284 10.1.2 Selection, 284 10.1.3 Crossover and mutation operations, 284 10.1.4 Fitness function and termination, 285 Simulated Annealing Algorithms, 285 10.2.1 The fundamentals of the simulated annealing algorithm, 286 10.2.2 Optimum design algorithm for simulated annealing, 287 Particle Swarm Algorithms, 287 10.3.1 General procedures, 287 10.3.2 Determination of parameters, 288 Ant Colony Algorithms, 289 Hill Climbing Algorithms, 290

283

ix

x

CONTENTS

10.5.1 General procedure, 290 10.5.2 Flow path generation with hill climbing algorithms, 290 References, 291 11

Optimization Methods Based on Surrogate Models

293

Yuehua Gao, Lih-Sheng Turng, Peng Zhao, and Huamin Zhou

11.1

11.2

11.3

11.4

11.5 11.6

PART IV 12

Response Surface Method, 294 11.1.1 RSM theory, 294 11.1.2 Modeling error estimation, 295 11.1.3 Optimization process using RSM, 295 Artificial Neural Network, 296 11.2.1 Back propagation network, 296 11.2.2 BPN training process, 298 11.2.3 Optimization process based on ANN, 298 Support Vector Regression, 298 11.3.1 SVR theory, 299 11.3.2 Lagrange multipliers, 300 11.3.3 Kernel function, 300 11.3.4 Selection of SVR parameters, 301 Kriging Model, 301 11.4.1 Kriging model theory, 301 11.4.2 The correlation function, 302 11.4.3 Optimization design based on the kriging surrogate model, 302 Gaussian Process, 304 Injection Molding Applications of Optimization Methods Based on Surrogate Models, 305 11.6.1 Application of the ANN model, 305 11.6.2 Application of the SVR model, 307 11.6.3 Application of the kriging model, 309 References, 312

PROCESS CONTROL

Feedback Control Yi Yang and Furong Gao

12.1 12.2 12.3

12.4

12.5

12.6

Traditional Feedback Control, 315 Adaptive Control Strategy, 316 Model Predictive Control Strategy, 318 12.3.1 GPC design for barrel temperature control, 320 12.3.2 GPC controller parameter tuning, 321 12.3.3 Experimental test results, 322 Optimal Control Strategy, 322 12.4.1 TOC for barrel temperature start-up control, 323 12.4.2 Simulation results, 324 12.4.3 Experimental test results, 329 Intelligent Control Strategy, 329 12.5.1 Fuzzy injection velocity controller, 330 12.5.2 Fuzzy feed forward controller, 333 12.5.3 Test with different conditions, 333 Summary of Advanced Feedback Control, 335 References, 337

313 315

CONTENTS

13

Learning Control

339

Yi Yang and Furong Gao

13.1 13.2 13.3

14

Learning Control, 339 13.1.1 Learning control for injection velocity profiling, 340 Two-Dimensional (2D) Control, 345 13.2.1 2D control of packing pressure, 346 Conclusions, 350 References, 352

Multivariate Statistical Process Control

355

Yuan Yao and Furong Gao

14.1 14.2

14.3 14.4

14.5

15

Statistical Process Control, 355 Multivariate Statistical Process Control, 356 14.2.1 Principal component analysis, 356 14.2.2 PCA-based process monitoring and fault diagnosis, 357 14.2.3 Normalization, 358 MSPC for Batch Processes, 358 MSPC for Injection Molding Process, 359 14.4.1 Phase-based sub-PCA, 360 14.4.2 Sub-PCA for batch processes with uneven operation durations, 361 14.4.3 Sub-PCA with limited reference data, 363 14.4.4 Applications, 365 Conclusions, 373 References, 373

Direct Quality Control

377

Yi Yang and Furong Gao

15.1 15.2

15.3

15.4 INDEX

Review of Product Weight Control, 377 Methods, 378 15.2.1 Weight prediction using PCR model, 378 15.2.2 Overall weight control scheme and feedback adjustment, 379 Experimental Results and Discussion, 380 15.3.1 Factor screening experiment, 380 15.3.2 PCR modeling of product weight, 382 15.3.3 Closed-loop weight control based on PCR model, 387 Conclusions, 389 References, 389 391

xi

PREFACE

When Wiley approached me to write a book on injection molding, I indicated that I would be interested in writing a book on the basics and applications of computer models, tools, and simulations in injection molding. This idea was embraced by Wiley, and the result is Computer Modeling for Injection Molding: Simulation, Optimization, and Control . No one doubts that injection molding is the most important process used in the manufacture of plastic products, which has consumed about 32 wt% of all plastics. Typical injection-molded products can be found everywhere, including automotive parts, household articles, and consumer electronics goods. Owing to growing plastics applications, increasing customer demand, and rapid growth of the global marketplace, quality requirements of injection-molded components have become more stringent, forcing companies in the constant struggle for just-in-time production with a zerofault quota. Effective quality assurance is therefore very necessary in enhancing the efficiency and competitiveness of the industry. Computer application has played a crucial role in the quality control of injection molding. In practice, a high quality component can be obtained only when various factors regarding the part and mold designs as well as the material selection and process setup have been considered. The quality of the injection-molded product is thus inherently difficult to predict and/or control without employing sophisticated computer simulation/optimization software during the design stage and/or frequent monitoring and intervention during the production stage. Instead of the past costly trial-and-error manufacture process, prediction and optimization of the product quality at the lowest cost has now become possible. Now it is unquestionable that a proper use of computer application can sharpen

a company’s completive edge in various aspects such as product design, process simulation, monitoring, control, and optimization. Computer modeling for injection molding has been an active research area for many years. In my opinion, computer models in injection molding can be generally organized into three categories, namely, simulation, optimization, and process control. Numerical simulation describes the physical process of injection molding directly, which is developed based on the first principle, involving the use of computer-aided engineering (CAE) software or mathematical models, whereas optimization employs various artificial intelligence-based models that should use expert knowledge, cases, empirical models, as well as simulation results, as their reasoning basis. These two approaches aim at establishing a reasonably accurate mapping between the influencing factors and part quality. On the other hand, process control of the injection molding machine attempts to repeat the molding process consistently with high accuracy, in order to ensure the repeatability and reliability of the product quality. Although there have been some introductory books on computer modeling of injection molding, none of them has involved all the above three essential ingredients in improving the product quality. As a result, the major problem for students and researchers who are desirous of an extensive knowledge in injection molding is that applications of the latest computer technology in quality improvement are scattered about, and rarely introduced comprehensively or systematically in postgraduate-level texts, forcing the students and researchers to wade through stacks of published papers looking for useful information. This book is intended to serve as a systematic and comprehensive introductory textbook on the computer xiii

xiv

PREFACE

modeling for injection molding, with important expansions into the successful application of the latest computer technology. It is based on the constant efforts of authors and colleagues in this area over the last few years, and provides what we have determined after years of working in this field to improve the product quality through computer modeling in simulation, optimization, and control. Students and researchers new to the field can get started with the basic information provided, and also, scientists and people involved in the polymer industry, institutes, and institutions seeking new ways to gain a competitive edge can work closely with the latest information provided in the book. Actually, the reader will obtain a comprehensive understanding and a lot of practical knowledge about how the latest computer technology can benefit the injection molding industry. I cannot acknowledge everyone who has helped in one way or another in the preparation of this manuscript. Above all, I am particularly grateful to have worked with an excellent group of contributors, especially Professors Lih-Sheng Turng and Furong Gao, and I thank them for

their efforts, patience, understanding, and support, and for all that they have taught me. Thanks are extended to the State Key Laboratory of Materials Processing and Die & Mould Technology at the Huazhong University of Science and Technology, and the National Natural Science Foundation Council of China for their continued support for the research, which served as the foundation for this book. I am indebted to Professor Dequn Li, for his guidance and suggestions, and Professors Jianjun Li and Xiaolin Xie, for their valuable advice. I thank all the graduate students of my research group who proofread, solved problems, and drew some of the figures. Mrs. Jing Zhao is acknowledged for her language polishing. Finally, I am very grateful to Mr. Jonathan T. Rose, Editor at John Wiley & Sons, Inc., who has always been there to help and support throughout the development of this book. Huamin Zhou Wuhan, Hubei, China Autumn 2011

CONTRIBUTORS

Lin Deng, State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

Fen Liu, Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong, China

Furong Gao, Department of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

Lih-Sheng Turng, Polymer Engineering Center, Department of Mechanical Engineering, University of Wisconsin–Madison, Madison, Wisconsin, USA

Yuehua Gao, School of Traffic & Transportation Engineering, Dalian Jiaotong University, Dalian, Liaoning, China

Yi Yang, Department of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang, China

Zixiang Hu, State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China Zhigao Huang, State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China Dequn Li, State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

Yuan Yao, Department of Chemical Engineering, National Tsing Hua University, Hsinchu, Taiwan Yun Zhang, State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China Peng Zhao, State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, Zhejiang, China Huamin Zhou, State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

xv

temp. (ºC)

temp. (ºC) 98.47

92.48 84.05

89.29

75.61

80.10

67.18

70.92

58.74

61.74

50.31

52.55

41.87

43.37

33.44

34.18 25.00

25.00

(a)

(b)

temp. (ºC) temp. (ºC)

98.32

99.19

89.15

89.92

79.99

80.64

70.82

71.37

61.66

62.10

52.49

52.82

43.33

43.55

34.16

34.27

25.00

25.00

(c)

(d)

FIGURE 5.22 The computed steady temperature field of case 4 by (a) full method, (b) direct rounding method, (c) combination method, and (d) splitting method.35

Computer Modeling for Injection Molding: Simulation, Optimization, and Control, First Edition. Edited by Huamin Zhou.  2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc.

(a)

(b)

(c)

FIGURE 6.32 Simulation results of warpage by (a) Moldflow fusion model, (b) Moldflow 3D model, and (c) present surface mesh model (OPT + RDKTM element).

(a)

(b)

FIGURE 8.9

Runner balancing for a family mold: (a) unbalanced design; (b) balanced design.

Pressure (MPa) 101.40 88.72 76.05 63.38 50.70 38.03 25.35 12.68 0.00

FIGURE 8.10

The distribution of pressure in the cavity when the filling ends.

FIGURE 8.12

The distribution of polymer temperature when the filling ends.

Shear rate (1/s) 26425.2 23122.0 19818.9 16515.8 13212.6 9909.47 6606.33 3303.19 0.04

FIGURE 8.13

The distribution of shear rate in the cavity when the filling ends.

Shear stress (MPa) 0.33 0.29 0.25 0.21 0.17 0.13 0.08 0.04 0.00

FIGURE 8.14

The distribution of shear stress in the cavity when the filling ends.

(a)

(b)

FIGURE 8.15 Modification of the design for elimination of the weld lines at the main surface: (a) original design with four gates; (b) modified design with three gates.

(a)

(b)

(c)

(d)

FIGURE 8.16 Modification of the design for elimination of the air traps: (a) the original design in which air trap appeared in the middle of the part; (b) the molded part with a hole according to the original design; (c) the modified design in which the thickness has been modified and the air trap has been eliminated; (d) the final molded part according to the modified design.

Shrinkage index 0.16 0.14 0.12 0.10 0.08 0.07 0.05 0.03 0.01

FIGURE 8.17

The shrinkage index distribution of the part.

Steady temperature (°C) 103.35 93.58 83.81 74.04 64.27 54.49 44.72 34.95 25.18

FIGURE 8.18

The temperature distribution in the cavity.

Cooling time (s) 39.04 34.27 29.50 24.73 19.96 15.19 10.42 5.65 0.88

FIGURE 8.19

The predication of cooling time.

Warpage (mm) 10.93 9.59 8.25 6.91 5.57 4.23 2.89 Z

1.55 0.21

FIGURE 8.20

The predication of warpage in the molded part.

Deflection, all effects: Z Componet Scale factor = 1.000 [mm] 0.0541 0.0271 0.0001 –0.0269 –0.0539 Z Scale (90 mm)

FIGURE 11.3

Y X

–39 –26 –30

Warpage of the cover after optimization.25

Y X

PART I BACKGROUND

1 INTRODUCTION Huamin Zhou State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

This book is primarily concerned with the computer modeling technology in the quality enhancement of polymer injection molding. This chapter outlines the injection molding process, factors that influence the quality of injection-molded products, and computer applications in injection molding. 1.1

INTRODUCTION OF INJECTION MOLDING

The past century has witnessed the rapid expansion of polymers and plastics (the term plastics describes the compound of a polymer with one or more additives) and their incursion into all markets. Although just over a century old, relatively new when compared to other materials, plastics are now among the most widely used materials, surpassing world’s consumption of steel, aluminum, rubber, copper, and zinc by weight (and volume, of course), as shown in Figure 1.1.1 Plastic materials and products cover the entire spectrum of the world economy in a position to benefit by a turnaround in any one of a number of areas: packaging, appliance, transportation, housing, automotive, and many other industries. Injection molding is regarded as the most important process used to manufacture plastic products. Today, more than one third of all thermoplastic materials are injection molded and more than half of all polymer processing equipment is for injection molding.2 1.1.1

The Injection Molding Process

Injection molding is a repetitive process in which melted (plasticized) polymer is injected (forced) into a mold cavity

or cavities, packed under pressure, and cooled until it has solidified enough. As a result, it duplicates the cavity of the mold (Fig. 1.2). Generally speaking, the mold consists of a single cavity or a series of similar or dissimilar cavities, connected with each other to flow channels or runners that direct the flow of the melt to the individual cavities. During this process, there are three basic operations: (i) heating the polymer in the injection or plasticizing unit so that it will flow under pressure; (ii) making the polymer melt to fulfill and solidify in the mold; and (iii) opening the mold to eject the molded product. The injection molding process is of great significance as it can produce finished, multifunctional, or complex molded parts accurately and repeatedly in a single, highly automated operation. It permits mass manufacture of a great variety of shapes, from simple to intricate three-dimensional ones, and from extremely small to large ones. When required, these products can be molded to extremely tight tolerances, very thin, and in weights down to milligrams. Typical injection moldings (molded products) can be found everywhere in daily life. Examples include automotive parts, household articles, consumer electronics components, and toys. 1.1.2

Importance of Molding Quality

In plastic industry, for years the so-called product innovation was the only rich source of new developments, such as reducing the number of molded components by making them able to perform a variety of functions. In recent years, however, the process innovation has also been moving into the forefront. The latter includes all the means that

Computer Modeling for Injection Molding: Simulation, Optimization, and Control, First Edition. Edited by Huamin Zhou.  2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc.

3

4

INTRODUCTION

FIGURE 1.1 World consumption of raw materials by weight.

FIGURE 1.2 Diagram of the injection molding process.

help tighten up the manufacturing process, understanding, and optimizing it. The core of all activities has to be the most efficient application of production materials, a principle that must run right through the entire process from polymer materials to the finished product. That is, the aim

is no longer merely to manufacture particular components, but to manufacture a finished product with the best quality and in the most rational way if possible. Other new factors also enjoy recognition, such as shorter development time, lower cost, and higher productivity.1

FACTORS INFLUENCING QUALITY

On the other hand, the quality of molded products will continue to be the major criteria determining the competitiveness and performance of an injection molding company. Owing to growing applications of plastics, increasing customer demand, and rapid growth of the global marketplace, the quality requirements of injection-molded components have become more stringent for various market sectors such as the automotive, computer, consumer appliances, medical, microelectromechanical systems (MEMS), optical, and telecommunication industries. At present, part quality is crucial to the survival and success of enterprises. Quality features include mechanical properties, dimensional accuracy, absence of distortion, surface quality, etc. Only with the beginning of a deeper understanding of process mechanisms and their underlying physical laws, could injection molding technology make any real progress and improve the final quality to the greatest extent. Unfortunately, it is clear that very little was known about what happens inside the molding process. In spite of what has been achieved so far, the industry has surmounted only the first hurdle of systematic development. The present should not be regarded as the last word in progress. On the contrary, there are great possibilities in development that must be recognized and examined with the close cooperation of theorists and technologists.

1.2

FACTORS INFLUENCING QUALITY

The mechanical properties and performance of a finished product is always the sequence of events. Manufacturing of a plastic part begins with part design and material choice in the early stages, followed by mold design and manufacturing, and then processing, at which time the material is not only shaped and formed but the properties that control the performance of the product are also set or frozen into place. In the development of any plastic product, it is important to understand that the entire manufacturing process and all involved factors in the links have an influence on the quality of molded products. These factors mainly include polymer properties and its performance during molding, product design and its characteristics, mold design and its configuration, process conditions (parameters), and injection molding machine and its process control. For example, various elements regarding the part and mold designs as well as the material selection and process setup have to be considered to ensure that the mold can be fulfilled; the inherent, nonuniform material shrinkage throughout the cavity due to cooling and crystallization (in the case of semicrystalline materials) is further affected by packing, mold cooling, constraints of mold geometry, and the possible presence of reinforcing fibers.

5

The following subsections will introduce these factors briefly. 1.2.1

Molding Polymer

Polymers (plastics) are a family of materials, including many thousands of different materials. Extensive compounding of different amounts and combinations of additives (colorants, flame retardants, heat and light stabilizers, etc.), fillers (e.g., calcium carbonate), and reinforcements (glass fibers, glass flakes, graphite fibers, whiskers, etc.) are used to produce new plastic materials, each having its respective melt behavior, product performance, and cost. Plastics can be classified according to several criteria. Our initial differentiation is between cross-linked and noncross-linked materials. Whatever are/is their properties or form, most plastics fall into one of two groups: thermoplastics (TPs, non-cross-linked) and thermosets (TSs, crosslinked). TPs, which are predominantly used, can go through repeated cycles of heating/melting and cooling/solidification. Different TPs have different practical limitations on the number of heating–cooling cycles before appearance and/or properties are affected. The TP resins consist of long molecules, either linear or branched, having side chains or groups that are not attached to other polymer molecules. Usually, TP resins are purchased as pellets or granules that are softened by heat under pressure allowing them to be formed. When cooled, they harden into the final desired shape. No chemical changes generally take place during forming. TSs, on their final heating (usually at least to 120 ◦ C), become permanently insoluble and infusible. During heating they undergo a chemical (cross-linking) change. The linear polymer chains are thus bonded together to form a three-dimensional network. Therefore, once polymerized or hardened, the material cannot be softened by heating without degrading some linkages. TSs are usually purchased as liquid monomer–polymer mixtures or a partially polymerized molding compound. In this uncured condition, they can be formed to the finished shape with or without pressure and polymerized with chemicals or heat. Most of the literature on injection molding refers entirely or primarily to TPs; very little, if any at all, refers to TSs. Considering that at least 90 wt% of all injection-molded plastics are TPs, this book mainly deals with injection molding of TPs, and the terms plastic and polymer used later in this book refer primarily to TPs. Injection-molded parts can, however, include combinations of TPs and TSs, as well as rigid and flexible TPs, reinforced plastics, TP and TS elastomers, etc. Polymers are said to be viscoelastic. The mechanical behavior of polymers is dominated by the viscoelastic parameters such as tensile strength, elongation at break,

6

INTRODUCTION

and rupture energy. The viscous attributes of polymer melt are important considerations during injection molding. The rheology of polymers deals with the deformation and flow of polymer melt under various conditions. Owing to the thermomechanical history experienced by the polymer during processing, macromolecules in injection-molded objects present microstructure and morphology influencing greatly the final performance of molded parts. In the case of TPs, some of the molecules can come closer together than others. These are identified as crystalline; the others are amorphous. The performance of these two microstructures varies to a great extent. There are no purely crystalline plastics; the so-called crystalline materials also contain different amounts of amorphous material. 1.2.2

Plastic Product

A plastic product must be designed to satisfy certain functional, structural, aesthetic, cost, and manufacturing requirements. One of the significant advantages of plastic parts is that a part that incorporates a multitude of features that might otherwise require machining and assembly of multiple parts can be molded. Therefore, the expectations in the plastic part and the pressure on the designer to satisfy the multiple functions present further challenges. Compounding this challenge is the need to combine these features while not overly complicating the tooling requirements that might reflect on the manufacturability of the product and its cost.3 So, in the product design stage, one has to comprehend factors such as the range of the material properties, structural responses, product performance characteristics, and available fabricating processes, as well as their influence on product performances. For structural applications a designer can use either standard design formulas (rough) or

finite-element structural analysis (more accurate) to calculate deflections and stress. Moreover, to simplify molding, whenever possible one should design the product with features that simplify the mold-cavity filling operation. Many such features can facilitate the molding process, improve the product’s performance, and/or reduce cost. An example is setting the mold-cavity draft angle according to the plastic being processed, tolerance requirements, etc. A too small draft of molded part will lead to poor mold release, distortion of molded part, and dimensional variations. And also, sharp transitions in part wall thickness and sharp corners will result in parts unevenly stressed, dimensional variations, air entrapment, notch sensitivity, and mold wear. Figure 1.3 shows a situation where it is possible to eliminate or significantly reduce shrinkage, sink marks, and other defects. Thus, in the design of any injection molded part, there are certain desirable goals that the designer should achieve. If neglected, problems can unfortunately develop. For example, the most common design errors usually occur in the following areas: • thick or thin sections and transitions resulting in warpage and stress; • parts too thin to mold properly (such as diaphragms); • parts too thick to mold properly; • flow path too long and tortuous; • orientation of polymer melt in flow direction; • hiding gate stubs; • stress relief for interference fits; • living hinges; • slender handles and bails; • thread inserts; • creep or fatigue over long-time stress.

FIGURE 1.3 Example of coring in products to eliminate or reduce shrinkage and sink marks.

FACTORS INFLUENCING QUALITY

1.2.3

Injection Mold

The mold is the central element of the injection molding process. Under pressure, hot melt moves rapidly into the mold. With TPs, temperature-controlled water circulates in the mold to remove heat; with TSs, electrical heaters are usually used within the mold to provide the additional heat required to solidify the plastic melt in the cavity. The mold basically consists of a sprue, a runner, a cavity gate, and a cavity. The sprue transports the melt from the plasticator nozzle to the runner. Next, melt flows through the runner and gate and into the cavity. The mold for producing a plastic part must be custom designed and built. The challenges in designing a mold include the following, among many others: the mold must accommodate delivery of the melt and accomplish automatic separation of runner and part; the cavity dimensions must be sized to account for the part’s shrinkage; the mold must provide adequate and uniform cooling and venting of gases; the mold must be strong enough to withstand cyclic internal loads from injection pressures and external clamp pressures; the mold components must be machinable. Many parts of an injection mold will influence the final product’s performance, dimensions, and other characteristics. These mold parts include the cavity shape, gating, parting line, vents, undercuts, ribs, hinges, etc., which are listed in Table 1.1. The mold designer must take all these TABLE 1.1

7

factors into account. At times, to provide the best design, the product designer, processor, and mold designer may want to jointly review where compromises can be made to simplify the process of meeting product requirements. With all these interactions, it should be clear why it takes a significant amount of time to prepare a mold for production. 1.2.4

Process Conditions

Different product requirements and material conditions are considered in choosing the most efficient injection molding process. It is well known that the process conditions have a direct influence on the performance of injection moldings. Mold filling involves both high deformation and high cooling rate. The process conditions are correlated with the internal structure of the plastic material, which represents the key for the behavior of the molded product, as shown in Figure 1.4. In order to have a stable and high-quality production, the following issues and relevant process parameters are worth investigating. The plasticization phase can be optimized by varying the screw rotation speed and back pressure so as to provide sufficient and uniform polymer melt. The injection velocity (speed) is critical to influencing the pressure drop, temperature difference after filling, shear rate (and thus orientation), etc. The switchover from filling to packing can be made based on smooth changes of pressure and filling

Examples of Errors in Mold Design

Faults Wrong location of gates Gates and/or runners too narrow Runners too large Unbalanced cavity layout in multiple-cavity molds Nonuniform mold cooling Inadequate provision for cavity air venting Poor or no air injection Poor ejector system or bad location of ejectors Sprue insufficiently tapered Sprue too long No round edge at the end of sprue Bad alignment and locking of cores and other mold components Mold movement due to insufficient mold support Radius of sprue bushing too small Mold and injection cylinder out of alignment

Possible Problems Cold weld lines, flow lines, jetting, air entrapment, venting problems, warpage, stress concentrations, voids, and/or sink marks Short shots, plastics overheated, premature freezing of runners, sink marks, and/or voids Longer molding cycle and waste of plastics Unbalanced pressure buildup in mold, mold distortion, dimensional variation between products (poor shrinkage control), poor mold release, flash, and stresses Longer molding cycle, high after-shrinkage, stresses (warpage), poor mold release, irregular surface finish, and distortion of part during ejection Need for higher injection pressure, burned plastic (brown streaks), poor mold release, short shots, and flow lines Poor mold release for large parts, part distortion, and higher ejection force Poor mold release, distortion or damage in molding, and upsets in molding cycles Poor mold release, higher injection pressure, and mold wear Poor mold release, pressure losses, longer molding cycle, and premature freezing of sprue Notch sensitivity (cracks, bubbles, etc.) and stress concentrations Distortion of components, air entrapment, dimensional variation, uneven stresses, and poor mold release Part flashes, dimensional variations, poor mold release, and pressure losses Plastic leakage, poor mold release, and pressure losses Poor mold release, plastic leakage, cylinder pushed back, and pressure losses

8

INTRODUCTION

FIGURE 1.4 Relationship between process conditions and properties of products.

To mold parts at the shortest cycle time, the molding machine would be set at the lowest temperature and highest pressure location on this diagram. If inferior quality appears, one has to move the parameters to higher temperature and/or lower pressure. This is a simplified approach to producing quality parts because only two variables are controlled here. Using this approach for making process windows, one can analyze all other process parameters. The process window for a specific plastic part can significantly vary if changes are made in its design, material choice, and/or the fabricating equipment used. Developing the actual data involves plenty of molding trials. 1.2.5

FIGURE 1.5

Illustration of process window.

rate. Optimizing the magnitude and duration of applied packing pressure can prevent sink marks, dimension out-oftolerance, and underweight. Cooling time depends on the melt temperature and part thickness. Attention must be paid to the mold and injection (barrel) temperature that influence both the quality and productivity. Process windows are the ranges of process conditions, such as injection speed, injection temperature, mold temperature, and holding pressure, within which a specific plastic can be molded with acceptable or optimum properties. A window is a defined “area” in the space of process variables. For example, by plotting injection temperature versus holding pressure, a molding area diagram that shows the best combinations of injection temperature and holding pressure to produce quality parts is developed, as shown in Figure 1.5. The size of the diagram denotes the molder’s latitude in producing good parts.

Injection Molding Machine

The injection molding machine is one of the most significant and rational forming methods that exist for processing plastic materials. There are different types of injection molding machines. The reciprocating screw injection molding machine is the most widely used one in plastics industry owing to its better reliability and overall performance, such as improved melting rates, closer tolerances on shot size, better control of temperatures, and simpler structure. A simplified general layout for an injection molding machine is shown in Figure 1.6. The injection molding machine has four basic components: the injection unit, the clamping unit, the control system, and the drive system. The injection unit, also called the plasticator, prepares the proper plastic melt and transfers the melt into the mold. The most important elements of an injection unit are (in the sequence of polymer flow) as follows: hopper, screw, homogenizing elements on the screw (in some cases), nonreturn valve (check valve) at the screw tip (in some cases), nozzle, and heater bands. The clamping unit opens the mold for demolding and closes it for the next shot. Because the polymer is pressed under high pressure into the mold, the clamping unit must also be

FACTORS INFLUENCING QUALITY

FIGURE 1.6

Schematic of an injection molding machine.

able to keep the mold tightly sealed during the filling and holding stages. At present, clamping units are available in three different forms in the market: mechanical, hydraulic, and hydraulic mechanical systems. The control system coordinates the machine sequences, keeps certain machine parameters constant, and optimizes individual steps in the process. All motion sequences of the machine, the correct order of these sequences, their initiation, the signaling of positions reached (such as by limit switches), and the reaction at predetermined times within a cycle have to be achieved, initiated, and coordinated. The temperature requirements during molding (including barrel, melt, and mold temperatures) are set up by the control system, and implemented by the tempering devices. The drive system provides power for the above components by the conventional way of hydraulic or by the recent developed ways of all-electric or hybrid-electric-hydraulic. At present, the hydraulic system is the most popular, while the electric one has the development tendency. The essential advantage of oil hydraulic systems is that the fluid can be distributed easily by hoses and pipes, and that no complicated mechanical transfer elements such as rods, cables, and toothed racks are necessary. Compared with electric systems, the main drawback is their higher energy loss. The injection molding machine performs certain essential functions: (i) plasticizing —heating and melting the plastic in the plasticator; (ii) injection —injecting from the plasticator under pressure a controlled-volume shot of melt into a closed mold; (iii) after-filling —maintaining the injected material under pressure for a specified time to prevent back flow of melt and to compensate for the decrease in volume of melt during solidification; (iv) cooling/ heating —cooling the TP molded part or heating the TS molded part in the mold until it is sufficiently rigid to be ejected; and (5) molded-part release —opening the mold, ejecting the part, and closing the mold so it is ready to start the next cycle. The type and size of an injection molding machine to be used are dependent on the dimensions and volume of the molded product.

9

The injection molding machine has extensive processcontrolling devices to maintain correct operating procedures. The physical values to be controlled (temperature, position, velocity, and pressure) are recorded with special sensors (thermocouples, displacement, and pressure transducers). These signals are then transformed and read in by the supervising computer. On the basis of these input data, the control program induces certain actions: for example, if the temperature of the plasticating unit is too low, the heater bands are switched on, or, if the screw has reached a set position during plastication, the control system shuts a valve, to switch off the screw rotation.4 Process control closes the loop between process parameters and appropriate machine control devices to eliminate the effect of process disturbances. Tighter operational controls permit production of high-quality products with less effort. In addition, the design of the control system has to incorporate the logical sequence of all basic functions, including injection speed, clamping and opening the mold, opening and closing of actuating devices, barrel temperature profile, melt temperature, mold temperature, cavity pressure, and holding pressure. 1.2.6

Interrelationship

As mentioned above, all factors involved in the entire manufacturing process affect the final quality of the molded products, including plastic properties, product characteristics, mold configuration, process conditions, and process control. This relationship can be illustrated as a fishbone diagram (Fig. 1.7). As an example, the dimensional accuracy of injection molding, which can be met, depends on such factors as properties of materials; accuracy of mold and machine performance; operation of the complete molding cycle; wear or damage of machine and/or mold, shape, size, and thicknesses of the product; postshrinkage (which can reach 3% for certain materials); and the degree of repeatability in performance of the machine, mold, material, etc. Moreover, there are strong and complicated interrelationships among these factors. For instance, it is well known that different plastics have different melt flow characteristics. What is used in a mold design for a specific material may thus require a completely different type of mold for another material. These two materials might, for instance, have the same polymer but use different proportions of additives and reinforcements. It is necessary to consider these interrelationships so as to fabricate a cost-performance effective molded product. Unfortunately, at present, the development stages of injection-molded parts are often handled sequentially and independently. A part designer will design a part with limited knowledge of mold, processing, and/or materials.

10

INTRODUCTION

FIGURE 1.7 The factors influencing the final quality of molded parts.

A mold designer will inherit this part and design and build a mold with limited understanding of processing and material behaviors during processing. The injection molder then inherits this mold and must try to find a process condition that can produce the required part. At this stage his options are very limited. In addition, we find that the processor often has had limited opportunity to take formal training that would allow him to understand the fundamental causeand-effect relationship of his actions on the molded part. Is the warpage problem which he is encountering dominated by part design, material, mold cooling, gating, process, or other factors? The attempts at solving problems are often based on trial and error, seat of the pants, gut feel, and intuition.3 On the basis of the above facts, it is of great importance to recognize that the best quality can only be achieved by overall optimization from the very beginning of a design concept through to production of injection-molded parts, and thus it is necessary to establish effective cooperation among part designers, mold designers, molders, and material engineers. The best approach may be to integrate computer modeling within an overview of the interrelated building blocks of an injection-molded part: product design, plastic material, mold design, process conditions, and the injection molding machine.

1.3

COMPUTER MODELING

One of the most revolutionary technologies to affect injection molding in the past decades certainly would be computer applications in the industrial production process. In the injection molding industry, computers permeate all aspects from the concept of a product design, mold manufacturing, raw material processing, marketing and sales, recycling, to administration and business, and so on. They provide word processing, databases, software, spreadsheets, design and manufacturing support, etc., while this book focuses on the computer’s service in improving the product quality. Most accept the fact that computers can, if properly used, improve efficiency, reduce costs, improve the quality of products, and reduce time for bringing new products to the marketplace. Mold costs can be reduced 10–40%, lead time cut by 20–50%, molding cycle time cut by 10–50%, material usage reduced by 5–30%, and product cycle time reduced by 50–80%.1 The advantages of computer modeling are, in particular, accentuated because in order to produce a single part to evaluate its performance first a custom-designed mold must be built, which may cost tens to hundreds of thousands of dollars. This is typically several million times the selling price of the product it is to produce. The process of designing and building a mold, and molding the first plastic

COMPUTER MODELING

parts can easily take 20 weeks. Not until this time can the actual size, shape, and mechanical properties of a molded part be known. It is rare that these first parts possess the required specifications. The next stage is typically a long, costly process of trying to produce parts that obey the specification, maybe involving changes to the mold, process, or the plastic material. This is in contrast to the development of machined products. Here, if the part does not satisfy expectations, it can be easily modified or a second part will be machined reflecting an altered design. So, if the first parts do not work, the investment in engineering and machine time is minimum compared to building a mold.3

1.3.1

Review of Computer Applications

The use of computers in manufacturing operations dates back to early work in the 1950s in which the dream was to control metal-cutting machine tools by computer. It was hoped that this would eliminate the requirement for many tooling aids, such as tracer templates, that favored the accuracy and repeatability of machining operations on the shop floor. During this period, the only types of computers available were extremely expensive “mainframe” computers. Programming was accomplished via a punched card medium and was tedious and time consuming to develop and debug. The only means to check cutter paths developed by the computer was to do a “prove out” run on the shop floor. The concept of using a graphic display device to visualize cutter paths was proposed and developed during the 1960s. During this same period, an important hardware progress was the development of microcomputers. This newcomer to the computer field brought in a totally new price and performance spectrum, which created a dramatic increase in the acceptance of computers (and also the concept of CAD/CAE/CAM) in general, particularly in the scientific, engineering, and manufacturing areas. The 1970s not only engendered a continued development of hardware and software products but also brought about a change in the business climate. The computer industry spawned the “turn-key” CAD/CAE/CAM suppliers that could supply both the computer hardware and user-friendly software, ready to run. The first predominant applications were in the area of two-dimensional printed circuit board (PCB) and integrated circuit (IC) design. Both of these applications were relatively easy to capitalize on, as they can be described by geometries on planar surfaces. During the following two decades, the rapid developments of CAD/CAE/CAM resulted in three-dimensional representations of objects. This implied a complete expansion in the capabilities of CAD/CAE/CAM systems, moving them from two-dimensional drafting tools into

11

true spatial mathematical modeling tools. The threedimensional modeling and the fast, smooth shading of surfaces help one to understand the shape geometry. Besides CAD/CAE/CAM, the computer applications for design and manufacture support in injection molding extended into computerized databases of plastics, trouble shooting, optimization, process control of molding machines, etc. In this new century, new software packages of CAD/ CAE/CAM continued to enhance their usefulness to part designers, moldmakers, and molders. The related technologies include two-dimensional drafting; three-dimensional modeling, design and assembly; finite-element analysis and simulation; visualization and virtual reality; (on-line and real time) optimization; numerical control programming; integrated, intelligent, Internet-based, and cooperative design; product data management (PDM); enterprise resource planning (ERP); manufacturing execution system (MES); product lifecycle management (PLM); etc. In the present time, the technologies of computer applications imply a completely different methodology of engineering design. The benefits that result from computer applications in injection molding are productivity improvement, quality enhancement, turnaround time improvements, more effective utilization of scarce resources, etc. Examples include (i) fewer errors in drawings, which improves mold quality and speeds up delivery time; (ii) better communication among part designers, mold designers and moldmakers; (iii) improved machining accuracy; (iv) standardization of parts and components, which reduces the amount of supervision required in a manufacturing facility; (v) improved speed and accuracy in the preparation of the quotation; and (vi) a faster response to market demand. 1.3.2

Computer Modeling in Quality Enhancement

Among all the benefits of computer applications, the quality benefits are perhaps the most underrated. Computer modeling has played a crucial role in the quality control of injection molding. Many of the analysis packages promote a better understanding of molding process and the interrelationships among correlated parameters. This contributes to a better ability to control previously mysterious phenomena (such as warpage). Instead of the past costly trial-and-error manufacture process, prediction and optimization of the product quality at the lowest cost has now become possible. The increased computer-aided process control has resulted in quick setup, automatic production, and an overall increase in part quality. It is unquestionable that a proper use of computer applications can sharpen a company’s completive edge in various aspects such as analysis, design, simulation, optimization, control, and monitoring. Here, we review the development process of injectionmolded products. During the early design stage, the

12

INTRODUCTION

material’s choice and product geometry are both decided mainly based on the functional requirements. After that, the mold is custom designed and manufactured. Once an injection mold is built and mounted on a machine, a molding engineer (or setup person) has to determine the process conditions (such as shot size, injection speed, pack/hold time and pressure, cooling time, back pressure, coolant temperature, and barrel temperature), depending on the material, product, and mold. Typically, these parameters can be set at the machine’s operating console. The machine control executes the commands set for moldings, and its performance has a direct impact on the final part quality. This development process is illustrated in Figure 1.8, and the relevant parameters are called design variables in this book. Instead of the design variables, numerous research efforts has showed that the thermomechanical histories during the injection molding process (referred to as processing variables here) finally determine the quality of the molded part (labeled as quality variables). The processing variables mainly denote the flow, temperature, and pressure within the polymer melt throughout all phases of injection molding such as melt temperature, melt pressure, melt shear rate, melt shear stress, and heating/cooling rate. The quality variables include quantitative and qualitative indices such as part weight and thickness, volume shrinkage, warpage, sink marks, weld lines, part strength, and part appearance. Because the processing variables are the true indicators of the conditions of the material inside the mold, they are more closely related to quality variables than are the design variables. Of course, these processing variables cannot be set up directly, depending on the collective effect of the specific resin and mold used, the machine setting, and the nonlinear, distributed, and time-varying process dynamics.5 Figure 1.8 describes the three-level hierarchy and dependency of the injection molding quality. The processing variables serve as the connection between the design variables and the quality variables. However, no generic quantitative models have been established for the connections from the design variables to the

processing variables and from the processing variables to the quality variables. The relationship between the design variables and the quality variables of molded parts can be expressed as a mapping in the following form: Q = f (m, p, d(m, p), c(m, p, d)) + v(c)

(1.1)

where Q is the collection of quality variables; m, p, and d are the collections of material properties, product characteristics, and mold configuration, respectively; c denotes the process conditions; v is the disturbances from the machine, affecting the execution of process conditions; and f is a mapping function without considering the disturbances. Unfortunately, f is typically complicated or unknown a priori . In practice, the expression of f has to be simplified to a certain extent in order to establish a reasonably accurate mapping between the influencing factors and part quality. The methods of mapping can be categorized into two approaches, namely, the numerical simulation approach and the optimization approach. The first approach describes the physical process of injection molding directly, which is developed based on the first principle, involving the use of computer-aided engineering (CAE) software or mathematical models. While the latter approach employs various artificial intelligence (AI)-based models such as case-based reasoning (CBR), artificial neural networks (ANNs), expert systems (ESs), fuzzy logic, genetic algorithm (GA), and design of experiments (DOE, using less AI technique). These AI methods should use expert knowledge, cases, and empirical models, as well as simulation results, as their reasoning basis. On the other hand, to achieve consistent quality, the machine controller should be able to repeat the process conditions consistently with high accuracy. However, there are plenty of unpredictable disturbances, including the mechanical and hydraulic deviations of machines and those coming from polymer pellets and melt, which are difficult to model and predict. Therefore, an accurate and robust process control of the injection molding machine also

FIGURE 1.8 Architecture of computer modeling in quality enhancement.

COMPUTER MODELING

plays an important role in ensuring the repeatability and reliability of the product quality. Besides the individual variable control of process conditions, newer works have attempted a direct (on-line) control of the final molded part quality (termed direct quality control ), but it is difficult to implement owing to the lack of an accurate quantitative description of the complex relation between quality characteristics and process conditions. In short, computer modeling for quality enhancement of injection molding could be organized into three categories, namely, numerical simulation, optimization, and process control, as shown in Figure 1.8. These are the focus of this book. In the following paragraphs, these three categories are reviewed briefly. 1.3.3

Numerical Simulation

Numerical simulation for injection molding is generally based on the rigorous, first-principle model that provides reasonably accurate descriptions and trends of the injection molding process. In the early stage, quite a few mathematical models (i.e., simplified numerical simulation) have been developed for describing the injection molding process.6 For example, Kamal and Kenig,7,8 Wu et al.,9 and Stevenson10 developed the mathematical models to describe the filling in a centergated disc; Toor et al.,11 Harry and Parrott,12 and Lord and Williams13 studied the one-dimensional filling behavior in rectangular cavity geometry; while Williams and Lord14 and Nunn and Fenner15 developed the mathematical models to describe the filling in a circular tube. These filling models are all limited to one-dimensional geometry. To apply these one-dimensional flow representations to simulate polymer flow in typically complex mold cavities, branching flow approach16,17 and network flow approach18,19 were proposed and implemented. These approaches involve laying flat and decomposing the cavity geometry into several conjectured flowpaths comprising a series of onedimensional segments such as strips, discs, fans, and/or tubes. With respect to mold cooling, Busch et al.20 and White21 derived mathematical models for the estimation of the cooling time. Tan and Yuen22 developed computer systems for calculating the process parameters and deriving an initial parameter setting for injection molding. Tan and Yuen23 proposed an analytical model for injection molding based on which the filling pressure, clamp force, shear stress, shear rate, and temperature at different time instants and locations can be calculated and used to determine suitable process conditions. In addition to the mathematical models, many numerical simulation models were developed to simulate the process behavior of injection molding. Hiber and Shen24 and Wang et al.25 employed a finite-element/finite difference

13

scheme for simulating filling of thin cavities of general planar geometry. These models were implemented based on the generalized Hele-Shaw flow for an inelastic, nonNewtonian fluid under the nonisothermal conditions. Chiang et al.26 developed a unified simulation model for the filling and postfilling stages on the basis of the hybrid finiteelement/finite difference numerical solution of the generalized Hele-Shaw flow for the compressive viscous fluid under the nonisothermal conditions. This 2.5-dimensional Hele-Shaw approach was extended or incorporated by other researchers to simulate mold cooling,27 fiber orientation,28 residual stresses,29 and shrinkage and warpage,30,31 as well as various special molding processes such as coinjection molding,32 gas-assisted injection molding,33 microchip encapsulation,34 injection/compression molding,35 reaction injection molding, and resin transfer molding.36 Zhou et al.37,38 presented a surface-model-based simulation which still used the Hele-Shaw assumption, but represented a three-dimensional part with a boundary mesh instead of the mid-plane. Some fluid behaviors at the free surface (flow front), near and at the solid walls, and at the merging of two or more fluid streams cannot be accurately predicted using the Hele-Shaw approximation.39 To date, several full three-dimensional simulation approaches for injection molding have been developed. Rajupalem et al.,40,41 Kim and Turng,42 Zhou et al.43 and Cheng et al.44 used equal-order velocity–pressure formulations to solve the Stokes equations in their three-dimensional mold filling simulation. Haagh and Van De Vosse45 implemented a finite-element program for injection molding filling, which employed a pseudoconcentration method. Hetu et al.46 employed the Petrov–Galerkin method to prevent these potential numerical instabilities. Chang and Yang47 developed a numerical simulation program for mold filling on the basis of an implicit finite-volume approach. Estacio and Mangiavacchi48 and Jiang et al.49 used the control-volume-based finite-element-method (CVFEM) to solve flow and heat transfer in injection molding. Considering the fact that product properties are, to a great extent, affected by internal structures (morphology),50 numerical simulation of the effect of operative conditions of injection molding process on the morphology distribution inside the obtained moldings has been performed, with particular reference to semicrystalline polymers.51 As for crystallization, the crystallization kinetics models include Avrami model,52 Nakamura model,53,54 Ozawa model,55,56 Mo model,57,58 Urbanovici–Segal model,59 – 61 and the flow-induced crystallization models.62 – 64 Evolution of crystallization morphology in injection molding is based on the nucleation and growth process.51,65 – 67 In the case of polymer blends, the molding process often gives rise to a heterogeneous microstructure that can be characterized by the size, shape, and distribution of the

14

INTRODUCTION

constitutive domains.68 Direct numerical simulations have been developed for single and multiple droplets behaviors in emulsions.69 – 73 Under the shear (or elongation) stress field in the cavity during processing, a skin–core structure is common in injection-molded parts.73,74 The study of orientation is closely related to the fact that orientation will inevitably lead to anisotropy in polymer properties, mainly including molecular orientation75 – 79 and fiber orientation.80 – 84 Some of the achievements in simulating the injection molding process were commercialized in the simulation packages such as Moldflow, Moldex 3D, HSCAE. Reliable CAE simulation tools could replace the traditional trialand-error approach and assist to select material, design the product and mold, and set up the molding conditions in a more effective manner. Moreover, some special CAE software could suggest optimal process conditions to achieve acceptable parts by using certain built in criteria and rules. For example, it is capable to carry out an automated DOE to determine a robust process window for producing “good parts.” And also, several process parameters can be first set step-by-step in the process setup stage, and further refined in the process optimization stage to achieve 100% yield by improving the process robustness and reducing the probability of producing defective parts.85,86 Nirkhe and Barry86 compared the software-based setup method with the manual approach and the results show that the former approach could obtain process conditions that lead to more consistent part weights and dimensions than the latter approach. Turng and Peic87 have integrated a CAE tool with various optimization algorithms to help identify the optimal process conditions to achieve a variety of optimization objectives while satisfying certain constrains. Lam et al.88 presented a simulation-based system to assist the determination of process parameters, allowing the designers to specify their intended quality measuring criteria such as minimum cavity pressure and shear stress, a uniform distribution of cooling time, end-of-fill temperature, and volumetric shrinkage. Although CAE software and mathematical models provide the developer with effective tools, their ability should not be overestimated. They also have many limitations. The underlying assumptions and simplifications in these first-principle models can sometimes lead to discrepancies between the real optimal scheme and those obtained from the models. And also, adequate training is important for proper use of these tools. The analyst should be trained not only in modeling and running the programs but also in traditional molding and design. 1.3.4

Optimization

Optimizations in injection molding have already been very popular with modern industries showing their substantial

power in competitiveness enhancement. Computer optimizations can be classified into two categories: noniteration methods (such as gray relational analysis, ES, fuzzy logic, and CBR) and intelligent optimization algorithms (including GA, simulated annealing algorithm, and particle swarm algorithm). And recently, surrogate modeling is often employed in optimization, including response surface method, ANN, and support vector regression. Only some of these methods are reviewed in this section. It should be noted that numerical simulation trials are sometimes used as data sources of optimization algorithms. DOE techniques, especially the Taguchi method, were widely used to generate meaningful experimental data and determine optimal process parameters for injection molding.89 – 92 These studies show that Taguchi parameter design can uncover subtle interactions among process variables with a minimum number of test runs. For instance, Liao et al.93 started with the process conditions suggested by a CAE tool and then optimized them with DOE to minimize the shrinkage and warpage of a cellular housing part. To improve the effectiveness of DOE, other techniques were incorporated with the Taguchi method. Yeung and Lau94 attempted to link quality function deployment (QFD) with DOE to establish a prioritization mechanism for setting the selected parameters with respect to all of the quality characteristics. Kuhmann and Ehrenstein95 combined the Taguchi method and the Shainin method to improve the robustness of the injection molding process. Amidst the diverse ways of building the AI models, ANN is one of the widely used methods.96 – 99 Generally, ANN approaches were applied far and wide in building a process model for quality control in injection molding.100 – 106 In these approaches, some indices of part quality, such as weight, thickness, warpage, shrinkage, flash, and/or strength, are established as the output of neural networks while the inputs are either the process conditions (such as injection speed, holding pressure, holding time, cooling temperature, and barrel temperature) or the processing variables (such as nozzle pressure, cavity pressure, and melt temperature), or a combination of them. For example, the ANN method was successfully used in predicting the shrinkage and warpage of injection-molded thin-wall parts.107 It is not surprising that networks based on processing variables could predict the part quality more accurately than those based on process conditions. Note that the ANN model has to be trained with a set of wellprepared data capable of describing the process sufficiently accurately. Otherwise, the model would only have little use. The ES simulates the human reasoning process by applying specific knowledge and inference. A typical ES consists of two major elements: the knowledge base and the inference engine. Quite a few ESs were developed to recommend the qualitative correction instructions108 and/or the quantitative change of molding parameters in response

COMPUTER MODELING

to the input molding defects. Jan and O’Brien109,110 developed an algorithm to calculate the decision indices that show the likelihood of the influencing variables responsible for defects. They were used to specify the assurance of possible remedies for the given injection molding problems.111,112 Kameoka et al.113 applied the multidimensional matrix technique to develop an ES called ESIM so as to realize the skilled operators’ inference procedures into the system. Kimura et al.,114 Dwivedi et al.,115 and Mok et al.116,117 developed integrated knowledge-based systems for mold design support in which flexible representation frameworks were studied for various types of expert knowledge. Bozdana and Eyercioglu118 developed a frame-based, modular and interactive ES (called EX-PIMM ) for the determination of the injection molding parameters of TP materials. Fuzzy logic was applied in the development of an ES, which can recommend the quantitative change of molding parameters.119 Tan and Yuen120 proposed a fuzzy multipleobjective approach to set up the process for minimizing injection molding defects. In their study, the defects were expressed by a scale number through fuzzy functions. The relationship between the severities of the defects and the machine variables was approximated by a set of quadratic polynomials via regression analysis. Chiang and Chang121 applied a gray-fuzzy logic approach for the optimization of machining parameters to an injection-molded part with a thin shell feature. Through the gray-fuzzy logic analysis, the optimization of complicated multiple performance characteristics can be converted into the optimization of a single gray-fuzzy reasoning grade. The basic idea of CBR is that a case-based reasoner solves a new problem by adapting the solutions that were used to solve the old problems. Kwong and colleagues developed a CBR system,122,123 a CBR system combined with fuzzy logic and neural networks,124 and an intelligent hybrid system,125,126 to determine the initial process conditions. In their study, the model of the process was in the form of a case library. The match of the current problem with the library was solved through fuzzy inference, and the case adaptation was implemented in neural networks. Shelesh-Nezhad and Siores127 also applied the CBR approach in deriving the first trial parameter setting of injection molding. Kwong128 developed a casebased system for process design of injection molding, which aims to derive a process solution for injection molding quickly and easily without relying on the experienced molding personnel. Huang and Li129 proposed a hybrid approach of CBR and Group Technology (GT) for injection mold design. GA approach has been applied to develop optimization systems for the process parameters of injection molding.130,131 Because optimization of process parameters for injection molding is not a static process, an optimization

15

system called Ibos-Pro has been developed on the basis of evolutionary strategies approach for on-line optimization of the process parameters.132 On the other hand, once a minimal region is identified during the search process of GA in process conditions optimization, it is inefficient, even sometimes impossible, in reaching its minimum. Here, the gradient method can help to guarantee a local minimum.133 Recently, a microgenetic algorithm (mGA)-based approach was presented to solve biobjective optimization of an injection mold design problem, such as gate positioning134,135 ; a distributed multipopulation GA was used to optimize injection molding with weld line design constraint136 ; and a multiobjective GA, denoted as Reduced Pareto Set Genetic Algorithm with Elitism, was applied to the optimization of the injection molding process.137 Deng et al.138 proposed a PSO (particle swarm optimization) algorithm for the optimization of multiclass design variables, including product characteristics (part thickness), process conditions (injection temperature, mold temperature, and injection speed), and mold configuration (gate location). The optimization is targeted at different aspects of molding quality, including part warpage, weld lines, and air traps. A computer program was developed that automates the steps such as adjusting the part thickness, the injection molding process parameters, and the gate location, activating the CAE software to simulate the injection molding process, retrieving the simulation results, and evaluating the objective functions. From the above applications, it can be seen that different AI methods were often combined together in injection molding optimization so as to exploit their respective advantages. As a more representative example, Chen et al.139 presented a hybrid approach for the process parameters optimization, which integrated Taguchi’s parameter design method, back-propagation neural networks, GAs, and engineering optimization concepts. The optimal design scheme and process conditions set by the above-mentioned methods are assumed to exist and remain constant for a specific combination of machine, mold, and material. However, this assumption may not be true in a real process, given various unexpected disturbances. Therefore, it requires some methodology to adjust the conditions in order to compensate for the disturbances and to improve the process control of the machine to minimize the disturbances. 1.3.5

Process Control

Process control and monitoring provide continual support toward achieving a higher level of technology implementation to meet performance demands at the lowest cost. Control systems available at present include feedback control, feed forward control, advanced control, learning control, etc. And monitoring may refer to the most lately

16

INTRODUCTION

statistical process control (SPC), multivariate statistics, and multiphase statistical process control. At present, there are many commercial control systems available in the market for the injection molding machine (e.g., Pro-Set,140 Xtreem XP,141 and MMI142 ). These systems usually include function modules for controlling position/velocity, pressure, temperature, and motion sequences. Well suited for injection molding machine control, programmable logic controllers (PLCs) were widely used in these systems.143 Typically, a PLC sequence program or logic program is created for controlling a motion sequence such as clamping close/open, ejection, injection unit forward/backward, and safety guard. The injection velocity, ram position, screw rotation speed, hydraulic system pressure at injection, barrel temperature, and coolant temperature can be controlled via a conventional PID (proportional–integral–derivative) controller embedded in the intelligent modules. Recently, some up-to-date machine controllers were built on industrial PCs (personal computers) directly,144 – 147 which brings about numerous benefits, including more powerful processing capabilities as well as more public and open resources available to machine control developers. Because of unpredictable disturbances, traditional PID control sometimes cannot guarantee high standard machine performance. Therefore, there have been continuous efforts in pursuing advanced control technologies to improve machine control.148 – 158 As the barrel temperature, injection speed, ram position, and hydraulic pressure are closely related to the injection stage, some advanced strategies have been developed to control these parameters.152 Having obtained the general state equation, numerous modern control algorithms such as linear quadratic optimal control159 and model predictive control160 can be applied to achieve better performance than PID. In particular, Bulgrin and Richards148 proposed a barrel temperature state control on the basis of a state equation obtained from a lumped heat capacity analysis. Considering the slow responses of injection molding barrel temperature, Yao et al.161 presented a combined strategy using a feedback controller and in iterative learning feedforward (ILFF) controller for barrel temperature control. Regarding injection velocity, there are many adaptive control schemes reported in the publications, such as the self-tuning regulator (STR) and generalized predictive control (GPC),154,162 sliding mode control (SMC),155 fuzzy logic control (FLC),156 iterative learning control (ILC),157,158 and on-line controller updating.163,164 Technically speaking, all of the above control schemes have a common key component—the process model—whose accuracy significantly affects the final performance, even though the proper controller design can partially compensate for a model mismatch.

When using the linear model structure proposed by Wang et al.,165 Huang et al.166 presented a predictive control scheme and analyzed the closed-loop properties. Furthermore, on the basis of a linear auto regressive with ex ogenous (ARX) input model, Yang and Gao154 applied adaptive GPC to ram velocity control and compared it with a self-tuning pole-placement controller enhanced by several measures: antiwindup estimation to eliminate the estimation windup, cycle-to-cycle adaptation to improve the model convergence, and adaptive feedforward and profile shift to improve the tracking speed. They concluded that the adaptive GPC controller performed well over a wide range of process conditions. Furthermore, the GPC design has inherently good tracking performance and excellent tolerance to model mismatch. As far as the nonlinear model based on physical principles is concerned,167 it is difficult to design a controller directly from the model owing to its complexity. However, the nonlinear model can be approximated by a simpler model structure, which is suitable for the control while the error resulting from the approximation can be taken into account in design. For example, Tan et al.155 developed an adaptive sliding mode controller, which was verified through simulation to be capable of achieving tight setpoint regulation. An on-line numerical simulation was developed recently that is capable of predicting state variables such as flow rate, melt temperature, shear rate, and melt viscosity by using real-time data from a nozzle pressure sensor.168 There are also some unconventional ways to represent the process models. For example, Tsoi and Gao156 used a fuzzy-logic-based model to control the injection velocity. Their experimental results revealed that the fuzzy logic controller worked well with different molds, materials, barrel temperatures, and injection velocity profiles, suggesting that the fuzzy logic controller has superior performance over the conventional PID controller in response speed, setpoint tracking, noise rejection, and robustness. Because of the cyclical nature of injection molding, it is well suited for an ILC strategy, which uses successive repetitions of the same action to refine the control input. ILC has been employed to control injection velocity by Gao et al.,157 and to control hydraulic pressure/ram position by Havlicsek and Alleyne,152 where the traditional openloop feedforward compensator was combined with feedback optimization control to achieve stable convergence and performance improvement akin to closed-loop control. Considering the difficulty of establishing a process model, a model-free self-organizing fuzzy controller (SOFC) was developed to control the molding machine.146 Experimental results demonstrated that the SOFC exhibits better control performance than the fuzzy logic controller or the PID controller in controlling the screw velocity and the holding pressure.

OBJECTIVE OF THIS BOOK

Advanced control algorithms such as adaptive control and model predictive control have been adopted to deal with the inherent process nonlinearity and time-varying characteristics. These control algorithms are all focused on single-cycle control performance. Recently, a multicycle, two-dimensional model predictive learning control has been developed for batch process control.169,170 This method has been applied to injection molding.171 – 173 Most of the work for process control of injection molding has tackled this challenging subject mainly through consistent machine operations or process condition control, that is, controlling the design variables. Recently, direct quality control has been proposed and developed, in which the quality variables, or the processing variables, are used as the setpoint parameters. The quality variables include part weight,174,175 flash,176,177 etc., whereas the processing variables include mold separation,178 cavity pressure,179,180 melt-front velocity,181 etc. One essential difficulty in direct quality control is the lack of a quantitative description of the complex relation between the final quality and the process conditions. Another difficulty is how to measure the part quality and processing variables on-line. Fung et al.182 applied an in-mold capacitive transducer for on-line part weight prediction. Chen et al.178 used a precision linear displacement transducer mounted on the outside of mold plates to monitor the momentary separation of the core and cavity plates. Chen et al.183 developed a soft-sensor scheme for melt-flow-length measurement during injection mold filling. Panchal and Kazmer184 developed a button cell type in-mold shrinkage sensor to measure in-mold shrinkage. In addition to the control of machine variables, there is a unique sequence control in injection molding, which is the switchover point from the injection phase to the pack/hold phase. Typical signals used for determining the switchover point include machine variables such as time, ram position, and/or hydraulic pressure, as well as process variables such as nozzle pressure and cavity pressure. Chang185 compared the process capability of five switchover modes where the part quality of weight and dimension were concerned. The research concluded that the desirable modes for switchover should be based on, in descending order, cavity pressure, hydraulic pressure, stroke, time, and speed. A similar sequence was also suggested in other independent works (e.g., Sheth et al.186 ). Owing to the abrupt changes that occur at the moment of switchover, large disturbances could be introduced into the system. Zheng and Alleyne187 developed a comprehensive model representing an injection molding machine with fillto-pack dynamics and proposed a bumpless transfer in order to achieve a smooth transition from filling to packing. As the nonlinear and time-varying characteristics are inherent in the injection molding machine, advanced adaptive process control technologies are capable of reaching higher

17

standard performance than conventional PID if the machine dynamics are well modeled.

1.4 OBJECTIVE OF THIS BOOK In conclusion, applications of plastic products have grown phenomenally during the last several decades, and injection molding is the predominant production process. Guaranteeing high quality of the final molded product is important but difficult to implement because so many interrelated factors are involved. It requires the integration of skills that include an understanding of the complex characteristics of plastic materials, product design, mold design, process conditions, and process control. In other words, the ideal developer should be a part designer, a mold designer, a process engineer, a materials specialist, and even a machine specialist. This problem will not be solved as long as one deals with injection molding. The best we can do is to acknowledge the problem and seek skills and tools to overcome it. Computer modeling technology is one tool that provides the engineer with a link through all the elements in the development of a plastic product. Computer modeling for quality enhancement of injection molding can be organized into three categories: numerical simulation, optimization, and process control. These three terms are taken from the latest computer technologies and have become synonymous with future competitiveness for injection molding firms. Although there have been some introductory books on computer modeling of injection molding, none of them has involved all the above three essential ingredients on the purpose of improving the product quality, only discussing either the fundamentals or a specific aspect. As a result, the major problem for students and researchers who are desirous of acquiring extensive knowledge in injection molding is that applications of the latest computer technology in quality improvement are scattered about, and rarely introduced comprehensively or systematically in postgraduatelevel texts, forcing the students and researchers to turn to wade through stacks of published papers looking for useful information. This book will serve as a systematic and comprehensive introductory textbook on the computer modeling for quality enhancement of injection molding, with important expansions into the successful application of the latest computer technology. It is based on the constant efforts of authors and colleagues in this area over the last few years. The objective of this book is to provide what we have determined after years of working in this field to improve the product quality through computer modeling in simulation, optimization, and control. Students and researchers new to the field can get started with the basic information provided, and also scientists and people involved in the polymer industry, institutes, and institutions

18

INTRODUCTION

seeking new ways to gain a competitive edge can work closely with the latest information provided in the book. In general, the reader will obtain a comprehensive understanding and much practical knowledge about how the latest computer technology can benefit the injection molding industry. REFERENCES 1. Rosato D.V., Rosato D.V., Rosato M.G., Injection Molding Handbook. 3rd ed. 2000, Norwell: Kluwer Academic Publishers. 2. Osswald T.A., Hern´andez-Ortiz J.P., Polymer Processing: Modeling and Simulation. 2006, Munich: Hanser Gardner Publications. 3. Beaumont J.P., Nagel R., Sherman R., Successful Injection Molding. 2002, Munich: Hanser Publishers. 4. Potsch G., Michaeli W., Injection Molding: An Introduction. 2008, Munich: Carl Hanser Verlag. 5. Chen Z., Turng L.S., A review of current developments in process and quality control for injection molding. Advances in Polymer Technology, 2005. 24(3): 165–182. 6. Mok S.L., Kwong C.K., Lau W.S., Review of research in the determination of process parameters for plastic injection molding. Advances in Polymer Technology, 1999. 18(3): 225–236. 7. Kamal M.R., Kenig S., The injection molding of thermoplastics. Part I: Theoretical model. Polymer Engineering and Science, 1972. 12(4): 294–301. 8. Kamal M.R., Kenig S., The injection molding of thermoplastics. Part II: Experimental test of the model. Polymer Engineering and Science, 1972. 12(4): 302–308. 9. Wu P.C., Huang C.F., Gogos C.G., Simulation of the moldfilling process. Polymer Engineering and Science, 1974. 14(3): 223–230. 10. Stevenson J.F., A simplified method for analyzing mold filling dynamics. Part I: Theory. Polymer Engineering and Science, 1978. 18(7): 577–582. 11. Toor H.L., Ballman R.L., Cooper L., Predicting mold flow by electronic computer. Modern Plastics, 1960. 38(4): 117–120. 12. Harry D.H., Parrott R.G., Numerical simulation of injection mold filling. Polymer Engineering and Science, 1970. 10(4): 209–214. 13. Lord H.A., Williams G., Mold-filling studies for the injection molding of thermoplastic materials. Part II: The transient flow of plastic materials in the cavities of injectionmolding dies. Polymer Engineering and Science, 1975. 15(8): 569–582. 14. Williams G., Lord H.A., Mold-filling studies for the injection molding of thermoplastic materials. Part I: The flow of plastic materials in hot- and cold-walled circular channels. Polymer Engineering and Science, 1975. 15(8): 553–568. 15. Nunn R.E., Fenner R.T., Flow and heat transfer in the nozzle of an injection molding machine. Polymer Engineering and Science, 1977. 17(11): 811–818.

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2 BACKGROUND Huamin Zhou State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

In the previous chapter, it has already been pointed out that the factors influencing the quality of the final molded product include polymer materials, product design, mold design, process conditions, and process control of the machine. This chapter reviews the basic background of these factors, providing essential fundamentals for computer modeling in the later chapters of this book.

2.1

MOLDING MATERIALS

There is a general accepted definition for plastics that goes like this: any one of a large and varied group of macromolecular materials consisting wholly or in part of combinations of carbon with oxygen, hydrogen, nitrogen, and other organic and inorganic elements that, while solid in the finished state, at some stage in its manufacture is made liquid, and thus capable of being formed into various shapes, most usually (although not necessarily) through the application, either singly or together, of heat and pressure.1 The great economic significance of plastics is intimately tied to their properties, such as low density, wide range of mechanical properties, easy to process, low thermal/electronic conductivity, high chemical resistance, and reusability. A fundamental feature of plastics is their variety. There are over 17,000 plastic materials available worldwide. Within the most common plastic families, there are five major thermoplastic types that constitute about two-thirds of all thermoplastics. Approximately, 18 wt% is low density polyethylene (LDPE); 17%, polyvinyl chloride

(PVC); 12%, high density polyethylene (HDPE); 16%, polypropylene (PP); and 8%, polystyrene (PS). In turn, each has thousands of different formulated compounds. The use of a virtually endless array of additives, colorants, reinforcements, fillers, alloying, etc. permits compounding the raw materials to impart specific performance qualities. Plastics can be made to be hard, elastic, rubbery, crystal clear, opaque, electrically conductive, strong, stiff, outdoor weather-resistant, electrically conductive, or practically anything that is desired, depending on the choice of starting materials and molding methods. The material, together with the molding process, determines the final quality of the molded part. For example, plastics generally have different rates of shrinkage in the longitudinal and transverse directions of melt flow. Moreover, these directional shrinkages can significantly vary because of changes under injection pressure, melt temperature, mold temperature, and part shape and thickness. Therefore, a specific plastic can be molded using different machine settings so that dimensional tolerances of the part vary after each molding, or the machine can be set so that extremely close tolerances are met repeatedly. 2.1.1

Rheology

Rheology is the science of the deformation and flow of matter under force. The rheological properties of a melt govern the way it deforms and flows in response to applied forces, as well as the decay of stresses when the flow is halted. These properties play a key role in the injection molding process. In mold filling, it is viscosity, along with thermal properties, that determines the ability of the

Computer Modeling for Injection Molding: Simulation, Optimization, and Control, First Edition. Edited by Huamin Zhou.  2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc.

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26

BACKGROUND

melt to fill the mold, that is, the pressure required to force the melt through the runner and gate and into the cavity. After filling, the relaxation of stress in the melt affects the residual stresses in the finished part, which has an important effect on its mechanical properties. For these reasons, it is important to know something about the rheological behavior of polymers. The polymer materials exhibit properties that combine those of an ideal viscous liquid (i.e., exhibiting pure viscous deformation) with those of an ideal elastic solid (i.e., exhibiting pure elastic deformation). Thus, the rheological behavior of polymer melt is termed viscoelasticity. The mechanical behavior of polymers is dominated by such viscoelastic characteristics as tensile strength, elongation at break, rupture energy, relaxation, and creep. The viscous attribute of polymer melt can be characterized by viscosity, which is an important consideration in polymer processing. 2.1.1.1 Viscosity Viscosity is a measure of the melt’s inner resistance to flow processes. In injection molding, the flow processes involve mainly shear of the melt, because the melt adheres to the adjacent cavity surfaces. Shear flow can be shown, in a simplified manner, by a two-plate model. As shown in Figure 2.1, one of two parallel plates is stationary, while the other moves in a straight line with a velocity V . The liquid layers in between slide correspondingly and the melt is sheared. The velocity distribution is given by v1 =

V x2 H

(2.1)

The shear rate (γ˙ ) is calculated from the difference in velocity between the upper and the lower side of the volume element in relation to its thickness, as γ˙ =

V dv1 = dx2 H

(2.2)

Thus, the shear rate is uniform throughout the fluid. If V does not change with time, we have a steady, simple shear flow. Shear stress is the shearing force necessary to deform the material divided by the area of the volume element. If F is the total force required to move the upper plate (equal to the force required to hold the lower plate stationary) and A is the surface area of the plate that is in contact with the liquid, the shear stress (τ ) is given by τ=

F A

(2.3)

Viscosity (η) is defined as the ratio of shear stress τ to shear rate γ˙ in a laminar flow, as η=

τ γ˙

(2.4)

This shear viscosity represents a criterion for the flow resistance in the sheared liquid. The higher the viscosity, the higher is the level of shear stress that has to be exerted at the same shear rate. Under unchanged shearing force, the shear rate, and thus the flow velocity, increases as viscosity decreases. In simple systems, viscosity is independent of shear rate, and the flow is called Newtonian. Low molecular weight, single-phase liquids such as water, thin mineral oil, glycerine, and syrup are examples of Newtonian fluids. However, most polymer melts and some liquids exhibit non-Newtonian flow response when force is applied. That is, their viscosity depends on the rate of shear. Deviations from ideal Newtonian behavior may be of different types. As shown in Figure 2.2, common types for shear rate dependence are as follows: • shear thinning (pseudoplastic): decreasing viscosity with increasing shear rate; • shear thickening (dilatant): increasing viscosity with increasing shear rate; • mixed : shear thinning and thickening at different shear rate ranges. Polymers have shear-thinning property, which means that their resistance to flow decreases as the shear rate increases. This is due to molecular alignments in the direction of flow and disentanglements. Different polymers generally have their own flow and rheological properties, so that their non-Newtonian curves are different. Viscosity of polymer melt also depends on other parameters besides shear rate, such as temperature, molecular weight, pressure, and fillers. Owing to the shear-thinning action, the advantages of an increased shear rate for polymers are a less viscous melt and easier flow. When water (a Newtonian liquid without the shear-thinning action) flows through an open-ended pipe by pressure, doubling the pressure doubles the flow rate of the water. However, in a similar situation, but using a polymer melt (a shear-thinning liquid), if the pressure is doubled, the melt flow rate may increase from 2 to 15 times depending on the polymer used. For example, linear low density polyethylene (LLDPE), with a low shearthinning action, experiences a low increase of flow rate, which explains why LLDPE may cause more processing problems than other PEs in certain equipment. The higher shear-thinning melts include PVC and PS. A disadvantage of higher shear rates is that viscous heating may increase the melt temperature, potentially causing problems in cooling, as well as degradation and discoloration. An extremely high shear rate can lead to a rough product surface because of melt fracture and other causes. So for each polymer, there is a maximum shear rate beyond which such problems will develop.

MOLDING MATERIALS

FIGURE 2.1

27

Schematic representation of simple shear flow between parallel plates.

melt flow index (MFI) is also an inverse but simple measure of viscosity. High MI implies low viscosity.

FIGURE 2.2

Types of flow illustrating shear rate dependence.

During injection molding, melt shear rate in the gate can be very high. In a pin gate, the melt pressure flow can produce a shear rate in excess of 100,000 s−l in extreme cases. Typically, it ranges from 1000 to 10,000 s−l in gates, from 10 to 1000 s−l in runners, and from 0.001 to 100 s−l in cavities. Local flow rates and shear rates constantly change during filling, in direct relationship to the channel depths and the cross sectional area of flow. Added to this variation is the highly transient nature of the process: the melt pool in the barrel is at rest before injection, but during “rest” periods the melt is rapidly accelerated or decelerated. For Newtonian fluids, simple observations are sufficient to establish a general equation that describes how the material responds to any type of deformation. However, for more complex materials such as molten polymers, the development of an equation is very complicated, requiring many different tests. Two basic test instruments are capillary viscometers and rotational viscometers. The results of viscosity measurements are usually represented in a graph of viscosity versus shear rate, as shown in Figure 2.3. Present modeling of rheological behavior of non-Newtonian polymers is largely empirical. The developed models for viscosity is discussed in detail in Chapter 3, as well as the models of the following other processing properties. In practice, the melt index (MI) or

2.1.1.2 Viscoelasticity For polymers, there are two types of deformation or flow: viscous, in which the energy causing the deformation is dissipated; and elastic, in which that energy is stored. The combination produces viscoelastic polymers. The response to stress of all polymers is viscoelastic, meaning that it takes time for the strain to accommodate the applied stress field. The time constants for this response, referred to as the relaxation time (λ) will vary with the specific characteristics of the material and processing techniques. In the rigid section of a block polymer, the response time is usually in the order of microseconds to milliseconds. With resilient rubber sections of the structure, the response time can be as long as one-tenth of a second to several seconds. This difference in response time is the cause of failure under rapid loading for certain plastics. A useful parameter often used to estimate the elastic effects during flow is the Deborah number (D e ), defined as De =

λ tprocess

(2.5)

where t process is a characteristic process time. As mentioned above, viscosity is a material’s resistance to viscous deformation. Its unit of measure is Pascalsecond (Pa s). Polymer melt viscosities range from 2 to 3000 Pa s (glass, 1020; water, 10−l ). The resistance to elastic deformation is the modulus of elasticity (E ), which is measured in Pascal (Pa). Its range for a polymer melt is 1000–7000 kPa, which is called the rubbery range. There are not only two classes of deformation but also two modes in which deformation can be produced: simple shear and simple tension. The actual action during melting and molding, as in a screw plasticator, is extremely complex, with all types of shear–tension combinations. 2.1.2

Thermal Properties

The properties of polymers are influenced by their thermal characteristics such as melt temperature, glass transition

28

BACKGROUND

2

6 4 2 8 6 4 2 10

FIGURE 2.3

Viscosity versus shear rate for a polymer material.

temperature, thermal conductivity, specific heat, thermal diffusivity. What is more, these thermal properties affect the manufacturability of materials and determination of the best useful process conditions to meet product performance requirements. Therefore, to select proper materials, engineers must be aware of their mechanical characteristics, rheological properties, and also their thermal properties, dimensional stability. The examples of thermal properties of common polymers are listed in Table 2.1. It should be pointed out that the material properties of polymers are not constant and may vary with temperature, pressure, or phase changes. 2.1.2.1 Melt Temperature The melt temperature (T m ) determines the lowest temperature required for injection molding. It occurs as a relatively sharp point for semicrystalline polymers. Amorphous polymers do not have a distinct T m , they simply start melting as soon as the heat cycle begins. In reality, there is no single melt point but rather a range that is often taken as the peak of a differential scanning calorimeter (DSC) curve. The melt temperature is dependent on the processing pressure and the time subjected to heat, particularly during a slow temperature change for relatively thick melts. Also, if the temperature is over T m but too low, the melt’s viscosity will be high and more power will be required to process it. 2.1.2.2 Glass Transition Temperature The glass transition temperature (T g ), also called the glass–rubber transition temperature, is the reversible change in phase of a polymer from a viscous or rubbery state to a brittle glassy state. T g is the point below which polymer behaves similar to glass, being very strong and rigid. Above this temperature, it is not as strong or rigid as glass, but neither is it

brittle. The amorphous thermoplastics have a more definite T g when compared with their crystalline counterparts. T g is usually reported as a single value. However, it occurs over a temperature range and is kinetic in nature. As can be seen from Table 2.1, the value of T g for a particular polymer is not necessarily low, which immediately helps explain some of the differences in polymers. For example, because at room temperature, PS and PMMA are below their respective T g values, we observe these polymers to be in their glassy state. In contrast, at room temperature, the HDPE is above its T g , so that it is very flexible. When cooled below its T g , the HDPE also becomes hard and brittle. 2.1.2.3 Thermal Conductivity The thermal conductivity (k ) is an important factor because polymers are often used as an effective heat insulator in heat-generating applications and in structures requiring heat dissipation. In general, thermal conductivity is low for polymers. To increase conductivity, fillers such as metals, glass, or electrically insulating fillers (i.e., alumina) can be incorporated. Conductivity is decreased by using foamed polymers. Amorphous polymers show an increase in thermal conductivity with increasing temperature, up to the glass transition temperature T g . Above T g , the thermal conductivity decreases with increasing temperature. For semicrystalline polymers, the high degree of molecular order makes the thermal conductivity higher in the solid state than in the melt state. In the melt state, their thermal conductivity reduces to that of amorphous polymers. 2.1.2.4 Specific Heat The specific heat (C p ) represents the energy required to change the temperature of a unit

MOLDING MATERIALS

TABLE 2.1 Polymers PP HDPE PA PET ABS PS PMMA PC

Examples of Thermal Properties of Common Polymers

Density, Melt Glass Transition Thermal Conductivity, Specific Heat, Thermal Diffusivity, Thermal Expansion g/cm3 Temperature, ◦ C Temperature, ◦ C 10−4 cal/s cm◦ C cal/g◦ C 10−4 cm2 /s Coefficient, 10−6 /◦ C 0.9 0.96 1.13 1.35 1.05 1.05 1.20 1.20

168 134 260 250 105 100 95 266

5 −110 50 70 102 90 100 150

2.8 12 5.8 3.6 3 3 6 4.7

mass of material by 1 ◦ C. It is usually true that the specific heat only changes modestly in the range of practical processing and design temperatures of polymers. However, semicrystalline thermoplastics display a discontinuity in the specific heat at the melting point of the crystallites, because more heat is required to melt the crystallites. 2.1.2.5 Thermal Diffusivity The thermal diffusivity, defined as α = k /(ρC p ), is the material property that governs the process of thermal diffusion over time. Although the density, thermal conductivity, and specific heat all vary with temperature, thermal diffusivity is relatively constant. 2.1.2.6 Thermal Expansion Similar to metals, polymers generally expand when heated and contract when cooled. The linear coefficient of thermal expansion is related to volume changes that occur in a polymer because of temperature variations, which can be well represented in the PVT diagram (discussed in the following section). Although the linear coefficient of thermal expansion varies with temperature, it can be considered constant within typical design and process conditions. It is relatively high for thermoplastics; that is, for a given temperature change, many thermoplastics experience a greater volumetric change than that of metals. Expansion and contraction are controlled in polymers by cross-linking, addition of fillers or reinforcements, orientation, etc. Fillers and/or reinforcements in materials can significantly reduce thermal expansion. The thermosets are much more resistant to thermal changes. The degree of cross-linking has a direct effect, with some thermosets exhibiting no change at all. 2.1.3

29

PVT Behavior

Besides the thermal expansion, polymers also have the characteristic of compressibility. All materials compress under load, but most not as much as polymers. When pressure is applied to polymers, polymers will compress significantly (i.e., reduce in volume) in proportion to the amount of pressure applied. This may be (within the range of molding operations) as high as 2% of the original volume.2

0.9 0.9 0.75 0.45 0.5 0.5 0.56 0.5

3.5 13.9 6.8 5.9 3.8 5.7 8.9 7.8

81 59 70 80 60 50 50 68

The dependence of density (or its reciprocal, specific volume) on pressure and temperature is called the thermodynamic material behavior or PVT behavior. This PVT behavior has a decisive influence on the process course in injection molding—especially the holding phase—and on the characteristics of the final part, particularly on shrinkage and warpage.3 As the polymer is injected, it is not only hot and, therefore, expanded but also under significant pressure, which reduces its volume. This makes it difficult to arrive at a true shrinkage factor, because the actual change in volume depends on the type of polymer, the melt temperature, the injection pressure required to fill the cavity space, and the temperature at which it will be ejected from the mold.2 The PVT behavior of amorphous polymers differs fundamentally from that of semicrystalline ones, as shown in Figure 2.4. For both types of material, the specific volume in the melting range changes linearly with temperature. As pressure increases, the specific volume decreases. In the temperature range where they are solids, these two types of polymers differ: amorphous plastics have a linear dependence of specific volume on temperature, but semicrystalline ones have an exponential dependence. When crystalline materials are cooled below their transition temperature, the molecules arrange themselves in a more orderly way, forming crystallites. The process of crystallization results in a more orderly way and hence denser packing for semicrystalline materials relative to that of amorphous materials, and thus in lower specific volume. Moreover, for semicrystalline materials, the specific volume in the solid range depends on the cooling rate. With slow cooling, high degrees of crystallinity are obtained, and thus in low specific volumes.3 If a plastic part is free to expand and contract, its expansion/contraction property will usually be of little significance. However, if it is restricted or attached to another material having a different PVT characteristic, just as it undergoes during injection molding, its movement will be restricted, and the potential to develop thermal stresses exists, which can cause product warpage and distortion.

30

BACKGROUND

FIGURE 2.4 PVT behavior of an amorphous material (ABS) and a semicrystalline material (HDPE).

2.1.4

Morphology

The term morphology means “the study of the form,” where form stands for the shape and arrangement of parts of the object. When referred to polymers, the word morphology is adopted to indicate the following: • crystallinity, which is the relative volume occupied by each of the crystalline phases, including mesophases; • dimensions, shape, distribution, and orientation of the crystallites; • orientation of amorphous phase and fillers (if any).

directions (a random configuration). Examples include PS, acrylic (PMMA), acrylonitrile-butadiene-styrene (ABS), polycarbonate (PC), and PVC. Polymer molecules that can be packed closer together are more likely to form crystalline structures in which the molecules align themselves in some orderly pattern. Examples of crystalline thermoplastics include PP, PE, polyamide (nylon) (PA), fluorocarbons (PTFE, etc.), polyester (PET, PBT), and acetal (POM). As symmetrical molecules approach within a critical distance, crystals begin to form in the areas where they are the most densely packed. A crystallized area is stiffer and stronger, while a noncrystallized (amorphous) area is tougher and more flexible. With increased crystallinity, other effects occur. For example, with PE, there is increased resistance to creep, heat, and stress cracking, as well as increased mold shrinkage. In general, crystalline types of plastics are more difficult to process, requiring more precise control during fabrication. They also undergo larger volumetric changes when melting or solidifying and tend to shrink and warp more than amorphous types during processing. They have a relatively sharp melting point. That is, they do not soften gradually with increasing temperature but remain hard until a given quantity of heat has been absorbed, and then change rapidly into a low viscosity liquid. Amorphous plastics soften gradually as they are heated. Table 2.2 compares the basic performance of crystalline and amorphous plastics. As commercially perfect crystalline polymers are not produced, these are noncrystalline (amorphous) regions in a crystalline plastic, typically accounting for up to about 20% of the plastic, as shown in Figure 2.5. Thus, they are identified technically as semicrystalline thermoplastics, but in this book they are also called crystalline (as is conventional in the plastics industry). Plastic acts as a composite of amorphous and crystalline polymers. Both regions contribute their characteristic properties to the overall behavior. Process conditions influence the morphology of plastics. For example, if thermoplastics that normally crystallize TABLE 2.2 Comparison of the Basic Performance of Crystalline and Amorphous Plastics Crystalline

The variety of plastic characteristics (mechanical, optical, electrical, transport, and chemical) derives not only from the chemical structure of molecules but also from the microstructure and morphology. For instance, crystallinity has a pronounced effect on the mechanical properties of the bulk material because crystals are generally stiffer than amorphous material, and also orientation induces anisotropy and other changes in mechanical properties.4 Amorphous is a term that means formless. Amorphous thermoplastics have molecules moving in all different

High density High tensile strength and modulus High shrinkage Low ductility and elongation High viscosity Sharp melting point Usually opaque Great effect of orientation Higher chemical resistance

Amorphous Low density Low tensile strength and modulus Low shrinkage High ductility and elongation Low viscosity Broad softening range Usually transparent Low effect of orientation Poor chemical resistance

PRODUCT DESIGN

(a)

(b)

31

(c)

FIGURE 2.5 Amorphous and crystalline structures of polymers: (a) amorphous, (b) crystalline, and (c) semicrystalline.

(a)

FIGURE 2.6

POM micrographs of (a) pure PP and (b) PP with 5% mass fraction of POE.

(semicrystalline plastics) are quenched extremely rapidly (the process of cooling the hot melt to solidify), the result is a complete (or far more) amorphous solid state. Its properties can be significantly different from those of materials cooled slowly (semicrystalline state). The effects of time are similar to those of temperature in the sense that any given plastic has a preferred or equilibrium structure into which it would prefer to arrange itself. However, it is unable to do so instantaneously or at least on “short notice.” If given enough time, the molecules will rearrange themselves into their preferred pattern. Heating speeds up this action. During this action, severe shrinkage and property changes could occur in the processed plastics. It is easy to determine whether the plastic is amorphous or crystalline by observing the sample using polarized light. Amorphous areas appear black, while crystalline areas are clear and have multicolored patterns. The reason of this difference is that crystallized molecules fold together in a uniformly orderly manner, whereas amorphous ones do not. The POM micrographs of PP and PP/POE blends are shown in Figure 2.6.5 The orientation of polymer molecules during processing is discussed in Section 2.4.3.

2.2

(b)

PRODUCT DESIGN

One must recognize that product design is as much an art as a science. The product design is mainly based

on the functional requirements. However, the moldability, productivity, and performance should also be taken into consideration. Regardless of the material used, there are certain basic design rules that apply to all plastics products, most of which can be presented concisely by describing design guidelines for plastic products.6 This idea is followed in the following section. 2.2.1

Wall Thickness

In injection molding, uniform wall thickness is strongly preferred. Thick sections near thin sections will cool more slowly and shrink away from the mold long after the gates are frozen. This can cause sink marks, residual thermal stresses, warpage and part distortion, and variations in color or transparency. Thick-to-thin wall transitions, if unavoidable, should be as gradual as possible and not vary more than 3 : 1 in ratio of thickness (Fig. 2.7). Figure 2.8 illustrate the proper design of corners in a plastic product. The inside radius should be a minimum of half the thickness of the primary wall. The outside radius should be the inside radius plus the wall thickness. This ensures that the wall thickness is constant throughout the corner. Structural stress considerations may determine the minimum wall thickness, which is supposed to achieve some target safety factor. Other factors are thermal, electrical, or acoustic insulation requirements. Impact resistance generally increases with thickness, but if one section is overly rigid, distribution of impact energy will

32

BACKGROUND

FIGURE 2.7 Effects of adjoining thick/thin wall sections and how to disguise sink marks at a thick T section.

(a)

(b)

FIGURE 2.8 (a) Use of the “ball” test to gauge whether a corner design is acceptable; panel (b) is a preferable design. Ri ≥ 1/2 t and Ro = Ri + t gives uniform wall around the corner.

be adversely affected. In addition, there are minimum and maximum recommended wall thickness ranges related to material and process. Most injection-molded parts range from 0.80 to 4.8 mm in thickness. Thicker walls generally are found in larger parts. Some materials flow easier (lower viscosity or greater shear sensitivity) than others, so their minimum recommended thickness is smaller. For example, recommended minimum wall thickness for a typical ABS is about 1.1 mm, while minimum wall thickness for an injection-molded LCP is only 0.20 mm. For injection molding, spiral mold flow curves, available from materials suppliers, show flow length versus injection pressure for given wall thicknesses and can be a guide to minimum wall thicknesses for an unfamiliar material. The minimum wall thickness is desirable to minimize material costs and cooling time, but this may lead to greater scrap. In addition, the wall should be thick enough to resist ejection pin forces. 2.2.2

Draft

As molded parts tend to shrink onto cores or male sections of the mold, they are almost always provided with a small taper in the direction of mold movement to allow part

FIGURE 2.9 Draft angles on molded parts. Parts shrink more over male cores so draft should be greater than on female cavities, as shown. Equal angles are used for core and cavity to provide uniform wall thickness. Angles shown are exaggerated for clarity.

ejection with minimum ejection force. Draft in the line of draw is shown on a part drawing as a draft angle of how many degrees per side. To maintain uniform wall thickness and to minimize adhesion of warm parts to the cavity or female mold sections, draft angles are generally equal for both male and female parts, as shown in Figure 2.9. The exception to the rule is when it is desired to have the part stay in the cavity when the mold opens; in this case, there is little or no draft on the cavity and knockout pins, or slight undercuts have to be added to the cavity side of the mold. For unreinforced thermoplastics, a common recommendation is a minimum draft angle of 1/2◦ . For glass-reinforced resins, which have much lower molding shrinkage, the recommended draft is at least 1.5◦ . 2.2.3

Parting Plane

Strictly speaking, the parting plane is a term in mold design (as discussed in Section 2.3.2). However, it must be thought ahead in the product design stage. Ideally, the parting plane should be a single flat surface to minimize mold-making costs, ensure optimum sealing against injection pressures, minimize flash, and simplify venting of air. Stepped or telescoped mold sections add to mold maintenance and flash removal costs, as shown in Figure 2.10. Where the parting

PRODUCT DESIGN

33

may be assembled after molding to produce a single part with an undercut or internal thread. Threads or other shapes can be molded around a removable insert, and the insert unscrewed manually after demolding. (a)

(b)

2.2.6

(c)

(d)

FIGURE 2.10 Parting planes for moldings: (a) a split parting plane costs more for the mold and to maintain than (b), which has a straight P.L.; a bead as in (c) or a deliberate mismatch as in (d) help hide parting line flash.

plane must be on a side wall, a bead or deliberate mismatch will disguise the inevitable mismatch of the two halves of the mold. The parting plane should be on the largest part dimension, if possible, and at a right angle to the direction of mold closing and opening. If properly aligned and operated, flash should be of the order of 0.02–0.03 mm thick. It is easiest to remove flash from a sharp corner, but this risks corner chipping; therefore, a common recommendation is to allow a small radius at the outer corner of the part. 2.2.4

Sharp Corners

A radius of 0.25 mm, while being very small, is sufficient to break a sharp internal or external corner without being noticeable to a casual observer. For highly loaded parts, more generous radius will reduce stress concentration effects. For highly reinforced parts or compression molded composites, more generous radii are needed to promote even flow of resin through the fibers and avoid appearance problems. Inside and outside radii should be proportioned to maintain uniform wall thickness where possible. 2.2.5

Undercuts

Various ingenious mechanisms have been developed for molding undercuts, side-opening holes, internal and external threads, and other part features requiring motions of mold parts perpendicular to the normal axial motion of the clamp side of the machine. Side-core mechanisms may be operated manually, hydraulically, or electromechanically. Sometimes, it is more economical or practical to avoid the need for side-acting mechanisms. For example, a hole can be drilled in a secondary operation or a thread tapped, utilizing the idle time of the machine operator. Or, two parts

Bosses and Cored Holes

Bosses provide cored holes that can be used for mechanical fastener locations. The boss distributes bearing loads and transfers fastener loads to the main structure. Excessively, high bosses tend to trap air, so boss height normally should not exceed twice the boss diameter. Slenderness ratio of a cored hole is its depth divided by diameter. If the core pin is supported in the mold at one end only, it will be more highly stressed in bending than if supported at both ends. Core pins present flow obstructions in injection molding and can cause serious weld line problems if gating is not appropriately chosen. An alternative that avoids weld lines is to core a shallow blind hole, which can be used as a drill starter for a secondary drilling operation of any depth to diameter ratio.

2.2.7

Ribs

Ribs are used for two main purposes: to improve melt flow into a section, such as a corner or a large boss, and to increase flexural stiffness of the part. There are several ways to increase stiffness or flexural modulus in plastic products. For example, for a rectangular cross section acting as a beam or as a plate, as the wall thickness W is increased, the section modulus increases in proportion to W 2 , while the weight increases only in proportion to W . Thus, a fourfold increase in stiffness entails doubling of the weight. Changing to a reinforced polymer can increase section modulus, and also increases material and processing costs significantly. The most economical approach to increasing stiffness may be the use of reinforcing ribs. For example, if a rectangular section of unit width B = 1 and thickness W = 0.1 is compared with the same section with one rib of unit depth H = 1 and thickness W , the section modulus is increased 25 times while the weight is only doubled. Adding stiffness not only reduces deflection and creep strain but also increases natural frequency and impact resistance. Figure 2.11 gives suggested proportions for ribs in thermoplastic injection molding. An alternative to ribbing for constant thickness parts is corrugation or forming of shallow pyramids in a sheet, as shown in Figure 2.12. Lengthwise corrugations are often seen in fiberglass panels for awnings; they stiffen panels in the longitudinal direction only. Pyramidal indentations add stiffness in two directions, and many forms of pyramids and prisms have been used for building construction with fiber-reinforced plastic panels.

34

BACKGROUND

FIGURE 2.11 Recommended proportions for ribs in thermoplastic moldings.

2.3

MOLD DESIGN

After materials and products have been decided, the next major step is to optimize the mold design so as to improve the manufacturing process and minimize possible molding defects. There are many rules for designing molds. It is usually helpful to follow the rules, but occasionally, it may work out better if a rule is ignored and an alternative way is selected. In this section, the most common rules are noted for productivity and performance.2 It should be pointed out that the pure structural design for injection molds, such as moldbase, ejection system, and alignment system, is not discussed here. 2.3.1

Mold Cavity

The mold cavity space is a shape inside the mold, “excavated” (by machining the mold material) in such a manner that when the molding material is forced into this space, it will take on the shape of the cavity space and, therefore, the desired product (Fig. 2.13). It has the following functions: • • • •

to to to to

distribute the melt, guarantee the part dimensions, withstand the melt pressure, and guarantee the surface quality of the part.

First, the cavity has to distribute the melt to fill all sections of the cavity. With regard to the second function, the shape of the cavity is essentially the “negative” of the

shape of the desired product, with dimensional allowances added to allow for shrinking of the plastic. Third, the high viscosity of the thermoplastic material requires high injection pressures, so the mold and all parts, such as cavity inserts in the mold, must be made from strong materials (i.e., steel), to resist high mechanical loads. If the mold has inserts, attention must be taken in mold design to prevent possible core displacement during the filling phase, because of unbalanced flow. Finally, the surface of the cavity must have the surface quality required for the molding. The shape of the cavity is usually created with chip-removing machine tools, or with electric discharge machining (EDM), with chemical etching, or by any new method that may be available to remove metal or build it up, such as galvanic processes. The cavity shape can be either cut directly into the mold plates or formed by putting inserts into the plates. Many molds, particularly molds for larger products, are built for only one cavity space (a single-cavity mold). But many molds, especially large production molds, are built with two or more cavities (Fig. 2.13b). The maximum number of cavities in a mold is determined by technical and economic restrictions. The technical criteria include the size of each cavity and the maximum mold dimension, the equipment of the available injection molding machine (multicavity molds may require larger machines with larger platen area, more clamping capacity, and larger injection unit), and the demands on part quality. Economic criteria are the date of delivery and the costs of part production. It takes only little more time to inject several cavities than to inject one, that is, the production increases in proportion to (approximately) the number of cavities. In addition, a mold with more cavities is more expensive to build than a single-cavity mold, but not necessarily in proportion to the cavity number. Today, most multicavity molds are built with a preferred number of cavities: 2, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, and 128. These numbers are selected because the cavities can be easily arranged in a rectangular pattern, which is easier for designing and dimensioning, for manufacturing, and for symmetry around the center of the machine. A smaller number of cavities can also be laid out in a circular pattern, even with odd numbers of cavities, such as 3, 5, 7, 9. It is also possible to make cavity layouts for any number of cavities. The basic demand on cavity arrangement is that all parts made within one shot should be identical. By convention, the hollow (concave) portion of the cavity space is called the cavity. The matching, often raised (or convex) portion of the cavity space is called the core. Most plastic products are cup-shaped. This does not mean that they look like a cup, but they do have an inside and an outside. The outside of the product is formed by the cavity, the inside by the core. The alternative to the cup shape is the flat shape. In this case, there is no specific convex

MOLD DESIGN

(a)

(b)

(d)

35

(c)

(e)

FIGURE 2.12 Methods of stiffening plastic structures: (a) hat section, (b) corrugations, (c) bidirectional corrugation, (d) crowning, and (e) doming.

(a)

(b)

FIGURE 2.13 Illustration of basic mold with (a) one cavity space and (b) two cavity spaces.

portion, and sometimes, the core looks like a mirror image of the cavity. Typical examples for it are plastic knives, game chips, or round disks such as records. Usually, the cavities are placed in the mold half that is mounted on the injection side, while the cores are placed in the moving half of the mold. The reason for this is that all injection molding machines provide an ejection mechanism on the moving platen (not on the injection side) and the products tend to shrink onto and cling to the core, from where they are then ejected. 2.3.2

Parting Plane

Figure 2.13 shows the cavity space inside a mold. To be able to produce a mold and to remove the molded pieces, we must have at least two separate mold halves, with the cavity in one side and the core in the other. The separation between these plates is called the parting plane. Actually, this is a parting surface or plane, but, by convention, in this context, it is also referred to as a part line (designated P.L.). In a side view or cross-section through the mold, this area is seen as a line, as shown in Figure 2.14.

FIGURE 2.14 parting line.

Illustration of schematic mold, showing the

The parting line can have any shape, but for ease of mold manufacturing, it is preferable to have it in one plane. The parting line is always at the widest circumference of the product, to make ejection of the product from the mold possible. If the parting line is poorly finished, the plastic will escape, which shows up on the product as an unsightly sharp projection, or “flash,” which must then be removed after demolding. There is even a danger that the plastic could squirt out of the mold and do personal injury. Sometimes, there are other parting (or split) lines than that separate the basic cavity and the core halves. If the cavity must separate (split or retract) to make it possible to eject the molded product as the mold opens, these are separating lines between these two (or more) cavity sections. Figure 2.15 shows a simple product but with rim and projection, where side cores are needed. These side cores, or split portions of the cavities, can represent just small parts of the cavity, or even only small pins to create holes in the side of the products, but they could also be

36

BACKGROUND

c P.L.

P.L. 2

b a

(a)

(b)

FIGURE 2.16 Illustration of schematic mold: (a) cold sprue and (b) cold runner. FIGURE 2.15 Schematic illustration of product but with rim and projection. Cavity is split, creating an additional P.L. 2.

sections molding whole sides of a product, for example, with beverage crates or large pails. As the polymer flows into the cavity space, the air trapped in it must be permitted to escape. Typically, the trapped air is being pushed ahead by the rapidly advancing melt front, toward all points farthest away from the gate. The faster the melt enters—which is usually desirable—the more the trapped air is compressed if it is not permitted to escape, or vented. This rapidly compressed air heats up to such an extent that the plastic in contact with the air will overheat and possibly be burnt. Even if the air is not hot enough to burn the plastic, it may prevent the filling of any small corners where air is trapped and cause incomplete filling of the cavity. Most cavity spaces can be vented successfully at the parting line, but often additional vents, especially in deep recesses or in ribs, are necessary. Another venting problem arises when plastic fronts flowing from two or more directions collide and trap air between them. Unless vents are placed there, the plastic will not “knit” and may even leave a hole in the wall of the product. This can be the case when more than one gate feeds one cavity space, or when the plastic flow splits into two after leaving the gate, because of the shape of the product or the location of the gate. Within the cavity space, plastic always flows along the path of least resistance, and if there are thinner areas, they will be filled only after the thicker sections are full.

2.3.3

Runner System

Besides the cavity spaces in a mold, there must be a runner system in order to bring the plastic into the cavity spaces. The flow passages are the sprue, from where the machine nozzle contacts the mold, the runners, which distribute the plastic to individual cavities, and the gates, which are

(usually) small openings leading from the runner into the cavity space. The geometric dimensions of the runner system should be such that the flow resistance is at a minimum. This reduces pressure drop and mechanical stress on the material. Sharp edges and large differences in cross- section should be avoided. On the other hand, if a conventional runner is used, its volume should be as small as possible to minimize material waste.3 There is a great variety of sprues, runners, and gates. Different runner systems are in use to meet different processing requirements. The most popular are cold and hot runners. With a cold runner, the melt flowing from the sprue to the gate solidifies by the cooling action of the mold as the melt in the cavity or cavities does. With a hot runner, the sprue to the gate is heated by heater bands and/or insulated from the chilled cavity or cavities, so that the melt remains hot and never cools; the next shot starts from the gate, rather than from the nozzle, as in a cold runner. Two sample methods of cold runners are illustrated in Figure 2.16. Panel (a) of this figure shows the simplest case of a single-cavity mold, with the plastic injected directly from the sprue into the cavity space. This method is frequently used for large products. It is inexpensive but requires the clipping or machining of the relatively large (sprue) gate. Panel (b) shows a typical (2-plate) cold runner system, with the plastic flowing through the sprue and the runner and entering the cavity space through relatively small gates, which break off easily after ejection. Instead of the two cavities as shown here, there can be any number of cavities supplied by the cold runners. For a multicavity mold, the runner system must be designed in such a way that melt of the same temperate and pressure fills the cavities simultaneously and uniformly; otherwise, moldings of different qualities and properties would be produced during one shot. This so-called balancing flow can be achieved by changing the runner size

MOLDING PROCESS

37

and length. Changing the gate size may provide a seemingly balanced filling. However, it affects the gate freeze-off time greatly, which is detrimental to part uniformity. Whenever possible, a naturally balanced runner system should be used to balance the flow of material into the cavities. If a naturally balanced runner is not possible, then the runner system should be artificially balanced. 2.3.4

Cooling System

In thermoplastics injection molding, the material is heated in the molding machine to its processing (melt) temperature. After injection, the plastic must be cooled so that the molded piece becomes rigid enough for ejection. Cooling may proceed slowly, by just letting the heat dissipate into the mold and from there into the environment. This is not suitable for large production. For a production mold, the cooling system is an essential mold feature, requiring special attention in mold design. (For thermosets, the material is heated in the mold for curing so that a heating system is needed.) Good cooling to remove the heat efficiently is very important for injection molding. Rapid and uniform cooling is achieved by a sufficient number of properly located cooling channels. Rapid cooling improves process economics (productivity) by shortening the molding cycle, whereas uniform cooling improves product quality by preventing differential shrinkage, internal stresses, and mold release problems. As illustrated in Figure 2.17, a mold cooling system typically consists of the following parts: (i) temperature controlling unit, (ii) pump, (iii) supply manifold, (iv) hoses, (v) cooling channels in the mold, and (vi) collection manifold. There are serial and parallel flow patterns in designing cooling channels. A serial pattern means that the flow is from channel to channel in sequence (Fig. 2.18a), whereas a parallel pattern means that the flow is split (Fig. 2.18b). In many multicavity molds, the cooling channels are arranged so that they are partly in parallel and partly in series (Fig. 2.18c). Sometimes, the mold may consist of areas too far away to accommodate regular cooling channels. Alternate methods of cooling these areas uniformly with the rest of the part involve the use of baffles, bubblers, spiral cores, and thermal pins. The physics and mathematics of mold cooling are quite complicated. Computer programs can evaluate the planned cooling layout only after the mold is designed. But the designer wants to know how to design the best cooling layout in the first place. There are several rules, based on experience, to help the designer. They are as follows: • Only moving coolant is effective for removing heat. Stagnant coolant at ends of channels, or in any pocket, does nothing for cooling.

FIGURE 2.17

A typical cooling system for injection molding.

• All cavities (and cores) must be cooled with the same coolant flow rate at a temperature that is little different from cavity to cavity (or from core to core). The coolant temperature will rise as it passes through cooling channels. If the temperature difference between cavities (or cores) is too large, the molded parts will lose accuracy. • The amount of heat removed depends on the flow rate of the coolant. The faster the coolant flows, the better it is, because a greater volume will flow through the channels, and there will be less temperature rise of the coolant. • The coolant must flow in a turbulent flow pattern, rather than in a laminar flow pattern. Turbulence within the flow causes the coolant to swirl around, thereby continuously bringing fresh, cool liquid in contact with the cooling channel walls and removing more heat. • The channel sizes (cross sections) must be calculated to ensure the same flow rate in all channels. • Heavy sections of the product (easy to cause potential shrinkage and sink marks) and the difficult-to-cool areas of the mold (such as thin and slender core pins, blades, and sleeves) must be considered first.

2.4 MOLDING PROCESS The objective of molding is to achieve complete filling without short shots and overpressurization while avoiding flash, sink marks, large dimensional deviation, warpage, high residual stresses, unfavorable orientation, sticking in the mold, and poor mechanical properties.

38

BACKGROUND

Cooling channels

(a)

FIGURE 2.18 cooling.

2.4.1

(b)

(c)

Schematic layout of (a) series cooling, (b) parallel cooling, and (c) parallel–series

The Molding Cycle

A typical example of an injection molding cycle is shown in Figure 2.19. Three main stages are recognized in the molding: filling, packing/holding, and cooling. During the filling stage, a hot polymer melt rapidly fills a cold mold reproducing a cavity of the desired product shape. During the packing/holding stage, the pressure is raised and extra material is forced into the mold to compensate for the effects that both temperature decrease and crystallinity development increase the density during solidification. During the cooling stage, the polymer melt cools down and solidifies sufficiently so that the product is stable enough for ejection.4 After cooling, the mold is opened, the part is ejected from the mold, and the mold is then closed again in readiness for the next cycle to begin. This stage is called the ejection stage. Although it is economical to have quick opening and closing of the mold, rapid movements may cause undue strain on the equipment, and if the mold faces come into contact at high speed, this can damage the edges of the cavities. Also, adequate time must be allowed for the mold ejection, depending on the part dimensions. For parts

FIGURE 2.19

to be molded with inserts, resetting involves the reloading of inserts into the mold. After resetting, the mold is closed and locked, thus completing one cycle.7 2.4.1.1 Filling Stage The filling stage denotes the part of the process between the beginning of mold filling and the point at which the machine is switched to holding pressure. Usually, the filling stage is performed under velocity-controlled conditions; that is, the screw forces the plasticated material into the cavity with a given velocity profile. Pressure is the driving force that overcomes the resistance of the polymer melt, pushing the polymer to fill the delivery system (the sprue, runner, and gate) and mold cavity. The injection pressure increases gradually with the filling process. The pressure distribution in the polymer melt along the flow path can be schematically illustrated as shown in Figure 2.20. In general, the injection pressure required to complete filling is correlated with relevant material, mold design, and process parameters, ranging from tens of megapascals to more than one hundred megapascals. Nowadays, higher injection pressure is needed to ensure

A typical example of an injection molding cycle.

MOLDING PROCESS

FIGURE 2.20 the cavity.

39

Pressure decreases along the delivery system and

that the mold is completely filled. The reasons include the following: 1. The thinner the wall thickness of the product, the more difficult it is to push the plastic through the cavity gap, thus demanding higher pressure. As material usually accounts for 50–80% of the total cost of a plastic product, it is highly desirable to reduce the weight (mass) of a part to a bare minimum. This usually means reducing the wall thickness as far as possible without affecting the usefulness of the product. 2. The colder the injected polymer melt, the higher its viscosity, and the more difficult it becomes to fill the mold. Higher temperature makes the melt easier to flow and fill the mold, but also require more time to cool down the plastic so that it can be ejected safely. This means more power (for heating and cooling) and longer cycles. Therefore, it is often better to inject at the lowest possible temperature, even if higher pressure is needed. The injection speed (injection time) depends chiefly on the type of molding and the material involved; however, it can be optimized based on pressure loss and temperature change. Very high injection speeds result in high pressure losses, because of the high volume flow, whereas extremely low injection speeds lead to low melt temperature (due to cooling) and a reduction of the free channel crosssection (due to solidification of the melt close to the wall), and thus higher pressure losses. The injection speed should be within the range of minimum pressure. Moreover, an optimized injection speed should assure the average material temperature to be constant over the molding for reasons of quality; that is, there is no difference between the temperatures at the beginning (injection temperature) and the end of the flow path. As the plastic fills the cavity space under high pressure, the pressure will tend to open the mold cavity at the parting line. The separating force is equal to the product of the

FIGURE 2.21

Typical cavity pressure trace.

pressure and the projected area at the parting line. From this, it becomes clear that the clamping force, the force exerted on the mold by the molding machine, must be at least as great as this separating force to keep the mold from opening (cracking open) during molding. 2.4.1.2 Packing/Holding Stage As soon as the cavity is filled, the pressure increases rapidly. The packing/holding stage starts at the switchover point and ends at the end of holding pressure exerted by the machine, and thus includes both the packing phase (compaction) and the holding phase (compensation). Switching to packing pressure is an important factor in avoiding pressure peaks, and thus the consequent overloading of the mold. The required packing/holding time depends on the properties of the materials being molded and the cooling rate. During the packing/holding stage, the melt cools and solidifies, and flow of material continues, at a slower rate, to compensate for any loss in volume of the material because of partial solidification and associated shrinkage. Unless pressure is maintained on the melt as it cools, it will shrink away from the mold surfaces or it will not have the density required for high strength. The packing/holding stage is performed under pressure control. Usually, this means that the screw is loaded with a pressure that can be adjusted in 5−10 different steps. The pressure profile has to be used to optimize shrinkage and warpage behavior of the part. After holding, the pressure in the mold cavity begins to drop. Figure 2.21 illustrates a typical cavity pressure trace in an injection molding cycle. Furthermore, the course of state described by PVT diagrams is helpful to understanding the physical processes in the cavity (Fig. 2.22).8 To draw this diagram, the pressure and temperature readings at a certain point in the cavity are entered isochronously onto the PVT diagram of the material.3 2.4.1.3 Cooling Stage To be more precise, cooling starts from the first rapid filling of the cavity and continues

40

BACKGROUND

removal from the cavity, the release of residual stresses results in shrinkage and warpage of the molding. At the cooling stage, once the injection unit is retracted, the plastication process for the next shot can begin. With a properly selected machine, the plastication process is completed before the cooling process of the molding is accomplished. In practice, which process finishes first depends mainly on the wall thickness of the molding and the volume of material being plasticated. If the plasticating performance of the machine is not adequate, the cycle time is determined by the plastication time, and production costs increase. 2.4.2

FIGURE 2.22

PVT diagram of the course of state.

during packing/holding. However, the cooling time must be extended beyond the packing/holding stage, as the molding normally has not yet cooled down sufficiently and is not stable enough for demolding (ejection). The length of the cooling stage is a function of the wall thickness of the part, the material used, and the mold cooling system. Because of the low thermal conductivity of polymers, the cooling time is usually the longest period in the molding cycle. At the point of holding pressure removal, the restriction between the mold cavity and the runner channels may still be relatively fluid, especially on thick parts with large gates. Because of the pressure drop, there is a chance for reverse flow of the material from the mold until the material adjacent to the gate solidifies (the sealing point is reached). Reverse flow can be minimized by proper design of the gates such that quicker sealing action takes place on plunger withdrawal. Following the sealing point, there is a continuous drop in pressure as the material in the cavity continues to cool. Apart from factors such as injection time, temperature, and holding time, the cooling time can have a significant effect on the properties of the molding. Following the holding phase, the molding remains in the mold for continued cooling. As long as the molding remains in the cavity, shrinkage and warpage are inhibited mechanically by the surrounding walls of the mold. Instead of deformation, residual stresses are built up within the molding during cooling. After its

Flow in the Cavity

As melt flows into the cavity, the situation cannot be described in terms of pressure flow between parallel plates with a gap equal to the mold clearance, because a frozen layer forms immediately at the cavity wall. Moreover, the melt flow during mold filling is much more complicated than that described in Section 2.1.1.1, because it is a nonisothermal, non-Newtonian, transient flow. However, it is desirable that the melt still moves in even “layers” and does not interfere with each other during processing. Nonlaminar flows may cause potential problems on fabricated products such as poor surface finish. An important phenomenon during filling the cavity is termed the fountain flow (Fig. 2.23). Because the melt velocity is the highest at the center and decreases from the center to the wall, the melt does not reach the wall or surface of the frozen wall layer by simple forward advance, but rather it tends to flow down the center of the cavity to the melt front and then flow out toward the wall. This can have an important effect on the direction of the flowinduced orientation of the polymer molecules. A weld line (also called a weld mark or a knit line) is formed when separate melt fronts meet. Weld lines can be caused by holes or inserts in the part, multiple gates, or variable wall thickness, where hesitation or race tracking occurs. Weld lines are generally undesirable when part strength and surface appearance are major concerns. If weld lines cannot be avoided, they can be positioned at low stress and low visibility areas by adjusting the gate position and dimension. The strength of weld lines can be improved by increasing the local temperature and pressure at their locations. A phenomenon that can lead to a complex pattern of weld lines is “jetting.” Jetting occurs when polymer melt is pushed at a high velocity through restrictive areas, such as the nozzle, runner, or gate, into open, thicker areas, without forming contact with the mold wall. The buckled, snakelike jetting stream causes contact points to form between the folds of melt in the jet, creating small-scale “welds” (Fig. 2.24).

MOLDING PROCESS

41

Mold

FIGURE 2.23

FIGURE 2.24

2.4.3

Melt flow pattern of the fountain flow.

Illustration of jetting.

Orientation

Molecular orientation, or simply orientation, arises in injection molding and may have desirable or undesirable consequences. It refers to the preferential alignment of macromolecules in the product. In injection molding, alignment is in the direction of flow of the melted polymer into the cavity. Tensile strength in the orientation direction is basically that of the carbon–carbon or other covalent bonds in the polymer backbone. In the transverse direction, the strength is provided by the secondary intramolecular bonds. Hence, uniaxially oriented materials will be much stronger and stiffer in one direction than in another. It is possible to limit the degree of orientation or to deliberately exploit it by controlling the process. First, it is necessary to understand its origin. During injection molding, the expanding flow front from the gate adheres to the cooler mold walls very quickly. Flow continues by stretching and sealing along the circumference. Shear stresses result from stretching, which changes the bond angles and uncoils the macromolecules. As the melt cools, viscosity increases and stress relaxation rate decreases. Therefore, some of the stretching or uncoiling becomes permanently frozen-in. Distribution and

orientation of the stretched molecules vary with the flow pattern and cooling rates in the cavity.6 The “fountain flow” effect keeps the bulk of the melt from freezing to the walls, but the outer layers will freeze as soon as they make contact (Fig. 2.25). These frozen outer layers are completely unoriented. However, some ends of the layers just below are attached to the frozen layer and are still moving with the melt front. These layers will be subject to the highest shear rates and will be the most highly oriented. Closer to the center of the section, the shear rates and stresses, and hence the orientation, will be lower, partly because the center has been thermally insulated from the walls by those layers that have frozen. When injection ceases, or the gate freezes, it is liable to have no more shear stress and the orientation relaxes. Orientation leads to anisotropy and residual stresses. In brief, orientation strengthens and stiffens in the direction of orientation at the expense of transverse properties. In injection molding, any variation in processing that keeps the molding resin hot throughout filling allows increased relaxation and, therefore, decreased orientation. Some of the steps that can be taken to reduce orientation are as follows: • Faster Injection (Up to a Point): less cooling during filling, hence a thinner initial frozen layer; low viscosity because of shear thinning; and less crystallinity all favor lower subsurface orientation. But excessively high injection speed can cause high surface orientation. • Higher Melt and Mold Temperatures: lower melt viscosity, easier filling, and greater relaxation favor reduced orientation. • Reduced Packing Time and Pressure: overpacking inhibits relaxation processes. • Reduced Gate Size: larger gates take longer to freezeoff and permit increased orientation. 2.4.4

Residual Stresses, Shrinkage, and Warpage

Shrinkage is inherent in the injection molding process. Shrinkage occurs because the density of polymer varies

42

BACKGROUND

(a)

(b)

(c)

(d)

FIGURE 2.25 Illustration of molecular orientation origin for a thin rectangular plaque having a single tab gate: (a) stretching of an originally spherical volume element, (b) effects of velocity, freezing and fountain flow, (c) shear stresses resulting from the flow patterns, and (d) orientation pattern across the section.

from the processing temperature to the ambient temperature. During injection molding, the variation in shrinkage both globally and through the cross section of a part creates internal stresses. These so-called residual stresses are mechanical stresses in the molding in the absence of the application of external force but act on a part with effects similar to externally applied stresses. Residual stresses are caused mainly by the different cooling rates in the various layers in the cross section of a molding,9,10 as illustrated in Figure 2.26. There is always a cold outer layer and a warm core area. The cooler-solidified material of the outer layer forms a solid shell that impedes the contraction of the core during its slower cooling process. If different layers were able to slide freely over one another, a profile of the molding similar to that in Figure 2.26b would emerge. The different layers are, however, mechanically linked to one another, so that they constrain one another during the process of thermal contraction. This causes tensile stresses in the core layers and compressive stresses in the outer layers (Fig. 2.26c). Over the cross section as a whole, the tensile and compressive stresses are in equilibrium. Apart from this principal cause of residual stresses, packing pressure (overpacking) and flow effects (fountain flow effects) within the melt also play significant roles. After the molding is ejected from the cavity, some of the generated stress is relieved by deformations, and the contraction process (shrinkage) that follows takes place without any external constraints. The shrinkage of molded plastic parts can be as much as 20% by volume, when measured at the processing temperature and the ambient temperature. Semicrystalline materials are particularly prone to thermal

(a)

(b)

(c)

FIGURE 2.26

Development of residual stresses.

shrinkage; amorphous materials tend to shrink less. Excessive shrinkage, beyond the acceptable level, can be caused by the factors such as low injection pressure, short pack-hold time or cooling time, high melt temperature, high mold temperature, and low holding pressure. The relationship of shrinkage to several process parameters and part thickness is schematically plotted in Figure 2.27. Excessive volumetric contraction also leads to either sink marks or voids in the molding interior. Controlling part shrinkage is important in part, mold, and process designs, particularly in applications requiring tight tolerances. Also, the mold design should take shrinkage into account in order to conform to the part dimension.

PROCESS CONTROL

FIGURE 2.27

43

Process and design parameters that affect part shrinkage.

An asymmetrical stress pattern in the thickness direction will cause the molding to warp, so the surfaces of the molded part do not follow the intended shape of the design. In addition, nonuniform shrinkage will also lead to warpage. If the shrinkage throughout the part is uniform, the molding will not warp and simply become smaller. But differential shrinkage of material in the molded part will cause warpage. Variation in shrinkage can be caused by molecular and fiber orientations, temperature variations within the molded part, and by variable packing, such as overpacking at gates and underpacking at remote locations, or by different pressure levels as material solidifies across the part thickness. 2.5

PROCESS CONTROL

Modern injection molding machines can facilitate highly automatic control system, which leads to higher efficiency and better product quality. The control of the complete cycle operation, as well as key process parameters, is nowadays normally done by the computer-based control system of the injection molding machine. The control system receives signals of various parameters (screw position, screw speed, mold stroke, holding pressure, barrel temperature, etc.), processes these signals, and generates specific output values, which initiate specific reactions (temperature increase, valve opening, pressure increase, etc.). Figure 2.28 shows a simplified overview of process control of injection molding machines. First, the control system has to realize the logical sequence of all principle functions, such as clamping and opening of the mold, as well as the so-called secondary functions, such as opening and closing of actuating cams, and also monitors the molding cycle so that it runs automatically. This task is relatively simple and easy to implement. What is more important is that to produce high quality products consistently, some key process variables closely

FIGURE 2.28 The simplified overview of process control of injection molding machines.

related to the product quality need to be properly set and precisely controlled, because the molding process involves physical, chemical, and mechanical changes, and has a complicated dynamics due to the thermodynamics and rheological behaviors of polymer melts. Furthermore, there are safety and administrative issues need to be considered. A sophisticated controller has to be a complicated multiobjective, multilayer system. On the basis of the categorization of the machine variables, the process variables, and the quality attributes,11 the overall process control system of injection molding can be illustrated as a hierarchical structure, as shown in Figure 2.29.12 An advanced process parameter control system constructs the basic layer and provides accurate and robust control of key process variables. The second layer consists of two modules: a profile setting layer determining the shapes of set point profiles for the key process variables, to achieve an evenly distributed part quality along the flow directions; and a process fault detection and diagnoses system monitoring the operation of the injection molding process. The third layer, a direct quality

44

BACKGROUND

FIGURE 2.29

Overall injection molding process control system architecture.

control layer, consists of a quantitative quality prediction model and a quality controller. This final layer provides a mechanism for closed-loop direct control of certain part quality. The quality of an injection-molded product, characterized in terms of its dimensions, appearance, and mechanical properties, is a strong function of the process conditions. In the meantime, there are different sources of disturbances in the molding process affecting the material, machine, and process variables, hence deteriorating the control performance. An accurate and robust control system for the key process variables is therefore necessary to ensure the repeatability and reliability of the product quality, and it forms the foundation layer of the overall control and monitoring system. Key process variables can be categorized into two classes, phase-based variables and nonphase-based variables. During filling, the injection velocity is the key variable that determines the flow pattern inside the mold cavity. In packing phase, the packing pressure is the most important, while in plastication, the screw rotation speed and back pressure are two critical variables determining the homogeneity of melting. Variables such as filling velocity, packing pressure, plastication screw rotation speed, and back pressure need to be controlled in certain phases, therefore are phase-based key variables. The barrel temperature and mold temperature need to be controlled throughout the molding cycle, to ensure a stable and evenly distributed melt temperature, and they belong to nonphase-based key variables. The key process variable control can ensure the repeatability and stability of the molding process, while the profile

setting module determines the optimal settings for key variables of each molding phase, that is, ram injection velocity in filling, packing pressure in packing, and melt temperature in plastication. Owing to the fact that the profile setting is not closely control related area, it is not the focus of this book. This part of work is reviewed and discussed extensively in Reference 13. The major function of process monitoring system is to detect, identify, and remove the changes in the trajectories of process variables that may cause the final product quality to deviate from its desired settings. As a typical batch process, in spite of using many advanced control methods in injection molding, some batch-to-batch variations in quality will still appear because of various process fluctuations, drifting of process conditions, changes in materials, and unknown disturbances. Online monitoring of process variables is of great importance not only for assessing the status of the process but also for ensuring the final product quality. When process monitoring indicates abnormal process operation, fault diagnosis can determine the source causes of this abnormal behavior. Traditional statistical monitoring techniques rely on the use of univariate statistical process control (SPC) tools, for example, Shewhart, cumulative sum (CUSUM), and exponentially weighted moving average (EWMA) charts, to monitor key process or quality variables.14 For multivariable processes, a number of these univariate control charts are used to monitor different variables. This one-variable-one-chart method, however, may become misleading and time consuming when monitoring a large number of correlated process variables. Multivariate

PROCESS CONTROL

statistical projection methods, which mainly rely on principal component analysis (PCA) and partial least squares (PLS), are therefore adopted because of its inherent advantages in multivariate batch process monitoring. Quality is the ultimate goal of the molding process. The molded product quality may be categorized into different groups such as surface properties, dimensional properties, and mechanical properties. In reality, it is difficult to control all quality attributes simultaneously. The selection of quality attributes for modeling and control is dependent on the process and product requirements. The product weight is selected in Chapter 15 as an example to demonstrate the quality modeling and closed-loop quality control.

2.5.1 Characteristics of Injection Molding as a Batch Process The following definition of a batch process is given by Shaw15 : “A process is considered to be batch in nature if, due to physical structuring of the process equipment or due to other factors, the process consists of a sequence of one or more steps (or phases) that must be performed in a defined order. The completion of this sequence of steps creates a finite quantity of finished product. If more of the product is to be created, the sequence must be repeated.” This definition clearly indicates that the injection molding process is a typical batch process. Compared to a traditional continuous manufacturing industry, batch processes such as injection molding have the following characteristics in terms of process dynamics and operation control: 1. Strong Time-Varying Characteristic. In batch process, raw material normally undergoes state transformation in different subphases. For example, in injection molding, polymer granules will be heated and melted from solid state to fluid during plastication. The viscosity of the material changes nonlinearly because of shear-thinning effect during filling phase. It finally solidifies again during packing and cooling. It is obvious that in each subphase, the physical properties of the processing material vary nonlinearly with cycle time. This kind of variation is the major reason for the time-varying characteristics of batch processes. It also prevents the traditional control strategies from successfully applying to the batch processes. 2. Strong Nonlinear Characteristic. Most batch processes involve complicated physical and chemical changes, which results in highly nonlinearity; for example, the nonlinear rheological properties of the molten polymer with respect to the processing temperature. In a continuous process, the traditional way

45

to deal with the nonlinearity is to use a steady state approximation. The batch process, however, normally operates in a wide operating range according to different requirements, so the traditional method cannot be applied directly. 3. Hybrid System in Nature. Most batch processes contain multiple subphases, for example, injection molding contains filling, packing/holding, and cooling (plastication) phases. The transition from one subphase to another may be decided based on various criteria, for example, the elapsing of time or logic check for certain conditions. In addition, there are several of constrains in the operation to ensure safety issues, so both analog (continuous) and logical (discrete) components are relevant and interacted in batch processes. The system with this kind of nature is referred to as hybrid system. 4. Repetitive Nature of the Operation. In a batch process, each cycle can make a finite amount of products. To reach a certain volume of production, the process needs to be repeated. During stable operation, the complicated dynamics can be fully or partially repeated between the neighboring cycles. This kind of repetitive nature provides a possibility to develop new control strategies for the batch process. Detailed discussion based on this point can be found in Chapter 13. Injection molding inherently has the characteristics as discussed above as it is a batch process. Although the sequence appears to be simple, injection molding is rather a complex nonlinear dynamic process during which the machine parameters, material properties, and process variables interact with each other. Many research works have discussed the relationship extensively and concluded that the quality of the molded part is a strong function of the process conditions. The key process parameters, therefore, need to be closed-loop controlled to ensure the repeatability and quality of the molding process.

2.5.2

Typical Control Problems in Injection Molding

Modern injection molding control system has four major functions: manipulation of machine sequence, control of major process/machine variables, monitoring of process status and performance, and quality control. The sequence control ensures the molding machine to operate according to the predetermined order and provides the safety locks through a sophisticated interlocking circuit or software. This part of the functions only follows a defined logic and has little influence on the product quality. The development of control strategies of key molding process variables, as shown in Figure 2.30, is the focus of this book.

46

BACKGROUND

The key molding process variables include particularly the following aspects: 1. Melt/Barrel Temperature Control . Temperatures, including mold temperature, barrel temperature, and melt temperature, are important variables for the injection molding process. They affect the material flow and thermal properties. Other key process variables such as injection velocity, packing pressure, plastication screw rotation speed, and cooling time are closely related to the temperatures. Many product quality attributes are thus strongly affected by the temperature settings and control performance. The melt temperature is a distributed parameter and difficult to be measured directly. It is therefore commonly accepted in molding industry to control the barrel temperature instead. Yao and Gao16 pointed out the difficulties in barrel temperature control as follows: (a) Barrel temperature has “integratorlike” dynamics in the relevant operation range. Any change in input, for example, a step input (or ‘heating on’), takes a long time to reach steady state. Model identification based on step responses or pseudo random binary sequences (PRBS) are thus time consuming and sometimes become practically infeasible. (b) The dynamics exhibits significant time delays in barrel temperatures because of the sensor position, the nature of barrel configuration, and the heat transfer characteristics. (c) There exists a control specification contradiction. The barrel temperatures are expected to arrive

FIGURE 2.30

at the set point fastest at one hand. It is also expected to have minimal overshoots, on the other hand, to avoid material degradation, as there is no cooling mechanism. (d) Nonlinearities exist because of heat radiation and shear heating caused by screw actions. A linear fixed model is, however, commonly used for simplicity. This often results in suboptimal even inadequate performance for other operating conditions. The molding conditions may have to be adjusted from time to time based on the materials and products to be molded. 2. Injection Velocity Control . Filling is the first stage of the molding. During filling, the polymer melt is forced into a mold cavity through the nozzle and runner system by a screw forward motion. The nature of the melt flow entering the cavity strongly influences the quality of the molded part, especially the mechanical properties, such as tensile strength, impact strength, heat distortion, and dimension stability. Ram injection velocity, often referred to simply as injection velocity, approximates but does not equal the polymer filling rate entering a mold cavity because of the melt compressibility and the plastic leakage through the check ring valve. Despite the fact that injection velocity differs from the polymer filling rate, it is still widely used as a controlled variable during injection phase, as it provides a better approximation than other variables such as cavity pressure or nozzle pressure. A proper setting and good control of injection velocity can achieve a more evenly

Typical control parameters in injection molding.

REFERENCES

distributed flow pattern and avoid overpressurization, high thermal stresses, and high residual flow stresses. It has been proved that the injection velocity has strong nonlinear and time-varying characteristics, mainly caused by the complicated mold geometry and nonlinear rheological properties of the molten polymer under different shear rates. 3. Packing Pressure Control . Packing/holding is another important phase in the injection molding process during which additional material is added to the mold cavity under certain pressure to compensate for the material shrinkage. Adding too much material to the cavity may result in highly stressed parts and mold flash. It may also cause ejection problems. Insufficient packing/holding may cause molding problems such as poor surfaces, sink marks, weld line, and shrinkage. Proper packing control is critical for producing molded parts with good mechanical properties, dimensional stability, and satisfactory surface characteristics. The degree of packing is indicated well by the cavity pressure, which is determined by the nozzle pressure, cavity geometry, and mold temperatures. Packing pressure exhibits strong time-varying and nonlinear characteristics because of the material cooling and solidification, and the nonlinear relationship between the control valve opening and the packing pressure response. 4. Mold Temperature Control . The purpose of mold temperature control is to manipulate the cooling pattern of the molten polymer inside the mold cavity, to achieve a satisfactory microscopic structure inside the final product. Similar to the barrel temperature, the mold temperature needs to be controlled throughout the cycle. In different subphases, the mold temperature has different control objectives, so it is both phase based and periodic. As cooling is normally the longest subphase in the molding cycle, proper control of the mold temperature can reduce the cycle time and improve the productivity significantly. The mold open/close operation and ambient temperature change are major disturbances for the mold temperature control. 5. Screw Rotation Speed and Back Pressure Control . The screw rotation speed and back pressure are two key variables during plastication phase. Different materials require different settings of screw rotation speed and back pressure to ensure a fast and uniform plastication. As the material undergoes phase change from solid to melt, the plastication dynamics also changes severely. The above control problems are typical and key issues for hydraulic reciprocating screw injection molding

47

machines. With the development of new sensor, signal processing technologies, and automatic equipments, there are various new control problems, for example, the multicavity molding filling control and multipoint pressure control, the melt-front-velocity control and the mold position control, etc. REFERENCES 1. Rosato D.V., Rosato D.V., Rosato M.G., Injection Molding Handbook. 3rd ed. 2000, Norwell: Kluwer Academic Publishers. 2. Rees H., Understanding Injection Mold Design. 2001, Munich: Carl Hanser Verlag. 3. Potsch G., Michaeli W., Injection Molding: An Introduction. 2008, Munich: Carl Hanser Verlag 4. Pantani R., Coccorullo I., Speranza V., et al., Modeling of morphology evolution in the injection molding process of thermoplastic polymers. Progress in Polymer Science, 2005. 30(12): 1185–1222. 5. Ying J.R., Liu S.P., Guo F., et al., Non-isothermal crystallization and crystalline structure of PP/POE blends. Journal of Thermal Analysis and Calorimetry, 2008. 91(3): 723–731. 6. Belofsky H., Plastics: Product Design and Process Engineering. 1995, Munich: Carl Hanser Verlag. 7. Boothroyd G., Dewhurst P., Knight W., Product Design for Manufacture and Assembly. 2nd ed. 2002, New York: Marcel Dekker, Inc. 8. Greener J., General consequences of the packing phase in injection molding. Polymer Engineering and Science, 1986. 26(12): 886–892. 9. Struik L.C.E., Orientation effects and cooling stresses in amorphous polymers. Polymer Engineering and Science, 1978. 18(10): 799–811. 10. Rezayat M., Stafford R.O., A thermoviscoelastic model for residual stress in injection molded thermoplastics. Polymer Engineering and Science, 1991. 31(6): 393–398. 11. Wang K.K., Zhou J., Sakurai Y., An integrated adaptive control for injection molding, in SPE Annual Technical Conference - ANTEC, Conference Proceedings. 1999, Germany: Society of Plastics Engineers. 12. Yang Y., Injection Molding Control: From Process to Quality, in Department of Chemical Engineering. 2004, Hong Kong: The Hong Kong University of Science and Technology. 13. Chen X., A Study on Profile Setting on Injection Molding. 2002, Hong Kong: The Hong Kong University of Science and Technology. 14. Montgomery D.C., Introduction to Statistical Quality Control. 2001, Hoboken: John Wiley & Sons, Inc. 15. Shaw W.T., Computer Control of Batch Processes. 1982, Cockeysville: EMC Controls, Inc. 16. Yao K., Gao F.R., Allgower F., Barrel temperature control during operation transition in injection molding. Control Engineering Practice, 2008. 16(11): 1259–1264.

PART II SIMULATION

3 MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION Huamin Zhou, Zixiang Hu, and Dequn Li State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

In this chapter, material properties and governing equations for fluid flow in the filling and packing phases of injection molding process will be introduced first, and then boundary conditions and some mold simplifications will be presented. At present, the mathematical description generally bases on the continuum assumption,1 – 3 on which the governing equations are premised, including mass conservation equation, momentum conservation equations, energy conservation equation, and constitutive equation reflecting the material properties. According to the characteristics of the problem, the governing equations can be simplified appropriately. Considering the appropriate initial conditions and boundary conditions, the above all create the mathematical model of fluid flow.

3.1 MATERIAL CONSTITUTIVE RELATIONSHIPS AND VISCOSITY MODELS The relationship between the viscous stress tensor τ and the rate of strain tensor ε˙ is called a constitutive equation. It is the property of material itself and is independent of the special conditions. For different polymers, different constitutive models are used to describe their material properties. Sometimes, because of the need to simplify the problem, different constitutive models would be used for the same type of polymers.

Common constitutive relationships of polymer melt are classified into Newtonian fluids, generalized Newtonian fluids, and viscoelastic fluids.2,4 In the following section, we introduce these three constitutive relationships and common viscosity models. 3.1.1

Newtonian Fluids

Newton inner friction law is given by τ = μγ˙

(3.1)

where τ is the shear stress tensor, the velocity gradient is called the shear rate; γ˙ is the rate of strain (or deformation) tensor; and the constant of proportionality μ is called the viscosity of the fluid , which is the index of the resistance of fluid to flow. Fluids for which stress is directly proportional to shear rate, that is Equation 3.1 holds, are called Newtonian fluids. For such fluids, the viscosity is usually constant at a given temperature. However, the viscosity may vary with temperature. Fluids for which Equation 3.1 does not hold are called non-Newtonian fluids. The Newtonian fluids include most pure liquids such as water and alcohol, light oil, low molecular weight compounds’ solution, and low speed gas. However, molten polymers are non-Newtonian fluids. As its viscosity is

Computer Modeling for Injection Molding: Simulation, Optimization, and Control, First Edition. Edited by Huamin Zhou. © 2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc.

51

52

MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

very complicated and nonlinear, it depends on its chemical structure, composition, and processing conditions.

3.1.2

Generalized Newtonian Fluids

In order to model the flow of molten polymers, a modification to Equation 3.1 is made to allow the viscosity to be a function of shear rate. Therefore Equation 3.1 becomes τ = η (T , γ˙ ) γ˙

(3.2)

where τ is called the viscous stress tensor, which may be written as a matrix of its components: ⎡ τxx τ = ⎣τyx τzx

τxy τyy τzy

⎤ τxz τyz ⎦ τzz

(3.4)

τ = 2ηε˙ + (λ∇ • u) I

(3.5)

or

where η is the dynamic viscosity, λ is the dilatational viscosity, I is the unit tensor, ε˙ is the rate of strain (or deformation) tensor, and σ is the stress tensor, which may always be written in the following form: σxx σ = ⎣σyx σzx

σxy σyy σzy

⎤ σxz σyz ⎦ = −pI + τ σzz



∂z

(3.7) And we have

σ = 2ηε˙ − (p − λ∇ u) I

⎡

⎤ ∂u ∂w + ∂z ∂x ⎥ ⎥ ⎥

⎥ 1 ∂v ∂w ⎥ ⎥ + 2 ∂z ∂y ⎥ ⎥ ⎥ ⎥ ⎦ ∂w

1 2

(3.3)

And η (T , γ˙ ) is called the viscosity function, also simply called the viscosity. Most starchy fluids, suspension and plastic melt in the nature are all these kind of fluids. In the case of the nonNewtonian fluids, the viscosity η (T , γ˙ ) decreases when temperature or velocity of flow increases. Furthermore, the plastic materials in molten process are more viscous and compressible. Generalized Newtonian fluids hold the generalized Newton viscous force law given by •

components: ⎤ ⎡ ε˙ xx ε˙ yx ε˙ zx  1 ∇u + (∇u)T ε˙ = ⎣ε˙ xy ε˙ yy ε˙ zy ⎦ = 2 ε˙ xz ε˙ yz ε˙ zz

⎡ ∂u 1 ∂u ∂v + ⎢ ∂x 2 ∂y ∂x ⎢ ⎢ ⎢

⎢ 1 ∂u ∂v ∂v ⎢ + =⎢ 2 ∂y ∂x ∂y ⎢ ⎢ ⎢



⎣ 1 ∂u ∂w 1 ∂v ∂w + + 2 ∂z ∂x 2 ∂z ∂y

(3.6)

For the isotropic fluid, viscous stress is a function of local deformation rate (or strain rate) of fluid. The microunit of the isotropic fluid has nine components of strain rate out of which six components are independent. Rate of strain (or deformation) tensor ε˙ may be written as a matrix of its

γ˙ = 2˙ε =∇u + (∇u)T ⎡ ∂u ∂u ∂v + ⎢ 2 ∂x ∂y ∂x ⎢ ⎢ ∂u ∂v ∂v ⎢ + 2 =⎢ ⎢ ∂y ∂x ∂y ⎢ ⎣ ∂u ∂w ∂v ∂w + + ∂z ∂x ∂z ∂y

⎤ ∂u ∂w + ∂z ∂x ⎥ ⎥ ∂v ∂w ⎥ ⎥ + ⎥ ∂z ∂y ⎥ ⎥ ∂w ⎦ 2 ∂z

(3.8)

Thus, Equation 3.5 becomes τ = ηγ˙ + (λ∇ • u) I

(3.9)

Notice that σ , ε˙ , and τ are all symmetric. In Equation 3.4, p = − σ ii /3, namely the hydrostatic pressure equals the arithmetic mean of the sum of three normal pressures and its direction is the opposite direction of the normal pressure. To ensure that the relationship is held by the compressible fluid, assume the following equation: 2 (3.10) λ+ μ=0 3 The fluid satisfying Equation 3.10 also is called Stokes fluid . At present, generalized Newtonian fluid is a widely used constitutive model in the polymer simulation. According to the different practical behavior, the generalized Newtonian fluids can be classified into two categories: time-dependent fluids and time-independent fluids. The former can also be organized into three kinds4,5 : Bingham plastics, pseudoplastic fluids, and dilatant fluids. The latter can also be further categorized into two kinds: rheopectic fluids and thixotropic fluids. Bingham plastics are fluids that remain rigid when its shear stress is smaller than yield stress but flow similar to a simple Newtonian fluid once the shear stress has exceeded

MATERIAL CONSTITUTIVE RELATIONSHIPS AND VISCOSITY MODELS

FIGURE 3.1 fluids.

Characteristic viscosity curve of pseudoplastic

this value. Different constitutive models representing this type of fluids were developed by Herschel and Bulkley,6 Oldroyd,7 and Casson.8 Pseudoplastic fluids have no yield stress threshold, and for these fluids, the ratio of shear stress to the rate of shear generally falls continuously and rapidly with increase in the shear rate. Very low and very high shear regions are the exceptions, where the flow curve is almost horizontal (Fig. 3.1). This shear thinning effect that the viscosity decreases with increasing rate of shear stress is an important phenomenon of pseudoplastic fluids. This is currently the most commonly used non-Newtonian viscous fluid model to simulate the polymer flow. The main characteristic of the dilatant fluids is that at low shear rate, the flow acts basically as a Newtonian fluid; when the shear rate is higher than a certain critical value, viscosity increases with increasing rate of shear (also termed shear thickening). Most dilatant fluids are multiphase mixture systems such as asphalt and liquid concrete. The viscosity of time-dependent non-Newtonian fluid has temporal correlation with the constant rate of shear strain. This is less used to describe the polymer melt. To model the injection molding process, a viscosity function (or model) is required. The viscosity curves of most thermoplastics have the same dependence on shear rates as shown in Figure 3.1. At lower shear rates, the viscosity is nearly a constant. This is the usually called as upper Newtonian region. Polymer chains are more uniformly aligned as the shear rate increases, so the viscosity decreases accordingly. This is called the shear thinning region. When all polymeric chains are fully aligned, the shear viscosity then becomes virtually insensitive to the shear rate. This is called as lower Newtonian region. The upper Newtonian region and shear thinning region can be observed in most polymers (the LCP

53

might be an exception). The lower Newtonian region is, however, not as obvious in most thermoplastics as it occurs with molecular degradation at ultrahigh shear rates. A number of well-known models are available, so it is important to choose a model that is both accurate over the processing range and for which data can be readily obtained. de Waele in 1923 and Ostwald in 1925 proposed a power law model,9 also known as Ostwald–de Waele relationship, which can successfully predict the shear thinning region. In 1977, Middleman10 proposed an advanced power law that overcomes a problem of the original model in which the stress is much higher than that at low rate of shear. The viscosity hardly changes when the rate of shear is very low or very high; namely, at this time, pseudoplastic fluid acts similar to a Newtonian fluid. At least four parameters are required to reflect this relationship between viscosity and shear rate. Cross model11 and Carreau model12 are proposed by Cross in 1965 and Carreau in 1972, respectively. All of them are believed to be better than the others in terms of reflecting the characteristics of a thermoplastic polymer. In addition, Sisko model13 proposed by Sisko in 1958 combined the characteristics of Bingham plastics and power law model. The other models such as Prandtl–Eyring model, Ellis model, Casson model, and fractional exponential model were also applied in some fields successfully. At present, the power law model and the Cross model are widely used in the field of polymer injection molding to describe the viscosity function.14 – 21 The following section introduces some common viscosity models. 3.1.2.1 form:

Power Law Model

This model has the following

η = η0 |γ˙ |n−1

Tb η0 = B exp T

(3.11) (3.12)

where η0 is the Newtonian viscosity or zero-shear rate viscosity, which is defined as the viscosity at zero-shear rate; B is a constant called the consistency index ; T b is a constant showing the temperature sensitivity of this material; T is the melting temperature; and n is the power law index with a value between 0 and 1 for polymer melts. When η0 = μ and n = 1, we obtain the relationship for a Newtonian fluid, that is, Equation 3.1. The effective shear rate |γ˙ | is given by 1 |γ˙ | = γ˙ij γ˙ij (3.13) 2 The zero-shear rate viscosity contains the effect of temperature and pressure on viscosity described by some different models. In the field of plastic injection molding process, the WLF model and Arrhenius model are commonly used.

54

MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

Arrhenius formula is given by

Tb η0 = B exp T



exp (βP )

(3.14)

−A1 (T − (D2 + D3 P )) η0 = D1 exp A2 + T − D2



ln (η) = (n − 1) ln (γ˙ ) + ln (η0 )

(3.16)

3.1.2.2 Ellis Model The Ellis model expresses the viscosity as a function of shear stress τ . It has the following form:

τ α−1 η0 =1+ (3.17) η τ1/2 where τ 1/2 is the value of shear stress for which η = η0 /2 and α − 1 is the slope of the graph ln[(η0 /η) − 1] versus ln(τ /τ 1/2 ). The model has the following

(n−1)/a  η = η∞ + (η0 − η∞ ) 1 + (K γ˙ )a

η0 |γ˙ | τ∗

1−n˜

(3.19)

where, τ * is the shear stress at the transition between Newtonian and power law behavior. If using WLF zeroshear viscosity model presented by Equation 3.15, we call Equation 3.19 as WLF Cross model, which is widely used in injection molding simulation. 3.1.3

The above equation shows that this three-parameter model reflects the observation that the viscosity function at medium-high shear rates is nearly a straight line in log–log plots. Therefore, the power law model can represent the behavior of polymer melts in the high shear-rate region. It is also quite easy to fit experimental data with this model and determine the constants η0 and n. The main disadvantage of the model is in the low shearrate range. Despite this disadvantage, the model has been widely used for modeling flow in injection molding. In the filling phase particularly, shear rates are frequently high enough to justify the use of the first-order model.

Carreau Model

η0

(3.15)

in which A1 , A2 , D 1 , D 2 , and D 3 are material parameters. Taking natural logarithms of both sides, we obtain

3.1.2.3 form:

The equation for the Cross model

1+

where B , T b , and β are material parameters. This model is suitable for semicrystalline materials. The WLF zero-shear viscosity model can reflect the variation of melt viscosity with temperature and pressure more exactly. It has the following form:

3.1.2.4 Cross Model is given by η=

(3.18)

where η∞ is the viscosity at infinite shear rate, η0 is the viscosity at zero-shear rate, n is a dimensionless constant with the same interpretation as that in Equation 3.11, and K is a material time constant parameter. The Bird–Carreau model is given by a = 2. For many shear thinning fluids, a ≈ 2.

Viscoelastic Fluids

Generalized Newtonian fluid model can describe the nonNewtonian characteristic that viscosity changes with the deformation rate tensor. But it cannot predict other phenomena such as recoil, stress relaxation, stress overshoot, and extrudate swell, which are commonly observed in polymer processing flows. These effects have a significant impact on the product quality in polymer processing, and they should not be ignored. The stress of viscoelastic fluid at any time depends on not only the movement and deformation at that moment but also the deformation history. Theoretically, all of these phenomena can be considered as the result of the material that has a combination of the properties of elastic solids and viscous fluids. Therefore, mathematical modeling of polymer processing flows should, ideally, be based on the use of viscoelastic constitutive equations. In the 1940s, Reiner22 and Rivlin23 made a breakthrough in the field of nonlinear viscosity theory and finite elastic deformation theory. The research on the constitute relationship of viscoelastic material had a great development in the following decades. In 1950, Oldroyd24 first introduced new concepts of coordinate system and derivative going with the material that improved the constitute theory of nonNewtonian fluid to a new level and had been considered as one of important foundations of the constitute theory of modern rheology. Oldroyd proposed Oldroyd 8-constant model25 in 1958 in which there is one zero-shear viscosity and seven time constants, and the stress term is kept linear. By letting different constants to be zero, Oldroyd 8-constant model can become Newtonian fluid model, upper-convected Maxwell model (UCM), Oldroyd-B model, second-order model, and Johnson–Segalman model.26 Among them Oldroyd-B is the most widely used. However, many experiments have also shown that for practical polymer flow, especially for the tensile deformation dominated flow, the results predicted by Oldroyd model and the actual results are quite different. In 1977, Phan-Thien and Tanner proposed PTT model with 4-constant stress term based on Lodge network theory,27,28 which can predict both shearing viscosity and

MATERIAL CONSTITUTIVE RELATIONSHIPS AND VISCOSITY MODELS

uniaxial extensional viscosity. In 1982, Giesekus proposed 3-constant Giesekus model29 based on molecular theory, which can predict the normal stress difference besides power law viscosity region of the non-Newtonian fluid. Warner introduced the dumbbell model by abstracting from the dilute polymer solution based on kinematical theory. Referring to this, in 1972, Warner30 proposed a nonlinear elastic dumbbell model named FENE (finitely extensible nonlinear elastic) based on which many scholars improved extended FENE model.31 The above-mentioned constitutive equations are all different equations. Integral constitutive relationship applies Boltzmann superposition principle to obtain stress components by accumulation of deformation history with appropriate functions. The simplest integral constitutive model is the one proposed by Lodge32 for rubberlike liquid in 1964 based on network theory; while the most wildly used model is K-BKZ model proposed by Kaya and Bernstein in 1962, and Kearsley and Zapas33 in 1963 as independent inventors. K-BKZ model is a nonlinear generalized model, including Rouse–Zimm model, Lodge model, Tanner–Simmons model, and Doi–Edwards model.26 The constitutive models of integral form can give stress formula explicitly but is not convenient as differential models. In addition, in general condition, they are suitable to be applied in Lagrangian frame, which is a reason why they are not widely used as the differential models. As the linear models cannot portray well the rheological properties, such as the shear thinning and nonquadratic first normal difference observed in typical melts, it needs to be extended to cover the nonlinear behaviors. The Pom–Pom model proposed by McLeish and Larson34 in 1998, was considered as a breakthrough in the field of the construction of viscoelastic constitutive equation. It is based on tube theory and a simplified topology of branched molecules. Pom–Pom model still has some disadvantages such as discontinuity phenomena in steady extensional flow, unboundedness of the equation describing the direction in high shearing rate flow, and inexistence of second normal stress difference in shear flow. In 2001, for these shortcomings, Verbeeten et al.35 proposed an extended Pom–Pom model, namely XPP model. The results of this model are consistent with the results from rheological tests of LDPE. Verbeeten et al.36,37 adjusted XPP model in 2002 and in 2004 for solving a problem that the principal axis value of tensile stress in radical sign might be negative and kept this method consistent with the other viscoelastic constitutive models in construction. Numerical experiments exhibit that XPP model has some characteristics that the others do not have and can better predict the polymer behaviors in complex flow.38 – 40 The following part will introduce commonly used viscoelasticity models, including upper-convected Maxwell

55

model, Oldroyd-B model, White–Metzner model, Giesekus model, and PTT model. 3.1.3.1 Upper Convected Maxwell Model (UCM) This model describes polymers with linear or quasilinear viscoelasticity, which combines the ideas of viscosity and elasticity. It has the following form:

 ∂τ + (u • ∇) τ − ∇u T • τ − τ • ∇u = ηγ˙ ∂t (3.20) The relaxation time λ is defined as the time required for stress to be reduced to half of its original value when the polymer stops deforming. A higher relaxation time indicates more memory (elastic) effect. τ + λ0

3.1.3.2 Oldroyd-B Model This model adds an additional Newtonian component to the UCM model as τ 1 + λ0

 ∂τ 1 + (u • ∇) τ 1 − ∇u T • τ 1 − τ 1 • ∇u = η1 γ˙ ∂t (3.21) τ 2 = η2 γ˙

(3.22)

τ = τ1 + τ2

(3.23)

η2 = rη0 ,

η1 = (1 − r) η0

(3.24)

If r = 0, the model reduces to the UCM model. The model predicts a constant viscosity and a quadratic first normal difference. It should be further noted that linear (or quasilinear) model has been widely used for viscoelastic flow calculations for its simplicity, but it is not suitable for injection molding process because of the lack of shear thinning characteristics. It is suggested to use those models in polymeric solutions for research purpose. 3.1.3.3 White–Metzner Model This model is modified from UCM model. It can illustrate reasonable profiles for shear-rate dependent viscosity h and relaxation time l . It is also easy to select the material parameters on the basis of the first normal stress difference and viscosity. The model is suitable in fast time-dependent motions.

∂τ + (u • ∇) τ − ∇u T • τ − τ • ∇u τ + λ (T , γ˙ ) ∂t = η (T , γ˙ ) γ˙ λ (T , γ˙ ) =

 (3.25)

η (T , γ˙ ) G

(3.26)

Viscosity η and relaxation time λ are both functions of temperature and shear rate. Relaxation time is referenced to the viscosity divided by modulus G.

56

MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

If λ uses the following form: λ0 aT

λ (T , γ˙ ) = 1+



λ0 aT k

|γ˙ |

1−n

(3.27)

where λ0 is the relaxation time at zero-shear rate and a T is the shift factor. We call it modified White–Metzner model. The model provides the flexibility by allowing different variation profile for the shear rate and is dependent on the viscosity and relaxation time. The variation of relaxation time is independent on the viscosity and can be fit by cross model. 3.1.3.4 Giesekus Model This model describes the rheological properties with nonlinear stress terms as follows:

 ∂τ αλ0 • τ τ + λ0 + (u • ∇) τ − ∇u T • τ − τ • ∇u τ+ η ∂t = η0 γ˙

(3.28)

where λ0 is the relaxation time at zero-shear rate, η0 is the shear viscosity at zero-shear rate, and α is the dimensionless mobility factor with a value between 0 and 1. The origin of the term involving α can be associated with anisotropic Brownian motion or anisotropic hydrodynamic drag on the constituent polymer molecules. Large decreases in viscosity and normal stress coefficients with increasing shear rate are possible, which is much more realistic than the linear models. 3.1.3.5 PTT Model This model describes the rheological properties with nonlinear stress terms as follows:

ελ0 tr (τ ) τ 1+ η

 ∂τ T• • • + λ0 + (u ∇) τ − ∇u τ − τ ∇u = η0 γ˙ ∂t (3.29) where λ0 is the relaxation time at zero-shear rate, η0 is the shear viscosity at zero-shear rate, and ε is the material property that controls nonlinear behavior. The model exhibits shear thinning and nonquadratic first normal stress difference, which is similar in its predictions to the Giesekus equation, except for some minor viscometric features.

3.2

THERMODYNAMIC RELATIONSHIPS

Several thermodynamic properties are required for simulation of injection molding. Thermoplastic generally undergoes a significant volumetric change over temperature and

pressure. It is therefore essential to characterize its pressurevolume-temperature (PVT) relationship in order to calculate the compressibility of the material during packing phase and final part’s shrinkage and warpage after ejection. An equation of state relates the three variables, pressure p, specific volume Vˆ , and temperature T . The specific volume is defined as the volume of thermoplastic per unit mass. For any material, we write the state equation in the following form:   f p, Vˆ , T = 0 (3.30) Given any two variables, it is possible to determine the third by using the equation of state. In particular, we can write Vˆ = g (p, T ) (3.31)

where g is some function. If we graph the PVT data of a material, we would obtain a PVT surface as shown in Figure 3.2. If the material at temperature T a undergoes a change in temperature while being held at constant pressure, the average change in volume over the temperature change is therefore, g (pa , Ta + T ) − g (pa , Ta ) Vˆ = T T Vˆ (pa , Ta + T ) − Vˆ (pa , Ta ) = T

(3.32)

In the limit as T → 0, we obtain the instantaneous change in volume for the material, which we denote by   ∂ Vˆ (3.33) ∂T p

where the subscript p indicates that the pressure is held constant. The coefficient of volume expansion of the material β is defined as follows:   1 ∂ Vˆ (3.34) β= Vˆ ∂T p

The coefficient of volume expansion is also called the expansivity of the material . It has units of reciprocal kelvin (K−1 ). Now consider the change in volume because of a change in pressure–temperature constant. This involves moving from point b to c in Figure 3.2. The average change in the volume because of a change in temperature is given by g (pa + p, Tb ) − g (pa , Tb ) Vˆ = T p =

Vˆ (pa + p, Tb ) − Vˆ (pa , Tb ) p

(3.35)

THERMODYNAMIC RELATIONSHIPS

57

Vˆ

T pa Δp

ΔT pc

Tb

ΔV Ta p

FIGURE 3.2

FIGURE 3.3 Specific volume–temperature curves of semicrystalline thermoplastic and amorphous thermoplastic.

PVT diagram.

Letting p → 0, we obtain the instantaneous change in volume, which we denote by 

∂ Vˆ ∂p

Vˆ = Vˆ0





where Vˆ and Vˆ0 are the densities of the material. Notice that here Vˆ represents the density instead of specific volume in this formula. It is intended to be more straightforward. In other PVT models, Vˆ represents specific volume instead.

T

∂ Vˆ ∂p



(3.37)

T

The negative sign indicates that the volume decreases with increasing pressure. Isothermal compressibility has units of one square meter per newton (m2 N−1 ). Different types of thermoplastic have different PVT behaviors across its transition temperature. Semicrystalline thermoplastic has a significant and abrupt volumetric change, while amorphous thermoplastic has only a change in slope in its specific volume–temperature curves without a sudden transition from melt to solid. Figure 3.3 depicts the difference between these two types of thermoplastics. For semicrystalline materials, the PVT data falls into three areas: low temperature, transition, and high temperature. Figure 2.4 represents the PVT plots of HDPE, a semicrystalline material, and ABS, an amorphous material. A good PVT model should characterize the dependence of specific volume on temperature and pressure, and the difference between these two types of thermoplastics. There are some PVT models introduced in the following section.

3.2.1

(3.38)

(3.36)

The isothermal compressibility κ is defined to be 1 κ=− Vˆ

This model has the following form:

Constant Specific Volume

The constant specific volume model assumes that the specific volume is independent of pressure and temperature. It corresponds to incompressible materials.

3.2.2

Spencer–Gilmore Model

This model is derived from the ideal gas law by adding a pressure and temperature correction term to the specific volume. This model has the following form: Vˆ = Vˆ0 +

ˆ RT P + P0

(3.39)

where Vˆ is the specific volume of the material and Vˆ0 is its reference specific volume at a given condition. There are ˆ and P 0 . T is the three parameters to be determined, Vˆ0 , R, material temperature.

3.2.3

Tait Model

The original version of Tait model is



P ˆ ˆ V = V0 1 − C ln 1 + B

(3.40)

Vˆ0 = b1 exp (−b2 T )

(3.41)

B = b3 exp (−b4 T )

(3.42)

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MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

There are five parameters to be determined, b 1 , b 2 , b 3 , b 4 , and C . The original version of Tait model cannot deal with the abrupt volumetric change of semicrystalline material. A modified Tair model can be written in the following form:   p  + Vˆt Vˆ = Vˆ0 1 − C ln 1 + (3.43) B  b1S + b2S T¯ if T ≤ Tt (3.44) Vˆ0 = b1L + b2L T¯ if T > Tt    b3S exp −b4S T¯ if T ≤ Tt   B= (3.45) ¯ if T > Tt b3L exp −b4L T    b7 exp b8 T¯ − b9 p if T ≤ Tt ˆ (3.46) Vt = 0 if T > Tt T¯ = T − b5

(3.47)

Tt = b5 + b6 p

(3.48)

where C = 0.0894 and 13 parameters are needed. This model is capable of describing the PVT relationship of both semicrystalline and amorphous materials. Equation 3.46 characterizes the abrupt volumetric change of a semicrystalline material around its melting point. With only linear PVT transitions, b 7 , b 8 , and b 9 are for amorphous materials. T t is used to characterize the abrupt viscosity change of the material around its transition temperature.

3.3

THERMAL PROPERTIES MODEL

Thermal properties such as thermal conductivity, specific heat capacity, and thermal diffusivity play critical roles in prediction of heat flow rate, part temperature distribution, and heat and cycle time estimation during filling and packing phases. According to Fourier’s law of unidirectional steadystate heat conduction in an isotropic medium, the thermal conductivity is defined by k=−

q ∇T

(3.49)

where q is the heat flux. The thermal diffusivity α determines the transient heat flow and the time-dependent temperature distribution in the polymer. It is related to the thermal conductivity through the following equation: k α= ρcp

(3.50)

where ρ is the density of the polymer and c p is the specific heat capacity. In general, thermal diffusivity is determined

FIGURE 3.4 A schematic diagram of linear interpolation function of thermal conductivity.

experimentally from the dynamic temperature response of the polymer. Thermal conductivity of thermoplastic appears to be a weak function of temperature, independent of molecular weight, and does not vary significantly from one thermoplastic to the other. The thermal conductivity of thermoplastic is relatively low compared to the mold metal in general. The low thermal conductivity reduces the heat that is transferred to the surroundings. Considering the heat dissipated by viscous forces of high viscosity thermoplastic, the temperature distribution across the thickness of thermoplastic is therefore quite nonisothermal. The simplest model of the thermal conductivity is the constant thermal conductivity model, namely k = k0

(3.51)

where k is the thermal conductivity and k 0 is its specified value. It assumes that the thermal conductivity is independent of temperature. This model is only supported in most of the commercial simulation software. The linear interpolation function is another common approximation of the dependence of thermal conductivity on temperature. Given thermal conductivities, k 1 and k 2 , at two different temperatures, T 1 and T 2 , as shown in Figure 3.4, we can obtain the following linear equation: k = aT + b, a =

k2 − k1 T2 k 1 − T1 k 2 ,b= T2 − T1 T2 − T1

(3.52)

The specific heat capacity is the amount of energy required to heat up a unit mass of the material for 1 ◦ C. If one neglects the effects of possible chemical or physical transformation in the material, the internal energy of the material is related to the material temperature through heat capacity.

GOVERNING EQUATIONS FOR FLUID FLOW

FIGURE 3.6 FIGURE 3.5 A schematic diagram of linear interpolation function of specific heat capacity.

Constant specific heat capacity model is generally a good approximation to assume heat capacity to be a constant, namely (3.53) cp = cp0 where c p is the heat capacity and c p0 is a given value. This model is also the only available model in most of the commercial simulation software. As similar to thermal conductivity, the linear interpolation function is another common approximation of the dependence of heat capacity on temperature. Given thermal conductivities, c p1 and c p2 , at two different temperatures, T 1 and T 2 , we can obtain the following linear equation: cp = aT + b, a =

cp2 − cp1 T2 cp1 − T1 cp2 ,b= T2 − T1 T2 − T1

(3.54)

The schematic diagram of this model is the same as shown in Figure 3.5. 3.4

GOVERNING EQUATIONS FOR FLUID FLOW

The governing equations for fluid flow consist of mass conservation equation, momentum conservation equation, and energy conservation equation. The three equations have the similar form, so a general transport equation can be generalized. These equations governing the flow of a compressible, viscous fluid are applicable to the flow of a polymer melt and are obtained using the principles of conservation of mass, momentum, and energy. We will derive them briefly. Consider a material volume of fluid, as shown in Figure 3.6, three measures of coordinate direction in Cartesian coordinate system are δx , δy, and δz . Six surfaces of the microunit are denoted by N , S , E , W , T , and B representing northern, southern, eastern, western, top, and bottom surfaces, respectively. The central point O is on the coordinate (x , y, z ).

59

A material volume of fluid.

All the fluid property parameters are functions of space and time, such as ρ(x , y, z , t), p(x , y, z , t), T (x , y, z , t), and u(x , y, z ) representing fluid density, pressure, temperature, and velocity vectors, respectively. The considered material volume of fluid here is small enough that it has enough accuracy to keep the first two terms of Taylor expansion of fluid property parameters on the surfaces. Taking the pressure p, for example, the pressure values on eastern and western surfaces can be represented as ∂p ∂x ∂p =p− ∂x

pE = p + pW 3.4.1

1 δx + o(x) ≈ p + 2 1 • δx + o(x) ≈ p − 2 •

∂p ∂x ∂p ∂x

1 δx 2 1 • δx 2 •

(3.55) (3.56)

Mass Conservation Equation

If V (t) is the material conservation of mass volume of fluid, in which there are no sources or sinks, then it means that the mass contained in V (t) does not change. That is, the mass increasing rate in the microunit equals net mass inflow flux. The mass increasing rate in fluid unit is ∂ ∂ρ • δx δy δz (ρ δx δy δz) = ∂t ∂t

(3.57)

The net mass flux through the unit surfaces is the product of density, surface area, and fluid velocity perpendicular to the surfaces. Known from Figure 3.7, the net mass flux through surfaces into fluid unit is 



∂ (ρu) • 1 ∂ (ρu) • 1 δx δy δz − ρu + δx δy δz ρu − ∂x 2 ∂x 2 



∂ (ρv) • 1 ∂ (ρv) • 1 + ρv − δy δx δz − ρv + δy δx δz ∂y 2 ∂y 2



 ∂ (ρw) • 1 ∂ (ρw) • 1 + ρw − δz δxδy − ρw + δz δxδy ∂z 2 ∂z 2 (3.58)

60

MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

The mass inflow rate is positive, and the outflow one is negative. Conservation of mass implies that Equation 3.57 equals Equation 3.58, after the simplification, we have ∂ (ρu) ∂ (ρv) ∂ (ρw) ∂ρ + + + =0 ∂t ∂x ∂y ∂z

(3.59)

The two terms on the left side of the equal to sign in Equation 3.60 represent the change rate of fluid mass per unit volume with time and the mass outflow rate per unit volume respectively, namely

(3.60)

∂ρ + ∇ • (ρu) ∂t

Writing it in a more compact form: ∂ρ + ∇ • (ρu) = 0 ∂t

where u is the speed vector. The Equation 3.59 or 3.60 is called the mass conservation equation for a compressible fluid, also known as the continuity equation or mass balance equation. For an incompressible fluid, the density is constant and so Equation 3.60 becomes ∂w ∂u ∂v + + =0 ∂x ∂y ∂z

(3.61)

∇ •u = 0

(3.62)

or

Sometimes it is useful to express the continuity equation in terms of the material derivative. Expanding out Equation 3.60, we get ∂ρ + ρ (∇ • u) + u • ∇ρ = 0 ∂t

(3.63)

Using the chain rule of differentiation, we find that the derivative of the density with respect to time is given by ∂ρ ∂ρ ∂x ∂ρ ∂y ∂ρ ∂z ∂ρ Dρ = + + + = + u • ∇ρ Dt ∂t ∂x ∂t ∂y ∂t ∂z ∂t ∂t (3.64) Using Equation 3.64, we may write Equation 3.60 or Equation 3.63 in the following form: Dρ = −ρ (∇ • u) Dt 3.4.2

And the change rate of ϕ with time per unit volume in fluid microunit is

∂ϕ Dϕ • =ρ + u ∇ϕ (3.67) ρ Dt ∂t

(3.65)

Introducing the characteristic variable ϕ into Equation 3.68, we simply obtain ∂ (ρϕ) + ∇ • (ρϕu) ∂t

Dϕ ∂ϕ = + u • ∇ϕ Dt ∂t

(3.66)

(3.69)

which represents the sum of the change rate of ϕ per unit volume in fluid microunit and net outflow rate of ϕ per unit volume in fluid microunit. Using the chain rule of differentiation and divergence theorem, Equation 3.69 can be written as

∂ϕ ∂ (ρϕ) • • + ∇ (ρϕu) = ρ + u ∇ϕ ∂t ∂t 

∂ρ (3.70) + ∇ • (ρu) +ϕ ∂t Referring to the mass conservation equation, the second term on the right side of the equal to sign in Equation 3.70 equals zero, so we have

Dϕ ∂ϕ ∂ (ρϕ) ρ =ρ + u • ∇ϕ = + ∇ • (ρϕu) (3.71) Dt ∂t ∂t The physical meaning of Equation 3.71 is that the increasing rate of ϕ in Equation 3.71 equals the sum of the increasing rate of ϕ in fluid microunit and net outflow rate of ϕ in fluid microunit. Let ϕ be the real velocity value to represent momentum change rate of the fluid microunit: x -component:

Momentum Conservation Equation

Conservation of momentum requires that the time rate of change of fluid particle momentum in a material volume, V (t), be equal to the sum of external forces acting on V (t). Replace ρ in Equation 3.63 with a characteristic variable ϕ, representing a certain characteristic parameter per unit mass in fluid microunit, then we obtain the change rate of ϕ with time per unit mass:

(3.68)

∂ (ρu) Du = + ∇ • (ρuu) Dt ∂t

(3.72)

∂ (ρv) Dv = + ∇ • (ρvu) Dt ∂t

(3.73)

∂ (ρw) Dw = + ∇ • (ρwu) Dt ∂t

(3.74)

ϕ = u, ρ y-component: ϕ = v, ρ z -component: ϕ = w, ρ

GOVERNING EQUATIONS FOR FLUID FLOW

FIGURE 3.7

61

The mass flux of a fluid microunit.

The combination of the above three equations can be written as ∂ (ρu) Du ρ = + ∇ • (ρuu) (3.75) Dt ∂t The forces on the fluid microunit include surface force and body force. The surface force includes the pressure and viscous force; and body force includes gravity, centrifugal force, electromagnetic force, etc. In general, the surface force is represented as independent stress components; meanwhile, the body force is put into the source term of the equations. As shown in Figure 3.8, the viscous force components are denoted by τ ij , the subscripts i and j represent the viscous force on the surface perpendicular to i -coordinate direction, whose direction is j -direction. There are nine viscous shear stress components in all, namely, τ xx , τ yy ,

τ zz , τ xy , τ xz , τ yx , τ yz , τ zy , and τ zx , out of which six components are independent. Referring to equivalent law of shearing stress, we have τxy = τyx , τxz = τzx , τyz = τzy

(3.76)

Taking x -direction forces, for example, as shown in Figure 3.8, the forces on the x -direction include the pressure p and the viscous force components τ xx , τ yx , and τ zx . According to Figure 3.7, the resultant forces on the eastern and the western surfaces are





∂τxx • 1 ∂p • 1 δx − τxx − δx δy δz p− ∂x 2 ∂x 2





∂p • 1 ∂τxx • 1 + − p+ δx + τxx + δx δy δz ∂x 2 ∂x 2

∂p ∂τxx = − + δx δy δz (3.77) ∂x ∂x The resultant forces on the southern and the northern surfaces are



∂τyx • 1 ∂τyx • 1 δy δzδx + τyx + δy δzδx − τyx − ∂y 2 ∂y 2 =

∂τyx δxδyδz ∂y

(3.78)

The resultant forces on the top and the bottom surfaces are



∂τzx • 1 ∂τzx • 1 − τzx − δz δx δy + τzx + δz δx δy ∂z 2 ∂z 2 FIGURE 3.8

Surface force components of a fluid microunit.

=

∂τzx δx δy δz ∂z

(3.79)

62

MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

Dividing the sum of the above three equations by the volume of the fluid microunit δx δy δz , we obtain the x direction resultant force of the surfaces on fluid microunit per unit volume: ∂ (−p + τxx ) ∂τyx ∂τzx + + ∂x ∂y ∂z

(3.80)

As mentioned earlier, the body force is denoted as f x . According to the law of conservation of momentum (also known as forces balance law ), the momentum conservation equation on the x -direction is ρ

∂ (ρu) Du = + ∇ • (ρuu) Dt ∂t ∂τzx ∂ (−p + τxx ) ∂τyx + + + ρfx = ∂x ∂y ∂z

 



 ∂uj ∂ui ∂p ∂ ∂ (ρui ) ∂ ρui uj μ + =− + + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi

∂uj ∂ λ + ρfi + (3.87) ∂xi ∂xj If the fluid is incompressible, we have Equation 3.68, μ is a constant and the dilatational viscosity has no effect. So Equation 3.87 becomes   ∂ (ρui ) ∂ ρui uj ∂p ∂ 2 ui =− + μ 2 + ρfi + ∂t ∂xj ∂xi ∂xj

(3.81)

Similarly, the momentum conservation equations on the ydirection and z -direction: ∂ (ρv) Dv = + ∇ • (ρvu) Dt ∂t   ∂ −p + τyy ∂τxy ∂τzy = + + + ρfy ∂x ∂y ∂z Dw ∂ (ρw) ρ = + ∇ • (ρwu) Dt ∂t ∂τzy ∂ (−p + τzz ) ∂τzx + + + ρfz = ∂x ∂y ∂z

For generalized Newtonian flow, the constitutive equation is Equation 3.4. Thus, the Navier–Stokes equation can be expressed as

(3.88)

or ∂ (ρu) + ∇ • (ρuu) = −∇p + ∇ • (η • ∇u) + ρf (3.89) ∂t

ρ

(3.82)

(3.83)

Substituting them into Equation 3.75, we obtain the momentum conservation equation ∂ (ρu) + ∇ • (ρuu) = ∇ • (σ ) + ρf ∂t

(3.84)

3.4.3

Energy Conservation Equation

The first law of thermodynamics states that the change rate of total energy in a material volume V (t) is equal to the difference between the work done on the volume and the heat loss of the fluid within the material volume. Let E (x , y, z , t) denote the total energy of a material volume V (t), then the change rate of E with time per unit mass is ∂E DE = + ∇ • (Eu) (3.90) Dt ∂t and the change rate of E with time per unit volume is

where σ is the stress tensor. For the incompressible case, Equation 3.84 can be written in the following form: ∂ (ρu) + ρu • ∇u = ∇ • (σ ) + ρf ∂t

(3.85)

ρ

∂ (ρE) DE = + ∇ • (ρEu) Dt ∂t

(3.91)

Let q denote the heat flux (Fig. 3.9). The rate of heat loss is x -component:

If the body force is just the gravity, then fx = 0, fy = 0, fz = −g

(3.86)

The viscous stress tensor τ in Equations 3.84 and 3.85 is unknown. In order to deal with fluids, it is necessary to have a constitutive relationship between the viscous stress tensor τ and the rate of strain tensor ε˙ . Referring to the earlier introduction, appropriate constitutive equation is selected, and the viscosity model substituted into Equation 3.84 or Equation 3.85 to obtain the corresponding Navier–Stokes equation.





 ∂qx • 1 ∂qx • 1 dx − qx + dx dy dz qx − ∂x 2 ∂x 2 =−

∂qx dx dy dz ∂x

(3.92)

y-component:



qy −



 ∂qy • 1 ∂qy • 1 dy − qy + dy dx dz ∂y 2 ∂y 2 =−

∂qy dx dy dz ∂y

(3.93)

GOVERNING EQUATIONS FOR FLUID FLOW

FIGURE 3.9

Heat flux components of a fluid microunit.

z -component:





∂qz • 1 ∂qz • 1 dz − qz − dz dx dy qz − ∂z 2 ∂z 2 =−

∂qz dx dy dz ∂z

(3.94)

Therefore, the rate of heat loss per unit volume is −

∂qy ∂qx ∂qz − − = −∇ • q ∂x ∂y ∂z

(3.95)

According to Fourier heat conduction law, we have q = −k • ∇T

63

(3.96)

where k is the thermal conductivity. So Equation 3.71 becomes (3.97) −∇ • q = ∇ • (k • ∇T ) The rate of work done on the material volume by external forces is due to both surface and body forces. The surface forces on x -direction are respected by Equations 3.77–3.79, whose work is       ∂ u (−p + τxx ) ∂ uτyx ∂ (uτzx ) + + dx dy dz ∂x ∂y ∂z (3.98) Similarly, the works on y-direction and z -direction are         ∂ v −p + τyy ∂ vτzy ∂ vτxy + + dx dy dz ∂x ∂y ∂z (3.99)

    ∂ w (−p + τzz ) ∂ (wτzx ) ∂ wτzy + + dx dy dz ∂x ∂y ∂z (3.100) So, the work of the surface force on the fluid microunit per unit volume is 



    ∂ (uτxx ) ∂ uτyx ∂ (uτzx ) ∂ vτxy + + − ∇ (ρu) + ∂x ∂y ∂z ∂x     ∂ vτzy ∂ vτyy + + ∂y ∂z    ∂ (wτzx ) ∂ wτzy ∂ (wτzz ) + + + (3.101) ∂x ∂y ∂z •

The total energy of fluid in a material volume is the sum of its kinetic energy e, internal energy i , and geopotential energy, which is always treated as work done by body force such as the gravity. According to energy conservation law and referring to Equations 3.92, 3.96, and 3.101, we have

∂ (ρE) ∂ (uτxx ) + ∇ • (ρEu) = −∇ • (ρu) + ∂t ∂x       ∂ vτyy ∂ uτyx ∂ (uτzx ) ∂ vτxy + + + + ∂y ∂z ∂x ∂y      ∂ vτzy ∂ (wτzx ) ∂ wτzy ∂ (wτzz ) + + + + ∂z ∂x ∂y ∂z + ∇ • (k • ∇T ) + ρu • g + SE

(3.102)

64

MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

where

or total specific enthalpy E =i+e =i+

 1 2 u + v 2 + w2 2

(3.103)

ρ gu is the work done by gravity, and S E is the source term. We can obtain the expression of change rate of momentum from Equations 3.81–3.83 as

 p 1 2 p u + v 2 + w2 = i + + e = E + 2 ρ ρ (3.108) The total enthalpy conservation equation can obtained from Equation 3.100 h0 = h +

(3.104)

∂ (ρh0 ) ∂p + ∇ • (ρh0 u) = ∇ • (k • ∇T ) + ∂t ∂t      ∂ (uτxx ) ∂ uτyx ∂ (uτzx ) ∂ vτxy + + + + ∂x ∂y ∂z ∂x     ∂ vτyy ∂ vτzy ∂ (wτzx ) + + + ∂y ∂z ∂x    ∂ wτzy ∂ (wτzz ) + + + ρu • g + Sh (3.109) ∂y ∂z

Internal energy conservation equation can be obtained by subtracting Equation 3.104 from Equation 3.100:

If it is isotropy generalized Newtonian fluid, referring to constitutive Equations 3.5 and 3.7, Equation 3.105 becomes

∂ (ρi) ∂u + ∇ • (ρiu) = −p • (∇ • u) + ∇ • (k • ∇T ) + τxx ∂t ∂x ∂u ∂u ∂v ∂v ∂v ∂w + τyx + τzx + τxy + τyy + τzy + τzx ∂y ∂z ∂x ∂y ∂z ∂x ∂w ∂w + τzz + Si + τzy (3.105) ∂y ∂z

∂ (ρi) + ∇ • (ρiu) = −p • (∇ • u) + ∇ • (k • ∇T ) +  + Si ∂t (3.110) where  is the dissipated power:  



∂u 2 ∂v 2 ∂w 2 ∂u ∂v 2 + =μ 2 + + + ∂x ∂y ∂z ∂y ∂x 



∂u ∂w 2 ∂v ∂w 2 + + + + λ(∇ • u)2 + ∂z ∂x ∂z ∂y (3.111)



∂τyx ∂τxx ∂τzx ∂ (ρe) • • + ∇ (ρeu) = −u ∇p + u + + ∂t ∂x ∂y ∂z

∂τyy ∂τzy ∂τxy + + +v ∂x ∂y ∂z

∂τzy ∂τzz ∂τzx +w + + + ρu • f ∂x ∂y ∂z

If the fluid is incompressible, then ∇ • u = 0, the destiny ρ is a constant, the internal energy i = cT in which c is the specific heat capacity. So Equation 3.105 becomes ∂u ∂u ∂T + c∇ • (ρT u) = ∇ • (k • ∇T ) + τxx + τyx ∂t ∂x ∂y ∂v ∂v ∂v ∂u + τxy + τyy + τzy + τzx ∂z ∂x ∂y ∂z ∂w ∂w ∂w + τzx + τzy + τzz + Si (3.106) ∂x ∂y ∂z

ρc

The specific heat capacity is defined to be the heat capacity per unit mass of material. The unit of specific heat is joules per kilogram per degree kelvin [J/(kg K)]. Specific heat capacity may be measured under conditions of constant volume or pressure and is denoted by c v or c p , respectively. Owing to the large stresses exerted on the containing vessel when heating a sample under constant volume, the use of c p is most common. If the fluid is incompressible, Equation 3.100 can be represented in the conservation form of enthalpy per unit volume (namely specific enthalpy). Specific enthalpy is defined as p (3.107) h=i+ ρ

 is a nonnegative function representing the effect of all viscous stress; and its physical meaning is the power done by viscous stress (deformation energy) in fluid. Equations 3.100, 3.105, 3.106, 3.109, and 3.110 are different forms of energy conservation equation. 3.4.4

General Transport Equation

Noticing Equation 3.60, the mass conservation equation, Equation 3.89, the momentum equation, and Equation 3.60, a typical form of energy conservation equation, we will find that although they are equations of different variables, they have a similar form. Let ϕ denote a general variable (or called characteristic variable), then the equations can be written as a unified form: ∂ (ρϕ) + ∇ • (ρϕu) = ∇ • (Ŵ • ∇ϕ) + Sϕ ∂t

(3.112)

where Ŵ is the diffusivity and S ϕ is the source term. Equation 3.110 is called general transport equation. The

BOUNDARY CONDITIONS

TABLE 3.1 Equation

Variables’ Selection in General Transport Ωmf

Equations

ϕ

Ŵ

Sϕ

Mass conservation equation

1

0

x -Component momentum equation

u

η

y-Component momentum equation

v

η

z -Component momentum equation

w

η

Energy conservation equation

i

k

0 ∂p − + ρfx ∂x ∂p + ρfy − ∂y ∂p − + ρfz ∂z +Si

Ωins

Ωinj

y

z Ωmf

x

first term on left side of equal sign is transient term. The second term is a convection term. The first term on right side of equal to sign is diffusion term. The physical meaning of the equation is that the sum of the change rate of ϕ with time and the convective outflow rate is equal to the sum of the diffusive increasing rate of ϕ and the source increasing rate of ϕ. Selecting the appropriate choice of ϕ, Ŵ, and S ϕ , we can obtain the mass conservation equation, the momentum equation, and the energy conservation equation as shown in Table 3.1. The integral taken over Equation 3.112 on the solution domain and time is 

t+t

 t+t  ∂ (ρϕ) dV dt + ∇ • (ρϕu) dV dt ∂t t t V V  t+t   t+t  ∇ • (Ŵ • ∇ϕ) dV dt + Sϕ dV dt = 

t

65

V

t

V

Ωwt

Ωem

Ωwb

FIGURE 3.10 Boundary conditions for injection molding simulation.

3.5 BOUNDARY CONDITIONS After obtaining the governing equations, we consider boundary conditions for the problem of injection molding. Unlike the governing equations, boundary conditions are quite specific for a particular problem. Therefore, it is possible to describe boundary conditions for our problem, which will hold for both the general equations and the simplified equations for injection molding. Figure 3.10 illustrates a simple mold cavity for which we discuss the required boundary conditions. There are several surfaces on which boundary conditions need be described:

(3.113)

Using Gauss’ Theorem (or the divergence theorem), the above equation becomes 

t

 t+t  ∂ (ρϕ) dV dt + n • (ρϕu) dS dt ∂t t S V  t+t   t+t  n • (Ŵ • ∇ϕ) dS dt + Sϕ dV dt =

t+t



t

S

t

V

(3.114)

in which S is the surface of V and n is the unit outernormal vector of S . For a steady problem, the transient term is zero, namely 

t

t+t



n • (ρϕu) dS dt = S



t

 +

t+t t



V

t+t



n • (Ŵ • ∇ϕ) dS dt

S

Sϕ dV dt

(3.115)

• inj is the surface through which the melt is injected into the cavity. • em is the surface defining the edge of the mold, which is in contact with the melt. • wt and wb is the top and bottom surface of the mold wall, respectively. • ins is the surface defining any insert in the mold. It is possible to have any number in a real cavity. • mf is the surface defining the position of the melt front. For the injection molding problem, boundary conditions relate to the solution of the pressure and thermal distributions in the cavity. In fact, it is possible to combine the continuity and momentum equations into a single equation for pressure. This pressure equation is coupled to the energy equation because the material’s viscosity, which affects pressure, is determined by both the temperature and shear rate. Solution to the injection molding problem therefore

66

MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

requires solution to the pressure and energy equations. The pressure and thermal boundary conditions will be discussed separately in the following section. 3.5.1

• The temperature, T , is prescribed on all mold boundaries. A different temperature can be prescribed on each boundary. That is,

The boundary conditions relating to pressure are as follows: • At any impermeable boundary, the pressure gradient in the normal direction to the boundary is zero. The impermeable boundaries are the mold edges, the mold walls, and any mold inserts. Therefore, we have ∂p = 0 on em , wt , wb , ∂n

and

ins

(3.116)

Physically, this means that it cannot flow through the mold walls or edges. • The melt flow rate q, which can determine the gate speed, or the pressure p, is specified at the surface where the melt enters the cavity (also known as gate). That is, q = qinj or p = pinj on inj

(3.117)

Engineers usually use a specified flow rate in the filling phase. This is obtained by dividing the volume of the part to be molded by the user-specified fill time. It is possible to vary this flow rate to model injection profiling. For the packing phase, it is common to specify a pressure at the points of injection gate. So, this may also vary from model packing profiles. • Assuming that the pressure datum is atmospheric pressure, the pressure is zero at the melt front. That is, (3.118) p = 0 on mf As most molds are vented that allow air ahead of the melt to escape, this is physically sensible. 3.5.2

T = Tem on em

(3.120)

T = Twt on wt

(3.121)

T = Twb on wb

(3.122)

T = Tins on ins

(3.123)

Pressure Boundary Conditions

Temperature Boundary Conditions

3.5.3

Slip Boundary Condition

At fluid–solid boundaries, the impermeable condition has been widely accepted, namely the melt has zero on the normal direction of the wall. This can be expressed as u • n = 0 on em , wt , and wb

• The temperature profile through the cavity thickness, T (z ), is prescribed for the surface through which the melt is injected. That is (3.119)

Most flow analysis software assumes that the melt temperature is uniform at the point of injection. In practice, this is not too critical because the melt is rapidly convected into the cavity or runner system, where thermal effects allow a temperature profile across the thickness to develop quickly because of shear heating and conduction.

(3.124)

where n is the normal vector of mold wall and edge. While for the tangential direction of wall, the no-slip condition41,42 is commonly used as u • t = 0 on em , wt , and wb

(3.125)

where t is the tangential vector of the wall. It is based on the following assumption: the shear stress on the interface is always less than the critical value required by the liquid to wet the solid wall. So the liquid adhesive to the wall moves (or keeps static) with the same velocity of the wall together. From Equations 3.124 and 3.125, we have u = 0 on em , wt , and wb

The boundary conditions relating to temperature are as follows:

T (z) = Tinj (z)

Recently, it has become common to first perform a thermal analysis of the mold and its cooling system, and then use the results from this analysis to define the temperatures on the mold for flow analysis.

(3.126)

As with most engineering approximations, the no-slip condition does not always hold in reality. Many scholars considered that in the polymer molding process, the shear stress on the cavity surface is always bigger than the critical value, as a result of which the fluid would slip on the cavity wall instead of being adhesive with it.43 On the other hand, the no-slip condition will cause stress singularity on the contact line, especially for the free surface tracing methods with free surface different equation. For the tracing methods with particles or volume function, the problem that stress changes tempestuously at contact points still exists, although it is not so bad as the former, which is difficult to treat in numerical simulation. If using

MODEL SIMPLIFICATIONS

nonslip condition, the stress singularity on the contact line is basically eliminated because the velocity near the contact line changes continuously.44 – 46 Hockin45 considered that the melt on the cavity wall is slipable and is effected by the friction. The friction depends on slip coefficient and the tangential velocity of the fluid on the cavity wall. Silliman and Scriven47 proposed the Navier slip condition according to the Navier linear–slip length model48 and established a quantitative relationship between the slip velocity and the shear stress of the wall shown as [βτ • n + (u − u b )] • t T = 0 (3.127) in which t and n represent the tangential and outer-normal unit vectors at boundaries, respectively, τ is the deviatoric stress tensor or viscous stress tensor, β is slip coefficient, and u and u b are the velocity of the fluid particle and the cavity wall. We can obtain the relationship between external force on the fluid at slip boundaries and the velocity of the fluid from Equation 3.127. In general case, the velocity of the cavity wall is zero, so the relationship can be written in the following form: ut (3.128) Tf = − β in which T f is the tangential external force on the fluid and u t is the tangential component of the fluid velocity on the cavity wall. Besides the linear–slip length model represented by Equation 3.127, the common slip boundary models include ultimate shear stress–slip model49,50 and nonlinear–slip length model.51,52 In linear–slip length model, it is considered that the slip occurs just when the shear stress exists at the fluid–solid interface and the slip coefficient (often represented by the slip length) is a constant. The ultimate shear stress–slip model considers that no slip occurs at low shear rate (or shear stress); the slip would occur only when shear rate (or shear stress) reaches an ultimate value. The nonlinear–slip length model considers that the slip coefficient is a constant at low shear rate, so it becomes linear–slip length model; while at high shear rate, the slip coefficient and the shear rate shows a strong nonlinear relationship, which can be described by the ultimate shear stress–slip model. The last two models are relatively complex and have some difficulties in theoretical analysis and numerical calculation. They are applied in research on micro–nano scale flow.

3.6

MODEL SIMPLIFICATIONS

The equations of governing the behavior of a fluid motion were derived in the preceding section of this chapter. They

67

are the continuity, momentum, and energy equations shown as ∂ρ + ∇ • (ρu) = 0 (3.129) ∂t ∂ (ρu) + ∇ • (ρuu) = ∇ • (σ ) + ρf (3.130) ∂t ∂u ∂u ∂T + c∇ • (ρT u) = ∇ • (k • ∇T ) + τxx + τyx ρc ∂t ∂x ∂y ∂v ∂v ∂v ∂w ∂u + τxy + τyy + τzy + τzx + τzx ∂z ∂x ∂y ∂z ∂x ∂w ∂w + τzz + Si + τzy (3.131) ∂y ∂z These equations are quite general and hold for all common fluids. With today’s computer hardware and numerical techniques, it is not feasible to solve them in complicated domains such as injection mold cavities. Our application is injection molding, and it is possible to simplify the equations for this purpose. Injection molding is a cyclic process with three fundamental steps: filling phase, packing phase, and cooling phase. In the filling stage, a polymer melt is injected into the cold-walled cavity, where it spreads under the action of high pressures and fills the mould. In the packing stage, after the mould is filled, high pressure is maintained, and additional melt flows into the cavity to compensate for density changes (shrinkage) during cooling. And in the last step, the melt is cooled, and the shaped article is ejected. We focus on different fluid characteristics during the filling and packing phases, so the assumptions, boundary conditions and further simplified numerical models are different in the simulation of these two phases. To make solution times and material data requirements more reasonable, a number of assumptions are made. In the following section, we discuss these assumptions and produce a simplified set of equations for analysis of the filling and packing phases, respectively.

3.6.1

Hele–Shaw Model

At present, the most widely used model simplification in polymer processing simulation is the Hele–Shaw model (named after Henry Selby Hele–Shaw). It applies to flows in small gap. Conventionally, most of plastic products have thin shell structures. The thickness of most plastic products is far beyond the planar directions, such that Lh 1 and ∂h h W 1, and the gap vary slowly such that ∂x 1 and ∂h ∂y 1. In addition, the long molecular chain structure causes a high viscosity, so the inertia force is much smaller than viscous shear stress. So the filling flow in a thin cavity can be approximated to Hele–Shaw flows. Figure 3.11

68

MATHEMATICAL MODELS FOR THE FILLING AND PACKING SIMULATION

FIGURE 3.11

A schematic diagram of a typical flow described by the Hele–Shaw model.

represents a typical flow described by the Hele–Shaw model. The governing equation of Hele–Shaw flows is identical to that of the inviscid potential flow and to the flow of fluid through a porous medium (Darcy’s law). It thus permits visualization of this kind of flow in two dimensions. Hele–Shaw approximation believes the following assumptions: 1. The filling flow of melt can be considered as extended laminar flow of incompressible fluid. The velocity in the thickness direction can be ignored and the pressure keeps constant. 2. Because of the high viscosity, the inertial force that is much smaller than viscous shear stress can be ignored. 3. Ignore the heat convection in the thickness direction. 4. In the direction of melt flow, heat conduction is slightly relative to heat convection, hence it can be ignored. 5. Do not consider the cross flow at the melt flow front, namely fountain effect. 6. Ignore the effect of gap on simulation.

And by substituting Equation 3.130 in Equation 3.131, we have ∇ • (ρuu) = ρ (∇ • uu) = ρu (∇ • u) + ρ (u • ∇u) = ρ (u • ∇u)

(3.133)

• During the filling phase, the melt can be represented as a Generalized Newtonian Fluid. This assumption implies that viscoelastic effects will be ignored, and we can use the constitutive relationship and viscosity models for the generalized Newtonian fluid. And because of the earlier assumption that the melt is incompressible, the deviatoric strain tensor and the strain rate tensor are equal. Using Equations 3.2, 3.6, and 3.8, we have ∇ • (σ ) = ∇ • (−pI + ηγ˙ ) = −∇p + ∇ • ηγ˙   = −∇p + ∇ • η ∇u + ∇u T (3.134)

And Equation 3.131 becomes Equation 3.110.

• During the filling phase, the thermal conductivity of the material is assumed to be constant. Despite the fact that the thermal conductivity k of polymers depends on temperature, this assumption is enforced because of the difficulty in obtaining material data.

The Hele–Shaw approximation (Shell model) is a good tool to simulate the thin parts injection molding processing. To model a solid part with thin shell structure, the geometry of the solid part is simplified into mid-plane model or the surface model.

Using the assumption the term involving k in Equation 3.131 becomes

3.6.2

Enforcing the assumptions above gives the following equations:

Governing Equations for the Filling Phase

• During the filling phase, the melt is assumed to be incompressible. This assumption means that the density is constant and the dilatational viscosity λ = 0. So Equation 3.127 becomes ∇ •u = 0

(3.132)

∇ • (k • ∇T ) = k∇ • ∇T = k∇ 2 T

∇ •u = 0

(3.135)

(3.136)

  ∂ (ρu) + ρ (u • ∇u) = −∇p + ∇ • η ∇u + ∇u T + ρf ∂t (3.137) ∂ (ρi) (3.138) + ∇ • (ρiu) = k∇ 2 T +  + Si ∂t

REFERENCES

3.6.3

Governing Equations for the Packing Phase

During the packing phase, it is essential to consider the effect of melt compressibility. Therefore, the relation ∇ • u = 0 may not be used to simplify the equations. We do, however, use all the other assumptions regarding material behavior and geometry as discussed in the previous section. The continuity equation in the packing phase is ∂ρ + ∇ • (ρu) = 0 ∂t

(3.139)

We know the term ∇ • (ρuu) in the momentum equation can be expanded as follows: ∇ • (ρuu) = ρ (u • ∇u) + (∇ • ρu) u

(3.140)

∂ ∂u ∂u ∂ρ u +ρ = − (∇ • ρu) u + ρ (ρu) = ∂t ∂t ∂t ∂t

(3.141)

And

We obtain ∂ ∂u (ρu) = ρ (u • ∇u) − ∇ • ρuu + ρ ∂t ∂t

(3.142)

Substituting the above equations in the momentum equation, we have ρ

∂u = −ρ (u • ∇u) + ∇ • σ + ρf ∂t

(3.143)

Using the assumption that the melt be considered as a generalized Newtonian fluid, Equation 3.142 becomes ρ

∂u = −∇p − ρ (u • ∇u) + ∇ • ηγ˙ + ρf ∂t

(3.144)

We use the same assumptions in the packing phase as that used in the filling phase except that the fluid is considered compressible now. The energy equation becomes



∂p ∂T + u • ∇T = βT + u • ∇p + ηγ˙ + k∇ 2 T ρcp ∂t ∂t (3.145) Equations 3.139, 3.144, and 3.145 are the continuity, momentum, and energy equations, respectively, governing the behavior of a fluid motion during the packing phase.

REFERENCES 1. Fung Y.C., A First Course in Continuum Mechanics. 1977, New York: Prentice-Hall. 2. Tanner R.I., Engineering Rheology. 1985, Oxford: Clarendon. 3. Liu I-S., Continuum Mechanics. 2002, Berlin: Springer.

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4. Nassehi V., Pactical Aspects of Finite Element Molding of Polymer Processing. 2002, Chichester: John Wiley & Sons. 5. Lenk R.S., Polymer Rheology. 1978, London: Kluwer Academic Pub. 6. Herschel W.H., Bulkley R., Flow of non-Newtonian fluids, in Encyclopedia of Fluid Mechanics. Rudraiah K.P.N., Editors. 1927, Gulf: Houston. 7. Oldroyd J.G., A Rational Formulation of the Equations of Plastic Flow for a Bingham Solid, in Proceedings of the Cambridge Philosophical Society. 1947, Cambridge: Cambridge Univ Press. 100–105. 8. Casson N., Rheology of Disperse Systems. Mill C.C., Editor 1959, London: Pergamon. 9. Bird R.B., Armstrong R.C., Hassager O., Dynamics of polymeric liquids, Vol. 1. Fluid Mechanics. 1987, New York: John Wiley and Sons Inc. 10. Middleman S., Greener J., Malone M., Fundamentals of Polymer Processing. 1977, New York: McGraw-Hill. 11. Cross M.M., Rheology of non-newtonian fluids: a new flow equation for pseudoplastic systems. Journal of Colloid Science, 1965. 20(5): 417–437. 12. Carreau P.J., Rheological equations from molecular network theories. Journal of Rheology, 1972. 16(1): 99–127. 13. Sisko A.W., The flow of lubricating greases. Industrial & Engineering Chemistry, 1958. 50(12): 1789–1792. 14. Liu C.T., Chen J.B., Wang L.X., et al., Numerical analysis of injection mold filling process. Mechanics and Engineering, 1999. 21(2): 37–39. 15. Cao W., Wang R., Shen C.Y., 3D melt flow simulation in injection molding. Journal of Chemical Industry and Engineering (China), 2009. 55(9): 1493–1498. 16. Geng T., Li D.Q., Zhou H.M., Numerical simulation of filling stage in injection molding based on a 3D model. China Plastics, 2003. 17(7): 78–81. 17. Chen J.L., The numerical simulation and application of the flow behaviour of the polymer in the injection process. Die and Mould Technology, 1997. (3): 29–33. 18. Wang L.X., Liu C.T., Shen C.Y., et al., Dynamic simulation of injection mould filling process. China Plastics, 1996. 10(5): 71–77. 19. Ma D.J., Chen J.N., Simulations of three-dimensional flow fields in metering section of screw during plasticization process. Journal of Petrochemical Universities, 2006. 19(2): 76–79. 20. Sun Y.P., Liu H.S., Xiong H.H., et al., Numerical simulation of gas assisted injection molding process. Plastics Science and Technology, 2000. (5): 38–40. 21. Lin L.F., Pen Y.S., Application of finite-element method to the solution of pressure field of filling in injection molding. Journal of Zhejiang University (Engineering Science), 2000. 34(1): 9–14. 22. Reiner M., A mathematical theory of dilatancy. American Journal of Mathematics, 1945. 67(3): 350–362. 23. Rivlin R.S., The hydrodynamics of non-Newtonian fluids. I. Proceedings of the Royal Society A, 1948. 193(1033): 260–281.

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24. Oldroyd J.G., On the formulation of rheological equations of state. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 1950, JSTOR: 523–541. 25. Oldroyd J.G., Non-Newtonian effects in steady motion of some idealized elastic-viscous fluids. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1958. 245: 278–297. 26. Bird R.B., Wiest J.M., Constitutive equations for polymeric liquids. Annual Review of Fluid Mechanics, 1995. 27(1): 169–193. 27. Thien N.P., Tanner R.I., A new constitutive equation derived from network theory. Journal of Non-Newtonian fluid mechanics, 1977. 2(4): 353–365. 28. Phan-Thien N., A nonlinear network viscoelastic model. Journal of Rheology, 1978. 22: 259–283. 29. Giesekus H., A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. Journal of Non-Newtonian fluid mechanics, 1982. 11(1–2): 69–109. 30. Warner H.R., Kinetic theory and rheology of dilute suspensions of finitely extensible dumbbells. Industrial and Engineering Chemistry Fundamentals, 1972. 11: 379–387. 31. Herrchen M., Ottinger H.C., A detailed comparison of various FENE dumbbell models. Journal of Non-Newtonian Fluid Mechanics, 1997. 68(1): 17–42. 32. Lodge A.S., Elastic Liquids. 1964, London: Academic Press. 33. Bernstein B., Kearsley E.A., Zapas L.J., A study of stress relaxation with finite strain. Journal of Rheology, 1963. 7(1): 391–410. 34. McLeish T.C.B, Larson R.G., Molecular constitutive equations for a class of branched polymers: The pom-pom polymer. Journal of rheology, 1998. 42: 81–110. 35. Verbeeten W.M.H., Peters G.W.M., Baaijens F.P.T., Differential constitutive equations for polymer melts: the eXtended Pom–Pom model. Journal of rheology, 2001. 45(4): 823–843. 36. Verbeeten W.M.H., Peters G.W.M., Baaijens F., Viscoelastic analysis of complex polymer melt flows using the eXtended Pom-Pom model. Journal of Non-Newtonian fluid mechanics, 2002. 108(1–3): 301–326. 37. Verbeeten W.M.H., Peters G.W.M., Baaijens F., Numerical simulations of the planar contraction flow for a polyethylene melt using the XPP model. Journal of Non-Newtonian fluid mechanics, 2004. 117(2–3): 73–84. 38. van Os R.G.M., Phillips T.N., Efficient and stable spectral element methods for predicting the flow of an XPP fluid past

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4 NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION Huamin Zhou, Zixiang Hu, Yun Zhang, and Dequn Li State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

In the previous chapter, we have got the governing equations in the filling and packing phases. In this chapter, we first introduce some common numerical methods for the mathematical problems to be solved. Then, some methods for tracking moving melt front are presented as an essential problem in the filling simulation of the injection mold. Finally, some methods for solving algebraic equations are discussed; meanwhile, parallel algorithms are also mentioned.

4.1

NUMERICAL METHODS

The filling and packing computer simulation of injection molding is mainly involved with the application of computational fluid dynamics (CFD) and numerical heat transfer (NHT). Often, a model that represents a polymer process is in the form of an algebraic equation, a set of nonlinear partial differential equations (PDEs) and/or an integral equation, which do not have analytical solutions. Numerical analysis will provide methods for obtaining useful solutions to those mathematical problems. Such methods will give an approximate but satisfactory solution to the problem, which can be used to interpret and understand the problem. In the past decades, thanks to the evolution of the high speed digital computers, numerical simulation has evolved together with the sciences of fluid mechanics, heat transfer, transport phenomena, and, of course, polymer rheology and processing.

So far, many numerical methods have been developed for governing systems of PDEs, which describe the conservation principle, such as finite differential method, finite volume method (FVM), finite element method (FEM), boundary element method, mesh-less method, and controlvolume finite element method. Because of limitations of numerical methods, some new methods have not been widely and maturely used in the field of injection molding simulation, such as boundary element method, finite analytic method, and nonstandard FEM including mixed FEM, moving FEM, discontinuous FEM, and spacetime FEM. Finite difference method (FDM) is probably the oldest numerical method for numerical solution of PDEs, which is believed to have been introduced by Euler in the eighteenth century.1 It is also the easiest method for flow and heat transfer problems for simple geometry. In this method the continuous domain is replaced with a finite difference mesh or grid constructed by numerable discrete points called nodes; functions of discrete variables defined at grid nodes are used to approximate functions of continuous variables in continuous solution domains; and difference quotient and integral sum are used to approximate derivation and integration in original equations and definite conditions, respectively; as a result, an algebraic equation is established in each node with an unknown variable at the present and neighborhood nodes. And then, the original differential equations and definite conditions can be replaced by system of algebraic equations, also known as system of difference

Computer Modeling for Injection Molding: Simulation, Optimization, and Control, First Edition. Edited by Huamin Zhou.  2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc.

71

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

equations. We can work out approximate solution in discrete nodes by solving the systems, from which the approximate solution in whole domain can be obtained by interpolation. In the case of structured mesh in a regular region, FDM is simple and efficient, and it is easy to introduce higher order scheme of convection term. However, it is hard to ensure the conservation property of discrete equations. Even worse, FDM has poor adaptability for irregular regions. As, injection molding parts usually have complex geometry, FDM is usually not used solely, but in combination with other methods. The FVM also gets mesh with numerable discrete points (namely nodes) to replace continuous domain, and defines control volume (CV) as an area around a node. This method considers governing conservation equations directly and integrates the equations on CVs as its starting point. During the integral process, different formats of the unknown function on interface (convective flux) and its first derivative (diffusion flux) are assumed, which leads to different schemes. The FVM can ensure conservation of discrete equations (only the requirement for interpolation on interface is the same for bilateral sides). Compared with the FDM, the FVM is highly adaptable to region area because it has been applied to unstructured mesh. The FVM is the most widely used numerical method, employed by most famous general CFD software. The FEM is similar to the FVM in many ways. It discretizes a continuous domain into a set of discrete subdomains, usually called elements (triangular elements or quadrilateral elements in two-dimensional domain, tetrahedron elements or hexahedron elements in threedimensional domain). For each element the FEM needs to choose a shape function (the simplest shape function is linear function), which is represented by the value of unknown variables of element nodes, and substitute the shape function into the governing equations before integrating the equations. The governing equations are multiplied by a weight function after integrating the equations and when the weighted average margin of the governing equations in the whole domain is zero. Thereby, a system of algebraic equations with unknown variables in wide-area nodes is formed. The adaptability to an irregular region is the advantage of FEM benefitting from the flexibility of element division. But it is not as mature as FVM in discretizing the convection term and in solving the primitive variable method for incompressible Navier–Stokes equations. Mesh-less method was introduced in the 1970s. After nearly three decades of development, some dozens of meshless methods have been proposed. The shape function of mesh-less method does not depend on the pregenerated mesh cell, which effectively avoids the dependency on cell shape of the traditional FEM results. And usually the shape function of the mesh-less method is much better than that

of the FEM, and is so smooth that it does not require postprocessing. The basis function of the mesh-less method can contain the function series representing the features of the unknown problem, which makes this method suitable for problems with high gradients or singularities. It is easy to analyze complex three-dimensional structures because in preprocessing only node position information is required, while mesh cell information is not required any more. But compared with FEM, the form of global matrix is relatively worse in the mesh-less method, computational efficiency is not high, calculation of the shape function is timeconsuming, some special approaches need to be used for boundary conditions, and it suffers from numerical stability problems. In the following part of this chapter, the main and current numerical methods that are used in polymer processing are presented.

4.1.1

Finite Difference Method

4.1.1.1 Difference Scheme The first step to be taken in finite difference procedure is to replace the continuous domain by the mesh. Generally speaking, the mesh is the collection of numerable discrete points called nodes. The difference mesh is usually structural mesh. The intervals among mesh nodes can be the same or not, and are called uniform or nonuniform mesh, respectively. Figure 4.1 shows typical one-, two-, and three-dimensional uniform difference meshes. Then, we can discretize equations in the given mesh. We will take the general transport equation as an example to discuss in the following part. For simplicity, we shall deal with the equation in a two-dimensional Descartes’ coordinate system: ∂ (ρϕ) ∂ (ρϕu) ∂ (ρϕv) + + =Ŵ ∂t ∂x ∂y



∂ 2ϕ ∂ 2ϕ + ∂x 2 ∂y 2



+ Sϕ

(4.1) This equation is a PDE, in which there are first and second derivatives. Essentially, the FDM discretizes a differential equation into a system of algebraic linear equations. The typical and simplest approach to this process can be obtained by using Taylor series expansions. Taylor series expansions allow the development of finite differences in a more formal basis. In addition, they provide tools to analyze the order of approximation and the error of the final solution. Here, we consider the two-dimensional nonuniform mesh, as shown in Figure 4.2. Suppose ϕ(x , y) is some function of x and y. The value of ϕ(x , y) at the node (x i , y j ) is denoted by ϕ i , j . If ϕ(x , y) is continuous, its value ϕ i + 1, j at the point (x i + 1 , y j ) may be written in terms of its value ϕ i , j at the point (x i , y j ) by the mean of a Taylor series

NUMERICAL METHODS

73

(a)

(b)

(c)

FIGURE 4.1

Typical 1D (a), 2D (b), and 3D (c) finite difference discretization.

(a)

(b)

FIGURE 4.2

1D (a) and 2D (b) nonuniform difference mesh.

expansion as follows: ϕi+1,j

  ∂ 2 ϕ  (x)2 ∂ϕ  = ϕi,j + (x) + ∂x i,j ∂x 2 i,j 2!  ∂ 3 ϕ  (x)3 + ··· (4.2) + ∂x 3 i,j 3!

and x = x i + 1 − x i . This may be rearranged to make the first derivative of ϕ i , j the subject as follows:    ϕi+1,j − ϕi,j ∂ 2 ϕ  (x) ∂ 3 ϕ  (x)2 ∂ϕ  − − −· · · = ∂x i,j ∂x 2 i,j 2! ∂x 3 i,j 3! (x) =

ϕi+1,j − ϕi,j + O (x) (x)

(4.3)

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

where O(x ) indicates the lowest order of the remaining terms. An estimate of the first derivative is obtained by truncating the series and ignoring the terms of O(x ) and higher. That is,  ϕi+1,j − ϕi,j ∂ϕ  ≈ ∂x i,j (x)

(4.4)

Equation 4.4 is called the forward difference scheme (FDS) for the first derivative. It is a first-order approximation. It ignores the first-order terms and is written in terms of the value of ϕ at a forward node (x i + 1 , y i ). If we choose the value of ϕ i − 1, j , we have a Taylor series:   ∂ϕ  ∂ 2 ϕ  (x)2 ϕi−1,j = ϕi,j − (x) + ∂x i,j ∂x 2 i,j 2!  ∂ 3 ϕ  (x)3 + ··· (4.5) − ∂x 3 i,j 3!

In the same way, we obtain the first-order backward difference scheme (BDS) to the first derivative:  ϕi,j − ϕi−1,j ∂ϕ  ≈ (4.6)  ∂x i,j (x) For equidistant spacing of the nodes, subtracting Equation 4.5 from Equation 4.2, we have

  ∂ϕ  ∂ 3 ϕ  (x)3 + ··· + 2 (x) ∂x i,j ∂x 3 i,j 3! (4.7) From which we obtain   ϕi+1,j − ϕi−1,j ∂ 3 ϕ  (x)2 ∂ϕ  + + ··· = ∂x i,j 2 (x) ∂x 3 i,j 3! ϕi+1,j − ϕi−1,j = 2

=

  ϕi+1,j − ϕi−1,j + O x 2 2 (x)

(4.8)

This leads to a second-order central difference scheme (CDS) to the first derivative:  ϕi+1,j − ϕi−1,j ∂ϕ  (4.9) ≈  ∂x i,j 2 (x) To approximate the second derivatives, add Equations 4.2 and 4.5 to obtain  ∂ 2 ϕ  (x)2 ϕi+1,j + ϕi−1,j = 2ϕi,j + 2 ∂x 2 i,j 2!  ∂ 4 ϕ  (x)4 + · · · (4.10) +2 ∂x 4 i,j 4!

After rearrangement we get    ϕi+1,j − 2ϕi,j + ϕi−1,j ∂ 2 ϕ  = + O (x)2 (4.11)  2 2 ∂x i,j (x) from which a second-order central difference approximation is found for the second derivative:  ϕi+1,j − 2ϕi,j + ϕi−1,j ∂ 2 ϕ  ≈ (4.12) ∂x 2 i,j (x)2

All the above-mentioned steps can be repeated for the y variable. If ϕ is a function of time t, namely, it is a transient problem, then we have the Taylor series expansion as follows:  n ∂ϕ n ∂ 2 ϕ  (t)2 n+1 n = ϕi,j + + ϕi,j (t) ∂t i,j ∂t 2 i,j 2! n ∂ 3 ϕ  (t)3 + ··· (4.13) + 3 ∂t i,j 3!

Rearranging we get

 n+1 n − ϕi,j ϕi,j ∂ϕ n + O (t) = ∂t i,j (t)  n+1 n ϕi,j − ϕi,j ∂ϕ n ≈ ∂t i,j (t)

(4.14)

(4.15)

The right superscript n denotes the value of the nth time step. n Equation 4.15 shows that when t = t n we know ϕi,j , n+1 then ϕi,j can be worked out directly. This is called explicit difference scheme, and is also a forward difference scheme. As there are values of two time levels t n and t n + 1 in Equation 4.15, it is also called a two-level first-order explicit scheme. Besides this scheme, there is a two-level secondorder explicit Lax–Wendroff scheme, etc. When choosing different schemes, some numerical issues are worth our attention, such as truncation error, compatibility, stability, and convergence of difference scheme. From the above discussion we understand that forward and back difference schemes have first-order truncation error, and CDS has a second-order truncation error, which represents that the CDS is precise. For problems represented by PDEs with more than one independent variable, the truncation error will be the sum of the truncation error for each first difference representation. For example, for a transient one-dimensional PDE, where we use a first-order approximation for the time derivative and a second order for the spatial derivative, we will have a truncation error, that is, O(t) + O((x )2 ), which can also be written as O(t, (x )2 ).

NUMERICAL METHODS

The consistency of a finite difference approximation is the behavior of this representation when the mesh is refined. In a one-dimensional case, for example, the mesh will indicate the value for x , which, as we discussed above, dictates the value of the truncation error. Thus, a finite difference representation of a PDE is said to be consistent if the truncation error approaches zero as the grid size reaches zero. The stability of a first difference representation deals with the behavior of the truncation error as the calculation proceeds in tune or marches in space, typically, transient problems and problems with convection–convection derivatives. A stable first difference scheme will not allow errors to grow as the solution proceeds in time or space. 4.1.1.2 The Discrete Equation When we discretize the control equation (Eq. 4.1) with a first-order forward difference for the time derivative and a second-order central difference at the new time for the spatial derivative, we will obtain an implicit finite difference scheme as follows: n+1 ϕi,j

−

n ϕi,j

t Ŵ = ρ

+ 

+

n+1 ϕi+1,j uni,j

−

n+1 ϕi−1,j

2x

+

n+1 ϕi,j +1 n vi,j

−

n+1 ϕi,j −1

2y

n+1 n+1 n+1 − 2ϕi,j + ϕi−1,j ϕi+1,j

2 (x)2 n+1 n+1 n+1 ϕi,j +1 − 2ϕi,j + ϕi,j −1

2 (y)

2



+

Sϕ ρ

(4.16)

If the spatial derivative is approximated by a second-order central difference formula at the old time instead of at the new time, we will obtain an implicit scheme in which the value at the new time is found explicitly from the value at the old time step. Explicit schemes are quite fast but suffer from the fact that the time step t must be sufficiently small. Therefore, the explicit scheme is not suitable to analyze a general unsteady problem. But if the time interval is appropriate, the explicit scheme is efficient for solving the simple diffusion problem. An FDM approximation provides an algebraic equation at each grid node similar to Equation 4.16, which contains the variable value at that node as well as values at the neighboring nodes. If the differential equation is nonlinear, the approximation will contain some nonlinear terms. The numerical process will then require linearization. For this case, after rearrangement, we obtain a system of linear algebraic equations having the following form: aP ϕP =



anb ϕnb + SP

(4.17)

nb

Here, the subscript P denotes the node at which the PDE is approximated and index nb runs over the neighbor

75

nodes involved in the approximation. The coefficient a nb depends on geometrical quantities, fluid properties, and, for nonlinear equations, on the variable values themselves. S P contains all terms that do not contain unknown variable values or what they are presumed to be known. The node P and its neighbor nodes involved in the approximation form the so-called computational molecule. Figure 4.3 shows examples of computational molecule. And for this case, the computational molecule is shown in Figure 4.3a. Owing to the shortcomings of the FDM, it is always used with or in other numerical methods. The FDM is usually employed to solve the temperature field. For instance, most of the present models that use the Hele-Shaw approximation apply an FEM description to the flow plane, and an FDM to the temperature field through the cavity thickness. Thus, we shall discuss the diffusion equation here using the onedimensional heat diffusion equation as follows: ∂T ∂ 2T =α 2 ∂t ∂z

(4.18)

where T is the temperature and α is the diffusivity. In the simulation of injection molding, the energy equation has a form similar to Equation 4.18. Using the same approximations with Equation 4.16, Equation 4.18 becomes  n+1  n+1 Ti+1 − 2Tin+1 + Ti−1 Tin+1 − Tin =α t (z)2

(4.19)

After rearrangement we obtain 

αt n+1 αt n+1 2αt Tin+1 − − T Ti−1 = Tin 1+ 2 2 i+1 (z) (z) (z)2 (4.20) When written for each node, Equation 4.20 yields a set of linear equations that may be expressed in a matrix form as follows: (4.21) AT n+1 = T n Here, the matrix A is a square matrix of known terms involving αt/(z )2 and constants, T n + 1 is a matrix of temperatures at the new time, and T n is a matrix of known temperatures at the old time. Equation 4.21 may then be solved for temperatures at the new time. Implicit methods do not suffer from the stability requirement unlike the explicit methods. They are stable for any size time step. However, if the time step is too large, they can converge the wrong solution albeit in a stable manner. This advantage must be balanced against the need to solve the matrix Equation 4.21 at each tune step. Typically, when there are many nodes, a numerical scheme such as Gauss–Seidel iteration is required.

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

(a)

(b)

FIGURE 4.3

(c)

Some examples of computational molecule.

4.1.1.3 Case Studies Zhao et al.2 proposed fast strip analysis (FSA) for fast prediction of injection pressure and minimal front temperature to optimize the process parameters for injection molding. They employed the FDM to simulate a nonisothermal and unidirectional flow in the strip after simplifying the original part’s geometric to a rectangular edge-gated strip on the basis of a geometric approximation method and calculating the dimensions of the rectangular strip by using the graph theory. In their FSA, the upwind differencing method and the central finite difference are employed in the x and z direction, respectively. Moreover, the time derivative is approximated to explicit finite difference scheme, so the analysis is quite fast but the time step must be sufficiently small. For the verification of the FSA model, a refrigerator drawer with a gate located at the center of the bottom surface was used as an example (Fig. 4.4), because of its relatively even thickness and the radial flow near the gate. Figure 4.5 illustrates the comparison of the injection pressures and minimal front temperature predicted by FSA model and Moldflow (injection molding simulation software using FEM) with different process parameters, including injection time, injection temperature, and mould temperature. It can be seen that the prediction results based on FSA model agree with those of Moldflow, indicating that FSA model has a good performance. On a computer with both Intel Core 2 2.0 GHz processor and 2 GB Ram, the running time of Moldflow is 627.6 s in contrast to the FSA model in which case it takes only 0.03 s, which is several orders of magnitude less than that of Moldflow.

FIGURE 4.4

Geometric model of a refrigerator drawer.

the construction of interpolation function that is essential in FEM. Thus, like FEM, FVM has the meshing flexibility so as to approximate the complex geometry region, and also has different schemes like FDM so that almost all design ideas and the related theoretical results of difference schemes can be used. 4.1.2.1 FVM Discrete Equations For the FVM also the solution domain needs to be subdivided into a finite number of discrete cells of limited size. In contrast to the FDM, the FVM defines the CV boundaries instead of computational nodes. The usual approach is to assign the computational nodes to the CV center. Figure 4.6 shows a typical twodimensional FVM mesh, nodes, and the CVs. The integral of the governing differential equation on a CV is the key step in FVM: ∂ (ρϕ) dV + ∇ • (Ŵ • ∇ϕ) dV ∇ • (ρϕu) dV = ∂t CV CV CV + Sϕ dV (4.22) CV

4.1.2

Finite Volume Method

The FVM originated in the 1960s mainly for the purpose of solving fluid flow and heat transfer problem. FVM was proposed and advanced based on FDM and some advantages of FEM. The difference between FVM and FDM is that the former uses cell-centered value (usually average value) instead of cell vertex value, and FVM avoids

Applying Gauss divergence theorem to Equation 4.22 results in   ∂ ρϕ dV + n(ρuϕ)dS = n(Ŵ • ∇ϕ)dS ∂t S S CV Sϕ dV (4.23) + CV

NUMERICAL METHODS

(a)

77

(b)

(c)

(d)

(e)

(f)

FIGURE 4.5 Comparison of the injection pressures and minimal front temperature predicted by FSA model and Moldflow with different process parameters, including (a,b)injection time, (c,d) injection temperature, and (e,f) mould temperature.

where V is the region volume of the CV, S is the closed boundary of the CV, n is a unit outward normal to S . If the problem is a transient one, the integral of Equation 4.23 on time interval t is required, which represents ϕ is still conservational in the time interval.

∂ t ∂t =

 t

ρϕ dV CV





dt +



t



div(Ŵ • ∇ϕ)dV dt +

CV

div(ρuϕ)dV dt

CV



t



Sϕ dV dt

(4.24)

CV

If the problem is a steady one, the term involved with time would be zero, so Equation 4.23 becomes

n(ρuϕ)dS = S



n(Ŵ ∇ϕ)dS + •

S



Sϕ dV CV

(4.25)

The FVM discrete process shows that, compared with other numerical methods, its starting point is the integral form of the governing equations and the conservation of ϕ in the CV. Every term in the FVM integral equation has its clear physical meaning, which can give the physical explanation. The node mesh is independent of the integral CV. Here, we use the mesh as shown in Figure 4.6 to discretize the two-dimensional steady convection–diffusion equation. In the figure, the shaded region is the CV of node P . Nodes W , E , N , and S are the adjacent nodes in the western, eastern, northern, and southern side of node P , respectively. The interface w, e, n, and s is defined in the middle of W–P , P–E , N–P , and P–S , respectively.

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

Here, values of ϕ e , ϕw , ϕ n , and ϕ s are calculated by interpolating the values of nodes P , W , E , N , and S . In the next section, some interpolation schemes of the convection term are introduced. When Equation 4.30 is discretized and arranged according to the node field variables, we have the following form:

P

aP ϕP = aW ϕW + aE ϕE + aN ϕN + aS ϕS + Su

(4.31)

It can also be written as aP ϕP = FIGURE 4.6 volume.

A typical two-dimensional FVM mesh and control

The general form of the two-dimensional convection– diffusion governing differential equation is     ∂ϕ ∂ ∂ϕ ∂ ∂ ∂ Ŵ + Ŵ +S + = (ρuϕ) (ρvϕ) ∂x ∂y ∂x ∂x ∂y ∂y (4.26) After the integrating Equation 4.26 in the CV of node P and applying the Gauss divergence theorem on it and referring to Equation 4.24, the following equation can be obtained:  

(ρuϕA)e − (ρuϕA)w + (ρvϕA)n − (ρvϕA)s       ∂ϕ ∂ϕ ∂ϕ = Ŵ A − Ŵ A + Ŵ A ∂x ∂x ∂y e w n   ∂ϕ (4.27) − Ŵ A + Sxy ∂y s S is the average value of the source term in the CV. According to Figure 4.6, we know that the interface length (or area in 3D problem) of the CV is Aw = Ae = y and An = As = x . Let Fe = (ρu)e Ae , Fw = (ρu)w Aw , Fn = (ρu)n An , Fs = (ρu)s As ,

(4.28)

Ŵe Ae Ŵw Aw Ŵn An Ŵs As De = , Dw = , Dn = , Ds = δxPE δxWP δyPN δySP The source term can be constant, or a function of constant variable, which is usually linearized in the FVM as SV = Su + SP ϕP

(4.29)

Substitute Equations 4.28 and 4.29 into Equation 4.27, then we have Fe ϕe − Fw ϕw + Fn ϕn − Fs ϕs = De (ϕE − ϕP ) −Dw (ϕP − ϕW ) + Dn (ϕN − ϕP ) − Ds (ϕP − ϕS ) + Su + SP ϕP

(4.30)



anb ϕnb + Su

(4.32)

nb

Here, nb denotes all neighbor cells of the node P , and,  generally, we have aP = nb anb − S ′P , equivalent source term S ′P = − (F − SP ). When the continuity equation holds, F = 0. The forms of a P and a nb lie on the schemes of convection term and diffusion term. Each node has one equation like Equation 4.32, all of which make up the system Aϕ = b. 4.1.2.2 Discrete Schemes of the Convection Term When we derive the FVM discrete equations, the value of field variable on the interface is interpolated by the values of adjacent nodes. In the previous section, the interface approximation equation of the diffusion term is       ϕE − ϕP ϕ ∂ϕ ∂ϕ ≈ = , ∂x e x e δxPE ∂x w   ϕP − ϕW ϕ = , ≈ x w δxWP       ∂ϕ ϕ ∂ϕ ϕN − ϕP ≈ = , ∂y n y n δyPN ∂y s   ϕ ϕP − ϕS = . (4.33) ≈ y s δySP From the perspective of difference method, this approximation equation is a central differencing scheme. For other parameters such as ρ and Ŵ, the method of calculation their interface values is same. In practice, when the diffusion term uses the central difference scheme, no significant difference is observed between the numerical and analytical solutions, even if the mesh is very coarse. Meanwhile, when the convection term uses the central differencing scheme, the numerical solution would be unreasonable in some conditions such as false diffusion, nonconservative, overshoot, or undershoot. So, we discuss the discrete scheme of the convection term in the following part. From the perspective of physical concept, the discrete scheme of the variable value on the CV interface should hold the following properties: conservation, boundedness,

NUMERICAL METHODS

i – 1/2

i + 1/2

i –2

i –1

i

i +1

i +2

WW

W

P

E

EE

e

w

FIGURE 4.7

1D FVM mesh.

and transport property. The conservation can hold only when the interpolation scheme ensures that the fluxes calculated from the two sides are the same. As for the boundedness, Scarborough in 1958 proposed a sufficient condition to judge whether the solving process converges to a reasonable solution by using the coefficient of the system, namely the boundedness criterion:   | |a ≤ 1 at all nodes nb nb (4.34) |aP | < 1 at one nodes at least Generally speaking, the boundedness criterion require all coefficients to be of the same sign (usually is positive) and the S ′P to be negative, or the solving process should not converge or get an unreasonable oscillatory solution. In 1972, Roache proposed that mesh (or cell) Peclet number can be used to measure the ratio of convection intensity and diffusion intensity of the field variable ϕ 3 . The Peclet number is defined as Pe =

ρu F = D Ŵ/δx

(4.35)

where δx is the mesh characteristic length. The bigger the Pe, the greater the effect of the convection in the transport of ϕ, and the greater the effect of the value of upstream nodes to downstream nodes. The formula of CDS used to determine the variable value on the interface of the CV is ϕi+1/2 =

ϕi + ϕi+1 2

(4.36)

Here, ϕ i + 1/2 is the value of ϕ at the interface of the CV, ϕ i + 1 and ϕ i are the value of ϕ at upstream and downstream nodes, respectively, as shown in Figure 4.7. If the CDS is adopted in Equation 4.31, we have aW = Dw + Fw /2, aE = De − Fe /2 aS = Ds + Fs /2, aN = Dn − Fn /2

(4.37)

The variable value at the common interface of adjacent CVs calculated by the CDS is compatible, so the scheme holds the conservation property. According to the Scarborough criterion, only when Pe < 2, the boundedness would be held. Therefore, the CDS is conditionally stable. The

79

CDS makes the field variable ϕ have the same effect on all adjacent nodes, which does not show difference between convection and diffusion, and so it does not have the transport property. The discrete equations with the CDS have the second-order truncation error. When Pe < 2 or the flow is diffusion dominated, the accuracy is high. But when the flow is convection dominated, the convergence is bad and the accuracy is low. To overcome these disadvantages, many advanced difference scheme have been proposed, some of which are introduced in the following part. 4.1.2.2.1 First-Order Difference Scheme To reflect the transport property, the value of the field variable ϕ at the upstream node is always selected as the value of ϕ in the CV interface. This scheme is called the first-order upwind scheme (or UDS for short), namely: ϕi+1/2 =



ϕi ϕi+1

u≥0 u 10, the diffusion term is zero; when |Pe| < 10, the flux through the interface is calculated by 5-power format. When the power law scheme adopted, in Equation 4.31 we have   aW = Dw • max 0, 1 − 0.1 |P ew |5 + max (0, Fw )   aE = De • max 0, 1 − 0.1 |P ee |5 + max (−Fe , 0)   aS = Ds • max 0, 1 − 0.1 |P es |5 + max (0, Fs )   aN = Dn • max 0, 1 − 0.1 |P en |5 + max (−Fn , 0) (4.41)

The results of the power law scheme is very close to the exact solution when |Pe| ≤ 20, the accuracy is high, and the workload is a little higher than that of the hybrid scheme. So, it is widely used. 4.1.2.2.2 Second-Order Upwind Scheme The stable discrete schemes mentioned in the previous section have only first-order accuracy. The upwind scheme is stable and holds the transport property; meanwhile, the numerical false diffusion has also been introduced. High order discrete schemes can reduce the error significantly by involving with more adjacent nodes. In Figure 4.7, for the CV of the node P , the second-order upwind scheme is ⎧ ϕi+1/2 = (3ϕi − ϕi−1 )/2, ⎪ ⎪ ⎪ ⎨

ϕi−1/2 = (3ϕi−1 − ϕi−2 )/2 u≥0 ⎪ ϕi+1/2 = (3ϕi+1 − ϕi+2 )/2, ϕi−1/2 = (3ϕi − ϕi+1 )/2 ⎪ ⎪ ⎩ u 8/3, but the solution might not be stable. For the sake of overcoming the disadvantages, QUICK does not hold the boundedness and the coefficient matrix of the discrete equations is not a tri-diagonal matrix, and many academicians have proposed advanced QUICK schemes. In 1992, Hayase et al.7 developed an advanced QUICK scheme, which according to some principles, can ensure the computational stability and fast convergence speed and the conservation of nonlinear problem and the iterative process of algebraic equations. The scheme is ⎧ ϕw = ϕW + (3ϕP − 2ϕW − ϕWW ) /8 uw ≥ 0 ⎪ ⎪ ⎪ ⎨ϕ = ϕ + (3ϕ − 2ϕ − ϕ ) /8 ue ≥ 0 e P E P W (4.48) ⎪ϕw = ϕP + (3ϕW − 2ϕP − ϕE ) /8 uw < 0 ⎪ ⎪ ⎩ ue < 0 ϕe = ϕE + (3ϕP − 2ϕE − ϕEE ) /8 The format of the discrete equation is

aP ϕP = aW ϕW + aE ϕE + S where aW = Dw + αw Fw aE = De − (1 − αe ) Fe 1 S = (3ϕP − 2ϕW − ϕWW ) αw Fw 8

81

(4.49)

1 + (ϕW + 2ϕP − 3ϕE ) αe Fe 8 1 + (3ϕW − 2ϕP − ϕE ) (1 − αw ) Fw 8 1 + (ϕEE + 2ϕE − 3ϕP ) (1 − αe ) Fe 8

(4.50)

When uw ≥ 0, αw = 1; when u e ≥ 0, α e = 1; when uw < 0, αw = 0; when u e < 0, α e = 0. The advantage of Hayasa’s scheme is that the coefficients of the discrete equation are positive. Thus, the conservation, the boundedness, and the transport property all hold; meanwhile, the accuracy is high. 4.1.2.3 Pressure–Velocity Coupling There is no explicit pressure gradient term in the convection–diffusion governing equations. Similar to incompressible fluid, the pressure gradient term is in the momentum equation in the form of a source term without an independent equation. For a compressible fluid, the relationship between pressure and density is determined by the state equation. Under the primitive valuable method whose unknown variables are velocity and pressure, for raveling out the difficulty that no independent pressure equation exists, many approaches have been developed, such as artificial compressibility method, penalty method, projection method, and pressure-correction method. These methods can be classified as shown in Figure 4.9. The most widely used method of them is the pressure-correction method; SIMPLE algorithm is a typical example. We introduce some pressure–velocity coupling solving methods in the following text.

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

FIGURE 4.9

Some methods for incompressible flow.

4.1.2.3.1 SIMPLE-Like Methods When solving the incompressible fluid problem, the velocity components and the pressure value can be worked out simultaneously using the solution of the momentum equation and the continuity equation. However, these direct methods or called coupled methods require plenty of computer memory, which is not suitable for most engineering applications. If segregated methods are adopted, namely, solving velocity components, then the pressure field would not be worked out because no independent pressure equation and the continuity equation will be considered. The pressure-correction method uses the mass conservation equation for the iterative calculation of the pressure field. SIMPLE-like methods are a class of typical pressure-correction methods, which can be traced back to the year of 1972 when Patankar and Spalding8 proposed semi-implicit method for pressure-linked equation, namely, SIMPLE algorithm. Since then, many similar algorithms have been proposed continuously such as SIMPLER and SIMPLEC. Considering a two-dimensional convective heat transfer problem in a rectangular coordinate system as an example, the governing equations are ∂ρ ∂ (ρui ) + =0 (4.51) ∂t ∂xi     ∂ui ∂ (ρui ) ∂ ρuj ui ∂p ∂ μ + Bi + =− + ∂t ∂xj ∂xi ∂xj ∂xj     k ∂T ∂ (∂T ) ∂ ρuj T ∂ = + ∂t ∂xj ∂xj cP ∂xj

(4.52)

u: ae ue =

nb

anb unb + b + Ae (pP − pE )

v: an vn =



Staggered mesh used in SIMPLE.

anb vnb + b + An (pP − pN )

(4.55)

nb

The mass conservation equation is    ρP − ρP0 xy + (ρu)e − (ρu)w Ae t 

+ (ρu)n − (ρu)s An = 0

(4.56)

The following are the solving steps of SIMPLE algorithm: 1. Assume the initial velocity fields u 0 and v 0 , on the basis of these variables calculate the coefficients a e , a n , a nb , and b. 2. Assume an initial pressure field p*. 3. Solve the discrete momentum Equations 4.60 and 4.61 to obtain u* and v *. 4. Calculate the pressure-correction value p’, provided (u* + u’) and (v * + v ’) with respect to (p* + p’) hold the continuity equation. Substitute them into the discrete format of the continuity equation, namely, Equation 4.56, then we have

(4.53)

aP p′P =



anb p′nb + b

(4.57)

nb

In the staggered mesh shown in Figure 4.10, the discrete momentum equations are

FIGURE 4.10

(4.54)

where aE = ρe Ae

Ae , ae

aN = ρn An

An an

(4.58)

NUMERICAL METHODS

a W and a S can be written in the same way. b is the residual mass flux of the CV. For the steady flow,    

 ∗   

 ρu w − ρu∗ e Ae + ρu∗ s − ρu∗ n An (4.59) 5. Calculate the velocity correction value u’ and v ’,    provided u∗e + u′e and vn∗ + v ′n still hold the momentum equations. Because the u∗e and vn∗ solved in step 3 hold the momentum equations,   ae u′e = (4.60) anb u′nb + Ae p′P − p′E

83

4. u* and v * worked out in step 3 may not hold the mass conservation, and (u* + u’) and (v * + v ’) worked out in step 5 may not hold the momentum conservation.

b=

nb

After the convergence the correction values of the pressure and the velocity are zero, so the first term on the right-hand side of Equation 4.60 can be discarded. Therefore, we can use p ′ to explicitly solve u′e = Similarly,

 Ae  ′ pP − p′E ae

 An  ′ v ′n = pP − p′N an

In 1980, Patankar improvised the first simplification and proposed SIMPLER (stand for SIMPLE Revised). In this algorithm, once the velocity field is given, the pressure field can be solved from the discrete momentum equations; the assumption of the iterative initial pressure field is not necessary any more. For the second simplification of SIMPLE, in 1984 Van Doormaal et al. proposed SIMPLEC (stand for SIMPLE Consistent). Its main calculation steps are the same to those of SIMPLE; the difference is that Equations 4.61 and 4.62 become   ′ Ae  (4.63) pP − p′E u′e = ae − anb nb

(4.61)

  ′ An  pP − p′N v ′n = an − anb

(4.64)

nb

(4.62)

p’ does not need under-relaxation factor.

6. Let (u* + u’), (v * + v ’), and (p* + α p p’) be the solutions of the present iteration, and begin the next iteration. Repeat steps 1–5 until the convergence has been achieved, meanwhile, the velocity results should hold both the continuity equation and the momentum equations. α p is the pressure relaxation factor whose value interval is between 0 and 1. The bigger the value is, the faster is the convergence speed, but the calculation process is likely to be unstable. The relaxation factor can also be introduced to the velocity. The under-relaxation factor of velocity is always put in the unknown equations. Patankar recommended in his monograph Numerical Heat Transfer and Fluid Flow that the value of the velocity under-relaxation factor can be selected as 0.5 and the pressure relaxation factor can be selected as 0.8. Demirdzic et al.9 and Hortmann et al.10 have also discussed the choice of α p and have given some suggestions.

4.1.2.3.2 PISO In 1986, Issa proposed a noniterative algorithm called PISO11 (stands for pressure implicit with splitting of operators) to solve the unsteady pressure–velocity coupling compressible flow problem; then, it was successfully used to solve the steady problem. In PISO, the time step contains one prediction step and two correction steps, which is expected to satisfy the coupling relationship of velocity and pressure. PISO can be seen to add another correction step into the SIMPLE. Three steps of the time step of PISO are as follows:

It can be seen from the above description that in SIMPLE the following assumptions and simplifications can be introduced:

2. The First Correction Step. In this step, ensure that the corrected pressure p* (p* = p (k ) + p’) and the corresponding velocities u * * and v * * satisfy the continuity equation. And u * * and v * * should hold the momentum equations explicitly, namely,   u ∗ (4.67) ae u∗∗ anb unb + bu + Ae pP∗ − pE∗ e =

1. The assumptions of the velocity field and the pressure field are independent of each other. 2. When calculating the velocity correction value, the effect of adjacent nodes can be ignored. 3. b in discrete momentum equations remains constant after velocity correction.

1. Prediction Step. This step is similar to that of SIMPLE, work out p (k ) of the k th time level. Solve the momentum equations implicitly   u ∗ (4.65) ae u∗e = anb unb + bu + Ae pP(k) − pE(k) nb

an vn∗

=

nb

  (k) v ∗ anb vnb + bv + An pP(k) − pN

(4.66)

nb

an vn∗∗ =

nb

  v ∗ ∗ anb vnb + bv + An pP∗ − pN

(4.68)

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

Subtract Equations 4.65 and 4.66 from Equations 4.67 and 4.68, respectively, then we have   an v ′n = An p′P − p′N (4.69) Similar to SIMPLE, this step solves   ae u′e = Ae p′P − p′N , aP p′P =



p

anb p′nb + bp

(4.70)

= uki + t (Ci + Di + Pi ) uk+1 i

nb

and we will obtain Ae ae An = v∗ + an

u∗∗ = u∗ + v

∗∗

  ′ pP − p′E ,   ′ pP − p′N

(4.71)

  Ae  ′′ 1 u  ∗∗ pP − p′′E anb unb − u∗nb + ae ae nb (4.72)  An  ′′  1 v  ∗∗ ∗ ∗∗ pP − p′′N anb vnb − vnb + = vn + an an nb (4.73)

u∗∗∗ = u∗∗ e + e

Substitute Equations 4.72 and 4.73 into the continuity equation p aP p′′P = anb p′′nb + bp (4.74) nb

After p ′′ is worked out, we have p∗∗ = p∗ + p′′

(4.76)

Classified by the distribution of the operator physical mechanism, the solving process can be divided into the following steps:

3. The Other Correction Step. On the basis of p*, u * * , and v * * obtained in the first correction, quest secondary correction value p * * , u * * * , and v * * * , which can stratify the continuity equation and the momentum equations better. Following the same steps, we can get

vn∗∗∗

4.1.2.3.3 Fractional Step Method Similar to predictor– corrector methods such as SIMPLE-like methods and PISO, the fractional step method is also a primitive valuable method. If we let C , D, and P to represent the convection term, the diffusion term, and the pressure gradient term, respectively, and discretize the time term with fist-order partial difference scheme, and if the discrete unsteady term is written partly, we have

(4.75)

This updatable approach can keep developing, but the experience shows that generally after two correction steps the continuity equation and the momentum equations hold, so the calculation can move on to the next iteration. Each iteration of PISO would solve the pressurecorrection equation twice; so the workload and the required memory are higher than that required for SIMPLE, but the iterative process is highly efficient, and the total calculation time is less than that required for SIMPE. Like SIMPLE, the under-relaxation factor needs to be used for PISO also to ensure calculation stability.

⎧ k ∗ ⎪ ⎨ui = ui + tCi ∗ u∗∗ i = ui + tDi ⎪ ⎩ k+1 ui = u∗∗ i + tPi

(4.77)

The fractional step method can choose different approaches as shown below: 1. According to the combination of the three operators, there can be three steps similar to those shown in Equation 4.77, or there can be only two steps by combining C i and D i together. 2. To solve the pressure gradient term, the iterative methods, the pressure Poisson equation method, or the stream function method can be used. 3. Solving the convection term and the diffusion term can use different discrete schemes. 4. The time advance process can employ the Runge– Kutta scheme, the Crank–Nicolson scheme, or an implicit scheme. Generally speaking, the fractional step method is usually used for unsteady flow, while the predictor–corrector method is more suitable for steady flow. 4.1.2.4 Case Studies Chang and Yang12 presented an implicit finite volume approach to simulate the threedimensional mold filling problems encountered during the injection molding. They employed the collocated FVM and the SIMPLE segregated algorithm to discretize and solve the Navier–Stokes equation. In addition, a bounded compressive high resolution differencing scheme is adopted to solve the advection equation to capture the interface on an Eulerian framework. Chang and Yang presented several computational examples. They simulated the fountain flow located near the advancing melt front in injection molding, which is ignored in the Hele-Shaw approximation. The results of the evolution of the predicted position of the melt–air

NUMERICAL METHODS

FIGURE 4.11

85

Evolution of flow front shape.12

FIGURE 4.12 Comparison of the simulated melt fronts from the Hele-Shaw and the three-dimensional models for example 1 (dashed line, 2.5D; meshed surface, 3D).12

interface at different times are shown in Figure 4.1112 . Figure 4.1212 illustrates comparison of the simulated melt fronts obtained by the Hele-Shaw model and the three-dimensional model. The part in the figure is a three-dimensional thin rectangular mold filling a cavity of 150 × 100 × 2 mm with a small rectangular volume of 40 × 20 × 2 mm coring out. A point gate is located at the right end center of the mold. The total CV number is 5840 and the gapwise direction is divided into eight CVs. The next cavity is considered to have a similar geometry as the first example except that the cavity is significantly thicker in the left region. The thickness in this region is increased from 2 to 6 mm. The mesh employed for the filling analysis contains 8448 CVs. The gapwise directions of the thin and thick regions are meshed with 8 and 16 CVs, respectively. Figure 4.1312 shows the comparison of the predicted filling patterns from Hele-Shaw and three-dimensional models at several distinct times as well as the velocity vectors predicted by the three-dimensional FVM model. Although the predicted melt front reveals similar trends for both

models, the correlation is not as good as that observed in the first example, particularly in the radial flow region, because the Hele-Shaw approximation does not function very well when thickness is changed. Figure 4.1412 shows the gapwise velocity component that was neglected in the Hele-Shaw model. Figure 4.15 presents the gear part of the computational mesh, a complex geometry, which comprises 8000 CVs. The mold was filled with two gates located at the two diagonal corners of the square tube inserted at the center of the cylinder. Figure 4.1612 demonstrates the shaded filling patterns at various filling stages. These computations were carried out on an IBM compatible PC with 64 MB RAM and an Intel Pentium III-450 CPU. Figure 4.17 summarizes the number of cells, number of time steps, the memory required, and the CPU time of each case according to Chang and Yang’s data. 4.1.3

Finite Element Method

Generally speaking, the injection molding parts have different complex geometries, so a general method of solution is required. The FEM, as a general method for solving engineering problems, is particularly well suited to this type of problem. The FEM originated from the need for solving complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to the work by Alexander Hrennikoff in 1941 and Richard Courant in 194213 . Hrennikoff’s work discretizes the domain by using a lattice analogy, while Courant’s approach divides the domain into finite triangular subregions for the solution of the second-order elliptic PDEs that arise from the problem of torsion of a cylinder. Courant’s contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh, Ritz, and Galerkin. Since the computer has been invented, its practical application began which is often known as finite element analysis (FEA).

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

(a)

(b)

(c)

(d)

FIGURE 4.13 Comparison of the simulated melt fronts from the Hele–Shaw and the threedimensional models (dashed line, 2.5D; meshed surface, 3D): (a) 10% filled, (b) 40% filled, (c) 70% filled, (d) 95% filled.12

FIGURE 4.14 direction.12

Simulated melt fronts viewed from gating

It was first developed in 1956 by Turner and Clough et al. to numerically analyze stress problems for the design of aircraft structures. In 1960, Clough first raised the name FEM and then used it in civil engineering. Soon, it was realized that the FEM could be applied to a wide range of problems. It has been modified to solve more and more general problems in solid mechanics, fluid flow, heat transfer, etc. 4.1.3.1 Weighted Residual Methods Unlike the FDM, this method is based on the equivalent integral form of differential equations instead of the differential equations and the corresponding definite conditions. The general form of equivalent integral is the weighted residual method (WRM), which is applied to universal equation form. Using the principle of the WRM, we can create many approximate schemes.

FIGURE 4.15

Numerical mesh of the gear mold.

For a general problem, the unknown function u should satisfy a certain differential equation set and the boundary conditions: ⎞ ⎛ A1 (u) ⎟ ⎜ A (u) = ⎝A2 (u)⎠ = 0 .. .

(in )

(4.78)

NUMERICAL METHODS

(a)

(b)

(c)

FIGURE 4.16 Simulated melt fronts at various percentages of fill: (a) 25% filled, (b) 45% filled, (c) 80% filled.12

8400

33 23

5300

560716

74,100

1.75

1.43

FIGURE 4.17 Number of cells, number of time steps, memory, and CPU times for each case (double precision is adopted).

87

88

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

Γe

Domain Ω A(u) = 0

Subdomain (element) Ωe

y

Boundary Γ B(u) = 0

Problem domain and boundary Ŵ.

⎞ B1 (u) ⎟ ⎜ B (u) = ⎝B2 (u)⎠ = 0 .. . ⎛

(on Ŵ)

(4.79)

in which the unknown function u can be either a scalar field or a vector field; Aand B represent the differential operators of the independent variables such as time coordinates or space coordinates. is the solution domain (volume, area, etc.), Ŵ is the boundaries of the domain as shown in Figure 4.18. The system of differential equations must hold at any point in the domain , so

υ T A (u) d ≡



where C , D, E , and F are different factors, in which the order of the derivative is lower than that of A, so the lower order of continuity of the function u can be accepted; meanwhile, the orders of the continuity of υ and υ¯ have to be higher. But it is not difficult because υ and υ¯ can be chosen. Created by appropriately increasing the continuity requirement of υ and υ¯ to reduce the continuity requirement of the field function u, the equivalent integral form is called the equivalent integral weak form of PDEs. In the solution domain , if the field function u is the exact solution, Equations 4.78 and 4.79 are all strictly satisfied, so are Equation 4.82 and its weak form Equation 4.83. However, for many complex problems, the exact solution is difficult to figure out, so we have to find an approximate solution with certain precision. Suppose the approximate function of the unknown function u is u, ˜ its form generally is

(υ1 A1 (u) + υ2 A2 (u) + · · ·) d ≡ 0

u ≈ u˜ =





(4.80) where υ = (υ1 υ2 · · ·)T is a set of arbitrary function vectors which have the same number of differential equations. Obviously, Equation 4.80 is the equivalent integral form of Equation 4.78. Similarly, for the boundary conditions, there is a set of arbitrary function υ¯ satisfying the following equation:

Ŵ



x

FIGURE 4.18

Notice that the integrability of Equation 4.82 is required, and so υ and υ¯ should be integrable monodrome function in and on Ŵ. If the highest order in the differential operator A is n, then it is required that in the domain the derivative of u is continuous from zero order to n − 1 order and the nth order derivative is integrable in the domain and has a finite number of discrete points. We can obtain the following form by integrating Equation 4.82 by parts: T ¯ F (u) dŴ = 0 (4.83) C (υ) D (u) d + E T (υ)

υ¯ T B (u) dŴ ≡

Ŵ



(4.81)

So, the integral form is



υ T A (u) d +



υ¯ T B (u) dŴ = 0

N iai = N a

(4.84)

i=1

in which a i is an undetermined parameter, N i is a known linear independence function selected from the complete sequence of functions and is called trial function (also known as basis function or shape function). Obviously, when n is finite, the approximation solution u˜ cannot satisfy Equations 4.78 and 4.79 completely. It will ¯ yield the margins R and R: A (N a) = R ,

(υ¯ 1 B1 (u) + υ¯ 2 B2 (u) + · · ·) dŴ ≡ 0

Ŵ

n

B (N a) = R¯

(4.85)

Here, R and R¯ are called residuals. Substitute the arbitrary functions υ and υ¯ in Equation 4.82 with n given functions: υ = W j,

(4.82)

υ¯ = W j

(j = 1–n)

(4.86)

Ŵ

That all of υ and υ¯ satisfy Equation 4.82 is equivalent to that the differential equations (Eq. 4.78) and the boundary conditions (Eq. 4.79) are satisfied. Thus, Equation 4.82 is called the equivalent integral form of differential equation.

to obtain the approximate equivalent integral form T T W j A (N a) d + W j B (N a) dŴ = 0 (j = 1–n)

Ŵ

(4.87)

NUMERICAL METHODS

which can also be written in the residual form T W Tj R d + W j R dŴ = 0 (j = 1–n)

(4.88)

Ŵ







  C T W j D (N a) d +



Ŵ

  E T W j F (N a) dŴ = 0 (j = 1–n)

(4.89)

The method, which solves the approximate solution of the different equations by letting the weighted integral of the residual be zero, is called the WRM. Obviously, any independent complete function set can be selected as the weight function. The difference in choosing the weight function results in different calculation approaches of the weighted residual. The staple choices of the weight function are collocation method, subdomain method, least square method, moment method, and Galerkin method. The Galerkin method is the most common method. The credit is to the Russian mathematician Boris Galerkin. It selects the basis function itself as the weight function, which is W j = −W j = −N j

W j = Nj,

(4.90)



N Tj A

 n i=1

N iai



d +



Ŵ

T NjB

 n

(j = 1–n)

i=1

N iai





Ŵ

C T (δ u) ˜ D (u) ˜ d −

When using the Galerkin method, the coefficient matrix of the unknown equations is usually symmetric, so it’s usually applied in the area of fluid mechanics FEM. 4.1.3.2 The Element and Its Shape Function In the previous section, we have discussed the approach to establishing the finite element equations using the WRM: namely, first wrote the field function in the polynomial form, then represented the undetermined parameters as the function of the node values of the field function, and the element geometry by using the node conditions, so the field function can be written in the interpolation form with the node values. In the following section, the choice of the element type and the shape function is discussed. Generally speaking, the choice of the element type and the shape function depends on the structure and geometry characteristics of the global solution domain, the function type, and the required solution accuracy. The shape functions depend on the element shape, node type, and number. According to the element geometry, the element type can be classified into one-, two-, and three-dimensional elements. As shown in Figure 4.19, a two-dimensional domain can be discretized by a series of triangular elements or quadrilateral elements. This is discussed in detail in the following parts. 4.1.3.2.1 One-Dimensional Elements The common onedimensional elements include the 2-node and 3-node linear elements (Table 4.1). For a one-dimensional n-node element, if the node parameters only have the node values of the field function y

e

(4.91)

e

o

x (a)

δ u˜ =

i=1

N i δa i

E T (δ u) ˜ F (u) ˜ dŴ = 0 (4.94)

Ŵ



dŴ = 0

Referring to Equation 4.84, the variational δ u˜ of the approximate solution u˜ can be defined as n



y

The approximate integral form can be written as

Here, δa i is totally arbitrary. Then, Equations 4.82 and 4.83 become T δ u˜ A (u) ˜ d − δ u˜ T B (u) ˜ dŴ = 0 (4.93)

W j and W j are called the weight functions. The meaning of Equations 4.87 and 4.88 is the residual to be zero in some average sense by selecting the undetermined coefficient a i . If the weighted integral of the residual is zero, an unknown system of equations is yielded to solve the undetermined coefficient a i , and then the approximate solution of the original problem can be worked out. The more the term number n of the trial function selected by the approximate function is, the higher is the accuracy of the approximate solution. When n tends to infinity, the approximate solution will converge into the exact solution. Similarly, the approximate form of the equivalent integral weak form is

89

(4.92)

o

x (b)

FIGURE 4.19 FEM discretization of the 2D domain: (a) triangular element; (b) quadrilateral element.

90

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

TABLE 4.1 Degree

Element Name

Linear

2-Node linear element Beam element 3-Node linear element

Quadratic

TABLE 4.2

The Common One-Dimensional Elements Schematic Diagram

The Common Two-Dimensional Elements

Degree Linear

Element Name

Schematic Diagram

3-Node triangular element 4-Node quadrilateral element

(taking the scalar field for example), then the field function in the elements can be written in the interpolation form as

u=

n

Quadratic

6-Node triangular element

8-Node quadrilateral element

Ni ui

(4.95)

i=1

Cubic

10-Node triangular element

where the shape functions N i (x ) have the following characteristics: n   Ni (x) = 1 Ni xj = δij ,

12-Node quadrilateral element

(4.96)

i=1

Here, δ ij is Kronecker delta. In global coordinates, the shape function can be selected as n − 1 power Lagrange polynomial li(n−1) (x): Ni (x) =

li(n−1) (x)

n 

=

j =1,j =i

x − xj xi − xj

(4.97)

In natural coordinate system, introduce the dimensionless local coordinates ξ=

x − x1 x − x1 = xn − x1 l

(0 ≤ ξ ≤ 1)

(4.98)

Here, l is the length of the element. Using the local coordinates, Equation 4.97 becomes

Ni (ξ ) =

li(n−1) (ξ )

=

n 

j =1,j =i

ξ − ξj ξi − ξj

(4.99)

Figure 4.20 shows a sample of 1D 2-node element.

u1

u

x1

x

u2

1

2

x

FIGURE 4.20

1D 2-Node linear element.

x2

4.1.3.2.2 Two-Dimensional Elements The common twodimensional elements include 3-node triangular linear element, 6-node triangular quadratic element, 10-node triangular cubic element, 4-node quadrilateral bilinear element, 8-node quadrilateral quadratic element, and 12node quadrilateral cubic element (Table 4.2). Local nature coordinates are generally used to create the shape function of the general triangular element. We use area coordinates as the nature coordinates. As shown in Figure 4.21, the position of the arbitrary point P in the triangle can be determined by three ratios, namely, P (Li , Lj , Lm ), where Li = Ai /A,

Lj = Aj /A,

Lm = Am /A

(4.100)

Ai , Aj , Am , and A are the areas of Pjm, Pmi , Pij , and ijm, respectively. Li , Lj , and Lm are called the area coordinates, and only two are independent of the relationship: (4.101) Li + Lj + Lm = 1 The coordinates of the points in the parallel line with the sides inside the triangle are the same as the corresponding coordinates of the corresponding node. After a simple derivation, it is easy to know that the coordinates of P (x , y) in a triangular element is x = xi Li + xj Lj + xm Lm , y = yi Li + yj Lj + ym Lm (4.102)

NUMERICAL METHODS

FIGURE 4.22 FIGURE 4.21

Area coordinates of the triangular element.

 1 x 1  Ai = 1 xj 2 1 x m

From

we have Li =

 y  yj  ym 

Ai 1 = (ai + bi x + ci y) A 2A

(i, j, m)

(4.103)

(4.104)

in which   x yj  ai =  j = xj ym − xm yj xm ym    1 yj    = yj − ym (i, j, m) bi = −  1 ym    1 xj    = −xj + xm ci =  1 xm 

with other coefficients obtained by a cycle permutation of subscripts in the order (i , j , m). Applying the compound function derivation law, immediately we obtain the following derivation formula: ∂ ∂Li ∂ ∂Lj ∂ ∂Lm ∂ = + + ∂x ∂Li ∂x ∂Lj ∂x ∂Lm ∂x   ∂ ∂ ∂ 1 (4.106) bi + bj + bm = 2A ∂Li ∂Lj ∂Lm   ∂ 1 ∂ ∂ ∂ (4.107) ci = + cj + cm ∂y 2A ∂Li ∂Lj ∂Lm

2D 4-node quadrilateral element.

lines connecting the midpoints of the opposite sides in an arbitrary quadrangle as r axis and s axis. Let the point of intersection of the two axes be the original point of the coordinate system. The measurement of the coordinate system is defined as the coordinates of the origin P , which is (0, 0); the formulas of the two couples of the opposite sides are r = ± 1 and s = ± 1, respectively. A kind of normal nature coordinate systems is established in this way. The coordinates of the four vertexes are (1, 1), (−1, 1), (−1, − 1) and (1, − 1). The shape function of the two-dimensional 4-node quadrilateral element is Ni =

(4.105)

1 (1 + ri r) (1 + si s) (i = 1, 2, 3, 4) 4

(i = 1, 2, 3)

(4.108)

The quadrilateral element usually uses the normal nature coordinate system. As shown in Figure 4.22, draw the

(4.109)

4.1.3.2.3 Three-Dimensional Elements The common three-dimensional elements include 4-node tetrahedral linear element, 10-node tetrahedral quadratic element, 8-node hexahedral linear element, and 20-node hexahedral quadratic element (Table 4.3). TABLE 4.3 Degree Linear

The Common Three-Dimensional Elements Element Name

4-Node tetrahedral element

8-Node hexahedral element

Quadratic 10-Node tetrahedral element

Thy 3-node triangular element is a linear element, its shape function represented by the area coordinates is Ni = Li

91

20-Node hexahedral element

Schematic Diagram

92

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

FIGURE 4.24 FIGURE 4.23 element.

Volume coordinate system of the tetrahedral

Referring to the triangular element, the nature coordinate system introduced in the tetrahedral element is the volume coordinate system. As shown in Figure 4.23, the volumes surrounded by points 243P , 134P , 142P , and 123P are V 1 , V 2 , V 3 , and V 4 , respectively. The coordinates of any point P (L1 , L2 , L3 , L4 ) in the element are Li =

Vi V

(i = 1, 2, 3, 4)

(4.110)

in which V is the volume of the tetrahedron. Only three out of four volume coordinates are independent, and we have the following relationship: L1 + L2 + L3 + L4 = 1

(4.111)

The transformation between the area coordinate system and the global Cartesian coordinate system is follows ⎧ ⎫ ⎡ ⎤⎧ ⎫ 1 1 1 1 ⎪ 1⎪ ⎪ ⎪L1 ⎪ ⎪ ⎪ ⎬ ⎢ ⎨ ⎪ ⎥ ⎨L2 ⎬ x x x x x 1 2 3 4 ⎥ (4.112) =⎢ L3 ⎪ y ⎪ ⎣y1 y2 y3 y4 ⎦ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ ⎭ ⎩ ⎪ z1 z2 z3 z4 z L4 or

⎧ ⎫ ⎡ L1 ⎪ a1 ⎪ ⎪ ⎪ ⎨ ⎬ 1 ⎢ L2 ⎢a2 = ⎣a3 L ⎪ ⎪ 6V 3 ⎪ ⎭ ⎩ ⎪ L4 a4

in which

3D 8-node hexahedral element.

b1 b2 b3 b4

c1 c2 c3 c4

⎤⎧ ⎫ 1⎪ d1 ⎪ ⎪ ⎨ ⎪ ⎬ x d2 ⎥ ⎥ ⎦ d3 ⎪ y⎪ ⎪ ⎩ ⎪ ⎭ d4 z

   x2 y2 z2  1      a1 = x3 y3 z3  , b1 = − 1 x4 y4 z4  1    x2 1 z2  x2      c1 = − x3 1 z3  , d1 = x3 x4 1 z4  x4

y2 y3 y4 y2 y3 y4

 z2  z3  , z4   1 1 . 1

(4.113)

with other coefficients obtained by a cycle permutation of subscripts in the order 1, 2, 3, and 4. The shape functions of the 4-node tetrahedral linear element are the corresponding volume coordinates, namely, Ni = Li

(i = 1, 2, 3, 4)

(4.115)

Referring to the tetrahedral element, the shape functions of the hexahedral element utilize the normal nature coordinate system as shown in Figure 4.24. Six crosspoints are obtained by connecting the middle points of the opposite sides in each face; define r axis, s axis, and t axis as the lines connecting the cross-points of the opposite faces. The cross-points of the three axes are the origin and their coordinates are (0, 0, 0). In the three opposite faces, we have r = ± 1, s = ± 1, and t = ± 1 respectively. Like this way, a kind of normal nature coordinate system of the hexahedral element is established. The nature coordinates of every vertex are given in Figure 4.24 as well. The shape functions of the 8-node hexahedral linear element are Ni =

1 (1 + ri r) (1 + si s) (1 + ti t) 8

(i = 1, 2, . . . , 8) (4.116)

4.1.3.3 FEM Discrete Equations Here, we consider the control equations for fluid flow to introduce the FEM discrete process. The form of the continuity equation and the Navier–Stokes equation can be represented as ∂uj =0 ∂xj

(4.117)

∂ui ∂ui 1 ∂p ∂ 2 ui + uj = fi − +ν 2 ∂t ∂xj ρ ∂xi ∂xj

(4.118)

The boundary conditions are (4.114)

ui = u¯ i

on Ŵ1

(4.119)

σij nj = p¯ i

on Ŵ2

(4.120)

NUMERICAL METHODS

93

n j are the components of the normal vector of the boundary Ŵ 2 , and σ ij are the components of the stress tensor σ , so we have   ∂uj ∂ui + σij = −pδij + μ (4.121) ∂xj ∂xi

Here, j and  k are the basis functions of element interpolation of u i and p, respectively, and the subscripts j and k represent the nodes of the element. Using the Galerkin method, the weight functions are δui = j

(4.129)

The Galerkin integral form of Equations 4.117 and 4.118 on a triangular element region (e) , as shown in Figure 4.19, is ∂uj δp d = 0 (4.122) (e) ∂xj

δp = k

(4.130)

    ∂p ∂ui ∂ 2 ui ∂ui + uj − fi + − μ 2 δui d ρ ∂t ∂xj ∂xi ∂xj (e) (4.123) Because the control equations are PDEs of the unknown variables u i and p, the weight functions should be chosen correctly. We choose interpolation basis functions of pressure p and velocities u i as the weight functions in the continuity equation and the momentum equation. Integrating by parts, Green’s theorem, and considering the boundary conditions, we can obtain their weak forms: ∂ u¯ n δp dŴ (4.124) uj (δp) d = (e) ∂xj Ŵ1 (e)  )  ∂ui ∂ui + uj δui ρ ∂t ∂xj (e) * 

 ∂uj ∂ ∂ui + δui d + −pδij + μ ∂xj ∂xi ∂xj = ρfi δui d (4.125) p¯ i δui dŴ +

(e)

Ŵ2

(e)

Here, u¯ n is the normal velocity component on Ŵ 1 and u¯ n = u¯ j n¯ j

(4.126)

The integral terms on Ŵ 1 or Ŵ 2 will exist only when the element (e) has the part (like a side or a face) belonging to Ŵ 1 or Ŵ 2 . In the integral expression, the velocity components u i and pressure p have different derivative order, so the corresponding element basis functions would not have the same order. Generally, the basis function of u i should have one more order than that of p, which can achieve higher accuracy. Suppose the approximate functions of u i and p in an element are ui = uij j

(4.127)

p = pk k

(4.128)

Then we can obtain the element FEM equations of the continuity equation and the momentum equation (some subscripts have been changed appropriately for a more convenient expression):

(e)

uβj j

∂k d = ∂xβ



(e)

u¯ nj j k dŴ

(4.131)

Ŵ1

)

   ∂  ∂  ρ uαj j + uβl l uαj j i ∂t ∂xβ (e) * 

  ∂i ∂i ∂  ∂  d +μ uαj j + uχj j −pm m δαβ ∂xβ ∂xβ ∂xα ∂xβ = ρfα i d (4.132) p¯ αm m i dŴ +



(e)

Ŵ2

(e)

Before discretization, we first divide the solution domain into finite elements. As shown in Figure 4.19a, we use the widely used 3-node triangular elements to discretize the solution domain to E element regions (e) (e = 1, 2, . . . , E ). The vertexes of every triangular element are the nodes whose index should be unified in a mesh. In this case, an element has three nodes, so the subscripts i , j , l , k , and m run over 1–3, meanwhile α and β run over 1–2 for a two-dimensional problem or 1–3 for a three-dimensional problem. For the minimization of nonzero elements bandwidth in the coefficient matrix of the final global finite element equations, the difference between node indexes in an element should be as small as possible. After substituting the sharp functions expression into Equations 4.131 and 4.132, we can obtain one element finite element equations having the following form A(e) ij

∂uαj (e) (e) (e) + Bi(e) βjl uαj uβl + Cαim pm + Dαi βj uβj = Eαi ∂t (4.133) (e) Fk(e) (4.134) βj uβj = Gk

When having every element finite element equation, we can generalize the equations to the whole region to assemble the global finite element equations by accumulating the coefficients of the element finite element equations into the corresponding coefficient matrix and right-hand vector of the global finite equation according to the corresponding relationship between element node indexes and global node indexes. Suppose the corresponding global node indexes of

94

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

the node indexes i , j , k , l , and m of the element (e) are n, r, h, s, and t, then the global finite element equations would be (e) Fhβr uβr = G(e) (4.135) h ∂uαr (e) (e) (e) (e) + Bnβrs uαr uβs + Cαnt pt + Dαnβs uβs = Eαn ∂t (4.136) If all the equations of a system are assembled, they can be written in the following form: A(e) nr

Ax = b

(4.137)

Now the system is singular. We should consider the boundary conditions to eliminate the singularity. There are two common treatment methods: deleting the rows and columns method and penalty method. In the present commercial software, nearly all of them use the former method. The main reason is that the latter is an approximate approach and the penalty number is not easy to determine; sometimes, solving would fail. The detailed implementation is introduced in Zienkiewicz and Tylor’s monograph The Finite Element Method . The coefficient matrix A is a symmetric matrix when using the Galerkin method. For the large-scale problem, it also has the sparsity, namely, most elements are zero and show zonal distribution (Figure 4.25). The next step is to solve the system; the methods are introduced later.

FIGURE 4.26

(a)

The part geometry for simulation.

(b)

(c)

FIGURE 4.27 Filling results at different filling time: (a) 25% filled, (b) 50% filled, (c) 75% filled.

4.1.3.4 Case Studies Yan14 developed a threedimensional filling simulation FEM program named HsCAE3D using the FAN (flow analysis network) method Band width A

Nonzero

FIGURE 4.28

Fountain phenomenon at the filling melt front.

0

0

FIGURE 4.25

Zonal distribution of the coefficient matrix.

for tracing the moving melt front. He chose a part as shown in Figure 4.26 to check the simulation results. The solution domain is meshed by the tetrahedron FEM element. The number of elements in the part mesh is 52,721, and the number of the nodes is 10,412. Figure 4.27 shows the filling results at 1/4, 1/2, and 3/4 of the filling time. Figure 4.28 illustrates the fountain phenomenon at the filling melt front. Figure 4.29 shows the comparison of pressure at the gate with the time solved by HsCAE3D and Moldflow Plastics Insight (MPI).

NUMERICAL METHODS

FIGURE 4.29

4.1.4

Pressure at the gate with respect to time.

Mesh-Less Methods

Since invented in the 1970s, mesh-less methods, also known as mesh-free methods, have become another class of numerical simulation algorithms in terms of solving PDEs. These methods construct an approximate system with a set of particles (either Lagrangian or Eulerian), so they can eliminate some or all of the traditional mesh-based view of the computational domain, which makes initial meshing and reconstruction unnecessary, and ensures the calculation accuracy. But the calculation workload is larger than that of the FEM, generally. Mesh-less methods can be classified into two categories in general: (i) particle methods based on Lagrangian approach such as smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS for short) developed TABLE 4.4

95

from SPH and (ii) gridless methods based on Eulerian approach such as gridless Euler/Navier–Stokes solution algorithm and element-free Galerkin method (EFG or EFGM). In Table 4.4 some common mesh-less methods are given. In 1992, Nayroles et al.15 introduced moving least square (MLS) into the Galerkin method, and proposed the diffuse element method (DEM). In 1994, Belytschko16 improved the DEM and introduced the EFG method. This method is one of the mature and widely used mesh-less methods. In this section we introduce the basic theory and its numerical realization of Galerkin-type mesh-less methods. Compared with the standard FEM, the discrete forms of the governing equations and the boundary conditions are the same, and are all based on the Galerkin weak form of the governing equations and the boundary conditions. In contrast, the approximate approach of the space interpolation for the field variable is different; namely, in FEM the approximate function is obtained using element interpolation, while in Galerkin-type mesh-less methods it is created using particle approaches such as MLS method, reproducing kernel (RK) approximation method, partition of unity method (PUM), hp cloud, radial basis function (RBF) method and point interpolation method (PIM). They are all based on the Galerkin method. Because of this, Galerkintype mesh-less methods have many advantages, such as no shackle of the mesh to some extent, but some shortcomings do exist like the difficulty in implementing essential boundary conditions and the slow calculation speed. 4.1.4.1 Mesh-Less Function Approximation After obtaining the Galerkin weak form of the governing equations, we can use many methods mentioned before to create

The Common Mesh-Less Methods

The Name

Approximation Approach

Element-free Galerkin (EFG) Finite point method (FPM) Reproducing kernel particle method (RKPM) Mesh-less point collocation method (PCM) Hp clouds Hp mesh-less clouds Partition of unity method (PUM) Local boundary integral equation (LBIE) Mesh-less local Petrov–Galerkin (MLPG) Smoothed particle hydrodynamics (SPH) Radial basis functions mesh-less method (RBF) Point interpolation method Least square collocation (LSC)

Moving least square Moving least square Reproducing kernel approximation Reproducing kernel approximation Hp clouds Hp clouds Partition of unity method Moving least square Moving least square Kernel function Radial basis function

Weighted least square mesh-less method (WLSM)

Moving least square

Point interpolation method Moving least square

Discretization Approach

Background Mesh

Galerkin method Point collocation method Galerkin method

Yes No Yes

Point collocation method

No

Galerkin method Point collocation method Galerkin method Petrov–Galerkin method Petrov–Galerkin method Point collocation method Point collocation method

Yes No Yes No No No No

Galerkin method Least square point collocation method Weighted least square method

Yes No No

96

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

the shape functions, in which the MLS method has become the main method. In the early 1980s, Lancaster and Salfauskas17 proposed MLS approximation systematically for the fitting of curve or curved surface. This method needs to only discrete the point model, not to mesh the domain. In 1994, Belytschko16 introduced MLS into the Galerkin FEM and proposed the EFG method, which swept a wave of enthusiasm for research on mesh-less method. In MLS, the approximate function uh (x ) of the function u (x ) is defined as uh (x ) =

m

pi (x ) ai (x ) ≡ p T (x ) a (x )

(4.138)

To ensure uh (x ) is the best approximation J should be the minimum. To achieve this, the partial derivative of Equation 4.140 with respect to a is ∂J = A (x ) a (x ) − B (x ) u = 0 ∂a

(4.145)

in which A = P T W (x ) P

(4.146)

B = P T W (x )

(4.147)

a (x ) = A−1 (x ) B (x ) u

(4.148)

So, we have

i

where ai (x ) is the coefficient, pi (x ) is a set of complete basis functions of monomial, and m is the number of terms of the shape functions. In a two-dimensional region, the first- (linear) and second-order basis functions  T = 1 x y , m = 3 and p T (x ) = are, respectively, p (x )

 2 2 1 x y x xy y , m = 6. For a three-dimensional problem, the first- (linear) and second-order basis functions  T = 1 x y z , m = 4 and p T (x ) = are, respectively, p (x )

 2 2 2 1 x y z x xy y yz z xz , m = 10. Suppose in the region of point x there are n nodes, of which the weight functions w (x − x I ) = 0, at these nodes the weighted sum of the squares of the difference between the approximate function uh (x ) and u (x ) is J =

n I

2

w (x − x I ) P T (x I ) a (x ) − u I

(4.139)

where the weight function w (x − x I ) is the same as the one in SPH, u I is the value of u (x ) at node x I . Equation 4.139 can also be written in the following form: J = (P a − u)T W (x ) (P a − u)

The approximate function uh (x ) can be represented as uh (x ) =

w (x − x I ) NI0 =  I w (x − x I )

I

Using Equation 4.150, we can obtain the partial derivative of NIk (x ): m

    −1  −1 −1 pj,i A B ji + pj A B ,i + A,i B (x ) = ji

(4.153)



u = u1 u2 · · · un

 a T = a1 (x ) a2 (x ) · · · am (x ) ⎡ ⎤ p1 (x 1 ) p2 (x 1 ) · · · pm (x 1 ) ⎢p1 (x 2 ) p2 (x 2 ) · · · pm (x 2 )⎥ ⎢ ⎥ P =⎢ . .. ⎥ .. .. ⎣ .. . . ⎦ .

p1 (x n ) p2 (x n ) · · · pm (x n ) ⎡ w (x − x 1 ) 0 ··· ⎢ 0 w − x · ·· ) (x 2 ⎢ W (x ) = ⎢ .. .. . .. ⎣ . . 0

(4.151)

which is also called the Shepard function, which satisfies zero-order consistency, namely, in (4.152) NI0 (x ) = 1

j =1

(4.149)

where k is the power of the basis function. Thus, the shape function is

 N k = N1k (x ) · · · Nnk (x ) = p T (x ) A−1 (x ) B (x ) (4.150) When k = 0, we have

k NI,i

T

NIk (x ) uI

i=1

(4.140)

where

n

0

···

(4.141) (4.142)

(4.143) ⎤

0 0 .. .

w (x − x n )

⎥ ⎥ ⎥ ⎦

(4.144)

in which −1 −1 A−1 ,i = −A A,i A

(4.154)

MLS interpolation satisfies the consistency condition. Implementation of MLS interpolation requires choosing the weight function initially. The weight function decides the calculation accuracy to a great extent. Commonly used weight functions are organized into two categories. First one is Gauss exponential weight functions:

 ⎧ exp − (dI /cI )2k ⎪ ⎪

⎪ ⎨ − exp − (r /c )2k  I I wI (x ) =

 0 ≤ dI ≤ rI (4.155) ⎪ 1 − exp − (rI /cI )2k ⎪ ⎪ ⎩ 0 dI > rI

NUMERICAL METHODS

Here, dI = x − x I is the distance between node x I and node x ; c I is the constant, which can control the shape of the weight function; and r I is the support region of the weight function wI , which is also the radius of the support region of node x I . The other category of the weight function is spline function: wI (x ) ⎧  2  3  4 ⎪ dI dI dI ⎨ 1−6 +8 −3 0 ≤ dI ≤ rI = rI rI rI ⎪ ⎩0 dI > rI

of the penalty function depends on experience. Coupled with the FEM means that mesh the region near the boundaries and implement FEM in the region, meshless method is applied in the rest region. Obviously, this method is contrary to the spirit of the meshless method. Besides, there are displacement constraint equations method, transformation method, and so on. In the following part we introduce some typical methods. 4.1.4.2.1 Lagrange Multiplier Method Suppose that essential boundary conditions are imposed on a fixed displacement boundary Ŵ u : u = u¯

(4.156)

4.1.4.2 Boundary Conditions How to impose essential boundary conditions exactly is one of the main difficulties in applying the mesh-less method. One reason is that the shape functions of the mesh-less method do not satisfy the property of the Kronecker delta. The more important reason is the shape functions of the mesh-less method are too rich. The physical quantity at the domain boundaries is related not only with the one at boundary nodes but also with the one at the whole nodes in the domain if the radius of the node effect region is big enough. Thus, unlike FEM, in mesh-less method merely providing the variable value of boundary nodes, which is known, is not persuading enough to ensure that the whole node values are known at the entire boundaries. The essentiality is that it is difficult to construct the trial function space (such as displacement approximation space), which can satisfy the essential boundary naturally, and the function space at essential boundaries (such as the approximation space of displacement variational), which can be zero naturally. At present, the approaches to impose essential boundary conditions can be classified into two categories, in general. One modifies the weak form of the problem, which is different from the weak form of the FEM, to be consistent with the original different problem, such as Lagrange multiplier method, penalty function method, and the Nitsche method. The other one ignores the factor “consistency” and uses the same weak form as the FEM and changes the form of the shape functions in mesh-less methods such as displacement constraint equation method and corrected collocation method to satisfy the essential boundary conditions only at nodes. Lagrange multiplier method was proposed earlier. It has high accuracy, but it has more unknowns and the discrete equation is not positive definite. Using the modified variational principle, the equations are symmetric and banded sparse and the accuracy is low, which can be improved by adding nodes. The equations obtained by the penalty function method are positive definite and the number of unknowns does not increase. But the choice

97

(4.157)

on Ŵu

Using the Lagrange multiplier method, introduce Equation 4.166 into the weak form of linear elastic problem and obtain  T T• δ u˜ T • t¯ dŴ− δ u˜ b d − ∇s δ u˜ : σ d −





δλT • (u − u) ¯ dŴ −

Ŵu



Ŵt

δ u˜ T • λ dŴ = 0

(4.158)

Ŵu

    where ∇s δ u˜ T represents the symmetric part of ∇ δ u˜ T , σ is Cauchy stress tensor, b is the body force vector, λ is Lagrange multiplier, u¯ and t¯ are the fixed value of u and external force t on boundaries, respectively, Ŵ t is the force boundaries, Ŵ u is the displacement boundaries, and Ŵ = Ŵ t + Ŵ u . Lagrange multiplier λ is decided by λ (x ) = NI (s) λI

x ∈ Ŵu

(4.159)

where N I (s) is the Lagrange interpolation shape functions, s is the length of the arc along the boundaries. The last two terms are the addition terms for imposing the boundary conditions. After space discretization, we can obtain the following system equation:

) * ) * K G U f = (4.160) G 0 λ q where U and λ are the vector of node values of displacement and Lagrange multiplier, respectively. Note that theoretically using the Lagrange multiplier method the essential boundary conditions can be imposed exactly, and it is not necessary to satisfy the essential boundary conditions in advance for the interpolation space constructed by the shape functions. But the coefficient matrix of Equation 4.160 is not banded and positive definite, so some commonly used solving methods are not available, which reduces the calculation speed. The Lagrange multiplier, which is introduced as an independent unknown, increases the scale of the problem. In addition,

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

the solving stability of the equations requires that the interpolation spaces of the displacement u and Lagrange multiplier satisfy the so-called Ladyzenskaja-BabuˇskaBrezzi (LBB) condition, or the numerical oscillation would occur. So choosing the appropriate interpolation space is not easy. 4.1.4.2.2 Modified Variation Principle Method To avoid the shortcomings of the Lagrange multiplier method, Belytschko et al. developed a modified variation principle method to introduce the essential boundary conditions. The main idea is as follows. The physical meaning of Lagrange multiplier is the surface force t on the boundaries Ŵ u , namely, (4.161) λ=t on Ŵu Substituting Equation 4.161 into the weak form of Lagrange multiplier method, we have   δ u˜ T • t¯ dŴ− δ u˜ T • b d − ∇s δ u˜ T : σ d −





δt T • (u − u) ¯ dŴ −



Ŵt

δ u˜ T • t dŴ = 0

(4.162)

Ŵu

Ŵu

After the space discretization, we can obtain the system equation in the following form from Equation 4.162 (K + K t ) U = P + P t

(4.163)

In contrast to the Lagrange multiplier method, the highlight is that the number of unknowns does not increase, and the matrix is still banded, symmetric, and positively definite. The shortcoming is that the accuracy is lower than the Lagrange multiplier method, and the modified weak form like Equation 4.162 cannot be written out for some engineering problems. So this method is not easy to be used widely. 4.1.4.2.3 Penalty Function Method Introducing a displacement constraint (Eq. 4.157) into the weak form using the penalty function method, we can obtain   δ u˜ T • t¯ dŴ δ u˜ T • b d − ∇s δ u˜ T : σ d −

+α





δt T • (u − u) ¯ dŴ = 0

Ŵt

(4.164)

Ŵu

where α is the penalty coefficient and is commonly selected 103 –107 times elastic modulus. The larger the penalty coefficient is, the higher is the accuracy. After the space discretization, we can obtain the governing equation   (4.165) K + K p U = P + Pp

The advantage of the penalty function method is that the system matrix is still banded and symmetric; if the penalty

is larger enough, the matrix is positive definite and no new unknown is introduced. On the other hand, its effectiveness intensively depends on the choice of the penalty coefficient. The accuracy increases with an increase in α, while the condition number rapidly increases with an increase in α and leads to singularity. Therefore, the choice of the penalty coefficient is a critical issue. 4.1.4.3 Numerical Integration of Mesh-Less Method In the FEM, element mesh is used not only for interpolation but also for integration. Because the solution domain has been discretized into a series of elements, the domain integral required by the Galerkin weak form can be transformed into the sum of element integral, which is easy to be implemented.

f (x ) d j = j

n

f (x i ) wi

(4.166)

i=1

where j represents the region of the element j , f (x ) is the integrand, n is the number of integral points in element j , x i is the i th integral point, and wi is the corresponding integral weight. The coordinates and weight of the element integral point can be decided with many approaches such as Gauss integration, Hammer integration, and Monte Carlo integration. In the Galerkin mesh-less method, the interpolation does not need the mesh, but the integral still needs the mesh. Because the solution domain is discretized with a series of computational nodes, the mesh cells do not exist, so some special approaches should be designed for the domain integral. On the other hand, the shape functions of meshless method are much higher than that of the FEM, so calculating the integral of the shape functions on the whole solution domain is difficult than is done in the FEM. The present approaches include integration with background cell structure, integration with background mesh, nodal integration, and MLS integration. 4.1.4.3.1 Integration with Background Cell Structure Figure 4.30 illustrates that in the mesh-less method the whole solution domain can be covered with a regular cell structure. The integral of the integrand on the solution domain can be transformed to the integral of the regular cell that is convenient to adopt various numerical integral approaches similar to the one shown in Equation 4.157. Gauss integration is the most commonly used approach, which is described as follows: 1. Generate the background cell structure covering the whole solution domain . 2. Do loop for all cells.

NUMERICAL METHODS

Gaussian points Background cell

Nodes

FIGURE 4.30

Integration with background cell structure.

(a) Generate integration points in Gaussian quadratude in the cell and determine the weight of each point. (b) Do loop for all Gaussian points. (i) Determine whether Gauss integral point is in the solution domain by the position coordinates of the point. (ii) If yes, calculate the numerical integral on the points and accumulate the results to a global system matrix or vector; if no, skip this Gauss integral point to the next one. (c) End loop for all points. 3. End loop for all cells. It is clear from Figure 4.30 that for any cell cell there are three relationships between the position of the cell and the solution domain : cell is completely in , cell is completely outside , and cell is partly in . Adopting this integration would cause a big error because of the third kind of cells. An improvement proposed by Kaljevic et al. is to divide these cells into several new cells any of which is completely in . So the accuracy can be increased to a certain extent. Note that in the mesh-less method the support boundaries of the shape functions is generally inconsistent with the background cell structure (Figure 4.31), namely, the support

99

boundaries are not superposed on the boundaries of cell. This leads to a big error. Dolbow et al.18 suggested that all nodes use rectangular support boundaries to construct the background cell structure. This makes their boundaries superposed so as to increase the integral accuracy. 4.1.4.3.2 Integration with Background Mesh As shown in Figure 4.32, similar to FEM, by using computational nodes the solution domain is discretized into a series of cells. So the integral on the domain can be transformed to the sum of the integral on each cell. The advantage is that all integral points are in the solution domain , the judgment on the position of the integral point is eliminated, so the efficiency and accuracy are increased. The shortcoming is that the computational mesh is required which goes against the characteristic of the mesh-less method. In FEM, the computational mesh (element mesh) is used not only to calculate the regional integral but also to interpolate the value of any point in the element with the values of the element nodes whose accuracy heavily depends on the mesh quality. Different from FEM, the background mesh in the mesh-less method is only used to calculate the regional integral; the creation of the approximate function is only based on the computational nodes that are independent of the computational mesh. So, it can eliminate the dependence on the mesh quality. For the coupling method of FEM and mesh-less method, this integration approach is better than others. In addition, the nodes of the background mesh need not be computational nodes, which increases the flexibility. But in practice, the same set of nodes is employed for simplicity. 4.1.4.3.3 Moving Least Square Integration The previous two methods need mesh, but they are not the truly meshless methods (or called pure mesh-less method ). Duflot et al.19 used partition of unit-based approach to calculate the integral of the integrand on the whole solution domain . Consider n functions ϕk (x ) with the following characteristics: 1. The value of the function ϕk (x ) is defined in / k , ϕk (x ) = 0; subdomain k , namely, when x ∈ 2. n overlapping k covers the whole solution domain , namely ⊂ ∪nk=1 k ; 3. The functions ϕk  (x ) satisfy the condition of partition of unit, namely, nk=1 ϕk (x ) = 1, ∀x ∈ .

Using n functions of partition of unit ϕk (x ), we can write the integral of the function f (x ) on the solution domain in the following form:



f (x ) d =

n k=1

∩ k

ϕk (x ) f (x ) d

(4.167)

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

FIGURE 4.31

Inconsistency of boundaries of supports and background cells.

Background integral cell

Nodes

FIGURE 4.32

Integration with background mesh.

Namely, the integral is transformed into the sum of the integral of the functions ϕk (x ) f (x ) on subdomains ∩ k . In general, the shape of subdomain k should be as simple as possible to facilitate the integral calculation. Obviously, the integration with background mesh can be seen as a special case of this method when the subdomains k are not overlapped and ϕk (x ) =



1 0

x ∈ k x∈ / k

(4.168)

then this integration becomes to be taken on the background mesh. Because of these three characteristics, ϕk (x ) in Equation (4.167) can be selected as the shape functions of MLS, then the approach is called moving least square integration. For increasing computational efficiency, Duflot et al. selected Shepard function as ϕk (x ), and the subdomains k are

rectangle. It should be noticed that, on the one hand, because the integral is taken on the subdomains ∩ k , it is necessary to judge whether the integral points of the subdomain k are in the solution domain which is similar to integration of the background mesh. On the other hand, the integrand f (x ) becomes a more complex function ϕk (x ) f (x ), and so more integral points are required to increase the accuracy, which results in the increase of the workload markedly. 4.1.4.4 Case Studies Mesh-less methods are not very mature yet, and there are a dozen ways proposed. Liu and his coworkers 20 – 22 developed parallel RKPM (reproducing kernel particle method) for large-scale CFD, and introduced multiscale RKPM into the analysis of viscous, compressible flows. Tsai et al.23 analyzed three-dimensional Stokes flows with dual reciprocity method (DRM) and the method of fundamental solution (MFS). Bernal et al.24 – 26 simulated the non-Newtonian Hele-Shaw flow for injection molding with RBF mesh-less method. They chose the asymmetric RBF collocation method (Kansa’s method) to compute the pressure distribution, which is truly mesh-free, and is one of the most frequently used methods due to its accuracy and ease of implementation. Martinez et al.27,28 studied injection processes involving short fiber molten composites with mesh-less method. Qian29 implemented mesh-less analysis on primary gas penetration during gasassisted injection molding. Au30 predicted two-dimensional flow front advancement in injection molding with the meshless method. Duan31 developed adaptive coupled FEM and meshless method, and used the generalized version of the characteristic Galerkin (CG) method in the ALE (arbitrary Lagrangian–Eulerian) framework to simulate the nonisothermal non-Newtonian viscous flow in mold filling process. He chose a typical symmetrical mold cavity to be filled. Figure 4.3331 shows the velocity distributions and the pressure contour within the filled molten polymer domains at different time.

TRACKING OF MOVING MELT FRONTS

101

FIGURE 4.33 Velocity distributions and the pressure contour within the filled molten polymer domains at different times.31

4.2 4.2.1

TRACKING OF MOVING MELT FRONTS Overview

The filling process represents a class of variable mass flow with moving free interface. For the three-dimensional simulation of filling process in casting and injection molding, it is significant to track accurately the flow front of melt, and to analyze the flow behavior of melt in the mold cavity and the interface characteristics of the flow front. However, the difficulty is to ensure that the free surface is a material surface with fluid particles at any time, and meanwhile, the normal stress and the tangential stress are always zero (regardless of the interfacial tension). In fact, the discussion about the moving interface problem can be traced back to the end of nineteenth century. In 1889, Stefan studied a typical case of so-called “moving boundaries” problems: ice–water problem (ice passing to water). Therefore, these problems are often called Stefan problems, which includes problems of change of boundary with respect to time frequently encountered in our lives, such as paraffin melting, water freezing, and the diffusion of oil drop when dropped on the water surface. But in the process such as casting and injection molding, they face

with “free surface problem.” The researches on numerical analysis of the free surface problem have now developed to the numerical simulation of tracking the moving interface. Depending on the selection of the referenced frame, the moving boundary problems can be tackled by Lagrangian or Eulerian algorithms32 . An updated ALE method has also been discussed and applied for mold filling33 – 35 . Simulations of the moving interface are divided into two types: interface tracking and interface capturing. The main difference between the two types is that the interface tracking uses the Lagrange algorithms, while the interface capturing uses Euler algorithms. 4.2.1.1 Lagrangian Algorithms Lagrangian algorithms focus on the material particle and the coordinate system used to describe the fluid fixed on fluid particles that are carried along with the fluid or body. Lagrangian algorithms can accurately track the free surface and easily impose boundary conditions, and meanwhile, the solving difficulty reduces with the disappearance of the convection term in the governing equations. However, these algorithms suffer the frequent mesh rezoning, which is not only time-consuming but also prone to reduction in the accuracy.

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

In 1970, on the basis of Lagrangian frame, Hirt et al.36 used FVM to simulate the two-dimensional transient incompressible viscous flow with free surface, which was named LINC (Lagrangian incompressible) algorithm. Then, Butler37 made some improvement on the LINC algorithm. In the LINC algorithm, the computational points (grid nodes) are completely determined by the material particles, which is only suitable for the case of no mesh distortion. In 1971, Crowley38 proposed a free Lagrangian algorithm to simulate incompressible ideal fluid, which introduced the mesh rezoning technology including adding, deleting, and reconnection of grids. Then, Fritts et al.39 made implementation and improvement on the Free Lagrangian algorithm. Crowley, Fritts et al. soon discussed about the mesh redistribution criteria and the mesh rezoning skills. Feng et al.40 put the Lagrangian algorithms into a solving framework of adaptive space–time least squares finite element that is used to simulate incompressible flows. In recent years, Idelsohn et al.41 developed the Lagrangian mesh-less methods and used it to tackle fluid–solid coupling problems. In the case of metal casting, Muttin et al.42 used the Lagrangian FEM to track the free surface and automatic mesh rezoning technology for avoiding mesh distortion. For three-dimensional flow with free surface, Khayat et al.43 developed the Lagrangian boundary element method to simulate the filling process of Newtonian flow. 4.2.1.2 Eulerian Algorithms Eulerian algorithms focus on spatial points and the coordinate system of the fluid is fixed in space domain, which is the most common method used to describe the fluid motion. Eulerian algorithms have good robustness, and they use fixed mesh to avoid a series of problems caused by mesh rezoning. So this is suitable to simulate complex cavity. However, in order to track the free surface, Eulerian algorithms need to calculate a flag variable, such as volume fraction and level set function. This increases the additional amount of computation, and meanwhile, the reasonableness of the calculation of the flag variable decides the effect of the tracking-free surface to a large extent. Floryan and Rasmussen44 organized Eulerian algorithms into four categories: fixed mesh methods, adaptive mesh methods, mapping methods, and special methods, among which the fixed mesh methods are most widely used. As its name, the mesh in the fixed mesh methods stays unchanged in the solution domain. The common fixed mesh methods include the surface tracking method, volume tracking method, and level set method. The surface tracking method45 uses a set of interpolation curves with a series of discrete points to represent the interface. Then, according to flow governing equations and free surface evolution equations, new locations of these discrete points and curve segments of each time step are

calculated so as to construct a new free surface shape. At each time step, the information of the related points’ locations and interrelations of them are stored, by which the interface within the Euler mesh can be described, and the accuracy is high. However, this needs to store a great deal of data; furthermore, automatically adding necessary discrete points is required to avoid inaccurate interface information caused by uneven distribution of the interpolate points on the interface. When dealing with overturn and merging of the free surface, the method is not effective for the multiplevalued problems, which leads to logical confusing. The volume tracking method does not directly store the interface shape, but store the flag variable reflecting the filling situation of each unit and reconstruct the interface based on the flag variable. This method cannot describe the details of the interface whose size is smaller than the size of mesh cells and cannot accurately determine the location, orientation, curvature, and other information of the interface. But because of its high robustness and because it uses less storage information to describe the interface, volume tracking methods have obtained a widespread use and success since the mid-1970s. Common volume tracking methods include MAC (Marker and Cell), FAN, SLIC (simple line interface calculation), PLIC (piecewise linear interface calculation), and VOF (volume of fluid). In 1965, Harlow and Welch46 proposed the MAC method. It is the first method to successfully treat problems involving complicated free surface motions. This method was also the first technique to use pressure and velocity as the primary dependent variables. MAC employed a distribution of marker particles to define fluid regions, and simply set free surface pressures at the centers of cells defined to contain the surface47 . This method introduces a group of imaginary massless marker particles in the region occupied by the fluid. These marker particles move with the fluid and the free surface is located in the cells containing these marker particles. However, the exact location, direction, curvature, and other details of the free surface cannot be given, and it is also difficult to introduce the boundary conditions of the interface. In addition, the information of the mesh points, as well as the information of marker particles, must be stored, which needs a great deal of memory and make the computational efficiency low. Tadmor et al.48 extended the MAC method in 1974 and proposed the FAN method, which uses the fluid volume fraction in the cells to describe the interface information, and they first introduced the concept of node CV. The volume fraction is determined by the flow rate on the boundaries of the CV. The method uses the volume fraction instead of the marker particles in MAC as the flag variable to determine the interface, which greatly reduces the computer storage requirements. In 1976, Noh and Woodward49 proposed the SLIC method. The interface within the cells is represented by a

TRACKING OF MOVING MELT FRONTS

straight line and the movement of interface is determined by the local fluid velocity. When reconstructed, the interface should be parallel to an axis at one time, so the interface is rough. In 1982, Youngs50 proposed the PLIC method. In this method, the interface is approximated by a plane surface of an appropriate inclination in each cell. The plane surfaces are not connected to each other at the cell faces. That is, the interface surface at each cell is determined independently of the neighboring interface, and their intersections with the cell faces need not necessarily be connected at the cell faces. The PLIC method has proved more satisfactory accuracy in practice. Hirt and Nichols47 proposed the VOF method in 1981. In the VOF method, evolution of the interface is described using a discrete function F , whose value in each cell of the computational mesh is the fraction of the cell occupied by the fluid. The value is equal to one in cells full of fluid, zero in empty cells, and a value between zero and one in cells containing mixed interface. This method makes use of PDE solvers that are not based on the characteristic equations to predict the evolution of the free surface. The method uses a value to describe the situations of the melt within the cells, so it can save memory. But each cell has only one piece of information to describe the interface, which makes it a little arbitrary when determining the shape of the interface, and it is impossible to describe the details of interface whose size is smaller than the cell size. Furthermore, unfortunately, it is difficult to calculate the curvature of the front from such a representation of the boundary. The main advantages are its robustness and capability to deal with the multi-material interface. VOF has been widely used and people have developed many different forms. Luoma et al.51 proposed a unified VOF method for tracking the free surface in the filling process of polymer melt. In 1988, Osher and Sethian52 proposed the level set methods, in which the moving interface is time dependent. Level set methods have become increasingly popular in a diverse group of applications including optimal design, computer-aided design (CAD), optimal control, and computer graphics. Research on interface propagation has led to the gradual development of level set algorithm, which is an efficient calculation tool for simulating the geometric topology changes of the closed moving interface along with the time evolution. The approach is completely different from the VOF technique where a discontinuous scalar field is transported. The level set method uses a continuous field instead, where the level “zero” corresponds to the flow front. The general idea behind the level set methods is to apply a function ϕ(x , t), called level set function, to the space which the interface inhabits, and the moving interface is defined as the zero isosurface (or isoline) of function ϕ, namely, ϕ(x , t) = 0. Then a scheme is used to approximate

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the value of ϕ(x , t) over small time increments. Let the function ϕ move at an appropriate speed, and ensure its zero isosurface is the material interface. At any time, if the function ϕ is known, work out its zero isosurface, then the active interface at that time can be determined. Then, the material-related physical quantities can be solved in the whole region, and it is easy to obtain the interface information such as the normal direction and curvature. One difficulty of level set methods, particularly concerning fluid flow interfaces, is the lack of mass conservation: existing methods conserve mass only in a global sense to a fraction of 1%. Some researchers proposed coupled level set and VOF method53 or hybrid particle level set method54 combining the original level set with the front tracking method to overcome the deficiency. The level set methods have been applied in filling simulation55,56 . Fedkiw and Aslam57 proposed Ghost fluid (GF) methods in 1999. GF methods must use an interface simulation method in combination like level set method, surface tracking methods, or volume tracking methods. The ideal of GF methods is to transform multiphase interfacial flow problem into several single-phase flow problems, then determine the interface and the values of the physical variables in the whole solution domain with the level set method or other methods. 4.2.1.3 Arbitrary Lagrangian–Eulerian (ALE) methods In the way of tracking the free surface, imposing boundary conditions, and maintaining the mesh form, Lagrangian algorithms and Eulerian algorithms have their own advantages and disadvantages, respectively. On the basis of the advantages of both, ALE methods have been developed. Different from the Lagrangian and Eulerian methods, the fundamental characteristic of ALE methods is to define the mesh points that are independent of particle and space point, and their movement metric—mesh velocity. Hirt58 and Noh59 first proposed the idea of ALE methods in the FDM. And Hughes et al.60 , Belyschk et al.61 , and Huerta et al.62 used and developed ALE methods in FEM. At present, the ALE FEM is widely applied to solve the problems having large overall moving boundaries (or interface), especially large amplitude liquid sloshing, fluid–solid coupling, processing, and contact and large deformation. Gastonet al.34 , Levis ea al.63 , and Han et al.64 had applied the ALE method for simulating the process of filling. In the ALE frame, the reference coordinate system (namely the mesh points), which describes the particle motion, can move without any restriction in principle, so there is no unified motion approach. When the reference coordinate system is fixed on the particles, the ALE description degenerates to the Lagrangian description; when the reference coordinate system is fixed on the space points,

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

the ALE description degenerates to the Eulerian description. Although the moving of ALE reference coordinate system is arbitrary, it should generally hold two principles: first, it should be easy to track the free boundaries so as to describe the deformation process as accurately as possible; second, it should be easy to correct the distortional cells and make them regularized so as to make the process of FEA smooth. There are two common ways to update the mesh, first of which is to apply the same movement to the mesh inside the solution domain as the one of the boundary meshes, such as the Lagrange–Euler matrix method60,62 . It describes the mesh movement by differential equations which are put into the control equation system. The positions of mesh points are determined by calculating the mesh velocity. The other way is to apply different movement (namely, the hybrid method) to the mesh inside the solution domain and boundary meshes, and track the position of the free surface on the boundaries while, in the solution domain, construct relatively regular mesh according to the mesh shape, then calculate the mesh velocity from the new positions of mesh points. For particles whose movement is complex, it is difficult to control the mesh shape by unified differential equations, and meanwhile, it also increases the solving scale when create the differential equations are created on the entire domain. So the hybrid is most used. In the ALE description, tracking the free surface accurately only requires normal velocity of the mesh on the free surface to be consistent with the normal velocity of corresponding fluid particles. In order to determine all velocity vectors of mesh points, one or two additional equations are required for the two- or three-dimensional problems, respectively. Huerta et al.62 developed an ALE Petrov–Galerkin finite-element technique to study nonlinear viscous fluids under large free surface wave motion. For tackling the fluid sloshing problems by the ALE method, Zeng and Wang65 fixed the mesh velocity on one direction so as to determine the velocity of another direction. Braess and Wriggers66 added an additional equation from the geometric view to ensure that the

(a)

FIGURE 4.34

arrangement of cells with the free surface is homogeneous. However, for complex cavity, it is difficult to give a unified additional equation. It is obviously not conducive to practical application if different additional equations are given for different cavity. Li et al.64 developed an adaptive method for introducing additional equation of the free surface, which is suitable for simulating tracking the free surface issues with a variety of cavity. The Lagrange–Euler matrix method adopts the same movement both on the internal mesh and boundary mesh, while most other algorithms are based on the mesh regularization principle for determining the movement of internal mesh. Donea proposed an empirical formula. Gadala and Wang67 proposed a transfinite mapping method. Wang et al.68 transformed the problem of the mesh node movement into an unconstrained optimization problem by defining the distortion potential. On the basis of the principle that the internal apex nodes should be as close as possible to the center of gravity of the adjacent cells, Gaston et al.34 proposed a mesh motion approach. Aiming at the filling characteristics, Li et al.64 introduced a new method in which the process of mesh generation was simplified to the process of polygon triangularization near the moving free surface, which greatly improved the speed of real-time mesh generation.

4.2.2

FAN

In 1974, on the basis of the MAC method, Tadmor et al. proposed the FAN method48 . VOF method and the pseudo–concentration method usually require solving the VOF transport equation, while the FAN method directly calculates the net flow of under-filling CV from velocity field to update the flow front. FAN algorithm can be seen as an explicit VOF algorithm based on node CV on a fixed mesh. By connecting the body-centered points, face-centered points, and edge midpoints, a tetrahedron can be divided

(b)

Subdomains of the element: (a) 2D triangular element, (b) 3D tetrahedron element.

TRACKING OF MOVING MELT FRONTS

into four subdomains as shown in Figure 4.34, in which each subdomain belongs to the corresponding tetrahedral nodes and each node CV is the sum of the corresponding subdomain of all tetrahedrons including the node. For each node, define the filling proportion function as fi =

Vfill Vcv

(4.169)

where Vcv is the volume of the CV, Vfill is the volume of the fluid filled within the CV of the node. Basically, when the node is empty (that is, no traced fluid inside the node’s CV), the value of fi is zero; if the node is full, we have fi = 1; and when the node is filling, we have 0 < fi < 1, which means the flow front interface cuts the cell and it is a flow front cell. During the process of filling simulation, update the filling proportion function of the node CV to track the flow front. First, calculate the flow rate among node CVs of the flow front cells; and the sum of flow rate is the total flow rate of the CV. Then, take the minimum filling time taken by the filling node in the flow front nodes to fill the CV as the current time step to update the filling proportion of the CV of the flow front node. And meanwhile, the cells that are about to be filled around the new filled cell become new flow front cells; and the nodes that are about to be filled are new flow front nodes. So nodes and cells update continuously until all of the node CVs are filled, that is, the filling process is complete. In each cell, the inflow of each node CV is the sum of the inflow through the common interfaces of the present CV and adjacent CVs (Figure 4.34). And in the cell, the flow between the CVs of node A and node B can be obtained by the surface integral of the ESOR. Let the time ti , when fi = 0.5, be the time when the interface of the flow front passes through the node i . This time ti is called the node filled time. So when the filled time of all nodes is known, at any time the interface of the flow front in the mesh cells can be known by interpolation.

4.2.3

VOF

Since proposed by Hirt and Nichols in 1981, the VOF method has been widely used. VOF method defines a discontinuous scalar function F named fluid fraction function whose value jumps from 0 to 1 when the argument moves into the interior of the traced phase. In a mesh cell, the value of F represents the fluid proportion in the cell. Then use the interface transport algorithm and interface reconstruction algorithm to track the free surface. The following parts will introduce the basic idea of the VOF method in detail and some common interface reconstruction algorithms.

105

4.2.3.1 Basic Ideal In each cell of a mesh it is customary to use only one value for each dependent variable defining the fluid state. The use of several points in a cell to define the region occupied by fluid, therefore, seems unnecessarily excessive. VOF method47 defines a function F whose value is unity at any point occupied by fluid and zero otherwise. The average value of F in a cell would then represent the fractional volume of the cell occupied by fluid. In particular, a unit value of F would correspond to a cell full of fluid, while a zero value would indicate that the cell contained no fluid. Cells with F values between 0 and 1 must then contain a free surface. The normal direction to the boundary depends on the direction in which the value of F changes most rapidly. Because F is a step function, its derivatives must be computed in a special way. When properly calculated, the derivatives can then be used to determine the boundary normally. Finally, when both the normal direction and the value of F in a boundary cell are known, a line cutting the cell can be constructed to approximate the interface there. This boundary location can then be used in the setting of boundary conditions. The time dependence of the fractional volume function F is governed by the transport equation ∂F + V • ∇F = 0 ∂t

(4.170)

If the fluid is incompressible, Equation 4.170 can be written as ∂F + ∇ • (V F ) = 0 (4.171) ∂t This PDE states that F moves with the fluid. In a Lagrangian mesh, Equation 4.171 reduces to the statement that F remains constant in each cell. In this case, F serves solely as an indicator identifying cells that contain fluid. In an ALE mesh, the flux of F moving with the fluid through a cell must be computed, but standard finite difference approximations would lead to a smearing of the F function and interfaces would lose their definition. Fortunately, the fact that F is a step function with values of zero or one permits the use of a flux approximation that preserves its discontinuous nature. In summary, the VOF method offers a region-following scheme with minimum storage requirements. Furthermore, because it follows regions rather than surfaces, all logic problems associated with intersecting surfaces are avoided with the VOF technique. The method is also applicable to three-dimensional computations, where its conservative use of stored information is highly advantageous. Thus, the VOF method provides a simple and economical way to track free boundaries in two- or threedimensional meshes. In principle, the method could be used to track surfaces of discontinuity in material properties, in

106

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

tangential velocity, or any other property. The particular case being represented determines the specific boundary condition that must be applied at the location of the boundary. For situations where the surface does not remain fixed in the fluid, but has some additional relative motion, the equation of motion (Eq. 4.170) must be modified. Examples of such applications are shock waves, chemical reaction fronts, and boundaries between single- and two-phase fluid regions. 4.2.3.2 VOF Interface Reconstruction Methods In volume tracking method, VOF is the most preferred method for the CFD researchers because of low memory requirement, simplicity and good sharpness, and quality of tracked interface. Along with the development of the VOF method, various interface reconstruction technologies have occupied the mainstream in VOF methods. The volume data is traditionally retained as volume fractions (denoted as F hereafter), whereby mixed cells will have a volume fraction F between 0 and 1, and cells without interfaces (pure cells) will have a volume fraction F equal to zero or unity. Because a unique interface configuration does not exist, once the exact interface location is replaced with discrete volume data, detailed interface information cannot be extracted until an interface is reconstructed. The principal reconstruction constraint is local volume conservation, that is, the reconstructed interface must truncate cells with a volume equal to the discrete fluid volumes. From the view of reconstruction, VOF methods can be organized to donor–acceptor, SLIC , FLAIR (flux linesegment model for advection and interface reconstruction), PLIC, and so on. The first two methods are the piecewise constant methods. The SLIC method proposed by Noh and Woodward49 reconstructs the interface in a cell by a straight line parallel to one of the coordinate directions. It is a direction-split algorithm. During each direction swap, only neighbor cells in the swap flux direction are considered to determine the interface. So the interface in a cell often has a different representation for different direction sweep. The original PLIC was proposed by Hirt and Nichols47 . In this method, the interface within each cell, similar to that in SLIC, is assumed to be a line aligned with one of the local coordinates. Unlike SLIC, all neighbor cells (eight for a two-dimensional case) are used to estimate the surface normal, and the interface is specified as horizontal or vertical depending on the relative magnitudes of the surface normal components. It is easy to be implemented and can be extended to three dimensions in a straightforward manner. However, it is only first-order accurate. Subsequent full reconstruction methods developed from the early used donor–acceptor-type zeroorder capturing methods with half-construction methods by Hirt and Nichols, in which the interface is finer and the

(a)

(b)

(d)

(c)

(e)

FIGURE 4.35 Interface reconstructions of actual fluid configuration: (a) the actual interface; (b) SLIC (x ); (c) SLIC (y); (d) VOF of Hirt and Nichols; (e) PLIC of Youngs.

accuracy is higher. Figure 4.35 schematically shows the results of the interface reconstruction using SLIC of Noh and Woodward with different sweep direction, VOF of Hirt and Nichol, and VOF of Youngs. We can see that the results of the SLIC method are very tough, and Youngs’ VOF are closer to the actual interface than the others, although they are much simpler. Youngs’ VOF is a good piecewise linear volume tracing method. As a pragmatic method, this method will be introduced in the following part. Seifollahi et al.69 developed an approach to implement the PLIC–VOF method. If the fractional volume F is given, the interface normal vector n (a unit vector perpendicular to the interface) needs to be determined for each cell. This is achieved using the gradient of F : ∇F n=− (4.172) |∇F | in which the gradient of F at each point is calculated using the values of F in its immediate 8 (in 2D square mesh) or 28 (in 3D hexahedron mesh) neighboring points. Here, we take a 2D staggered mesh for example. The interface in two-dimensional problems is assumed to be a straight line (a plane surface in 3D problems; Figure 4.36) for the x -momentum cell. In this figure, the thick solid inclined line is the approximate interface location in the x -momentum cell located at (i + 1/2, j ). Here, we replace the interface functions Sx , i , j , k and Sx , i , j + 1, k with the twodimensional interface functions Lx , i , j and Lx , i + 1, j . We now need to calculate the values of Lx , i , j and Lx , i + 1, j . We can use Equation 4.172 to reconstruct the interface. The vector m is defined as the gradient of Fi , j . To obtain m numerically, we first approximate its value at the cell

TRACKING OF MOVING MELT FRONTS

107

Therefore, mx,i,j =

my,i,j

mx,i+1/2,j +1/2

1 = 2

my,i+1/2,j +1/2 =

1 2

 

Fi+1,j − Fi,j Fi+1,j +1 − Fi,j +1 + xi+1,j − xi,j xi+1,j +1 − xi,j +1



(4.173) 

Fi,j +1 − Fi,j Fi+1,j +1 − Fi+1,j + yi,j +1 − yi,j yi+1,j +1 − yi+1,j

(4.174) Then the x and y components of m at (i , j ) are obtained by means of averaging:

mx,i,j =

my,i,j =

1 mx,i−1/2,j −1/2 + mx,i−1/2,j +1/2 4  + mx,i+1/2,j −1/2 + mx,i+1/2,j +1/2 1 my,i−1/2,j −1/2 + my,i−1/2,j +1/2 4  + my,i+1/2,j −1/2 + my,i+1/2,j +1/2

FIGURE 4.37

(4.175)

(4.176)



+

Fi,j − Fi−1,j Fi,j +1 − Fi−1,j +1 + xi,j − xi−1,j xi,j +1 − xi−1,j +1

+

Fi+1,j −1 − Fi,j −1 Fi+1,j − Fi,j + xi+1,j −1 − xi,j −1 xi+1,j − xi,j

Fi,j −1 − Fi−1,j −1 Fi,j − Fi−1,j + xi,j −1 − xi−1,j −1 xi,j − xi−1,j

Fi+1,j − Fi,j Fi+1,j +1 − Fi,j +1 + + xi+1,j − xi,j xi+1,j +1 − xi,j +1

FIGURE 4.36 Interface in an x -momentum cell (dotted line) and interface function Sx or Lx .

corners, for example at position (i + 1/2, j + 1/2), we have

1 8

1 = 8





(4.177)

Fi−1,j − Fi−1,j −1 Fi,j − Fi,j −1 + yi−1,j − yi−1,j −1 yi,j − yi,j −1

+

Fi,j − Fi,j −1 Fi+1,j − Fi+1,j −1 + yi,j − yi,j −1 yi+1,j − yi+1,j −1

+

Fi,j +1 − Fi,j Fi−1,j +1 − Fi−1,j + yi−1,j +1 − yi−1,j yi,j +1 − yi,j

+

Fi,j +1 − Fi,j Fi+1,j +1 − Fi+1,j + yi,j +1 − yi,j yi+1,j +1 − yi+1,j



(4.178)

And nx and ny components of the unit normal vector n are nx,i,j = − + ny,i,j = − +

mx,i,j

(4.179)

m2x,i,j + m2y,i,j my,i,j

(4.180)

m2x,i,j + m2y,i,j

Once the normalized unit vector n is calculated, a straight line (the thick solid declined lines in Fig. 4.36) is positioned perpendicular to it in such a way that it matches with the value of F in the cell. It is shown in Figure 4.37 that depending on the orientation of interface, eight different cases may occur. The normal vector angle θ (Fig. 4.38) can take any

Different configurations for an interface in a cell for the 2D case.

108

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

Flim,1

− → product n × ab is positive. To calculate the components of vectors a and b, we need to specify the limiting values of F for a particular n.

Flim,2 b

b n

Flim,1 =

q b

nmin 2nmax

and

Flim,2 = 1 − Flim,1

where

a

   nmin = min |nx | , ny 

and

Then,

y a

a

Flim,1 =

hmin 2hmax

and

   nmax = max |nx | , ny  (4.182)

Flim,2 = 1 − Flim,1

(4.183)

where

x FIGURE 4.38 the 2D case.

(4.181)

Different locations of an interface in a cell for

and

    hmin = min |nx | x, ny  y     nmax = max |nx | x, ny  y

(4.184)

The components of vectors a and b in Figure 4.38 can be defined as • For F ≤ Flim, 1 (triangle): ax =

,

2F xy

ny nx

ay = 0 bx = 0 by =

2F xy ax

(4.185)

• For Flim, 1 ≤ F ≤ Flim, 2 (quadrilateral): ax = F x +

ny y 2nx

ay = 0 bx = F x − FIGURE 4.39

L for a unit length cell for the 2D case.

value between 0 and 2π . When n is in the first octant (0 ≤ θ ≤ π /4), different cases may occur, which are shown in Figure 4.38. All the other cases can be obtained by mirroring the equivalent situation with the first octant on the x -axis, the y-axis, and the bisector between them. a and b vectors in Figure 4.38 are used to determine L (Fig. 4.39). In order to calculate L, the locations of a and b, the two ends of the straight line in each cell, need to be determined. a and b are determined such that the cross

ny y 2nx

by = y

(4.186)

• For F ≥ Flim, 2 (pentagon): ax = y , nx ay = y − 2(xy − F ) ny bx = x −

2(xy − F ) y − ay

by = y

(4.187)

TRACKING OF MOVING MELT FRONTS

bx = 0

b

Lx = by

y a

ay

Ly = ax x

FIGURE 4.40 Lx and Ly for a unit length cell for collocated grid for the 2D case.

Lx and Ly can now be calculated using a and b. For instance, for the cell shown in Figure 4.39, Lx is Lx = ay + 0.5ax

by − ay ax − bx

(4.188)

Similarly, for the same cell Ly is Ly = bx + 0.5by

ax − bx by − ay

(4.189)

If the grid is collocated, the momentum cells and the F cell coincide at one location for each node point (i , j , k ). Therefore, Sx , i , j , k , Sy, i , j , k , Sy, i , j , k are already available. For the two-dimensional case, the interface functions S s become Ls, and, thus, the values of Lx and Ly (Fig. 4.40) can be obtained from Equations 4.188 and 4.189: Lx,i−1/2,j = by

and

Ly,i,j −1/2 = ax

(4.190)

109

(2, 1)). The results are shown in Figure 4.41 in which (b) and (g) are for SLIC, (c) and (h) for Hirt–Nichols’ VOF, (d) and (i) for FCT (flux-corrected transport)–VOF and (e) and (j) for Youngs’ method. The figure shows that SLIC maintains good interface shape in all cases in which advection is aligned with a coordinate direction, but is unacceptably poor when advection inclines to the coordinate axes. This is particularly obvious in advection of a coordinate-aligned square, which develops large interfacial ripples with an amplitude of several mesh cells. Interface disturbances start at the upwind corner of the square and propagate along the sides. Only Hirt–Nichols’ VOF produced good results for the coordinate-aligned square advected in a coordinate direction, in all other cases there was significant interface distortion and some interface spreading. The somewhat poorer performance of Hirt–Nichols’ VOF compared with SLIC is surprising considering that the interface reconstruction is multidimensional (even though the flux calculation is direction split). FCT–VOF performs well in most cases, the exceptions being the angled square for both velocity fields. Youngs’ method gives the most consistent results as far as the interface shape is concerned, and although the case handled best by the other three methods is not handled quite as well by Youngs’ method, the error is very small. Zalesak’s problem, in which a slotted circle is rotated through one or more revolutions, is widely used as a test for scalar advection methods. The initial conditions and results after one rotation for each of the four methods are shown in Figure 4.4270 . Both SLIC and Hirt–Nichols’ VOF generate quantities of jetsam and both give quite poor interface shapes after one rotation. Hirt–Nichols’ VOF also shows some slight spreading of the interface. In contrast, FCT–VOF and Youngs’ method give sharp interfaces without jetsam and both give rise to acceptable interface shapes, although the sharp corners at both ends of the slot become rounded. Shearing flow is used to test the topological change in the solution. Rudman chose the following simple velocity field: u (x, y) = cos (x) sin (y) v (x, y) = − sin (x) cos (y)

(4.191)

70

4.2.3.3 Results Comparison Rudman introduced an algorithm for volume tracking based on flux-corrected transport (FCT), and examined three other methods including SLIC, Hirt–Nichols’ VOF and Youngs’ method. Simple advection tests were also made. Figure 4.4170 shows the representations of these methods in the condition of a two-dimensional unidirectional field. Three different scalar fields are considered: a hollow square aligned with the coordinate axes, a hollow square at an angle of 26–57◦ to the x -axis and a hollow circle. Each of the three scalar fields is separately advected with two velocity fields ((0, 1) and

with x , y ∈ [0, π ]. Figure 4.4370 shows the results of the shearing field. For steps less than 1000, FCT–VOF maintains a slight advantage over both SLIC and Hirt–Nichols’ VOF, although by a smaller margin than the simple advection tests. Youngs’ method is obviously the best one which is up to an order of magnitude more accurate for all steps. For 2000 steps, all methods have begun to break down because the shearing flow starts to stretch the fluid circle into thin filaments only one or two cells wide. Notably, Youngs’ method does a fairly good job, nevertheless. On

110

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

FIGURE 4.41

Advection with unidirectional velocity fields (0, 1) (top) and (2, 1) (bottom).70

the basis of the simple advection problems discussed previously, FCT–VOF appeared to be a suitable alternative to both SLIC and Hirt–Nichols’ VOF, with better interface shapes and errors typically three times smaller. Although FCT–VOF still outperforms both SLIC and Hirt–Nichols’ VOF, the advantages are less clear compared with all other methods except Youngs’ having severe problems, and returning the initial interface for steps more than 1000. It is seen in this example that after the application of a significant period of shearing, all methods break down to an unacceptable level. This is due in part to the inability of the mesh to resolve the fine features of the interface, and it reinforces the necessity of shearing flow test. Rudman also discussed the efficiency and order of accuracy of these methods. More test details such as the mesh size can be obtained from Rudman’s paper.

4.2.4

Level Set Methods

The level set method71 is a computational technique for tracking a propagating interface over time, which has proven more accurate, in many problems, in handling topological complexities such as corners and cusps, and

in handling complexities in the evolving interface such as entropy conditions and weak solutions. It is a robust scheme that is relatively easy to implement. 4.2.4.1 Level Set Function and Level Set Equation Level set methods define a function ϕ(x , t) to implicitly represent the interface, where x is a point in the space and t is a point in time. The function is initialized at t = 0, and then a scheme is used to approximate the value of ϕ(x , t) over small time increments. The first step in applying the level set method is to pick a mesh, or a grid of points, that covers the image. In general, the finer the mesh is, the more accurate is the level set method. However, our digitized image puts a limit on how fine a mesh can be used. As the image consists of several thousand pixels, the mesh must be at least as coarse as the individual pixels, and optimally even coarser. Once a mesh is chosen, the next step is to initialize the value of ϕ(x , t) at each point of the mesh. Not only level set function can be defined but also differential equations can be defined based on the requirement of the problem. A typical choice for ϕ is the signed distance from the interface. For any point x in the mesh (which in our case

TRACKING OF MOVING MELT FRONTS

SLIC

Initial conditions 3.5

Hirt–Nichols

3.5

3.5

3.0

3.0

2.5

2.5

2.0

2.0

3.0 1.0

1.5

2.5

2.0

2.5

3.0

1.0

1.5

1.0

1.5

2.0

2.5

3.0

3.5

3.5

3.0

3.0

2.5

2.5

2.0

2.0

1.0

FIGURE 4.42 SLIC

1.5

2.0

2.5

2.5

3.0

2.5

3.0

Youngs

FCT–VOF

2.0

2.0

3.0

1.0

1.5

2.0

Zalesak’s problem for solid body rotation.70 Hirt–Nichols

FCT–VOF

Youngs

3.0

3.0

3.0

3.0

2.0

2.0

2.0

2.0

1.0

1.0

1.0

1.0

(a)

0.0 0.0

1.0

2.0

3.0

0.0 0.0

1.0

2.0

3.0

0.0 0.0

1.0

2.0

3.0

0.0 0.0

3.0

3.0

3.0

3.0

2.0

2.0

2.0

2.0

1.0

1.0

1.0

1.0

1.0

2.0

3.0

1.0

2.0

3.0

1.0

2.0

3.0

1.0

2.0

3.0

(b)

0.0 0.0

1.0

2.0

3.0

0.0 0.0

1.0

2.0

3.0

0.0 0.0

1.0

2.0

3.0

0.0 0.0

3.0

3.0

3.0

3.0

2.0

2.0

2.0

2.0

1.0

1.0

1.0

1.0

(c)

0.0 0.0

1.0

2.0

3.0

0.0 0.0

1.0

2.0

3.0

0.0 0.0

1.0

2.0

3.0

0.0 0.0

3.0

3.0

3.0

3.0

2.0

2.0

2.0

2.0

1.0

1.0

1.0

1.0

(d)

0.0 0.0

1.0

2.0

3.0

0.0 0.0

1.0

2.0

3.0

0.0 0.0

1.0

2.0

3.0

0.0 0.0

FIGURE 4.43 Results for shearing field: (a) after 1000 steps forward; (b) after 1000 steps forward followed by 1000 steps backward; (c) after 2000 steps forward; (d) after 200 steps forward followed by 2000 steps backward.70

111

112

NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

is a point in the plane), we have ϕ(x, t) = ±d

(4.192)

where d is the distance from the point x to the interface and it is called signed distance function (SDF). The positive sign is used if the point x is inside the interface (in the melt), and the negative sign is used if the point x is outside the interface. If d = 0, namely, zero level set function ϕ(x , t) = 0, the point x is on the very interface. Thus the name of the level set method is explained: at any time t, the evolving interface corresponds to the locus of all points x such that ϕ(x , t) = 0, and that locus is a level surface or curve of the ϕ function. The locus of all points x , such that ϕ(x , t) = c, contour around the original surface or curve, where c is an arbitrary positive or negative constant. A PDE of ϕ is used to evolve the interface. For any point x at any time t on the interface, we have ϕ(x , t) = 0. So the governing equation for ϕ is ∂ϕ dϕ = + V • ∇ϕ = 0, dt ∂t

V =

dx dt

(4.193)

For a specific question, Equation 4.193 has a special form. If the governing equation is an N–S equation, Equation 4.193 becomes ∂ϕ + u • ∇ϕ = 0 ∂t

(4.194)

where u is the fluid velocity. Let n be an outward directed normal to the surface or the curve. Then n can be rewritten in terms of ϕ as follows: n=

∇ϕ |∇ϕ|

(4.195)

4.2.4.2 Reinitialization In order to make the solving process convenient, it is very important for level set function ϕ(x , t) to maintain the SDF all the while. Because of the internal property of the numerical methods, after several steps, ϕ(x , t) does not satisfy the property of SDF. Thus, ϕ(x , t) should be modified in order to let it be SDF again. This process is the reinitialization of ϕ. Supposing at time t level set function ϕ 0 has been known, the reconstructed function ϕ should hold the following two conditions: 1. ϕ satisfies the SDF. 2. The zero isosurface of ϕ is the same as that of ϕ 0 . For this, we can solve the stable solution of the following initial value problem:  ϕτ = sign(ϕ0 ) (1 − |∇ϕ|) (4.196) ϕ(x, 0) = ϕ0

In order to facilitate solving, smooth the sign function sign(ϕ 0 ): ϕ0 (4.197) signε (ϕ0 ) = + ϕ02 + ε2 The general steps of level set method are as follows: 1. Initialization. Initialize all unknown physical variables and level set function ϕ(x , t). In the following steps the physical variables and ϕ(x, tn ) at time tn are supposed to be known. 2. Solving level set equation. Solve the special form of Equation 4.193 to obtain level set function ϕ(x, tn + 1 ) at tn + 1 . So the new moving surface is ϕ(x , tn + 1 ) = 0. But now ϕ(x , tn + 1 ) is not an SDF anymore. 3. Reinitialization. Replace ϕ 0 in Equation 4.196 with ϕ(x , tn + 1 ), work out the stable solution of Equation 4.196 by iterative solution as ϕ(x, tn + 1 ) after reinitialization. 4. Solving the governing equation of the physical variable. Solve the governing equation of the physical variable with the value of ϕ(x, tn + 1 ) to work out the value of the physical variable at tn + 1 . On the interface where ϕ(x, tn + 1 ) changes sign some special processes should be carried out. 5. Repeat steps 2–4 for the next time step. 4.2.4.3 Case Studies Figure 4.44 presents simulation results of the process of a water droplet falling into water. In this case, the water droplet with radius 0.15 m fell from 0.42 m height. We can see that the results obtained by using level set method are very satisfactory, although it is a simple example. It proves the capacity of this method. Pang et al.72 used a solution algorithm (SOLA) particle level set method to simulate the process of water filling into a box through a lateral inlet. The results are in good agreement with those of experiment, even so are some details. Hao et al.73 simulated the casting filling process with gas–liquid two-phase based on level set method, and compared the simulated results with the results obtained by Fluent, which is business software using VOF. Zhang et al.74 proposed projection level set method to simulate casting’s mold filling process, and compare the results of their method with those of HZCAE, a casting simulation package using single-phase flow method and VOF. In Zhang’s studies, a process of water filling into a rectangular cavity with an inlet located at the bottom was involved75 . Figure 4.45 illustrates the comparison of simulation results based on the particle level set method and those of hydraulic simulation, where the vectors indicate the velocity field and the color indicate the pressure field. We can see that the overall interface shape of simulation is similar as that of the experiment.

METHODS FOR SOLVING ALGEBRAIC EQUATIONS

(a)

(b)

(c)

(d)

FIGURE 4.44

Process of water drop falling into water: (a) 0.23 s; (b) 0.28 s; (c) 0.32 s; (d) 0.36 s.

FIGURE 4.45 and 1.47 s.

Results comparison of hydraulic experiment and level set method at 0.65 s, 0.87 s

4.3 METHODS FOR SOLVING ALGEBRAIC EQUATIONS The numerical solution for PDE consists three steps: 1. Function approximation. In this step, approximate functions represented by a series of discrete node values are used to approximate the unknown field function. Thus, a continuous problem with infinite degree of freedom can be turned into a discrete problem with finite degree of freedom. 2. Forming a system of linear algebraic equations. In this step, the residual error caused by governing equations operator within some points (using point collocation methods) or some subdomains (using weak integration methods) in the solution domain is eliminated. 3. Solve the system of linear algebraic equations. Then obtain the values of discrete points, sequentially the approximate solution on the overall solution domain is known. In the process of problem solving, it takes a lot of time to solve the linear algebraic equations. If inappropriate solving

113

methods are adopted, not only the expense of calculation is much higher but also it might cause instability or failure in solving procedure. In this chapter, we discuss some methods for solving the system of algebraic equations. 4.3.1

Overview

Since late the 1980s, increasing attention has been paid to research on efficient methods for solving algebraic equations based on nonlinear discrete problems. Methods for solving linear simultaneous equations can be roughly sorted into two categories: direct methods and iterative methods. Cramer’s rule and Gaussian elimination method are the typical examples of direct methods. However, they are not efficient. If the algebraic equations have certain characteristics (such as coefficient matrix is positive definite or symmetrical), some more efficient methods can be applied, for example, LU decomposition solution, Crout decomposition solution, LDLT decomposition solution, or LLT decomposition solution. Jacobi iterative method, Gauss–Seridel iterative method and overrelaxation or under-relaxation method are common examples of iterative methods. Workload or the number of calculation step can be known in advance in direct methods, which means the workload for certain problem is definite. For iterative

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methods, the workload is uncertain; even it is unknown of whether the solving process can converge. But such methods usually take less storage, thus making it possible to solve a big problem with relatively small capability of computing power. For a given system of algebraic equations, which method works better depends on the scale and property of the system. Generally speaking, if the number of equations is abundant enough, iterative methods would take less time than direct methods. If the system of equations is linear, direct methods would be better. But if it is nonlinear, only iterative methods are applicable, because every intermediate result between the iterative steps does not need much precision. Mostly the system of discrete algebraic equations in the problem of filling and packing simulation of injection molding is nonlinear, and the fine mesh used to obtain more details leads to a very large computing scale, so it is efficient to adopt iterative methods. After having used numerical methods such as FVM or FEM to discretize governing equations of unknown variables, we usually obtain the form of the system of algebraic equations as shown in the below equation: Aϕ = b

in SIMPLE algorithm, the convergence acceleration of the inner iterative process eventually appears as the convergence acceleration of the nonlinear problem. So we usually use the decreased speed of the unbalanced margin to reflect the features of different solving methods. The other characteristic is that when discussing the acceleration methods of iterative convergence of the algebraic equation, we usually take pressure equation (or pressure-correction equation) as an example, and it is usually discussed with specific algorithms such as SIMPLE and SIMPLER. In this chapter, we firstly review some of the direct methods to solve algebraic equations, namely Gauss elimination method, TDMA (tri-diagonal matrix algorithm), LU decomposition, and PDMA (pentadiagonal matrix algorithm). They would be used in iterative methods for solving the multidimensional heat transfer and flow problem. After that, some of the iterative methods, especially iterative matrix methods of increasing attractiveness are introduced, including strong implicit procedure (SIP), Modified SIP (MSIP), conjugate gradient (CG), and some CG-like iterative methods for solving the system of nonsymmetrical algebraic equations. The iterative convergence acceleration methods mentioned in this chapter generally represent the methods used in inner iteration.

(4.198)

where A is the coefficient matrix, ϕ is the column vector of unknown variable, b is the right-hand column vector. No matter which form is to be chosen, the coefficient matrix A would be a large-scaled sparse matrix, because the nodes in solution domain usually appears to be farther than those adjacent nodes introduced by the algebraic equations. Except heat conduction problems, the coefficient matrix is usually asymmetric formed by convective heat transfer problems. Namely, element values of the coefficient matrix A themselves depend on the unknown variable ϕ. So, although Equation 4.198 itself is a system of linear algebraic equations, elements of the coefficient matrix A remain improving until a convergent solution has been worked out. In addition, for pressure-correction algorithm used to solve incompressible flow problems, calculation practice shows that it takes a far longer time to solve the pressure equation or pressure-correction equation than to solve a momentum equation or energy equation. These above-mentioned methods represent two characteristics of the research on algebraic equation solution in CFD or NHT field. Namely, iterative methods are always adopted to solve the algebraic equation. It should be noticed that, for a nonlinear physics problem, the solving process includes inner iteration (solve the linearized algebraic equations) and outer iteration (renew the coefficients of the algebraic equations). In this chapter, we mainly discuss how to accelerate the convergence of the inner iterative process. But when discussing the pressure-correction equation

4.3.2

Direct Methods

4.3.2.1 Gauss Elimination Method Gauss elimination method is the most basic direct method. Other important methods are all inspired by it. In the Gauss elimination method the upper triangular matrix of the coefficient matrix is formed by elimination. Consider the general form of n-order system of algebraic equations (Eq. 4.198). n-order system of linear algebraic equations needs n − 1 eliminations. When a cyclic reduction algorithm is adopted, the mth elimination uses the mth row of the matrix after the m − 1th elimination as the princi(m−1) pal component row, and Amm is the principal component; eliminate the i th row (i > m) using the expression shown below: (m−1) A(m) − ij = Aij

bi(m) = bi(m−1) −

A(m−1) im

A(m−1) mj

(4.199)

(m−1) bAm

(4.200)

A(m−1) mm

A(m−1) im A(m−1) mm

the superscript (m) denotes that the value of this element is the one after m eliminations. The mth elimination is based on m − 1th elimination. After n − 1 eliminations, we can obtain an upper triangular matrix. Now, the value of the last element can

METHODS FOR SOLVING ALGEBRAIC EQUATIONS

be obtained by

then Equation 4.205 becomes ϕn =

bn(n−1)

(4.201)

A(n−1) nn

ϕ3 =

and the rest can be obtained by back substitution from n − 1th row according to the following equation: ϕi =

bi(n−1)

−

n

(4.202)

where the first row and the last row, namely ϕ 1 and ϕ n + 1 , are known as boundary conditions. The general format of any variable in the system of equations is (j = 2, 3, . . . , n)

(4.204) The system of equations that arise from Equation 4.204 can be solved by forward elimination and back substitution. The process of forward elimination begin from eliminating ϕ 2 from ϕ 3 :

ϕ3 =

α3 ϕ4 + D3 − β3 Dα22

Let a2 =

β3



β2 D2 ϕ1

D3 −



b2 D2 β3 Dα22

+

(4.207)

β3 b′2 + b3 α3 , b′3 = D3 − β3 a2 D3 − β3 a2

(4.208)

then Equation 4.207 becomes

4.3.2.2 TDMA Algorithm TDMA, also known as Thomas algorithm, was proposed by Thomas in 1949, and is an effective method for tri-diagonal equations (there are only three nonzero elements in each equation). Its essence is a special case of Gauss elimination method. It can be used to solve high dimensional problem similar to iterative steps. Tri-diagonal equations are often referred to as discrete equations of one-dimensional FVM. Let the coefficient matrix be ⎡ ⎤ 1 0 0 0 ··· 0 ⎢−β2 D2 −α2 0 ··· 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ −β D −α · · · 0 3 3 3 ⎢ ⎥ (4.203) A=⎢ . . . . . . .. .. .. .. .. ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎣ 0 ··· · · · −βn Dn −αn ⎦ 0 ··· ··· ··· ··· 1

βj bj αj ϕj +1 + ϕj −1 + Dj Dj Dj

β3 b′2 + b3 α3 ϕ4 + D3 − β3 a2 D3 − β3 a2

And let a3 =

(n−1) ϕk k=i+1 Aik (n−1) Aii

The computation times of Gauss elimination method are proportional to n 3 /3, mainly taken by elimination (the times of back substitution are n 2 /2). In NHT, the Gauss elimination method is rarely implemented, because it is time-consuming and not easy to be parallelized, but it is a fundamentally essential method for one-dimensional TDMA.

ϕj =

115

+ b3

β2 b2 α2 , b′2 = ϕ1 + D2 D2 D2

(4.205)

(4.206)

ϕ3 = a3 ϕ4 + b′3

(4.209)

Eliminate as shown above until ϕ n − 1 has been eliminated from ϕ n . The formula for back substitution is ϕj = aj ϕj +1 + b′j

(j = 2, 3, . . . , n)

(4.210)

where aj =

αj , Dj − βj aj −1

b′j =

βj b′j −1 + bj Dj − βj aj −1

(4.211)

According to boundary conditions, we have a1 = 0, b′1 = ϕ1 , an+1 = 0, b′n+1 = ϕn+1

(4.212)

The elimination process begins from ϕ n , until ϕ 2 has been worked out. For discrete equation results from FVM in twodimensional or three-dimensional problems, we can assume that all the other node values are known at first, then solve the equations in one selected dimension, and successively scan in another dimension with the same method, until all the node values are known. After that, such an iteration method is repeated until the differences between two adjacent iteration results in all nodes are small enough. The astringency of the calculation process is related to the propagation speed of the boundary conditions in the solution domain. So alternating direction scanning can be adopted to accelerate the speed of iteration. Different prior calculating directions in different planes are also available. 4.3.2.3 PDMA Algorithm If some high order schemes such as QUICK are adopted, discrete equation results from FVM are related not only to neighbor nodes but also to farther neighbor nodes. Take two-dimensional problems for example, if deferred correction method is not used, the coefficient matrix is a nondiagonal matrix. This method makes it possible to use TDMA by losing the convergence speed. PDMA can increase the quantity of direct methods in the process of solving algebraic equations; as a result, the convergence speed is increased. In addition, PDMA

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

would be adopted when coupled equation line solver (CELS) algorithm is implemented. In the following part, the implementing process of PDMA in two-dimensional problem is introduced. Like TDMA algorithm, let x - direction be the first solving direction, then node values in y-direction are assumed to be known; substitute them in the right-hand term b, then we have the formula: Ai ϕi =Ai+2 ϕi+2 + Ai+1 ϕi+1 + Ai−1 ϕi−1 + Ai−2 ϕi−2 + bi (4.213) The above formula omits the subscript j in y-direction. PDMA also has two steps: elimination and back substitution. After two elimination steps, we can get upper triangular equations ϕi = ai ϕi+2 + b′i ϕi+1 + ci

(i = 1, 2, . . . , n)

(4.214)

where the recursive equations of the coefficients ai , b′i , and ci are ai =

Ai+2 , Ai − Ai−2 ai−2 − b′i−1 d

b′i =

Ai+1 + ai−1 d , Ai − Ai−2 ai−2 − b′i−1 d

ci =

bi + Ai−2 ci−2 + ci−1 d Ai − Ai−2 ai−2 − b′i−1 d

(4.215)

(4.216)

When i = 1, we have Ai+2 Ai



, b′1 =

1



Ai+1 Ai



1

, c1 =



bi Ai



(4.217) 1

Likewise, we have a2 =

(Ai+1 )2 + (Ai−1 )2 a1 (Ai+2 )2 ′ , ′ , b1 = − b (Ai )2 (Ai−1 )2 1 (Ai )2 − (Ai−1 )2 b′1

c2 =

b2 + (Ai−1 )2 c1 (Ai )2 − (Ai−1 )2 b′1

(4.218)

After Ai , Bi , and Ci (i = 1, 2, . . . , n) are worked out, we can solve ϕ n and ϕ n − 1 : ϕn = cn , ϕn−1 = b′n−1 ϕn + cn−1

(4.219)

Then we can back substitute one by one according to Equation 4.214 until ϕ 1 is worked out.

LY = b

(4.221)

Uϕ = Y

(4.222)

Here, the relations among the elements in L and U and elements in A are ⎞ ⎛ j −1  lik ukj ⎠ i > j i = 1, 2, . . . , n ) lij = ⎝Aij − k=1

i−1 k=1

d = Ai−1 + Ai−2 b′i−2



After U and L worked out, the solving process can be divided into the following two steps:

uij = Aij −

in which

a1 =

4.3.2.4 LU Decomposition Algorithm LU decomposition algorithm (also called LU factorization algorithm) is a variant of Gauss elimination method. It writes a matrix A as the product of an identity upper triangular matrix U and a lower triangular matrix. LU decomposition is a decomposition of the form A = LU (4.220)

 lik ukj i ≤ j

(4.223)

j = 2, 3, . . . , n ) (4.224)

Once the lower and upper matrices L and U of matrix A are worked out, then Y would be solved according to Equation 4.221. This calculation process is similar to back substitution in TDMA. But here the substitution process is implemented from top to bottom, so it is called forward substitution. When Y is known, introduce it into Equation 4.222 then we can work ϕ out. For a two-dimensional problem, LU decomposition algorithm is equivalent to decomposing the problem into two onedimensional problems to solve. 4.3.3

Iterative Methods

In a sense, the direct methods can be seen as a limiting case of iterative methods: using such methods, only one iteration of calculation is needed to obtain the precise solution of the algebraic equations, but the computing cost is extremely high. In iterative methods, however, in contrast to direct methods, the computing process consists of many iterations and the computing cost in each iteration is declined substantially. With a proper iterative method, the computing would be more efficient, which is just the situation in CFD and NHT involved in filling and packing simulation1 . Especially, when there are many nodes to calculate, even an iterative method of slow convergence may be more effective than an elimination method76 . There are two major issues in researches on iterative methods. One is convergence, namely, whether the iteration form we construct can lead to the solution of algebraic

METHODS FOR SOLVING ALGEBRAIC EQUATIONS

equations; the other is, if the method is convergent, how to accelerate the convergence speed, as a result of which many acceleration methods appear. Algebraic equations (Eq. 4.198) can be written in the following form when iterative methods are adopted M ϕ (n+1) = N ϕ (n) + B

(4.225)

The formation of matrix M ,N , and column vector B depends on the iterative scheme that is used. When the iteration is convergent, we have ϕ (n+1) = ϕ (n) , so the relation between M , N , B and A, b in Equation 4.198 is A = M − N, B = b (4.226) Equation 4.225 can also be written in the following form:   ϕ (n+1) = M −1 N ϕ (n) + M −1 B = H ϕ (n) + c

(4.227)

where H is called the iterative matrix . Let error vector be εn , then we have ϕ =ϕ (n) +ε (n) M ε (n+1) = N ε (n)

or

ε (n+1) = H ε (n)

(4.228) (4.229)

Iteration convergence means that this condition holds: lim ε (n) = 0. Therefore, the necessary and sufficient n→∞ condition of iteration convergence is that the eigenvector of iterative matrix H must be less than 1, namely, the spectral radius of the iterative matrix must be less than 177 . The sufficient condition of convergence of Jacobi iteration and Gauss–Seidel iteration is that the coefficient matrix is strictly diagonally dominant or irreducible diagonal dominant. For SOR/SUR (successive overrelaxation/successive under-relaxation) iteration, under the condition of irreducible diagonal dominant, the iteration process must converge if the relaxation factor is between 0 and 2. For convection–diffusion equations, the characteristics of the discrete equations’ coefficient matrix are relevant to the adopted differential scheme of the convection term. Using the upwind differential scheme can ensure the formation of a diagonally dominant coefficient matrix. But it is noteworthy that the iterative convergence of the arithmetic equations does not mean that the solution is reasonable physically. Iterative methods for solving systems of algebraic equations can be organized into three categories: point iterative method, block iterative method, and alternativedirection iterative (ADI). Each method can be classified into Jacobi method (using solutions of the last iteration as adjacent node values) and Gauss–Seidel method (using new values in the present iteration as adjacent node values). The convergence speed of Jacobi iterative method is the slowest.

117

Taking this as a standard, we can classify the convergence acceleration methods as shown Figure 4.46. The major method in accelerating the iterative convergence is to introduce the influence of the boundary conditions into the computed domain as soon as possible, while the Jacobi point iteration method makes the influence of the four boundary conditions to be introduced only by one cell after an iteration. When the G–S point iteration method is used, the influence of one end is introduced into the calculation domain, but the influence of the other end is still introduced into only one cell. In the ADI linear iteration method, the boundary conditions of the two ends influence all inner nodes along this line, and the influence of all four boundaries can be introduced into the calculation domain after one iteration by alternating direction method. So its speed of convergence is the fastest. This proves that, in iterative calculation, increasing the speed of iterative convergence by increasing the quantity of implicit direct solving is helpful. 4.3.3.1 Fundamental Iterative Methods A system of algebraic equations like Equation 4.225 will be solved in each iteration whose coefficient matrix is M , so we naturally hope that M has some characteristics, such as diagonal, upper triangular, diagonal block, or upper triangular block matrix, which make the system to be handled easily. On basis of this idea, split the matrix A in different ways according to Equation 4.226, then we can construct many consistent iterative methods. Suppose A is divided into blocks as

An×n

⎡

A11 ⎢A21 ⎢ =⎢ . ⎣ ..

Ak1

A12 A22 .. .

··· ··· .. .

Ak2

···

⎤ A1k A2k ⎥ ⎥ .. ⎥ . ⎦

(4.230)

Akk

and let D = diag (A11 , A22 , . . . , Akk ) ⎡

0 ⎢A21 ⎢ CL = −⎢ . ⎣ .. CU

··· ··· .. .

0 0 .. .

Ak1 · · · ⎡ 0 A12 ⎢ ⎢0 0 = −⎢ ⎢ .. .. ⎣. . 0 0 L = D −1 C L ,

Ak,k−1

(4.231)

⎤ 0 0⎥ ⎥ .. ⎥ .⎦

0

⎤ A1k .. ⎥ . ⎥ ⎥ ⎥ Ak−1,k ⎦ 0

(4.232)

U = D −1 C U

(4.233)

··· ··· .. . ···

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

FIGURE 4.46 Classification of convergence acceleration methods for solving systems of algebraic equations.

Then we have

The iterative scheme is

A = D − C L − C U = D (I − L − U )

Four fundamental iterative methods are introduced in the following text. 4.3.3.1.1 Jacobi Iterative Method Jacobi iterative method is also known as simple iterative method . This method splits A into A = MJ − NJ

Dϕ (n+1) = (C L + C U ) ϕ (n) + B

(4.234)

(4.235)

(4.238)

4.3.3.1.2 Gauss–Seidel Iteration Method Gauss–Seidel iteration method splits A into A = MG − NG

(4.239)

Here, M G = D − C L,

NG = CU

(4.240)

The iterative matrix is

Here, M J = D,

NJ = CL + CU

(4.236)

H G = (D − C L )−1 C U = (I − L)−1 U

(4.241)

The iterative matrix is −1 H J = M −1 (C L + C U ) = L + U = I − D −1 A J NJ = D (4.237)

The iterative scheme is (D − C L ) ϕ (n+1) = C U ϕ (n) + B

(4.242)

METHODS FOR SOLVING ALGEBRAIC EQUATIONS

4.3.3.1.3 Relaxation Iteration Method (SOR Iteration Method) SOR iteration method splits A into A = Mω − Nω

(4.243)

Here, Mω =

1 D − C L, ω

NG =

1−ω D + CU ω

(4.244)

ω is a nonzero real number called relaxation factor. We introduce it to accelerate the convergence process. The iterative matrix is −1 [ωU + (1 − ω) I ] (4.245) H ω = M −1 ω N ω = (I − ωL)

The iterative scheme is (D − ωC L ) ϕ (n+1) = [ωC U + (1 − ω) D] ϕ (n) + ωB (4.246) If the value range of relaxation factor is 0 < ω < 1, the method is called the under-relaxation iteration method . If ω = 1, SOR iteration method becomes the Gauss–Seidel iteration method. And if 1 < ω < 2, it is called SOR. When the value of ω is appropriate, the convergence speed of the SOR iteration method is faster than that of the Gauss–Seidel iteration method. If the value of ω is between 0 and 2, the convergence can be ensured.

Compared with SOR, SSOR can make full use of information when data are exchanged between memory and external storage. This will reduce exchange times and increase the computational efficiency, and the choice of ω is not as sensitive as in the case of SOR. For some special problems, because the SOR method is not convergent, the SSOR iteration method that is convergent can be constructed. 4.3.3.2 Conjugate Gradient Method Conjugate gradient method is a type of iterative implicit method for solving the algebraic equations proposed by mathematicians in the early 1950s78 . But owing to the limitation of the computer memory then, it was not implemented in practice in the solving of CFD and NHT problems unlike ADI linear iteration method. With the development of CFD and NHT in grid generation, pressure and speed coupling method, and differential scheme of convection term, the number of nodes was increasing. Meanwhile, as the computer industry also developed rapidly, beginning from the mid-1980s, methods that use direct method in every iteration (like CG method) are gradually being widely used. When the coefficient matrix in Equation 4.198 is a symmetric positive definite matrix, the solving of the algebraic equation (Eq. 4.198) equals to solving the minimum value of the following quadratic functional: F=

4.3.3.1.4 Symmetric Successive Overrelaxation (SSOR) In the SSOR (symmetric successive overrelaxation) iteration method, C L and C U are treated equally based on SOR, and SOR is implemented twice. SSOR splits A into A = MS − NS

(4.247)

Here, 1 (D − ωC L ) D −1 (D − ωC U ) ω (2 − ω) 1 [(1 − ω) D + ωC L ] D −1 NS = ω (2 − ω) [(1 − ω) D + ωC U ] (4.248)

MS =

The iterative matrix is HS =

M −1 S NS

= (I − ωU )

1 T ϕ Aϕ − ϕ T b 2

[ωL + (1 − ω) I ] H ω (4.249)

The iterative scheme is (D − ωC L ) ϕ (n+1/2) = [ωC U + (1 − ω) D] ϕ (n) + ωB (4.250) (D − ωC U ) ϕ (n+1) = [ωC L + (1 − ω) D] ϕ (n+1/2) + ωB (4.251)

(4.252)

The simplest method for getting the minimum value of a function is the steepest descent method. The steepest descent direction of function F at point ϕ (k) is its negative  (k) . Meanwhile, ∇F is the gradient direction −∇F ϕ   margin of Aϕ (k) − b . If the margin is zero, ϕ is the (k) as very solution of the equation. Thus, we can see ϕ  (k) an approximate solution and seek ϕ making F ϕ reach the minimal value along the negative gradient direction of F beginning from ϕ (k) . However, this approach is very impractical because its convergence speed is very slow. As an improved approach, the search for the minimal value does not always follow one direction, but is along several directions and one direction each time. For this, the relation among these directions must hold as follows: p 1 • Ap 2 = 0

−1

119

(4.253)

where p 1 and p 2 are two different search direction. If Equation 4.253 holds, the vectors p 1 and p 2 are conjugate. CG method developed on the basis of the steepest descent method. It searches along different direction, but it is required that every direction must conjugate to the former direction. This is why this method is called conjugate gradient.

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

The CG method algorithm for solving systems of algebraic equations can be proposed as follows: 1. Input A, b, and initial field ϕ (0) , calculate that r (0) = b − Aϕ (0) ,

ρ (0) = r (0)T • r (0) ,

p (0) = r (0)

2. Iteration loop (k = 0, 1, 2, . . . ), until the margin r (k) is less than tolerance w (k) = Ap (k) α (k) = ρ (k) /p (k)T w (k) ϕ (k+1) = ϕ (k) + α (k) p (k) r (k+1) = r (k) − α (k) p (k) β (k) = ρ (k+1) /ρ (k) p (k+1) = r (k+1) + β (k) p (k) If ρ (k ) < ερ (0) , the output is ϕ, or continue the iteration loop. As a practical iteration method, the CG iteration method can make full use of sparsity of matrix A, and does not need to predict some parameters like what the relaxation method does. Furthermore, computation in each iteration is mainly vector computation, which is easy to be parallelized. The convergence speed of the CG method depends on the condition number of matrix A, namely, the ratio of the maximum of matrix eigenvalue to the minimum. The bigger the condition number is, the slower the convergence speed is. For the discrete algebraic equations of convective heat transfer problems, the condition number is roughly proportional to the square of the number of nodes on a certain direction. As a result, the eigenvalue of the coefficient matrix are evenly distributed in a very long interval, which leads to a very slow convergence speed. In order to make the CG method applicable to solving large-scale systems of algebraic equations formed after discretization of elliptic PDEs, preconditioned conjugate gradient method79 was proposed based on the CG method. This method assumes a nonsingular matrix C , which makes the distribution of eigenvalue of A = C −1 AC −T concentrated. Then, the CG method can be used in the following equation and the convergence speed would be fast: Ax =b (4.254)

Here, x = C T x and b = C −1 b. Let M = C C T , which is called preconditioned matrix . In every preconditioned CG (PCG) iteration, a system similar to M z = r would be solved. So, an excellent preconditioned matrix M should have some characteristics like symmetric positive definite, making the distribution of the eigenvalue of M −1 A (namely A) concentrated.

There are many ways to decide the preconditioned matrix M , so different PCG implementation methods can be created. Incomplete Cholesky decomposition conjugate method79 is the commonly used method. The main aim of creating preconditioned matrix in this method is Let M = LDLT

(4.255)

where L is a lower triangular matrix, whose distribution of nonzero diagonal elements is the same as A. LT is the corresponding transposed matrix and is a upper triangular matrix. D is a matrix whose main diagonal elements are all nonzero values and the rest of the elements all have zero value. The CG method mentioned above is only propitious to the systems whose coefficient matrix is symmetric. When discretize diffusion equations with nonuniform mesh or convection–diffusion equations with mesh of any type, the coefficient matrix of the system is not symmetric. So, the CG method for solving the wide-area symmetric positive definite linear systems should be extended for general linear systems. These methods all have two kind of approaches of convergence acceleration: (i) introduce a preconditioned matrix M to overcome the shortcoming of slow convergence speed caused by big condition number of the coefficient matrix A and (ii) introduce methods for accelerating the convergence speed of the iteration process of the nonsymmetric matrix. Peters80 and Deng et al.81 classify CG-like schemes for nonsymmetric coefficient matrix into four categories: (i) Generalized minimal residual algorithm (GMRES). This method needs much memory, and calculation workload is also large. So Peters proposed that it should be recalculated after k iterations, which is denoted as GMRES(k ). (ii) Lanczos-type scheme, which simplifies the matrix A to a tri-diagonal matrix by matrix transformation to solve eigenvalue problems76 . In the process AT , the transposed matrix of A, would be required. This scheme includes Bi CG, QMR, etc. (iii) Transpose-free Lanczos-type scheme, which avoids calculating AT , which costs lots of time. It includes CGS, Bi -CGSTAB, TFQMR, etc. (iv) Normal equation CG method, which applies CG method to normal equations. This method includes CGNR, CGNE, etc. As a method for solving large-scale sparse nonpositive linear systems, CGN still has some advantages of symmetric and positive systems, such as sparsity of A can be made full use of and it is easy to be parallelized. However, as the condition number of AT A is the square of the condition number of A, when A is morbid, this method may make the convergence speed slow. An important approach to increase the convergence speed is to extend the precondition method in symmetric situation to nonpositive situation.

METHODS FOR SOLVING ALGEBRAIC EQUATIONS

4.3.4

Parallel Computing

With the development of modern injection products, the complexity of products and requirement for performance are increasing. This leads to an increasing scale of injection molding process simulation. The time spent on solution of large-scale nonlinear equations accounts for 80% of the total simulation time. Such a large-scale nonlinear computation places extreme demands on computing power. For many numerical problems, only parallel solutions are possible if the problem size is reasonably large. Using parallel computing to solve large-scale and complex systemic problem is an important and popular research field of scientific and engineering computing. Parallel computing is the concurrent use of parallel computers in which there are multiple processors (cores or computers) with shared memory systems to do computational work. Some time-independent operations can be broken up into multiple subtasks and can be assigned to different processors to be executed simultaneously. Separate processors can execute different portions without interfering with each other, thereby decreasing the calculation time or enhancing the problem scale. Besides parallel computers, parallel computing usually requires parallel programming. The programmer has to figure out how to break the problem into pieces, and has to figure out how the pieces relate to each other. 4.3.4.1 Parallel Computer System Staring from the forties, the development course of modern computer can be divided into two ages clearly: serial computing age and parallel computing age. Each age begins with the architecture development and then system software and applications develop, and finally it reaches a peak with the development of solving environment. The main reason for using parallel computing is that it is one of the best ways against the computing speed bottleneck of single processor. Parallel computer is composed of a set of processor units that can complete a massive task together at a faster speed by intercommunicating and cooperating. Thus, computational nodes and mechanisms of internode communication and cooperation are the two key components. The development of parallel computer architecture is embodied in improving the performance of computing nodes and internode communication technology. The computer architecture can be classified into four categories based on the number of concurrent instruction (or control) and data streams available in the architecture. It is popularly known as Flynn’s taxonomy proposed by Flynn82,83 in 1966. • Single Instruction, Single Data (SISD) stream • Single Instruction, Multiple Data (SIMD) streams • Multiple Instruction, Single Data (MISD) stream

121

• Multiple Instruction, Multiple Data (MIMD) streams. Relatively, SIMD and MISD architectures are more suitable for special computing. Among commercial parallel computers, MIMD is the most widely used architecture, SIMD ranks second, and the least is MISD. As of 2006, all of the top 10 and most of the top 500 supercomputers are based on an MIMD architecture. At present, there are five main physical parallel computer structural models: • • • • •

Parallel vector processor (PVP) Symmetric multiprocessor (SMP) Massively parallel processor (MPP) Cluster of workstations (COW) Distributed shared memory (DSM).

They can be classified into two categories: shared memory computer, including PVP, SMP, and DSM, and distributed memory computer, including MPP and COW. 4.3.4.2 Parallel Algorithm Parallel algorithm is suitable for being implemented on parallel computers. A good parallel algorithm should be able to fully utilize the computing power of parallel computers. Some indicators used for program performance evaluation are introduced first here. • Grain size: the measurement of the task size that each processor can execute independently in parallel. Large grain size reflects that the processor is able to execute a large amount of instructions in parallel, which is also known as coarse granularity. Instruction-level parallelism (ILP) is a small grain size in parallel, also known as fine granularity. • Execution time (wall time): the time measured by timer such as a clock from the parallel program begin to run, until all processes are finished. Execute time contains CPU calculation time, CPU communication time, synchronization time, and process idle time owing to synchronization. • CPU calculation time: the time spent in executing the instruction by processor. It can be divided into the time spent in executing program (user time) and the time spent by the operating system (system time). • Speedup ratio: If the serial execution time is T s , the parallel execution time with q processors is T p (q), then the speedup ratio is Sp (q) =

Ts Tp (q)

(4.256)

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

Efficiency is defined by the following equation: Ep (q) =

Sp (q) q

(4.257)

• Performance: If the computation of a problem solution is W , execution time is T , then the performance is measured by the following equation: Perf =

W T

(4.258)

In the 1980s, FLOP/s was used as the unit. In the 1990s, MFLOP/s and GFLOP/s were used. At present, GFLOP/s and TFLOP/s are widely used, and PFLOP/s is also increasingly used. • Amdahl’s law: considering a given calculation problem, if the percentage of the serial part is α, then the parallel speedup radio is Sp (q) =

1 α + (1 − α) /q

(4.259)

Amdahl’s law indicates that when q increases, S p (q) also increases as a result. Meanwhile, it has an upper bound. That is, no matter how many processors are used, the speedup multiple cannot be greater than 1/α. If in each processor the percentage of the serial part is β, then the parallel speedup radio with q processor is Sp (q) = β + q • (1 − β)

(4.260)

Parallel algorithm is the foundation of parallel computing. It provides solution for efficient use of parallel computers by combining with implementation technology. The fundamental principles are summarized as follows: • Combine with the architecture. • Be scalable. It is an important criterion, which evaluates the efficiency of a parallel algorithm whether the algorithm is capable of speeding up linearly or nearly linearly with the increase of the number of processor. That is, if the speedup radio of a parallel algorithm is S p (q) = O(q) or S p (q) = O(q/(1 + log(q))), we can say it is scalable. • Be coarse-grained: Normally, the size of the grain is better and bigger. It is because each processor has many tasks to calculate that coarse-grained processor can fully utilize the function of a multiprocessor. Generally speaking, the parallel speedup ratio is not very high in the fine-grained problem. This is why parallel computing is used to solve large-scale problems.

• Reduce the communication traffic: Communication is very important in an efficient parallel algorithm. The key point to improving performance is to reduce the communication traffic and frequency, in which the frequency usually is the determinant. • Optimize performance: Effectiveness of the algorithm depends not only on the results of theoretical analysis but also on the technologies used during the implementation process. Performance mainly depends on the percentage of computing power utilized by a single processor, followed by the parallel efficiency. There may be many factors that affect the efficiency of the parallel algorithm. The above given are just the main factors. Therefore, in the algorithm design process, if the above five fundamental principles are considered carefully, a very good result can be achieved. 4.3.4.2.1 The Basic Iterative Algorithm Jacobi, Gauss–Seidel, and SOR are basic iterative algorithms for solving linear equations. The parallel computer makes the difference in performance of these algorithms a focus of attention. Jacobi iteration has evidently internal parallelizability because of the independence of each component’s modification. Its main advantage is simplicity, but the convergence speed is low. On basis of the research on improving the convergence speed, Missirlis proposed parallel Jacobi iteration in 1983, and discussed the convergence. As for Gauss–Seidel iteration, it has no parallelizability because it makes full use of the last value to improve the convergence speed. In terms of SOR iteration, the internal parallelizability is inferior to Jacobi iteration, because the calculation of each component is dependent. So generally, it is considered to be not suitable for parallel computing, because SOR iteration is most frequently used to solve large-scale FDM or FEM sparse equations, it is an effective approach to implement parallel SOR iteration by using red-black ordering or multicolor ordering, considering the special distribution of zero or nonzero element in the coefficient matrix. If there are p processors, and 2p can be divisible by n, let t = n/2p and divide A into 2p × 2p blocks, then the equations will have the following form: ⎤⎡ ⎤ ⎡ ⎤ ⎡ b1 x1 A11 · · · A1,2p ⎢ ⎥ ⎢ ⎥ ⎢ .. . . . .. .. ⎦ ⎣ .. ⎦ = ⎣ ... ⎥ (4.261) ⎦ ⎣ . A2p,1

···

A2p,2p

x 2p

b 2p

where Aij (i, j = 1, . . . , 2p) are all matrices of order t × t, x i , and b i are both t-dimensional column vectors, and we have the following matrix splitting: Aii = D i − Li − U i

(4.262)

METHODS FOR SOLVING ALGEBRAIC EQUATIONS

where D i , −Li , and −U i denote the diagonal matrix, the lower, and upper triangular matrix of Aii respectively. Split the matrix A as A=M −N

(4.263)

with ⎡

(i = j ) , (i, j = 1, . . . , p)

(4.264)

According to the aforementioned splitting format, we can have the iteration formula x (k+1) = M −1 N x (k) + M −1 b

α (k) = r (k−1)T r (k−1) /p (k−1)T p (k−1)   r (k) = r (k−1) − α (k) M −1 A p (k)

β (k+1) = r (k)T r (k) /r (k−1)T r (k−1)  T p (k+1) = M −1 A r (k) + β (k+1) p (k)

In this calculation, the key point is the parallel T computation of M −1 Ap (k) and M −1 A r (k) . Before calculating M −1 Ap (k) , Ap (k) should be worked out first, which is a simple product of matrix and vector. Then solve the system M y = Ap (k) (4.267)

⎤

M 11 · · · M 1p ⎢ .. .. ⎥ .. M =⎣ . . . ⎦ M p1 · · · M pp 

O D 2i−1 − L2i−1 M ii = D 2i − L2i A2i,2i−1 

O O M ij = A2i,2j −1 O

123

(4.265)

Obviously, the necessary condition for the above equation is that D i should be a nonsingular matrix. When p = 1, this algorithm becomes the Gauss–Seidel algorithm. Introducing the relaxation factor to ω improve the convergence speed, and the iteration formula becomes   x (k+1) = ω M −1 N x (k) + M −1 b + (1 − ω) x (k) (4.266)

4.3.4.2.2 Parallel Modification of CG Method As a practical iteration method, CG method can fully utilize the sparsity of the matrix, and some parameters need not be estimated. In addition, each iterative computation is mainly a vector operation, which is convenient for parallelization. By reasonable selection of preconditioned matrix, traditional PCG method can be fit for parallel computing. Assuming A to be unsymmetric, the preconditioned matrix uses the format of Equation 4.273. For implementing parallel algorithm, distributed storage should be implemented first. If there are p processors for n order matrix A, and n = pm, then a (i−1)m+j,k , x (0) (i−1)m+j , and b (i−1)m+j are stored at the i th processor pi , where j = 1, 2, . . . , m and k = 1, 2, . . . , n. Parallel PCG algorithm is as follows: 1. Input the initial iterative value x (0) , calculate r (0) = b − Ax (0) and p (1) = AT r (0) . 2. Iterative loop (k = 0, 1, 2, . . . ),until the margin r k is less than tolerance

Considering the structure of M , its parallel computation in factis the parallel iterative solving mentioned before. As T for M −1 A r (k) , we have  T  −1 (k)  −1 T (k) r M A r = AT M −1 r (k) = AT M T (4.268)  −1 (k) Solving M T r is equal to solving the system M T z = r (k)

(4.269)

Each processor stores blocks of the coefficient matrix by column, namely, stores M ij and N ij (i , j = 1, 2, . . . , p) in the i th processor Pi . 4.3.4.2.3 Parallel Multisplitting Iterative Method A good parallel algorithm requires computation independence among processors, namely, good parallelism, for reducing the communication and the synchronization spending as much as possible. On the basis of this, O’Leary and White proposed the parallel multisplitting iterative method in 1985, which has been considered as an important parallel numerical method. The purpose is to solve the large-scale system of linear equations Ax = b (4.270) Here, A is an m × n nonsingular matrix. Let M l , N l , and E l all are n × n matrix, and l = 1, 2, . . . , α, if the following conditions hold: • A = M l − N l , and M −1 exists; l α • l=1 E l = I , I is an n × n identity matrix and E l is nonnegative diagonal matrix. Then the triple (M l , N l , E l ) l = 1, 2, . . . , α is called multisplitting of A. The iterative scheme of the multisplitting method is x (k+1) = H x (k) + Gb

k = 0, 1, 2, . . .

(4.271)

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NUMERICAL IMPLEMENTATION FOR THE FILLING AND PACKING SIMULATION

Case 2 (113978 elements) 8400

Case 1 (7586 elements) 33 33

5300

Number of elements

Case 3 (560716 elements)

560,716

74,100

42,300

Speedup ratio 1.75 1.58 1.43

FIGURE 4.47

Performance comparison of filling analysis.

2. Solve y l (l = 1, 2, . . . , α)

with H =

α l=1

E l M −1 l N l, G =

α

E l M −1 l

M l y l = N l x (0) + b

(4.272)

l=1

H is called the iterative matrix of the multisplitting iterative method. On the basis of the concepts defined above, the iterative algorithm of the multisplitting method is as follows: 1. Select the value of x (0) arbitrarily, and repeat the following two steps until the convergence is achieved.

3. Calculate x (k+1) =

α

E lyl

(4.273)

(4.274)

l=1

The multisplitting iterative method naturally has parallelism. Each (Eq. 4.273) of y l is independent, so if there is one parallel computer with one host computer and α processors, then the above algorithm can be executed as follows:

REFERENCES

each processor executes a local iteration (Eq. 4.273), and in each step the global iterative value x (k+1) , namely, Equation (4.274), from the weighted average of local iterative value y l is obtained, then x (k+1) is sent back to every processor for the next iteration. If a diagonal element in E l is zero, the corresponding component of y l need not be calculated so as to save the workload. This shows that E l has the effect of deploying the workload of each processor. E l should be chosen to let the workload in the processors to be balanced to reduce time spent on waiting for synchronization. 4.3.4.3 Case Studies In practice, the computing speed for solving the preconditioned matrix and a sparse matrix in CG method is fast, respectively, so the effect on acceleration of parallel algorithm is not very significant84 . Yang et al.85 applied domain decomposition method (DDM) to guarantee maximum degree of parallelization. DDM divides the whole model, including part model, runner system, cooling channels, and mold base, into a few smaller subdomains of equal or similar size, then distributes each subdomain to each CPU for calculation. During computation, each CPU will only calculate the subdomain assigned to it. At the same time, necessary data exchange across interfaces is required to transmit the results of one subdomain to its surrounding subdomains. Finally, the results from subdomains are collected into the whole results. They performed a filling simulation with simple part geometry to verify their parallel method, which was analyzed on single, two, and four processors. The testing platform is an SMP system with Dual Intel Xeon 2.4 GHz CPU and Microsoft Windows XP Professional OS. It is evident that the prediction results given by sequential and parallel computations are the same. Figure 4.47 lists the parallel computation performance with three different models. We can see that 60–75% acceleration can be achieved, although speedup performance partially depends on the model geometry. The results demonstrate a big reduction in the overall analysis time for flow analysis, which is usually the most time-consuming step in injection molding simulation.

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50. Youngs D.L., Time-dependent multi-material flow with large fluid distortion, in Numerical Methods for Fluid Dynamics, Mortom K.W., Baines M.J., Editors. 1982, New York: Academic. 449–468. 51. Luoma J.A., Voller V.R., An explicit scheme for tracking the filling front during polymer mold filling. Applied Mathematical Modelling, 2000. 24(8–9): 575–590. 52. Osher S., Sethian J.A., Fronts propagating with curvaturedependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, 1988. 79(1): 12–49. 53. Sussman M., Puckett E.G., A coupled level set and volume of fluid method for computing 3D and axisymmetric incompressible two phase flows. Journal of Computational Physics, 2000. 162(2): 301–337. 54. Enright D., Fedkiw R., Ferziger J., et al., A hybrid particle level set method for improved interface capturing. Journal of Computational Physics, 2002. 183(1): 83–116. 55. Holm E.J., Langtangen H.P., A unified finite element model for the injection molding process. Computer Methods in Applied Mechanics and Engineering, 1999. 178(3–4): 413–429. 56. Soukane S., Trochu F., Application of the level set method to the simulation of resin transfer molding. Composites Science and Technology, 2006. 66(7–8): 1067–1080. 57. Fedkiw R.P., Aslam T., Merriman B., et al., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). Journal of Computational Physics, 1999. 152(2): 457–492. 58. Hirt C.W., Amsden A.A., Cook J.L., An arbitrary Lagrangian–Eulerian computing method for all flow speeds. Journal of Computational Physics, 1974. 14(3): 227–253. 59. Noh W.F., CEL: A Time-Dependent, Two-SpaceDimensional, Coupled Eulerian-Lagrange Code. 1963, Lawrence Radiation Laboratory, University of California, Livermore. 60. Hughes T.J.R., Liu W.K., Zimmermann T.K., Lagrangian– Eulerian finite element formulation for incompressible viscous flows. Computer Methods in Applied Mechanics and Engineering, 1981. 29(3): 329–349. 61. Belytschko T., Flanagan D.P., Kennedy J.M., Finite element methods with user-controlled meshes for fluid–structure interaction. Computer Methods in Applied Mechanics and Engineering, 1982. 33(1–3): 669–688. 62. Huerta A., Liu W.K., Viscous flow with large free surface motion. Computer Methods in Applied Mechanics and Engineering, 1988. 69(3): 277–324. 63. Lewis R.W., Navti S.E., Taylor C., A mixed Lagrangian– Eulerian approach to modeling fluid flow during mould filling. International Journal for Numerical Methods in Fluids, 1997. 25: 931–952. 64. Han X.H., Li X.K., An ALE free surface tracking and mesh generation scheme for injection molding process. Journal of Dalian University of Technology, 2005. 45(5): 633–639. 65. Zeng J.H., Wang Z.L., Arbitrary Lagrangian–Eulerian finite element simulation for viscous flow with large-amplitude

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81. Deng G.B., Piquet J., Queutey P., et al., Navier Stokes equations for incompressible flows: Finite-difference and finite volume methods, in Handbook of Computational Fluid Mechanics, Peyret R., Editor. 1996, Washington, DC: Academic Press Ltd. 25–97. 82. Flynn M.J., Some computer organizations and their effectiveness. IEEE Transactions on Computers, 1972. 100(9): 948–960.

83. Duncan R., A survey of parallel computer architectures. Computer, 1990. 23: 5–16. 84. Zhang Y., Acceleration Approaches of the Computation for Plastic Injection Molding Simulation. 2009, Wuhan, PRC: Huazhong University of Science and Technology. 85. Yang W.H., Peng A., Liu L., et al., Parallel true 3D CAE with hybrid meshing flexibility for injection molding, in SPE Annual Technical Conference - ANTEC, Conference Proceedings. 2005, Boston, Massachusetts, USA: Society of Plastics Engineers. 56–60.

5 COOLING SIMULATION Yun Zhang and Huamin Zhou State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

5.1

INTRODUCTION

The injection molding process can be generally classified into three stages, namely, filling, packing (or postfilling), and cooling. The cooling stage takes about 80% of the total cycle, and it is the only stage that can be considerably reduced by using better mold design. In the plastic injection molding, the cooling system design affects not only the production efficiency but also the quality of final products. A rapid cooling improves process economics by shortening the cycle time. Meanwhile, a uniform cooling improves product quality by preventing differential shrinkage, internal stresses, part warpage, and mold release problems, as well as shorten the molding cycle. There are various ways to provide coolness to the injection mold. The cooling system mainly considered in the present study is the widely employed cylindrical cooling channels that provide passages of coolant such as water and synthetic oil. The coolant is continually running along cooling channels, in general, or flows only after the polymer melt fills the cavity; thus, when the melt enters the mold, it is not subjected to a fast cooling shock. Generally speaking, the major concern of the cooling system design lies in finding the best arrangement of the cooling channels and the best combination of cooling conditions, such that a rapid, uniform cooling can be achieved. Accordingly, the parameters involved in the cooling system design mainly include the size, location, and connectivity of the cooling channels, as well as the temperature, flow rate, and thermal properties of the coolant.

Fortunately, numerical simulation has been proved to be an efficient engineering tool for mold designers to optimize mold cooling system design before the mold is built. The goal of cooling analysis is to accurately simulate the cooling process that takes place within an injection mold. It can help the mold designers to optimize all cooling configurations, and thus (i) balance the cooling in the mold and diminish the thermal stresses that cause warpage, (ii) increase the part production by cutting down the cooling time, (iii) reduce operating costs by providing the data needed to select the optimum coolant flow, pressure, and temperature. Even though the injection molding process can be generally classified into three stages, the polymer melt actually starts cooling as soon as it gets into the cavity. Hence, strictly speaking, the mold cooling process is a three-dimensional, cyclic, transient heat conduction problem with convective boundary conditions on the mold and cooling channel surfaces. The interaction between the polymer melt and the mold at all these stages should be taken into account for a complete analysis. However, it can be found in most literatures that most analyses for the filling stage assume a constant mold-wall temperature to account for the cooling effect. This assumption seems to be acceptable because the filling time only occupies a small part of the total cycle time such that the discrepancy due to neglecting the mold-wall temperature change is negligible. Thus, most cooling analyses assumed that the polymer melt has fully filled the cavity and ceased flowing. However, as is discussed later, the result of the temperature distribution in the hot polymer melt from the filling analysis can still be employed as the input data for the cooling

Computer Modeling for Injection Molding: Simulation, Optimization, and Control, First Edition. Edited by Huamin Zhou.  2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc.

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analysis in order to give a more complete accounting for the cooling. The reported cooling analysis can be divided into two kinds: (i) the cycle-averaged approach and (ii) the transient approach. The cycle-averaged approach is most widely used because it is simple, computationally efficient, and yet sufficiently accurate for mold design purposes in standard plastic injection molding process. Owing to the complex geometry of the mold introduced by the mold cavity and the significant difference in thermal properties between the part and the mold, the cycle-averaged approach decouples the problem and separately performs the heat conduction analysis of the part and the mold. In this approach, the mold heat conduction is modeled to a steady-state heat transfer approximation, and thus a cycle-averaged temperature field can be obtained by performing a cyclic steady-state cooling analysis. It is based on the fact that the mold cavity temperature oscillates in a relative small range, and the averaged temperature of the mold does not change with time during the continuous processing. On the other hand, as the polymer part is usually very thin and the dimension along the thickness direction is much smaller than that in other two directions, the heat conduction can be regarded to be predominant in the thickness direction of the part. Thus, the cycle-averaged heat flux over the part surface, which is obtained by a one-dimensional transient analysis along the part thickness, can be regarded as the second boundary condition over the mold cavity for the mold cooling analysis. In such a manner, the heat conduction process in the mold and the part can be solved in a separate way. They are connected via the conformability of the heat flux over the mold–part interface. The three-dimensional Laplace’s equation for steady-state heat conduction of the mold can be solved by different methods. The BEM was thought to be particularly well suited for the steadystate heat conduction problem of the mold, and it was adopted in most cycle-averaged cooling simulations. A major advantage of the BEM is that it reduces the spatial dimensions of computational domain by 1, and there is no need to mesh the whole mold solid zone, only the boundary surfaces of the mold. Thus, the standard and modified BEM plays a dominant role in mold cooling analysis, and this chapter mainly focuses on its discussions. The cycle-averaged approach has been in development for 20 years and is on the state of tending to mature. As Kwon1 demonstrated the fundamentals and some features of the cycle-averaged cooling modeling, most researches on cooling simulations during the following decade are based on this modeling. Himasekhar et al.2 performed a comparative study for the three-dimensional mold cooling analysis based on cycle-averaged approach. Himasekhar et al.2 and Chen et al.3 performed mold cooling analysis by coupling the cycle-averaged mold analysis and the

transient part analysis by an iterative procedure. Chen et al.4 also developed an alternative formulation by separating the mold temperature into two components: a steady component and a time-varying component, and the time-varying component was evaluated by the difference between instantaneous heat flux and cycle-averaged heat flux on the mold–part interface. Rezayat and Burton5 developed a special boundary integral equation for threedimensional complex geometry, in which the mold–part interface is replaced by the mid-surface of the part, and the numerical quadrature over the surface of the cooling channels is reduced to integration over the axis of cylindrical segments, followed by Park et al.,6 – 8 etc. Hioe et al.9 developed software capable of simulating the thermal state of the part and mold for multiple injection molding cycles. Although an cycle-averaged temperature distribution can be obtained from the cycle-averaged cooling analysis, and it might be sufficient for optimizing certain parts of the cooling configurations, such as the location, diameter of the cooling line, coolant properties, and mold materials, the transient temperature fields from an accurate threedimensional transient cooling analysis together with the mold and the part are needed for several reasons. First, although the temperature in the mold interior is relatively steady, the temperature on the mold cavity suddenly jumps up when the polymer melt toughs it and plunge down in the following few seconds. The temperature changes on the mold cavity in a molding cycle can enhance the understanding of shrinkage and warpage of plastic parts. Second, the molded part is assumed to be very thin, and the heat conduction is thought to occur only along the thickness direction in the cycle-averaged cooling analysis. However, one-dimensional transient simulation for a complicated part may also not be accurate at sharp corners or may cause sudden change in thinness. Moreover, the variotherm mold was rapidly developed to improve the part quality and the efficiency of injection molding in the past decade. Unlike the standard injection molding process with constant mold temperature control, the mold is heated by various methods, and the temperature on the mold cavity is kept above the polymer-softening temperature before the polymer melt is injected into the cavity, and then be cooled down rapidly until the part is ejected out.10 For the more complicated dynamic mold temperature control switching between heating and cooling cyclically, the conventional cycle-averaged approach may introduce non-negligible prediction errors because of the significant temperature variation.11 The research on the transient approach began in the 1990s. Hu et al.12 performed a transient cooling simulation using the dual reciprocity boundary element method. Tang et al. simultaneously computed the transient temperature distributions in the mold and the polymer part by Galerkin

MODELING

finite element method using a matrix-free Jacobi conjugate gradient scheme.13,14 Qiao15 developed a fully transient mold cooling analysis using the boundary element method based on the time-dependent fundamental solution. Lin et al.16 and Chiou et al.17,18 performed fully the threedimension simulations of mold temperature variation based on the FEM and FVM. Cao et al.19 solved the mold transient temperature distributions using a 3D finite element method by specifying the heat-flow rate at the interface between mold and part. Until now, the cooling analysis for the rapid heat cycle molding is available in many commercial simulation softwares, such as Moldflow and Moldex3D. In contrast to the cycle-averaged approach, the temperature distributions in the mold and the polymer part should be solved simultaneously in a coupled model, in which the Fourier thermal conductivity theory is satisfied on the cavity surface.

5.2 5.2.1

MODELING

131

into two parts: the cycle-averaged temperature field during one cycle and the temperature fluctuation field. In general, the fluctuating component of mold temperature is relatively small compared to the cycle-averaged component (or steady component). The temperature fluctuation is greatest near the cavity surface and diminishes away from the cavity surface. The localization of the temperature fluctuation may be explained by the fact that the diffusion time scale of the mold [(length of mold)2 /(heat diffusivity of mold)] is roughly 2 orders of magnitude higher than that of the polymer material for thin part thickness [(part thickness)2 /(heat diffusivity of polymer melt)]. Therefore, cooling systems far enough from the cavity surface affect mainly the cycle-averaged temperature field, but do not significantly affect the transient fluctuation. In this regard, the steady cycle-averaged temperature field may well represent the overall cooling behavior via cooling system, in the cycle-averaged sense, during this continuous process. In addition to this approximation, the following simplifications and assumptions have been made in obtaining a solution to this problem:

Cycle-Averaged Temperature Field

The heat transfer in injection molding is complicated, as four types of heat transfer behavior are involved: (i) heat exchange within the polymer melt, (ii) heat flow from the polymer to the metal mold, (iii) heat transfer between the mold and the coolant, and (iv) heat escape from the mold outside surfaces to the ambient air. Shown in Figure 5.1 is the typical mold temperature response at a point near the cavity surface during the continuous injection molding operation. It can be seen that after a certain transient period at the beginning of molding, the continuous operation results in a steady-state cyclic heat transfer behavior in the mold. The temperature field during such a continuous operation can be decomposed

1. The thermal and other material properties of the polymer, mold material, and, coolant, such as the mass density, thermal conductivity, and specific heat, are assumed to be constants. 2. Heat conduction in the polymer is assumed to be in the gap-wise direction only since most injectionmolded parts have a sheetlike geometry, that is, the thickness is much smaller than other dimensions of the part. 3. The polymer melt and the mold wall are assumed to be in perfect thermal contact with no air gaps resulting from possible shrinkage of the polymer melt. In other words, the interfacial temperature is the same for both materials. 4. The volumetric flow rate of the coolant is assumed to be sufficiently large, where the coolant temperature can be treated as a constant in the calculation. (Nevertheless, the calculation still provides a cycleaveraged temperature increment as a check on this assumption.) Under these assumptions, the Fourier’s equation governing the temperature in the mold thus reduces to the Laplace’s equation with appropriate boundary conditions. So, a cycle-averaged approach is computationally more efficient than a fully transient technique and a periodic analysis. The Laplace’s equation governing the mold cycle-averaged temperature distribution is given by

FIGURE 5.1 Schematic representation of cyclic, transient variation of the mold temperature.

∂ 2T ∂ 2T ∂ 2T + + = 0, for (x, y, z) ∈  ∂x 2 ∂y 2 ∂z2

(5.1)

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COOLING SIMULATION

As mentioned previously, heat conduction in the polymer is assumed to be in the gap-wise direction only. Therefore, a local one-dimensional transient analysis in the thickness direction is usually adequate for the polymer. The governing equation for the melt temperature is

Γp Γo

∂ 2T ∂T =a 2 ∂t ∂s Ωp Γe

Ω

FIGURE 5.2 A schematic drawing of a cross-sectional view of an injection mold: -mold region, pp -plastic part region, (Ŵ c )cooling system surface, (Ŵ p )Ŵ p -mold cavity surface, and (Ŵ e )Ŵ e mold exterior surface.

where T is the temperature,  is the mold region, and x , y, and z are the Cartesian coordinates.

5.2.2

Cycle-Averaged Boundary Conditions

Shown in Figure 5.2 is a schematic drawing of a crosssectional view of an injection mold.  denotes the mold region, whose boundary consists of mold cavity surface (Ŵ p ), cooling channel surface (Ŵ c ), and exterior of mold boundary surface (Ŵ e ). p denotes the mold cavity region. To complete the model for mold cooling, all boundary conditions need to be specified.

(5.4)

where t denotes time, T is the melt temperature, s is the local coordinate along the gap-wise direction, and a is the thermal diffusivity, and it can be expressed as a = K e /ρC , where ρ, K e , and C are the density, effective thermal conductivity, and specific heat, respectively. When the polymer melt touches the mold cavity, its temperature is assumed to be identical in the thickness direction and is denoted as T 0 . The temperatures on each side of the mold cavity surface are T w1 and T w2 , respectively. The changes in the part temperature distribution at the whole cooling stage are illustrated in Figure 5.3. The final temperature distribution of the part when it is ejected out can be calculated by either an analytical method or an implicit finite-difference method. Equation 5.4 can be directly solved by the separate variable method. Let T (t, x ) = U (x ) + V (x , t), and then

T0

T0 t1

5.2.2.1 Mold Cavity Surface The cycle-averaged heat flux over the mold cavity surface (the surface adjacent to the polymer) is given by

t2

t3

∂T = q¯ −Km ∂n

(5.2)

t4 t5

where n is the normal to the surface, K m is thermal conductivity of the mold material, and q¯ is the cycleaveraged heat flux, which is defined as q¯ =

 tc 0

q(t)dt

tcycle

Tw2

Tw1

(5.3)

where t cycle is the cycle time, t c is the cooling time, q(t) is instantaneous heat-flux values from the polymer to the mold. It should be mentioned that cooling happens during the entire cycle time, and the cooling time t c referred to in this chapter is defined as the time taken from the melt injecting to the mold opening, namely, t cycle = t c + t d , where t d is the demolding time.

2s

FIGURE 5.3 A schematic drawing of heat conduction along the thickness direction in the polymer part: t 1 –t 5 —different cooling times, s —half of part thickness, T 0 —initial plastic melt temperature, and T w1 , T w2 —temperature of mold cavity surface on each side.

MODELING

Equation 5.4 becomes ∂ 2U =0 ∂x 2 ∂V ∂ 2V = ∂t ∂x 2

(5.5) (5.6)

133

Besides, an implicit, finite-difference method with a variable mesh can also be used to solve Equation 5.4. Discretize the thickness of the part into L nodes, that is, 0 = s 1 < s 2 < · · · < s L = H , and discretize the cooling time into M steps, that is, 0 = t 1 < t 2 < · · · < t M = t c . Let Tin denote the part temperature of s i at time t n . Equation 5.4 can be expressed in an implicit finite-difference form:

The boundary conditions are U = Tw1 , U = Tw2 ,

for x = 0 for x = 2s

(5.7)

and V = T0 − U , for t = 0 V = 0, for x = 0 and x = s

(5.8)

respectively. Equations 5.5 and 5.6 are solved, and thus the solution of T (t, x ) is obtained, as follows: T (t, x) = Tw1 + ×e

x (Tw2 − Tw1 ) + s

2 2 − an 2π t s

∞  n=1

 nπ x  2 sin nπ s

  (T0 − Tw1 ) − (T0 − Tw2 ) (−1)n

(5.9)

In the above equations, the thermal diffusivity, a, is treated as a constant. In reality, the thermal diffusivity is a function of temperature, and it is preferable to use an average value of thermal diffusivity. The effective thermal conductivities for plastics used commonly in injection molding can be found in Reference 20. Thus, the cycleaveraged heat flux q¯ can be calculated as q¯ = =

 tc 0

q(t)dt

tcycle

=

 tc 0

∂T (t,x) ∂x dt

tcycle ∞

 2s  1 T0 − Tw1 (Tw2 − Tw1 ) + 2 2 s an π tc n=1

2 2   − an π2 tc n s (5.10) 1 − exp − (T0 − Tw2 ) (−1)

The average temperature of the polymer part cross section along the thickness direction, which is meaningful for estimating the cooling time, can be calculated by 1 T˜ (t, x) = s



s 0

∞ 

T (t, x)dx =

1 (Tw1 + Tw2 ) 2

 2 − an22π 2 t  1 − (−1)n e s 2 2 n π n=1   (T0 − Tw1 ) − (T0 − Tw2 ) (−1)n (5.11) +

n+1 n+1 − 2Tin+1 + Ti−1 Ti+1 ρC Tin+1 − Tin =0 − (s)2 t (Ke )n+1 i (5.12) Because the amount of internal radiation depends on the temperature of the glass, the effective thermal conductivity K e varies and depends on position and time. Therefore, the above finite-difference equation is nonlinear, which should be solved by iteration technique to obtain the part temperature distribution.

5.2.2.2 Cooling System Surface On the cooling system surface (the surface adjacent to the coolant), a heat transfer coefficient is imposed and defined as Km

∂T = −hc (T − T∞ ) ∂n

(5.13)

where h c represents the effective heat transfer coefficient between the mold and the coolant at a temperature of T ∞ , n is the normal to the surface, and K m is thermal conductivity of the mold. The cylindrical cooling channel is the most widely employed way that provides passages for the coolant to remove heat from the injection mold. As heat transfer of flows in circular ducts arises in many types of heatexchanger equipment, it has been studied extensively in many literatures. The heat transfer coefficient, combined with a Nusselt number, was found to be a function of the flow Reynolds number and the Prandtl number of the fluid. That is hc D = f (Re, P r) (5.14) Nu = kc where k c represents the thermal conductivity of the coolant with D being the diameter of the cooling channel. In particular, flow with a Reynolds number less than approximately 2100 is characterized as laminar flow, in which the flow can be regarded as various layers moving at different velocities. Owing to the slow motion of fluid near the channel surface, heat is removed essentially by conduction, which turns out to be very inefficient. On the other hand, fully developed turbulent flow occurs for Reynolds numbers greater than about 10,000. As there is a component of velocity perpendicular to the flow direction, the heat transfer is dramatically improved. Flows between these two Reynolds number regions exhibit transient characteristics intermittent between laminar flow

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and turbulent flow. The Prandtl number, which is defined as the kinematic viscosity of the fluid divided by the thermal diffusivity of the fluid, is a measurement of how rapidly momentum is dissipated compared to the rate of diffusion of heat through the fluid. Owing to the great benefit of maintaining a turbulent flow in the cooling channel (at the expense of increased pumping requirements), it has been tacitly assumed that the flow of coolant in the cooling channel is turbulent. The mold cooling analysis should provide the Reynolds number for each channel as a check on this assumption. By examining several equations for calculating the heat transfer coefficient in turbulent flows, we observe that the equation presented by Dittus and Boelter seems to be suitable for the present problem. This Dittus–Boelter equation states that Nu =

hc D = 0.023Re0.8 P r 0.4 kc

the Nusselt number of the air. For vertical mold exterior surfaces, N¯ u is calculated by N¯ u = 0.677P ra1/2 (0.952 + P ra )−1/4 Gr 1/4

where Pr a and Gr are Prandtl number and Grashof number of the air, respectively. For horizontal mold exterior surfaces, the convective heat transfer coefficient is expressed as (5.20) N¯ u = C(GrP ra )m where C and m are constants. C = 0.54, m = 0.25 for upward surfaces and C = 0.58, m = 0.2 for downward surfaces. Pr a , Gr are calculated by Gr =

(5.15)

4ρQ hc = 0.023 π μD

0.8

cp μ kc

0.4

kc D

(5.16)

where ρ, μ, and c p represent the mass density, absolute viscosity, and the specific heat of the coolant, respectively, and Q is the volumetric flow rate. The heat transfer coefficient in Equation 5.16 is the boundary condition defined on the cooling channel surfaces. 5.2.2.3 Mold Exterior Surface It is assumed that there are no heat conduction between the nozzle and mold. For the mold surfaces in contact with the ambient air, heat is mainly transferred by convection and radiation. However, the heat transferred by radiation is relatively small because of its moderate temperature, and thus is not taken into considerations. Similar to the cooling system surface, the boundary condition on the mold exterior surface can be given, expressed as Km

∂T = −ha (T − Ta ) ∂n

gβ(Twa − Ta )L3 νa2

(5.21)

νa αa

(5.22)

and P ra =

The above equation is valid for 10,000 < Re < 120,000 and 0.7 < Pr < 120. It can also be rewritten as

(5.17)

where h a represents the convective heat transfer coefficient between the mold and the air at the temperature T a . The convective heat transfer coefficient is prescribed based on the correlations available in the literature for a steady-free convective heat transfer on exterior surfaces21,22 and is expressed as ¯ • ka Nu (5.18) ha = L where k a is the heat transfer coefficient of the air, L is the characteristic length of the mold surface, and N¯ u is

(5.19)

respectively, where g is the acceleration due to gravity; T a and T wa are temperatures of the air and the mold surface, respectively; and β, ν a , and a a are the volume thermal expansion coefficient, kinematical viscosity coefficient, and thermal diffusion coefficient of air, respectively. 5.2.3

Coupling Calculation Procedure

To perform an analysis for cycle-averaged mold temperature, one needs to know all the boundary conditions as described above. Among them, heat transfer conditions for the cooling system surfaces and mold exterior surfaces can be predicted by correlations discussed above. However, for the mold cavity surface, the cycle-averaged heat flux has to be calculated using Equation 5.3, which in turn needs the cycle-averaged mold surface temperature distribution and the cooling time t c . This implies that the present simulation procedure is iterative. The convergence is based on the consistency of solution, that is, to maintain compatibility of temperature and heat flux at the mold–melt interface. As shown in Figure 5.4, the following iterative procedure is developed for this purpose: Step 1. To begin the iteration, assume a mold–melt interface temperature distribution and a cooling time t c . For the mold–melt interface temperature distribution, the average value of polymer melt and cooling fluid temperature may be a good initial value. The estimation of the cooling time t c will be discussed later in detail. Step 2. Carry out the melt/part analysis to determine heatflux variation with time along the mold–melt interface. Using the heat-flux variation, the cycle-averaged flux values along the mold–melt interface are determined.

MODELING

Step 3. Using the resulting cycle-averaged flux distribution along the mold–melt interface and the heat transfer conditions for the cooling system surfaces and mold exterior surfaces, perform the mold steady heat transfer analysis to obtain the cycle-averaged temperature distribution along the mold–melt interface. Step 4. If the resulting melt–mold interface temperature distribution and the cooling time calculated by the mold analysis are in close agreement with what is assumed for the melt analysis, the iterative procedure is stopped; otherwise proceed to step 2 and repeat.

5.2.4

Calculating Cooling Time

It should be noticed that to perform a cycle-averaged cooling analysis, the cooling time should be known to calculate the cycle-averaged heat flux using Equation 5.3. However, the cooling time is actually undetermined because it relates to the temperature field of the part. This means that the cooling time t c should be estimated first and determined iteratively in parallel with the mold cavity temperature, as illustrated in Figure 5.4. An accurate initial guess of the cooling time is important to reduce the outer iteration number. The cooling time t c is defined as the period of time required to cool a part from its injection temperature to reach a specified ejection temperature, T e , at which the part can be removed from the mold yet still preserve its shape. The cooling time is critical to the productivity because that the cooling process is the longest stage in the whole

injection process. The part is excessively cooled to preserve its shape if the cooling time is too long. On the other hand, if the cooling is insufficient, the part temperature would be too high at the moment it is ejected. The solidified layer of the part is too thin to preserve its shape, and it might significantly deform under the ejection force. For these reasons, the cooling time should be carefully selected to balance the productivity and the part quality. The optimum cooling time is the minimum cooling time needed to satisfy the required dimensional accuracy of the mold part. Certainly, it depends on the temperature distribution of the part at cooling stage. However, it is difficult to give a criterion to determine the optimum cooling time by the part temperature: temperatures of how many and which portions of the part reach the ejection temperature T e can be regarded as the watershed that the part can be ejected within a acceptable deformation? Simple equations are thought to be straightforward to mold engineers and can be derived from semitheoretical and theoretical models of heat transfer in the plastic part itself.23 – 26 White26 considered that the cooling time is the time span from injection ending to the moment when the mid-plane temperature at maximum cavity thickness reaches the ejection temperature, and it can be estimated by tc =





4S 2 4 T i − Tw ln αeff π 2 π Te − Tw

The flow chart of the coupling calculation

(5.23)

where S is half of the maximum cavity thickness, a eff is the effective thermal diffusivity, T i is the melt temperature at injection, T e is the ejection temperature, and T w is the mold temperature. According to Equation 5.23, all polymer melts solidifying for the cooling time if the mold temperatures are uniform. However, Equation 5.23 usually overestimates the cooling time. Liang et al.23 declared that the calculation of cooling time based on a cycle-averaged temperature over the cross-section for the molded part might be better than doing calculation based on the mid-plane temperature. He proposed a judicious selection of the representative ejection temperature, and the cooling time could be estimated by



8 T i − Tw 4S 2 ln tc = αeff π 2 π 2 Te − Tw

FIGURE 5.4 procedure.

135

(5.24)

where symbols have same meanings as in Equation 5.23. Taking the assumption that the part can be ejected from the mold yet still preserve its shape when its solidification layer reaches one-third thickness over the whole surface, a reasonable cooling time can be calculated by  √

  2 3 T i − Tw 4s 2 tc = max ln (5.25) αeff π 2 π Te − Tw where s is half of the cavity thickness.

136

COOLING SIMULATION

5.3 NUMERICAL IMPLEMENTATION BASED ON BOUNDARY ELEMENT METHOD

−

The boundary element method consist of two techniques: (i) Transforming the partial differential equation that describes the behavior of the unknown inside and on the boundary of a domain into an integral equation relating only boundary values, and (ii) finding out the numerical solution to this boundary integral equation. As all numerical approximations take place only on the domain boundaries, the dimensionality of the problem is reduced by 1, and it is efficient and economical for solving potential problems, whose governing equation is the Laplace’s or Poisson’s equation. Moreover, as we mentioned in the beginning of this chapter, because discretizations are necessary only on the domain boundaries in numerical approximations, it particularly suits for porous domains or complicated domains that are difficult to be discretized over whole bodies, such as the plastic injection mold.

−T (x )]

Ŵ2

q(x )w(y, x )dŴ(x ) −



[T (x ) Ŵ1

∂w(y, x ) dŴ(x ) ∂n

(5.30)

where i = 1, 2, 3 and Einstein’s summation convention for repeated indices is implied. Integrate by parts once more, 2 ∇ w(y, x )T (x )d(x ) = − q(x )w(y, x )dŴ(x ) Ŵ1



− −

Ŵ2

q(x )w(y, x )dŴ(x ) − u(x )

Ŵ1



T (x ) Ŵ2

∂w(y, x ) dŴ(x ) ∂n

(5.31)

or, generally, 2 ∇ w(y, x )T (x )d(x ) = − q(x )w(y, x )dŴ(x ) Ŵ



5.3.1

∂w(y, x ) + T (x ) dŴ(x ) ∂n Ŵ

Boundary Integral Equation

The mold cycle-averaged temperature distribution is governed by Equation 5.1. We are seeking an approximate solution satisfying this equation and all boundary conditions, yields ∇ 2 T (x ) = 0 in  (5.26) with essential and natural boundary conditions T (x ) = T (x ) on Ŵ1

(5.27)

q(x ) = q(x ) on Ŵ2

(5.28)

∂T (x ) ∂n

is the gradient on the boundary and where q(x ) = the terms with overbars indicate known values of T (x ) or q(x ). Now, we will derive the boundary integral equation of Equation 5.26 using the weighted residual technique. The error introduced by approximate solutions of T and q can be minimized using the following weighted residual statement: 2 ∇ T (x )w(y, x )d(x ) = [q(x ) − q(x )]w(y, x )dŴ(x ) 

Ŵ2

∂w(y, x ) dŴ(x ) − [T (x ) − T (x )] ∂n Ŵ1

∂w(y, x ) dŴ(x ) ∂n

(5.29)

where w is a weighting function of variable y and x . Integrate Equation 5.29 by parts with respect to x i gives ∂T (x ) ∂w(y, x ) d(x ) = − − q(x )w(y, x )dŴ(x )  ∂xi (x ) ∂xi (x ) Ŵ1

(5.32)

Now, we will introduce the fundamental solution of Laplace’s equation. If a function T ∗ (y, x ) satisfies the following equation ∇ 2 T + δ(y, x ) = 0

(5.33)

T ∗ (y, x ) is called the fundamental solution of Laplace’s equation. In the above equation, y is a point in the interior of the domain  and δ(y, x ) is the Dirac delta function. The Dirac delta function has the following properties: δ(y, x ) = 0,

for y = x

δ(y, x ) = ∞,

for y = x

and for any function u(x ), u(x )δ(y, x )d(x ) = u(y)

(5.34)

(5.35)



Thus, the function T ∗ (y, x ) satisfies ∇ 2 T ∗ (y, x ) = −δ(y, x )

(5.36)

for two-dimensional problems T ∗ (y, x ) =



1 1 ln 2π r(y, x )

(5.37)

for three-dimensional cases 1 T (y, x ) = 4π ∗

1 r(y, x )



(5.38)

NUMERICAL IMPLEMENTATION BASED ON BOUNDARY ELEMENT METHOD

where r(y, x ) denotes the distance between the point y and x . Substituting Equation 5.36 into Equation 5.32, and according to the properties of Dirac delta function, it gives ∗ T ∗ (y, x )q(x )dŴ(x ) T (y) + q (y, x )T (x )dŴ(x ) =

Γe

or

y

Ŵ

T (y) = ∗



Ŵ

Ŵ

137

(5.39)

 ∗  T (y, x )q(x ) − q ∗ (y, x )T (x ) dŴ(x ) (5.40)

r=e

∂T ∗ (y,x ) . ∂n

where q (y, x ) = Thus, we obtain the integration equation about an arbitrary point y in the interior of the domain . It can be seen that the value of function T (y) at an interior point y can be represented by the integration of T and its directional derivative q over the boundary. Taking account of that if the point y is on the boundary rather than in the interior of the domain , two different procedures can be employed to deduce the boundary integral equation. One is through the physical consideration that a constant potential applied over a closed body produces no flux. The other is by assuming that the body under consideration can be augmented by a small region Ŵ ε , which is part of a circle of radius ε centered at point y on the boundary Ŵ (Fig. 5.5). The latter approach is herein presented for three-dimensional problems. Without loss of generality, assume that the second boundary condition is satisfied on whole boundary, which now consisting of two parts: the new hemisphere surface Ŵ ε and the old boundary excluding the projection of the new hemisphere Ŵ  − ε . y is now an interior point by introducing the small region Ŵ ε , thus the integral equation (Eq. 5.39) yields qT ¯ ∗ dŴ T q ∗ dŴ + T q ∗ dŴ = T + Ŵ−ε

Ŵε

Ŵε

+

Ŵ−ε

qT ¯ ∗ dŴ (5.41)

When the radius ε approximates 0, the integrations in the left of Equation 5.41 becomes T q ∗ dŴ = T q ∗ dŴ (5.42) lim ε→0 Ŵ−ε

Ŵ

 ∂ 1 lim T q dŴ = lim T dŴ ε→0 Ŵε ε→0 Ŵε ∂n 4π r −1 = lim T dŴ (5.43) ε→0 Ŵε 4π ε 2

∗



Using the mean value theorem for integrals to Equation 5.43, and assuming that the boundary is smooth at the point y, we have −1 T (ξ )2π ε2 (5.44) T q ∗ dŴ = lim lim ε→0 Ŵε ε→0 4π ε 2

FIGURE 5.5 region Ŵ ε .

Three-dimensional body augmented by a small

where ξ is a certain point on the hemispherical surface. When ε → 0, T (ξ ) → T (y), thus lim

ε→0



1 T q ∗ dŴ = − T 2 Ŵε

(5.45)

Similarly, the integrations in the right of Equation 5.41 yield lim



ε→0 Ŵ−ε

lim



ε→0 Ŵε

qT ¯ ∗ dŴ =



Ŵ

qT ¯ ∗ dŴ

qT ¯ ∗ dŴ = 0

(5.46) (5.47)

Introducing Equations 5.42, 5.45–5.47 into Equation 5.41, the boundary integration equation can be obtained as 1 T (y) + 2



Ŵ

T (x )q ∗ (y, x )dŴ(x ) =



q(x )T ∗ (y, x )dŴ(x ) Ŵ

(5.48) In the above deduction of the boundary integration equation, the second boundary condition over the whole boundary is assumed. However, Equation 5.48 is still valid for other two boundary conditions. If the boundary is not smooth anymore at the point y, the boundary integration equation should be modified as ∗ c(y)T (y) + T (x )q (y, x )dŴ(x ) = q(x )T ∗ (y, x )dŴ(x ) Ŵ

Ŵ

(5.49) where c(y) is the solid angle coefficient at the point y and c(y) =

α 4π

(5.50)

for three-dimensional problems. α is the solid angle formed by the tangent planes at the point y and α = 2π if the boundary is smooth.

138

COOLING SIMULATION

5.3.2

Numerical Implementation

5.3.2.1 Boundary element model In general, it is impossible to obtain analytical solutions to Equation 5.49 with particular geometries and boundary conditions in actual applications. The boundary element method produces a suitable reduction of the equation to an algebraic form that can be solved by a numerical approach. It generally consists of the following steps27 : 1. The boundary Ŵ is discretized into a series of elements over which the potential and its normal derivative are supposed to vary according to interpolation functions. The geometry of these elements can be modeled using straight lines, circular arcs, parabolas, and so on. 2. Using the method of collocation, the discretized equation is applied to a number of particular nodes within each element, where values of the potential and its normal derivative are associated. 3. The integrals over each element are carried out using, in general, a numerical quadrature scheme. 4. By imposing the prescribed boundary conditions of the problem, a system of linear algebraic equations is obtained. The solution of this system of equations, which can be effected using direct or iterative methods, produces the remaining boundary data. Considering that the mold body is three-dimensional, its boundary is divided into series of “triangles” or “boundary elements,” as shown in Figure 5.6. The points where the unknown values are considered are called nodes and taken to be in the center of each triangle for the so-called “constant” elements. Linear elements, in which the nodes are at the adjacent vertexes between elements, and quadratic elements in which an extra node on the adjacent edges between elements are generally used element types. For simplicity, discretizations with three-dimensional constant

boundary elements are discussed in this chapter, while other element types are not involved. Considering the constant element case, the boundary is discretized into N elements, of which N 1 are assumed to belong to Ŵ 1 and N 2 belong to Ŵ 2 . The values of u and q are taken to be constant on each element and equal to their values at the center node of the element. Note that in each element, the value of one of the two variables (T and q) is known, and the other one is to be evaluated. Taking y as the node i on which the fundamental solution is applied, that is, T (y) = T i . Similar consideration applies to the coefficient c(y). Dropping y, x notation which is written between brackets for simplicity, Equation 5.49 can be discretized as follows: ci Ti +

N  j =1

Ŵj

T q ∗ dŴ =

N  j =1

T ∗ q dŴ,

Ŵj

for i = 1, 2, . . . , N

(5.51)

Notice that for constant elements, the boundary is always “smooth,” hence the c i coefficient is identically equal to 1/2. Ŵ j is the area of element j . Equation 5.51 represents, in discrete form, the relationship between the node i , at which the fundamental solution is applied, and all the j elements (including the one i = j ) on the boundary, as shown in Figure 5.7. As the values of T and q inside the integrals in Equation 5.51 are constant within each element, and consequently they can be taken out of the integrals. This gives N

 1 Ti + 2 j =1



Ŵj

 N  q dŴ Tj = ∗



j =1

Ŵj

∗



T dŴ qj ,

for i = 1, 2, . . . , N

(5.52)

 The integrals q ∗ dŴ relate the i node with the element j over which the integral is carried out. Hence, these integrals

Element Node

FIGURE 5.6

Three-dimensional constant boundary elements.

FIGURE 5.7 Relationship between the fundamental solution at boundary node i and the boundary elements.

NUMERICAL IMPLEMENTATION BASED ON BOUNDARY ELEMENT METHOD

will be called Hˆ ij . It can be calculated by

Hˆ ij =

j

q ∗ (y, x )ds =



j

∂T ∗ (y, x ) ds ∂n

q= (5.53)

where j denotes the j th triangular integral element.  Similarly, the integrals of type T ∗ dŴ will be called G ij , calculated by Gij =



j

T ∗ (y, x )ds =



j

1 ds 4π r(y, x )

1 Ti + 2

j =1

N 

Hˆ ij Tj =

Gij qj ,

j =1

· · · + Hij Tj + · · · = · · · + Gij qj + · · · = · · · +Gij fc (Tj − T∞ ) + · · ·

1 2

for i = 1, 2, . . . , N

· · · + (Hij − Gij fc )Tj + · · · = · · · + Gij fc (−T∞ ) + · · · (5.62) The above method is also adopted for the mold exterior surface. For the mold cavity surface, Equation 5.2 gives

(5.55) q¯ ∂T =− ∂n Km

when i = j when i = j

(5.56)

Equation 5.55 can be written as N  j =1

Hij Tij =

N  j =1

Gij qj , for i = 1, 2, . . . , N

(5.57)

which is a set of equations and can also be expressed in matrix form as H T = Gq,

(5.61)

It follows that

If we define  Hˆ ij Hij = Hˆ ij +

(5.60)

where fc = − Khmc . Using Equation 5.60, the extended form of Equation 5.58 can be given as

(5.54)

Hence, Equation 5.52 can be written as N 

hc ∂T =− (T − T∞ ) = fc (T − T∞ ) ∂n Km

139

H , G ∈ R n×n ,

T , q ∈ Rn

(5.58)

Note that N 1 values of u and N 2 values of q are known on boundary Ŵ, hence there are only N unknowns in Equation 5.58, which now can be reordered in accordance with the unknown under consideration. Reorder Equation 5.58 with all the unknowns on the left-hand side and a vector on the right-hand side obtained by multiplying matrix elements by the known values of T¯ and q. ¯ This gives a system of linear equations as Ax = b (5.59) where x is the vector of unknown T or q and b reflects boundary conditions. Equation 5.59 can be solved computationally by the Gauss elimination method or other iterative solvers. Once Equation 5.59 is solved, all unknowns T or q of the nodes on the boundary are obtained. Now, we introduce the boundary conditions described in Section 4.2.2 into Equation 5.58 to obtain a system of linear algebraic equations. For the cooling system surface, Equation 5.13 can be rewritten as

(5.63)

By defining ⎧ ⎪ ⎨Hij − Gij fc on the cooling system surface Ŵ c ˜ Hij = Hij − Gij fa on the mold exterior surface Ŵ e ⎪ ⎩ Hij on the mold cavity surface Ŵp (5.64) ⎧ f (−T ) on the cooling system surface Ŵ ⎪ ∞ c ⎪ ⎨ c f (−T ) on the mold exterior surface Ŵ a a e pj = ⎪ q¯ ⎪ ⎩− on the mold cavity surface Ŵ p km (5.65) b = Gp (5.66) a system of linear algebra equations is arisen: H˜ T = b

(5.67)

It is worth to notice that H˜ is a fully unsymmetrical populated matrix of order N . Once Equation 5.67 is solved, all the values of T and q on the boundary are known, and one could calculate the value of temperature at any interior point using Ti =



Ŵ

qT ∗ dŴ −



T q ∗ dŴ

(5.68)

Ŵ

It can be written in discretized form as Ti =

N  j =1

qj Gij −

N  j =1

Tj Hˆ ij

(5.69)

140

COOLING SIMULATION

The values of internal heat fluxes can be calculated by differentiating Equation 5.68 as was done in Equation 5.50. Hence,

∂T ∂xl



=



Ŵ

q

∂T ∗ dŴ − ∂xl



u Ŵ

∂q ∗ dŴ ∂xl

y

(5.70)

where x l are the coordinates, and l = 1, 2, 3 for threedimensional cases. As the main interest in the cooling analysis lies in finding the temperature and heat flow over the boundary surface, the direct result of solution of Equation 5.67 would be just sufficient for the present design purpose. It should be remembered that as the compatibility of temperature and heat flux at the mold–melt interface must be maintained, the whole cooling simulation procedure is iterative. This means that Equation 5.67 must be resolved for many times with different boundary conditions assumed on the mold cavity. Thus, the matrices H˜ and G must be assembled and maintained during the whole computational process. The flow chart of the cycle-averaged cooling simulation base on boundary element method is shown in Figure 5.8. It may be noted that there are nested iterations in the solution. The inner iteration is used to solve the linear algebra equations, and the outer one is employed to

D (y, x) r

y

n

2 l23

x

0

x

a

l12

3 l31

z

1

FIGURE 5.9

Illustration of the integral over a field element.

update the boundary conditions and therefore the algebra equations. 5.3.2.2 Integration Techniques The coefficients Hˆ ij and G ij , which means integrals over the element j related the node i , can be calculated using simple Gauss quadrature rules for all elements except one where i = j . Let D(y, x ) denote the vertical distance between y and the j th element j , as shown in Figure 5.9, in this case, ∂r r • n D(y, x ) = = cos α = • | r | | n| ∂n r(y, x )

(5.71)

where α is the angle between n and r . Let us recall that the fundamental solution T ∗ (y, x ) = 1 4π r(y , x ) to Laplace’s equation for three-dimensional problems as expressed in Equation 5.38, it follows that ∗

∂



1 4π r(y,x )



∂T (y, x ) = ∂n ∂n −D(y, x ) ∂r(y, x ) −1 = = 2 ∂n 4π |r(y, x )| 4π |r(y, x )|3

q ∗ (y, x ) =

(5.72)

Using Gaussian integral formula, Equations 5.53 and 5.54 can be rewritten as Aj D(y, x ) ˆ Hij = q ∗ (y, x )ds = − 2π j

FIGURE 5.8 The flow chart of the cycle-averaged cooling simulation based on boundary element method.

m  k=1

wk  3/2 2 (xk − xi ) + (yk − yi )2 + (zk − zi )2

(5.73)

141

NUMERICAL IMPLEMENTATION BASED ON BOUNDARY ELEMENT METHOD

Gij =



j

m  k=1

Aj 1 ds = 4π r(y, x ) 2π

I2 I21

wk  2 (xk − xi ) + (yk − yi )2 + (zk − zi )2

I24

(5.74) I22

where m is the number of integral points, (x k , y k , z k ) is the coordinate of integral points, wk is the associated weighting factor, (x i , y i , z i ) is the coordinate of the source point y, and Aj is the area of the field element j . The integral points can be obtained by ⎧ ⎪ ⎨xk = x1 φk (x) + x2 φk (y) + x3 φk (z) yk = y1 φk (x) + y2 φk (y) + y3 φk (z) ⎪ ⎩ zk = z1 φk (x) + z2 φk (y) + z3 φk (z)

(5.75)

where [φ k (x ), φ k (y), φ k (z )] is the coordinate coefficient of integral points; (x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 ), and (x 3 , y 3 , z 3 ) are the vertex coordinates of the triangular field element. Usually, the number of integration points should be sufficient to provide the required accuracy. As the mold cavity usually has large planes but a much smaller thickness, uniform mesh in the discretizations will inevitably create a huge number of elements. When using the Gaussian integral formula, the number of Gaussian integral points affects the accuracy and efficiency of the calculation greatly. Insufficient integral points cannot guarantee the accuracy, while too many will significantly increase the computation cost. Therefore, it is necessary to use fewer integral points, while the accuracy remains satisfactory. Balancing the integration accuracy and efficiency, a dynamic allocation method of integral points can be used. In this method, the size of a triangular field element is evaluated by its averaged side length, that is, L = (l 12 + l 23 + l 31 )/3, where l 12 , l 23 , and l 31 are the lengths of three sides, respectively, as shown in Figure 5.9. For different value ranges of Ratio = r/L, different integral orders will be adopted as follows: 1. If Ratio > 24, the integral order of 2 is adopted, that is, the number of Gaussian integral points is 4. 2. If 6 < Ratio ≤ 24, the integral order of 3 is adopted, that is, the number of Gaussian integral points is 7. 3. If 1.5 < Ratio ≤ 6, the integral order of 4 is adopted, that is, the number of Gaussian integral points is 16. 4. If Ratio ≤ 1.5, higher integral order should be adopted. According to experience the triangular field element will be divided into four subelements, as shown in Figure 5.10. Accordingly, the integration over the field element will also be transformed into the sum of integrations over the four subelements, that is, I = I 1 + I 2 + I 3 + I 4 , and for each subelement,

I23

I3 I1

I4

FIGURE 5.10

Illustration of the element division.

recalculate the Ratio and follow the above rules. This procedure should be repeated if further division is required, such as I 2 = I 21 + I 22 + I 23 + I 24 . The coefficients Hˆ ij and G ij cannot be calculated using simple Gauss quadrature scheme while i = j , as the integrals corresponding to the elements are singular. The H ii coefficients are equal to 1/2 for constant elements. The G ii coefficients, which contain integrable singularities, can be evaluated analytically by employing polar coordinates (Fig. 5.11), Gii =



θ1 0

+



R1 (θ)

dR dθ +

0

2π θ1 +θ2





θ1 +θ2

θ1



R2 (θ)

dR dθ 0

R3 (θ)

dR dθ

(5.76)

0

3

a3

l31

l23

R1(q) q1

q2 R2(q)

q q3 R3(q)

a2

a1 1

l12

2

FIGURE 5.11 Geometrical definitions of integration while the source point is in the field element.

142

COOLING SIMULATION

which after evaluation yields

 tan[(θ1 + α2 )/2] 1 ln r23 tan(α2 /2) 

tan[(θ2 + α3 )/2] 1 ln + r31 tan(α3 /2) 

tan[(θ2 + α3 )/2] 1 ln + r31 tan(α3 /2)

2Ai Gii = 3

where M is the preconditioning matrix. Thus, the solution of Equation 5.67 can be obtained from



T = M −1 y

(5.77)

5.3.2.3 Solving the linear systems arising from BEM As the matrix of system arising from the boundary element method is fully populated and nonsymmetric, the solution of the system is usually obtained using direct solvers such as Gauss elimination, in which O(N 3 ) operations, and O(N 2 ) memory are required. These limit the BEM only practicable to medium-size problems. The memory requirement and the computation efficiency are two bottleneck problems in the BEM-based cooling simulation for injection molding. As Equation 5.67 must be resolved many times because there is an outer iteration employed to update the boundary conditions, seeking for a stable solver for Equation 5.67 are significant to improve the computation efficiency. Compared with the Gauss elimination method, a fast convergent iterative method, which would require the number of iterations much less than N , will decrease the computational cost to O(N 2 ). However, the classical iterative methods such as Jacobi or Gauss–Seidel, even with relaxation, either do not converge or present very low convergence rates.28 In recent years, the Krylov subspace iterative methods are attractive for solving linear systems arising from the BEM formulation, especially combining with the preconditioning technique.28 – 33 The dense nonsymmetric linear systems (Eq. 5.67) are usually ill-conditioned, as the mixed boundary conditions are complex, especially for large-scale problems. It is well known that the convergence rate of an iterative method for solving a linear system greatly depends on the spectral properties of coefficient matrix. The iterative solver might converge very slowly or even diverge for ill-conditioned linear equations. Preconditioning is an essential technique to accelerate the convergence rate of iterative solvers by improving the spectral properties of coefficient matrix. Right preconditioning is superior to left preconditioning, in general, for linear equations arising from the boundary element method.32 Let MT = y

(5.78)

Equation 5.67 can be rewritten as H˜ M −1 y = b

(5.79)

(5.80)

Equation 5.79 can be resolved more easily than Equation 5.67, as the spectral radius of its coefficient matrix H˜ M −1 is diminished. In practice, the preconditioning matrix should meet that (i) H˜ M −1 is better conditioned than H˜ and (ii) M − 1 is inexpensive to apply to an arbitrary vector. Jacobi preconditioning matrix is the simplest one and consists of only the diagonal elements of the coefficient matrix H˜ . The Jacobi preconditioner is convenient to use for no extra storage beyond that of the matrix H˜ itself is needed and M − 1 can be easily calculated. Xiao et al.28 performed numerical experiments of preconditioned Krylov subspace methods solving the dense nonsymmetric systems arising from the BEM. It was proved that Jacobi preconditioning prevails over incomplete factorization preconditioning for solving BEM equations. Biconjugate gradient (Bi-CG), generalized minimal residual (GMRES), conjugate gradient squared (CGS), quasi-minimal residual (QMR), and biconjugate gradient stabilized (Bi-CGStab) are several typical Krylov subspace iterative solvers in use. In general, GMRES, QMR, and BiCGStab are more effective in comparison with other Krylov subspace methods for solving BEM equations.28 Thus, Jacobi preconditioned GMRES, QMR, and Bi-CGStab would be preferable to solve Equation 5.67. Right preconditioned GMRES(m) for a linear system H˜ T = b is briefly described as follows, and more details about GMRES, QMR, and Bi-CGStab can be found in Reference 34. Right Preconditioned GMRES(m) Algorithm Step 1. Give an initial solution vector T 0 ∈ R n and a limited tolerance ε > 0. Step 2. Compute the residual error: r 0 = b 0 − H˜ T 0 , β = r 0 2 , v 1 = r 0 /β. Step 3. For a positive integer m ≪ n, construct the Hessenberg matrix H¯ m ∈ R (m+1)×m in an iterative manner, as follows: for j = 1, 2, . . . , m for i = 1, 2, . . . , j

w j = H˜ M −1 v j

hij = (w j , v i ) w j = w j − hij v i end for i   hj +1,j = w j 2 , if h j + 1, j ≡ 0, then set m = j and go to Step 4 wj v j +1 = hj +1 end for j

ACCELERATION METHOD

FIGURE 5.12

143

Comparisons on convergences of several iterative solvers.

Step 4. Solve the least square problem: y m = arg minm

TABLE 5.1

The Space Requirement of the BEM Matrix

y∈R

||βe 1 − H¯ m y||2 , and obtain the approximate solution by T m = T 0 + z m , where z m = M −1 V m y m . Step 5. Compute the residual error rm = ||r 0 − H˜ z m ||2 , if r m < ε, the computation is convergent and the solution is T = T m , or else update the initial solution vector by T 0 = T m , go to Step 2.

Numerical experiments have been performed to compare convergence properties between GMRES and successive over-relaxation (SOR) method. Comparisons on convergences between them are shown in Figure 5.12. The testing data arises from Equation 5.67, and the number of unknowns is 13,772. The error used to evaluate the convergence is defined as ε = b − M T 2 / b2 . Besides, an under-relaxation factor (ω ≤ 1) is adopted to guarantee convergence in the SOR. Indeed, we found that SOR might not converge even if an under-relaxation factor is employed while solving Equation 5.67 in many cases. It can be seen that the convergence rate of preconditioned GMRES, either left or right preconditioned, are both superior to SOR. Right preconditioned GMRES converge quickly than left preconditioned GMRES while the error is very small. This agrees with the numerical results in Reference 32. 5.4 5.4.1

ACCELERATION METHOD Analysis of the Coefficient Matrix

In the simulation, H˜ and G are two basic matrices that are both nonsymmetric and dense. These dense matrices require a great storage space during solution, which are

Number of Elements

Space Requirement (MB)

5,000 10,000 20,000 30,000 40,000 50,000

100 400 1,600 3,600 6,400 10,000

different from the sparse matrix generated by the finite element method. The space requirement of each matrix is O(N) = 4 × N 2

(5.81)

where N is the number of elements. A float-point number is stored as 4 bytes (or 8 bytes in double precision) in the computer memory because the current computer hardware and operating system are popularly 32-bit systems. It can be seen that the memory space required by the BEM is directly proportional to the square of the number of elements. It increases rapidly with the number of elements. Considering that the matrix G is only used to generate the linear algebra equations in the outer iteration, it can be stored in the hard drive. The space requirement of H˜ is shown in Table 5.1. On the other hand, although the matrix H˜ is a nonsymmetrical dense matrix, the absolute values of most elements in H˜ are very small and approach 0. Equation 5.73 indicates that the value is inversely proportional to the square of the distance between the source point and the field element. And also the element is 0 if the source point is in

144

COOLING SIMULATION

TABLE 5.2

The Numbers of Elements with Absolute Values in Different Intervalsa

≥ ri ≥ 5 × ri ≥ 10 × r i ≥ 50 × r i ≥ 100 × r i Case Number Total Number Number Ratio, % Number Ratio, % Number Ratio, % Number Ratio, % Number Ratio, % 1 2 a Ref.

9,420 27,756

1,542 5,808

16.4 20.9

1,014 2,577

10.8 9.3

822 1,856

8.7 6.7

554 1,038

5.9 3.7

419 885

4.5 3.2

35.

the same plane as the field element because of D(y, x ) = 0. Although the elements over the cooling system surface and the mold exterior surface are modified using Equation 5.64, the changes are relatively small. Moreover, the number of elements on the cooling system surface and the mold exterior surface is much fewer than that on the mold cavity surface. Therefore, the feature of decaying with distance remains valid. Two examples are included as follows to illustrate the above distribution feature. Define the averaged value of the i th row of H˜ as Sumi (5.82) ri = N

From Table 5.1, it can be seen that if traditional methods are used to solve the resulted algebra equations, the current computer hardware configuration is unable to meet the memory demand of large and complicated parts. If the memory requirement of the analysis exceeds the physical RAM (random access memory), the computer must use the inefficient virtual memory or swap space, which is set aside on the hard drive. The use of virtual memory would dramatically increase the solution time. So how to speed up the solution is a big bottleneck in the application of the simulation system.

where r i and Sumi are the averaged value and the sum of the absolute values of the i th row, respectively. In general, the maximum value of each row in H˜ is the element of the principal diagonal, 0.5, and the averaged value is much smaller, about O(1/N ). The numbers of elements with absolute values in different intervals are listed in Table 5.2. It can be found that the number decreases rapidly with the value. The elements with absolute values larger than 10 times of r i are less than 9%. Figure 5.13 illustrates the element ratio with absolute values larger than a given value for case 2. The statistical distribution of the number of elements of case 2 between the interval [− 100 × r i ,100 × r i ] is shown in Figure 5.14.

5.4.2

The Approximated Sparsification Method

The basic idea of the approximated sparsification method is to transform some coefficients in the BEM influence matrix to 0. This kind of approach can transform the dense matrix into a sparse matrix so that it can be solved in the RAM memory. 5.4.2.1 Direct rounding method Considering that many of the absolute values of the matrix H˜ are very small, if the coefficient in the matrix is directly assigned to 0 when its absolute value is less than a specified value ε, the matrix will be transformed to a sparse matrix. The critical point in the sparsification is the selection of ε. In this chapter, the 10,000

Element number

8000 6000 4000 2000 0 –100

FIGURE 5.13 Illustration of the element ratio with absolute values larger than a given value of case 2.35

–50

0 Times of ri

50

100

FIGURE 5.14 The statistical distribution of the number of elements of case 2.35

ACCELERATION METHOD

145

consisting of elements with large values and a remnant matrix consisting of other elements. Because the elements with large values are scarce, the dominant matrix will be sparse so that can be stored in the RAM memory. The remnant matrix is still a dense matrix. That is H˜ = M + Q

(5.84)

where M consists of elements with   ˜  Hij  > ri × Seff

(5.85)

(M + Q)T = b

(5.86)

M T = b − QT = bˆ

(5.87)

bˆ = b − QT

(5.88)

where S eff is the sparse coefficient, normally, 1 ≤ S eff ≤ 50. The other elements compose Q. With Equation 5.84, Equation 5.67 becomes FIGURE 5.15

Illustration of the topology relationship.

It follows that following criteria is used   Seff ˜  Hij  < Sumi • N

(5.83)

where H˜ ij is the coefficient in the BEM matrix H , Sumi is the sum of the i th row, S eff is the sparse coefficient, normally, 1 ≤ S eff ≤ 50, and N is the element number. 5.4.2.2 Combination method Direct rounding method will directly assign the matrix coefficient with small absolute value to 0. If we do not round off the coefficient, but combine it into another coefficient, the approximation error will be reduced. First, the concept of adjacent elements is introduced. After the boundary surface is discritized into triangular elements, the topology relationship will be set up among all the elements. As shown in Figure 5.15, for the i th element, the elements signed of , , and are its first-, second-, and third-loop adjacent elements, respectively. During the calculation of the BEM matrix, for each row, the i th field element relates to the i th column. If the coefficient is so small as to meet the sparsification criteria, this coefficient will be combined with the maximal coefficient (nonzero) of its adjacent elements. The combination procedure will check the first-loop adjacent elements first. If all the coefficients of the first-loop elements are 0, then the second-loop elements are checked, and so on.

5.4.3

The Splitting Method

The basic idea of the splitting method is to transform the dense BEM matrix into two matrices: a dominant matrix

where

In the above equation, QT is relatively small because of the attribute of Q. Considering that b has to be updated in the outer iteration and QT is relatively small, the item QT can be transformed from the inner iteration to the outer iteration. That is, the coupling simulation procedure described in Figure 5.8 is modified, as shown in Figure 5.16, where the modified steps are drawn in a shadow background. As mentioned earlier, for complicated parts with large number of elements, H˜ are too large to be stored in the RAM memory, but M is so sparse that it can be done. Q is still a dense matrix that is allocated on the hard drive and only used in the outer iteration. Consequently, the solution time of M T = bˆ is much shorter than that of H˜ T = b. The selection of S eff is a critical point that decides the sparsification ratio. If S eff is too small, M will be very large and consequently the inner iteration slows down, especially in the condition of exceeding the RAM memory. On the other hand, if S eff is too large, the values of elements in Q will correspondingly be large, and consequently, the iterative number of the outer iteration increases greatly. In extraordinary instances, larger S eff may cause the divergence of the solver. After the BEM matrix is sparsified, the Yale Sparse Matrix Format is used for the sparse matrix M , which yields huge savings in memory.36 And the Jacobi right preconditioned Krylov subspace iterative methods can be employed to solve the resulting linear system of equations.

146

COOLING SIMULATION

FIGURE 5.17 Comparison of the CPU time used by the conventional BEM and FMM-BEM.

FIGURE 5.16 The modified flow chart of the coupling procedure in the splitting method.

5.4.4

The Fast Multipole Boundary Element Method

As we discussed earlier, computational efficiency and memory requirement are two drawbacks in the BEM-based cooling analysis. Almost most acceleration methods reported concerning the BEM-based cooling analysis assume that the difference in temperature is moderate over the mold surface. This assumption is just valid for solving mold thermal conduction problems. On the other hand, these two drawbacks are also inherent problems existed in general BEM applications. In the past decade, an acceleration approach called fast multipole boundary element method (FMM-BEM) has been developed to overcome such drawbacks in the traditional BEM, and it has been successfully applied to solve elastostatics problems, potential problems, fluid mechanics problems, and so on. The use of FMM-BEM reduces the memory requirements to O(N ) and the operation count to O(N ). This makes that solving large-scale problems with several million degrees of freedom by BEM on a desktop computer within hours is possible. Figure 5.17 illustrates comparisons between the CPU times used by the conventional BEM and FMM-BEM for solving 2D heat conduction problems.37 It can be found that FMM-BEM is extremely efficient compared to the conventional BEM, and it seemed to be attractive to BEM-based cooling analysis, especially for solving large mold thermal conduction problems.

The fast multipole method (FMM) was introduced by Rokhlin38 and was extended to reduce the computational cost for the pair-wise force calculation in three-dimensional N -body problems. It was nominated as one of the top 10 algorithms developed in the past century. In the past decade, the FMM has been applied to various numerical analyses, including BEM. In the following sections, the FMM-FEM will be briefly introduced. We begin the introduction with several important concepts in the fast multipole method. 5.4.4.1 Multipole expansion Assuming x0 is a point  →  − y  is satisfied close to the field point x , and − x 0 x  < x→ (Fig. 5.18), the fundamental solution of Laplace’s equation can be expanded into a series of products of functions of y and those of x as follows: T ∗ (y, x ) =



−→ φ i (− x→ 0 y)ψi (x 0 x )

(5.89)

i

where ψ i is a regular function near the point x 0 , φ i is a regular function at infinity. It should be noted that y and x are separated now because of the introduction of an “intermediate point” x 0 . Let Ŵx0 be a subset of boundary Ŵ and consists of a union of boundary elements l (l = 1, 2, . . . , N )  → Ŵ− y  (x ∈ Ŵx0 ). The in which all points satisfy − x 0 x  < x→ integral in Equation 5.50 can be deduced to

Ŵx

0

T ∗ (y, x )q(x )dŴ =

where Mi (x 0 ) =

N  l=1



Ŵl

φ i (− x→ 0 y)Mi (x 0 )

(5.90)

i

ψi (− x→ 0 x )q(x )dŴ

(5.91)

Mi (x 0 ) is called the multipole moment centered at x 0 .

ACCELERATION METHOD

Γ x0 y ′0

x

x0

y0

and it is called the coefficient of the local expansion centered at y 0 . The translation described by Equation 5.96 is called multipole moment to local expansion (M2L) translation. 5.4.4.4 L2L translation about the point y ′0 as y→ ξ j (− 0 y) =

x ′0

y Multipole expansion



Local expansion

Related points in the fast multipole method.

5.4.4.2 M2M translation Suppose that the function ψ i can be expanded to the following form: (5.92)

i

j −−→ where x ′0 is a new location near x 0 and αi (x ′0 x 0 ) is the expansion coefficient. By substituting Equation 5.92 into Equation 5.91, the multipole moment centered at x ′0 can be translated from the multipole moment centered at x 0 , if the point x 0 is moved to the new location x ′0 , expressed as



j −−→ αi (x ′0 x 0 )Mi (x 0 )

(5.93)

i

This translation is called multipole moment to multipole moment (M2M) translation. 5.4.4.3 Local expansion Let a point near the   →y0 be−− collection point y and satisfy − y 0 y  < x 0→ y 0 , the function φ i can be expanded about the point y 0 (e.g., using the Taylor series expansion) as follows:  → −→ x→ βli (x−− (5.94) φ i (− 0 y) = 0 y 0 )ξl (y 0 y) l

where ξ l is a function and βli is a certain expansion coefficient. Substituting Equation 5.94 into Equation 5.90,  T ∗ (y, x )q(x )dŴ = ξ l (− y→ (5.95) 0 y)Ll (y 0 ) Ŵx

l

0

where Ll (y 0 ) =

 l

→ βli (x−− 0 y 0 )Mi (x 0 )

(5.97)

i

L2L translation

Mj (x ′0 ) =

−→ j −→ j − χi (y ′0 y 0 )ξi (y ′0 y)

j

M2L translation

 j −−→ −→ ψj (x ′0 x ) = αi (x ′0 x 0 )ψi (− x→ 0x )

Expanding the function ξ l

where χi is the expansion coefficient. The following formula can be evaluated:

M2M translation

FIGURE 5.18

147

(5.96)

Li (y ′0 ) =



−→ j − χi (y ′0 y 0 )Ll (y 0 )

(5.98)

l

The above formula means that the coefficients of the local expansion centered at y ′0 can be translated from these centered at y 0 . The translation defined by Equation 5.98 is called local expansion to local expansion (L2L) translation. The FMM-FEM can be regarded as matrix-free technology in conjunction with iterative solvers for linear system. Massive paired collection point to source point calculations in the BEM are translated into cluster-to-cluster calculations, and thus accelerated by the FMM. The main procedures are as follows: Step 1. Perform discretizations over the boundary in the same manner as in the conventional BEM. Step 2. Construct the tree structure of the mesh. First, find a cube that can contain the entire problem domain and is called the cell of level 0. Then divide the level 0 cell into eight equal child cells of level 1. Continue this dividing until a cell only contains elements less than a given number, and this cell is called a leaf . Step 3. Upward. Giving a right vector, compute the multipole moments starting with all leaves using Equation 5.91, and x 0 is the centroid of the cell. Then, compute the multipole moments of a nonleaf cell from these of its eight children using M2M translation. This procedure is repeated, tracing the tree structure of cells upward, and stops at level 2. Step 4. Downward. Compute the coefficients of local expansion tracing the tree structure of cells downward. The local expansion associated with a cell C is the sum of two parts. One is calculated from moments associated with cells, which are not adjacent but near enough to the cell C (called the interaction list of C, in strictly speaking), using the M2L translation. Another is calculated from the moment of its parent cell by using the L2L translation, namely shifting the expansion point form the centroid of its parent cell to that of itself.

148

COOLING SIMULATION

that codes need be rewritten because there are significant changes in FMM-BEM compared to traditional FMM. 5.4.5

FIGURE 5.19 Computational flowchart of the FMM-FEM (solved by GMRES).

Step 5. Evaluation of the integrals in Equation 5.51. For a collocation point y on an element in a leaf C, the integrals in Equation 5.51 can be computed in following manners. Contributions from elements in the leaf C and its adjacent cells are evaluated directly as in the conventional BEM. Contributions from all other cells are evaluated using the local expansion; that is, shifting the expansion point from the centroid of C to the collocation point y. Step 6. Iterations of the solution. Update the right vector, and back to Step 3 for the matrix and unknown vector multiplication until the solution converges. The iteration steps are illustrated in Figure 5.19. Applications of FMM to BEM for solving threedimensional Laplace’s equation have been investigated by Greengard et al.,39 Nishimura,40 and other researchers.41 – 45 The details about the multipole expansion, M2M, M2L, and L2L in three-dimensional Laplace’s equation can be found in Reference 43. However, to the author’s knowledge, there are no publications and applications previously concerning the cooling analysis implemented in FMM-BEM. It might be due to that the traditional BEM-base cooling simulation can perform an analysis in 2 h in most cases, which meets the needs of practical use. Another reason might be

Results and Discussion

To demonstrate the usefulness of the simulation system, six typical cases with different element numbers were considered. Table 5.3 shows their element numbers, geometries and cooling configurations. The computer with a 2 GB RAM and an Intel Pentium 2.8 G CPU is employed in the numerical experiments, running Windows XP. All these cases adopted the same material and processing conditions, listed in Table 5.4. What listed in Table 5.5 are the associated material properties required for the numerical simulation. The time taken for generating the BEM matrix is compared in Fig. 5.20. It is revealed that comparing with the fixed integral order, the automatic selection of the integral order can save as much time as two-thirds. And all the errors are within 1%. The solution times and memory requirements of the cases are listed in Table 5.6. It can be seen that the memory requirement and solution time of the full matrix increase rapidly with the number of elements. Once the required memory exceeds the physical RAM, and a certain proportion of virtual memory has to be used, the solution time will increase tremendously, as shown in Figure 5.21. The direct rounding, combination methods, and splitting methods all can transform the full matrix into a sparse matrix. The nonzero coefficient ratio after sparsification is about 5–10%, which reduces the required memory by 90–95%. Because all of the sparse matrix can be stored in the RAM, the solution time increases more gradually with the number of elements. The solution errors of all cases are listed in Table 5.7. It is also found that the errors of the direct rounding are apparent and great to be accepted, so this method is of no practical use. The combination method and the splitting method both reduce the error significantly. Its averaged error is about 2 ◦ C, within the error limit of engineering design. Figure 5.22 shows the simulation results of the case 4 by the full, rounding, combination, and splitting methods, respectively. For the combination method, combining some coefficients of adjacent elements into one coefficient is equivalent to combining some adjacent elements into one element. The temperature of a representative element is used as their averaged temperature. Considering that there is a continuity of the mold temperature on the cavity surface, and the temperature shifts very smoothly from position to position, this kind of treatment is reasonable, and the error is acceptable. For the splitting method, strictly speaking, it is equivalent to the original full method. However, as the temperature distribution is included in the right-hand side of Equation 5.87 and it should be updated in the outer

ACCELERATION METHOD

TABLE 5.3

149

The Numbers of Elements, Geometries, and Cooling Configurations of the Casesa Number of Elements Mold Cavity Cooling Channel Mold External Surface

Case Number

Total

1

9,420

8,318

506

596

2

16,584

15,664

264

656

3

27,756

26,404

820

524

4

34,566

33,240

690

636

Geometry and Cooling Configuration

(continued)

150

COOLING SIMULATION

TABLE 5.3

(Continued )

Total

Mold Cavity

Number of Elements Cooling Channel

Mold External Surface

5

41,014

39,876

562

576

6

49,110

47,932

598

580

Case Number

a Ref.

35.

TABLE 5.4

The Material and Processing Conditionsa

Parameter Polymer material Mold material Coolant material Injection temperature, ◦ C Ejection temperature, ◦ C Coolant temperature, ◦ C Coolant flow rate, m3 /min a Ref.

Geometry and Cooling Configuration

Value ABS PA757 Nickel–cobalt alloy Water 232 80 25 ◦ C 7 × 10−3

35.

TABLE 5.5 The Materials Properties of the Polymer, Mold, and Coolanta Properties Material type Mass density, kg/m3 Specific heat, J/(kg·◦ C) Thermal conductivity, W/(m·◦ C) a Ref.

iterations, there are errors arising from the inconsistency of the convergence criterion. The quantitative distribution of errors using splitting method in case 4 is shown in Figure 5.23. It can be found that most of the errors are limited to 0.1 ◦ C. 5.5 SIMULATION FOR TRANSIENT MOLD TEMPERATURE FIELD In the cycle-averaged temperature field modeling, it is supposed that the mold temperature is relatively stable during the continuous injection process. Thus, the cavity temperature is assumed to be a constant at the filling

Polymer

Mold

Coolant

ABS PA757 1083.0

Nickel–cobalt alloy 8890.0

Water 996.95

2474.0

460.0

4180.5

0.23

60.6

0.609

35.

and packing stages, accordingly the thermal and motion problem for the melt during these stages can be solved in a separate way. Indeed, the cavity surface temperature varies acutely during filling stage and plays an important role in melt-flow behavior. To improve the accuracy of the filling simulation, the cavity surface cycle-averaged temperature distribution obtained from cooling simulation can be regarded as a boundary condition for filling simulation. Similarly, the melt temperature distribution at the moment when the melt fills the cavity can be a boundary condition for cooling simulations.

SIMULATION FOR TRANSIENT MOLD TEMPERATURE FIELD

FIGURE 5.20 TABLE 5.6

Case Number 1 2 3 4 5 6

151

The required time for generating the BEM matrix of the cases.

The Solution Results of the Cases

Full Memory Solution Size, MB Time, s 305 1,100 3,081 4,779 6,729 9,647

158 977 2,865 14,857 26,693 52,463

Direct Rounding Method Memory Sparse Solution Size, MB Ratio, % Time, s 27 59 208 321 332 478

8.72 5.40 6.74 6.72 4.94 4.96

163 193 659 1,554 3,052 5,547

Combination Method Memory Sparse Solution Size, MB Ratio, % Time, s 31 79 264 432 424 694

10.05 7.15 8.57 9.05 6.30 7.19

TABLE 5.7

Case Number 1 2 3 4 5 6

FIGURE 5.21 the cases.

The required memory size and solution time of

A high cavity surface temperature at the filling stage has a lot of advantages, such as enhancing the flow ability of the polymer melt, reducing welding lines, and improving the product quality. However, a high mold temperature will increase the cooling time to cool down part and

184 355 1,860 3,221 4,673 6,913

Splitting Method Memory Sparse Solution Size, MB Ratio, % Time, s 27 59 206 320 330 482

8.72 5.40 6.74 6.72 4.94 4.90

138 731 2,420 3,863 7,244 13,119

The Solution Errors of the Cases

Direct Rounding Method

Error, ◦ C Combination Method

Splitting Method

−9.30∼1.00 −10.96∼0.00 −19.27∼11.94 −13.49∼13.41 −10.54∼7.77 −5.21∼10.65

−0.39∼0.38 −1.29∼1.44 −0.33∼1.85 −1.10∼2.21 −2.83∼1.67 −0.65∼0.69

−0.01∼0.22 −0.24∼0.21 −2.05∼3.67 −4.03∼1.83 −2.79∼0.51 −0.15∼2.69

accordingly increase the cycle time. In the traditional injection molding process, the mold temperature must be kept in applicable range to balance product quality and production efficiency. The variotherm mold is a new injection molding technology and is rapidly developed in the past decade. In the variotherm mold, the mold cavity surface is first heated to a high mold temperature before melt injection, usually higher than the glass transition temperature of the polymer, second kept at the high

152

COOLING SIMULATION

temp. (ºC)

temp. (ºC) 98.47

92.48 84.05

89.29

75.61

80.10

67.18

70.92

58.74

61.74

50.31

52.55

41.87

43.37

33.44

34.18 25.00

25.00

(a)

(b)

temp. (ºC) temp. (ºC)

98.32

99.19

89.15

89.92

79.99

80.64

70.82

71.37

61.66

62.10

52.49

52.82

43.33

43.55

34.16

34.27

25.00

25.00

(c)

(d)

FIGURE 5.22 The computed steady temperature field of case 4 by (a) full method, (b) direct rounding method, (c) combination method, and (d) splitting method.35 (See insert for color representation of the figure.)

temperature during filling and packing stages, and finally cooled down rapidly to solidify the shaped polymer melt in cavity for demolding.46 This technology includes the advantage of high mold temperature during filling stage and, meanwhile, ensures that the cooling time will not increase, thus the cycle time will not rise accordingly. As the mold temperature variation is usually significant in variotherm injection molding, the mold temperature should be accurately controlled. There are a lot of kinds of variotherm processes, such as rapid heat cycle molding (RHCM) process,10 pulsed cooling process,47 and coolant temperature switch.48 Therefore, predicting the transient mold temperature field by performing a coupling mold and part transient cooling simulation is significant and has been developed in

the past few years for the following reasons: (i) It predicts the temperature distribution of polymer part accurately and thus makes a better understanding on melt-flow behaviors; (ii) As the mold temperature changes rapidly during a process cycle, the cycle-averaged temperature model is not suitable any more, whereas the transient temperature distribution should also be studied to accurately control the mold temperature. Considering the mold region , the transient thermal conduction is given by



 ∂ ∂ ∂Tm ∂Tm ∂Tm Km + Km = ∂t ∂x ∂x ∂y ∂y

 ∂Tm ∂ Km + q(t) (5.99) + ∂z ∂z

ρm Cm

SIMULATION FOR TRANSIENT MOLD TEMPERATURE FIELD

FIGURE 5.23 The quantitative distribution of errors of case 4 by the splitting method.

where ρ m , C p,m , K m , and T m denote the density, specific heat, thermal conductivity, and temperature of the mold, respectively, and q(t) is the heat generation term of the embedded heaters in the mold. The boundary conditions defined over cooling channel surface (Ŵ c ) and exterior of mold boundary surface (Ŵ e ) are same as these in cycleaveraged temperature model, namely, defined by Equations 5.13 and 5.17, respectively. The boundary condition over mold cavity surface (Ŵ p ) is defined by −Km

∂Tp ∂Tm = Kp = qˆ ∂n ∂n

(5.100)

and Tm = Tp

(5.101)

where the K p and T p denote the thermal conductivity and temperature of the polymer melt, respectively. The two equations above mean that the heat flux and temperature of the mold must be in agreement with the part on the cavity surface, during the filling, packing, and cooling stages. On the other hand, the energy equation of the polymer melt is



 ∂Tp ∂Tp ∂Tp ∂Tp ∂Tp ∂ = Kp +u +v +w ρp Cp ∂t ∂x ∂y ∂z ∂x ∂x



 ∂Tp ∂Tp ∂ ∂ Kp + Kp + ηγ˙ 2 (5.102) + ∂y ∂y ∂z ∂z where C p is the specific heat of the polymer melt, η is the sheer viscosity, γ˙ is the effective sheer rate, and u, v, w are components of velocity in x , y, and z directions, respectively. To perform a coupling transient cooling simulation, Equations 5.99 and 5.102 should be

153

simultaneously solved or be solved in an iterative manner to satisfy the boundary condition described by Equations 5.100 and 5.101. As the injection process is a fully continuous operation, and the mold heat transfer behavior becomes a cyclic variation after a certain transient period at the beginning, the coupling transient molding simulation should start from the injection operation beginning and terminates when a stationary state is reached. It is assumed that the mold temperature is initially equal to the coolant temperature when the operation begins, and the mold temperature distribution at the end of the previous cycle is used as the initial condition for the new cycle. As we discussed in Chapter 4, there are several models and numerical implements for the filling and packing simulation. Consequently several coupling transient cooling simulations have been reported.13,15 – 19,46,47,49 Cao et al.19 presented a coupled method that determines the interface temperatures by filling and cooling analyses simultaneously. In their works, the heat conduction problem was solved by a three-dimensional finite element method and the filling and packing simulation was implemented based on dual-domain technology. The coupling procedure is briefly described as a tutorial in the following text. In the dual-domain technology, the melt pressure in incompressible filling stage is governed by



 ∂p ∂ ∂p ∂ S + S =0 (5.103) ∂x ∂x ∂y ∂y with S=



b 0

(b − z)2 dz η

(5.104)

where z is the coordinate along the thickness direction, b is the half thickness, and p is the pressure. Similarly, in the compressible packing stage, the governing equation can be expressed as



 ∂ ˜ ∂p ∂ ∂p ∂p − = −F (5.105) − S˜ S G ∂t ∂x ∂x ∂y ∂y ¯ and F are defined in Reference 50. Assume where G, S, that heat conduction in the polymer occurs only in the gapwise direction and the melt flows in the laminar motion. The energy equation during filling and packing stages can be modified as 

∂Tp ∂Tp ∂ 2 Tp ∂Tp = Kp 2 + ηγ˙ 2 (5.106) +u +v ρp Cp ∂t ∂x ∂y ∂z After the polymer melt flow ceases, the convection and viscous heat terms in the above equation disappear. The thermal conduction becomes the only form for polymer parts, and the above equation can be simplified as ρp Cp

∂ 2 Tp ∂Tp = Kp 2 ∂t ∂z

(5.107)

154

COOLING SIMULATION

During the mold opening stage, the whole mold is explored in the air except the channels filling with flowing coolant. Thus, the boundary condition on the cavity surface can be described by Equation 5.17, and the mold thermal problem does not need to couple with the polymer part. At each time step, the pressure distribution is first determined by solving Equation 5.103 or 5.105, then the melt velocity and flow front are updated, and finally the part temperature field and the mold temperature field are calculated in an iterative manner. Namely, while solving the part temperature field, the mold cavity temperature is used as the boundary condition, in other words, the mold surface temperature is assumed to be known (determined by solving from the mold thermal conduction problem). Consequently, the melt heat flux over the mold cavity surface is calculated, and the boundary condition described in Equation 5.100 is updated by an under-relax scheme with previous iteration values. Then the mold temperature field is calculated by solving Equation 5.99 again. The above procedure is repeated until the temperature distribution over cavity surface is convergent. The finite element method can be used to determine the pressure from Equation 5.103 or 5.105, while the finitedifference approach is employed to solve Equations 5.106 and 5.107 and obtain melt temperature distributions inside the cavity. What is more, the implementation details can be found in the filling simulation chapter. For the mold heat conduction problem, Equation 5.99 can be solved by the Galerkin finite element method. Supposing the mold domain is discretized into Ne finite elements, the temperature field is represented by means of the shape function of the element nodes. After the spatial and time discretization for Equation 5.99,51 it gives 

ξK + C T t+t = ξ R t+t + (1 − ξ )R t t 

C − (1 − ξ )K T (5.108) + t with K =

 Ne



B Tθ Km B θ d +



N T ha N dŴ

Ŵe

 + N T hc N dŴ

(5.109)

Ŵc

C =



R=



Ne

Ne

 N ρm Cm N d T





N T q(t)d +

T N hc Tc dŴ − + Ŵc



(5.110) N T ha Ta dŴ

Ŵe T

Ŵp

N qˆ dŴ



(5.111)

where ξ is an adjustable parameter of time, N is the shape function, and B e is the partial derivative of the shape function, and other symbols are the same as defined above. If there are no embedded heater in the mold, the heat generation term q(t) disappears from Equation 5.99, the mold heat conduction problem can be also solved using the boundary element method.15 Tang et al.13 assumed that the filling time is relatively short, thus the convection and viscosity heat exchange in the polymer part is neglected. Consequently, the cooling simulation in plastic injection molding becomes a heat conduction problem, and the transient temperature distributions in the mold and the polymer part can be simultaneously computed by Galerkin finite element method described earlier. To decrease the memory requirement, a matrix-free Jacobi conjugate gradient scheme was designed to solve linear equations arising from the Galerkin finite element method. In conclusion, despite a lot of benefits that can be acquired from the transient cooling simulation, there is still a severe problem to be solved: how to improve the simulation efficiency and decrease the computational time, as it must couple the filling and packing simulation and cover dozens of molding cycles. At present, the transient cooling simulation is more often applied in variotherm injection molding process, rather than the traditional injection molding.

REFERENCES 1. Kwon T.H., Mold cooling system design using boundary element method. Journal of Engineering for Industry, 1988. 110(4): 384–394. 2. Himasekhar K., Lottey J., Wang K.K., CAE of mold cooling in injection molding using a three-dimensional numerical simulation. Journal of Engineering for Industry, 1992. 114(2): 213–221. 3. Chen S.C., Hu S., Simulations of cycle-averaged mold surface temperature in mold-cooling process by boundary element method. International Communications in Heat and Mass Transfer, 1991. 18(6): 823–832. 4. Chen S.C., Chung Y.C., Simulations of cyclic transient mold cavity surface temperatures in injection mold-cooling process. International Communications in Heat and Mass Transfer, 1992. 19(Compendex): 559–568. 5. Rezayat M., Burton T.E., A boundary-integral formulation for complex three-dimensional geometries. International Journal for Numerical Methods in Engineering, 1990. 29(2): 263–273. 6. Park S., Kwon T., Thermal and design sensitivity analyses for cooling system of injection mold, Part 1: thermal analysis. Journal of Manufacturing Science and Engineering, 1998. 120: 287.

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7. Park S., Kwon T., Thermal and design sensitivity analyses for cooling system of injection mold, part 2: design sensitivity analysis. Journal of Manufacturing Science and Engineering, 1998. 120: 296. 8. Park S., Kwon T., Optimization method for steady conduction in special geometry using a boundary element method. International Journal for Numerical Methods in Engineering, 1998. 43(6): 1109–1126. 9. Hioe Y., Chang K., Zuyev K., et al., A simplified approach to predict part temperature and minimum “minimum safe” cycle time. Polymer Engineering and Science, 2008. 48(9): 1737–1746. 10. Kim B.H., Donggang Y., Shia-Chung C., Rapid thermal cycling of injection molds: an overview on technical approaches and applications. Advances in Polymer Technology, 2008. 27(4): 233–255. 11. Chen S., Wang I., Chiou Y., et al. An investigation on the temperature behavior in mold embedded with heater, in SPE Annual Technical Conference - ANTEC, Conference Proceedings. 2007, Cincinnati, Ohio, USA: Society of Plastic Engineers. 717–721. 12. Hu, S.Y., Cheng N.T., et al., Effect of cooling system design and process parameters on cyclic variation of mold temperatures—simulation by DRBEM. 1995, Institute of Materials: London, ROYAUME-UNI. 13. Tang L.Q., Pochiraju K., Chassapis C., et al., Threedimensional transient mold cooling analysis based on Galerkin finite element formulation with a matrix-free conjugate gradient technique. International Journal for Numerical Methods in Engineering, 1996. 39(18): 3049–3064. 14. Tang L.Q., Chassapis C., Manoochehri S., Optimal cooling system design for multi-cavity injection molding. Finite Elements in Analysis and Design, 1997. 26(3): 229–251. 15. Qiao H., Transient mold cooling analysis using BEM with the time-dependent fundamental solution. International Communications in Heat and Mass Transfer, 2005. 32(3–4): 315–322. 16. Lin Y.W., Li H.M., Chen S.C., et al., 3D numerical simulation of transient temperature field for lens mold embedded with heaters. International Communications in Heat and Mass Transfer, 2005. 32(9): 1221–1230. 17. Chiou Y.C., Chou Y.Y., Chiu H.S., et al. Integrated true 3D simulation of rapid heat cycle molding process, in SPE Annual Technical Conference - ANTEC, Conference Proceedings. 2007, Cincinnati, Ohio, USA: Society of Plastic Engineers. 2577–2580. 18. Chiou Y.C., Wang H.C., Chiu H.S., et al., Thermal feature of variotherm mold in injection molding processes, in SPE Annual Technical Conference - ANTEC, Conference Proceedings. 2009, Chicago, Illinois, USA: Society of Plastic Engineers. 2491–2495. 19. Cao W., Shen C., Li H., Coupled Part and Mold Temperature Simulation for Injection Molding Based on Solid Geometry. Polymer-Plastics Technology and Engineering, 2006. 45(6): 741–749.

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20. Yu C.J., Sunderland J.E., Determination of ejection temperature and cooling time in injection molding. Polymer Engineering & Science, 1992. 32(3): 191–197. 21. Gebhart B., Editor. Heat Transfer. 2nd ed. 1971, New York: McGraw-Hill. 22. Eckert E.R.G., Drake R.M., Editors. Analysis of Heat and Transfer. 1972, New York: McGraw-Hill. 23. Liang J.Z., Ness J.N., The calculation of cooling time in injection moulding. Journal of Materials Processing Technology, 1996. 57(1–2): 62–64. 24. Stelson K.A., Calculating cooling times for polymer injection moulding. Proceedings of the Institution of Mechanical Engineers Part B-Journal of Engineering Manufacture, 2003. 217(5): 709–713. 25. Zarkadas D.M., Xanthos M., Prediction of cooling time in injection molding by means of a simplified semianalytical equation. Advances in Polymer Technology, 2003. 22(3): 188–208. 26. White J.L., Computer Aided Engineering for Injection Molding. 1983, New York: Hanser. 27. Brebbia C.A., Telles J.C.F., Wrobel L.C., Editors. Boundary Element Techniques-Theory and Applications in Engineering. 1984, Berlin: Springer. 28. Xiao H., Chen Z., Numerical experiments of preconditioned Krylov subspace methods solving the dense non-symmetric systems arising from BEM. Engineering Analysis with Boundary Elements, 2007. 31(12): 1013–1023. 29. Saitoh A., Kamitani A., GMRES with new preconditioning for solving BEM-type linear system. IEEE Transactions on Magnetics, 2004. 40(2 Part 2): 1084–1087. 30. Fischer M., Perfahl H., Gaul L., Approximate inverse preconditioning for the fast multipole BEM in acoustics. Computing and Visualization in Science, 2005. 8(3): 169–177. 31. Valente F.P., Pina H.L., Conjugate gradient methods for threedimensional BEM systems of equations. Engineering Analysis with Boundary Elements, 2006. 30(6): 441–449. 32. Chen Z., Xiao H., Preconditioned Krylov Subspace Methods Solving Dense Nonsymmetric Linear Systems Arising from BEM, in 7th International Conference on Computational Science. 2007, Beijing: Springer. 33. Rui P.L., Chen R.S., Sparse approximate inverse preconditioning of deflated block-GMRES algorithm for the fast monostatic RCS calculation. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 2008. 21(5): 297–307. 34. Young D.M., Iterative solution of large linear systems, 2003, New York: Dover Publications. 35. Zhou H., Zhang Y., Wen J., et al., An acceleration method for the BEM-based cooling simulation of injection molding. Engineering Analysis with Boundary Elements, 2009. 33(8–9): 1022–1030. 36. Bank R., Douglas C., Sparse matrix multiplication package (SMMP). Advances in Computational Mathematics, 1993. 1(1): 127–137.

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6 RESIDUAL STRESS AND WARPAGE SIMULATION Fen Liu Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong, China

Lin Deng and Huamin Zhou State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

During the molding process, shrinkage and warpage of products are generally constrained by the surrounding walls of mold within the cavity. Meanwhile, residual stresses are built up within the mold. After the ejection, the product is removed from the cavity and exposed to a completely different set of thermal and mechanical boundary conditions. As a result, part of the generated stress will be relieved by deformation, and the product is allowed to contract without any restrictions or external constraints, which leads to shrinkage or warpage. Flow stresses during the filling stage will deteriorate the surface optical property of products,1 while warpage and deformation due to thermal stresses produced in postfilling stages are detrimental to the dimension of products and even result in local cracking. With an increasing demand on dimension stability and quality of products in plastic industry, warpage has become an important criterion in evaluating the quality of injection-molded products. Prediction of residual stresses and warpage therefore provides a starting point for enhancing the quality of molded products through the optimization of processing conditions.

6.1

RESIDUAL STRESS ANALYSIS

Residual stresses can be defined as mechanical stresses in molding without the application of external force,2

which remain after the ejection and cool down to the ambient temperature. An accurate description of residual stresses definitely requires a clear understanding of the whole injection molding process. A typical profile of residual stresses in the gapwise direction of injected molded products is shown in a parabolic form, with tensile stresses at the center and surface and compressive stresses in the subskin region. Residual stresses in injection molding will affect the usability, such as the dimensional accuracy and mechanical strength. 6.1.1

Development of Residual Stress

The principal cause of residual stresses may be the different cooling rates throughout the mold.2 Other causes include the holding pressure and flow effects. According to different origins, residual stresses in injection molding can be classified into two groups: flow-induced stresses and thermal stresses. Owing to the fountain effect at the melt front during the filling stage, the polymer melt flows from the cavity center toward the mold wall, and an orientated frozen layer is gradually deposited on the cold wall,3 which retains the maximum elongational orientation and generates extensional stresses on the surface of the mold. As flow stresses are associated with molecular orientation, the frozen-in birefringence may be used in measuring flow stresses indirectly. Research from Baaijens,4 Douven,5

Computer Modeling for Injection Molding: Simulation, Optimization, and Control, First Edition. Edited by Huamin Zhou.  2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc.

157

158

RESIDUAL STRESS AND WARPAGE SIMULATION s

s

(a)

FIGURE 6.1

(b)

Stress profile for (a) free quench products and (b) injection-molded products.

Zoetelief et al.,6 and Kamal et al.7 observed that the flow residual stress is generally 1–2 orders of magnitude lower than the thermal residual stress. Wimberger-Friedl et al.8 compared the residual stresses of injection-molded PS (polystyrene) and PC products, and examined their origins and mechanisms. He reported a lower flow residual stress in the material with a low molecular weight and shorter relaxation time. The schematic drawing of residual stresses for free quench parts and injection-molded products is shown in Figure 6.1. In the simplest case of free quench, when an amorphous plate is molded, there are different cooling rates in the thickness, and neighboring layers may hinder the activity of each other. Typically, the outer layer is cooled faster. As this layer solidifies, it will contract and form the compressive stress on the core layer. This compressive stress may soon relax because of the viscous deformation of polymer melts. With the elapse of time, the core layer begins to solidify, but the contraction is impeded by the solid outer shell. The internal constraint of deformation among different layers, therefore, leads to tensile stress in the core and compressive stress in the outer layers. In the gapwise direction of a molded product, the residual thermal stresses are in equilibrium. The case of an injection-molded product is, however, much more complicated. During postfilling stages, the material is solidified under a holding pressure. And there exists a large temperature gradient across the thickness of the sample because of the low thermal conductivity of polymers,9 which in turn results in an inhomogeneous temperature distribution. As the material cools down, the sample contracts. As a result of different cooling rates throughout the body, the contraction extent is also differential. However, the deformation of each part is restrained by adjacent elements, thus producing thermal stresses. These stresses will remain as residual thermal stresses in the sample. Owing to different compressibility of the outer solidified layers and the glassy-transforming material, different levels of stress may be induced, resulting in the nonuniform distribution of residual stresses. Specifically, the core region may still be

under pressure once the product is removed, leading to an expansion of the mold after ejection until an equilibrium state of stresses is obtained. The principles of the development of residual thermal stress will be illustrated by means of a schematic representation of an injection molding experiment, analogous to the example of Struik for the free quench experiment.10 Cooling is idealized in five steps in which the pressure also varies as a function of time (Fig. 6.2). As the cooling front moves inward, the temperature drops from T h >T g to T l < T g , where T g is the glass transition temperature. It is assumed that the material behaves as an ideal fluid whenT > T g , thus σ = − pI , and turns linear elastic when T < T g. At t = t 0 , the mold is completely filled, and the residual thermal stresses develop as follows: • t = t 0 : full mold, the temperature is homogeneous and equals T h , the pressure is zero, and the material is free of stresses. • t = t 1 : contraction of the outer layers is hindered by the nonslip condition at the mold walls. A small tensile stress σ xx (z ) is introduced in the solidified outer shells. • t = t 2 : a holding pressure p acts on the melt, resulting in compressive stress σ = − p h , compressing the rigid shell. As a result, the stress in these layers is decreased by σ = vp h /(1 − v ) (assuming that all displacement in the x -direction in the shell are suppressed), with v Poisson’s ratio. • t = t 3 : during the holding stage, the pressure remains constant, while a small layer of the material solidifies during the time interval [t 2 , t 3 ]. Contraction of the newly cooled material decreases the compressive stress in the surface layers. • t = t 4 : the pressure is set to zero and the stress in the melt disappears. The stresses in the rigid shell increases by σ . • t = t 5 : finally, the product is released from the mold. Further cooling is now similar to a free quench. This results in tensile stresses in the core, which are in equilibrium with the stresses in the outer layers.

RESIDUAL STRESS ANALYSIS

t2

159

t3

T

FIGURE 6.2

Residual stress development in injection-molded products.

z=h

analysis of residual stresses should therefore take into account these two main types of residual stresses.

z

z = z1 z = h/2 y

6.1.2 x

z=0

FIGURE 6.3

Rectangular section of the flat specimen.

Instead of compressive surface stresses in the case of a free quench experiment, tensile stresses come at the surface of the sheet. These tensile stresses are in the order of 10 MPa and depend mainly on the magnitude and duration of the holding pressure. The residual thermal stresses are induced by a hydrostatic pressure in combination with rapid cooling, so the stresses in the x -direction must equal those in the y-direction (see Fig. 6.3 for definitions). Hastenberg et al.10 measured the residual stresses of amorphous plates in injection molding with the layer removal method and reported a high tensile stress in the skin layer, along with a compressive stress in the subskin layer and a tensile stress region in the core. Zoetelief et al.6 showed that the residual stress distribution was dependent on the holding pressure instead of the differential shrinkage in the gapwise direction. Final residual stresses of molded products are the result of these mechanisms and can be coupled together. Although the flow-induced residual stress is much lower than the thermally induced one, it can affect the properties of the polymer, and thus the thermal stresses.9 A complete

Model Prediction

6.1.2.1 Review of Stress Models Research of residual stress models has begun since the 1960s. Isayev and Crouthamel,11 Bushko and Stokes,12 and Kamal et al.7 provided good reviews on various stress models in injection molding. 6.1.2.1.1 Flow-Induced Stress Flow stresses are mainly developed in the filling and packing stages in injection molding. The governing equations, including the mass, the momentum, and the energy equations, should therefore be solved first in order to obtain the transient filling field of the velocity, pressure, and temperature. To help solve these equations, the rheological constitutive equations and the compressibility of the polymer will be combined.9 Earlier work simplified this work by studying the flow between two parallel plates, and a generalized nonisothermal Newtonian constitutive equation was employed.13 And flow-induced normal stresses could be obtained from the components of the viscoelastic stress tensor. The compressible Leonov model and the compressible K-BKZ model are both widely employed in the calculation of flow-induced residual stresses. The Leonov model takes the form of a viscoelastic differential constitutive equation and the K-BKZ model are shown in the viscoelastic integral constitutive equation. Research from Isayev and Hieber14 showed that the Leonov’s constitutive equation15 could predict the flow stress and birefringence in PS samples.

160

RESIDUAL STRESS AND WARPAGE SIMULATION

Considering a multimode model containing a number (m) of viscosities ηi and relaxation times θ i , the compressible Leonov model can be written as16 σ = −P I + σed + σp σed =

m  i=1

ηi d B , σp = 2ηr D d θi ei

(6.1) (6.2)

where P h is the hydrostatic stress or the negative value of stress spherical tensor and I represents the unit tensor. This model can be linearized for small strains. The parameters ηi , θ i , ηr , etc. can be determined by measuring the linear viscoelastic material functions. 6.1.2.1.2 Thermoelastic Model The thermally induced residual stress model was originated from the free quenching organic glass. Isayev11 gave a detailed introduction to the development of various thermally induced residual stress models for the organic glass. Struik17 adopted a simple analytical model that considered the stress relaxation but ignored the pressure effects in the cavity. This model is therefore not suitable for the injection molding process. In order to calculate the residual stress in injection molding, many researchers have established the free quenching model, which neglected the melt pressure, interaction between the part and the wall, and the geometry of the mold. Many experiments have shown that the residual stress profile in injection-molded samples is quite different from the well-known parabolic profile of free quenching. This suggests that other factors in injection molding should be accounted for in the calculation of stresses. Bartenev11 considered an infinite large slab of inorganic glass as being perfect elastic at temperature below its glass transition temperature T g and as perfect plastic at temperatures above that. The glass was therefore separated into two regions: one with perfect fluid behavior and the other with perfect elastic behavior. It leads to the following expression of the residual stress profile in the slab: σ (y) =

 βE  φ(y) − φ 1−ν

(6.3)

where σ (= σ xx = σ zz ) is the normal stress in the x–y plane, β is the linear coefficient of thermal expansion, E and ν are Young’s modulus and Poisson’s ratio of the glassy state, respectively, φ is the fictive temperature, and φ is the gapwise-averaged value of the φ. Obviously, Bartenev’s theory only accounted for the temperature gradient in the plastic region and ignored the elastic region. Indenbom11 later modified this theory and proposed a new formulation based on the Bartenev’s assumptions, calculating the residual stress in the elastic region with the predetermined elastic strain.

Rigdahl18 assumed that thermally induced residual stress was due to the constrained thermally induced shrinkage by the mould. He combined the elastic stress equation σ = EαT with the energy equation and performed a finite element analysis. Titomanlio et al.19 and Brucato20 employed a thermoelastic model and set up two models for calculating thermally induced residual stresses combining the pressure effects. Results from these models showed the commonly reported stress profile with three distinct regions: surface stress induced by the pressure in the skin layer and a parabolic distribution in the core region. Comparison with experiments showed an overprediction of residual stresses. Boitout et al.21 applied the thermoelastic model and accounted for the effects of mould deformation and solidification of gates. The research indicated that these factors have a significant effect on final residual stresses. Jansen and Titomanlio22,23 calculated the residual stress during the solidification of injection molding with a simple elastic model. They defined the pressure number N p as the ratio of the strain under the maximum pressure to the thermal strain and suggested that the actual shrinkage in the flow plane and the thickness were determined by N p . Analytic expressions for the residual stress and shrinkage were proposed based on the thickness of the frozen layer, free shrinking stress along with the pressure at solidification. 6.1.2.1.3 Thermoviscoelastic Model Polymers, either amorphous or crystalline, are typically viscoelastic materials,24 whose mechanical behavior and physical properties would vary with respect to both time and temperature. A model for describing the development of residual stresses should therefore combine both elastic and viscous responses of polymers in order to characterize the behavior of material in the injection molding process more accurately. Rezayat25 modified the model proposed by Titomanlio, considering the effect of convection and radiation in the cooling analysis after ejection. A thermoviscoelastic model was employed in calculating the deviatoric components of stress and strain, and a thermoelastic model was used in the spherical components simulation. Baaijens4 and Douven5 simulated the thermally induced residual stresses in the PS and PC injection-molded samples, with a linear thermoviscoelastic model derived from the linear compressive Leonov model. The material was assumed isotropic, and the packing pressure effect was included through the continuity equation and PVT equation. Results showed that there was a tensile stress in the surface, followed by a compressive stress. Kabanemi and Corchet26 assumed a viscoelastic behavior of materials and simulated the thermally induced stress with a threedimensional finite element method. The effect of mould constraints and packing pressure was excluded. Their

RESIDUAL STRESS ANALYSIS

results showed a parabolic profile of residual stresses in the gapwise direction, with compressive stresses at surface and tensile stresses in the core. Santhanam27 also employed a thermoviscoelastic model in calculating the residual stress. The packing pressure effect was accounted for by the initial strain at the end of filling, and the initial strain was only determined by the initial melt pressure of specified materials, without considering the evolution of pressure during the packing or the impact of thickness. Analysis showed high tensile stresses in the surface layer, followed by a compressive stress valley inward and a parabolic tensile stress in the core region. Comparison with the experimental data by Crouthamel11 showed that the surface stress value was overpredicted and the measured stress was more sensitive to the mold constraint. Bushko and Stokes12,28,29 applied the viscoelastic model in the residual stress analysis of simple slabs. Their model considered the effect of packing pressure and the compensation during the packing stage. The effects of processing and boundary conditions on the residual stress and shrinkage were also systematically studied. Ghoneim and Hieber30 conducted a quantitative study on the effect of density relaxation on the stress profile of PS samples, exploring a significant role of density relaxation (spherical components of the viscoelastic materials) in the evolution of residual stresses during the injection molding process. Zoetelief et al.6 also adopted the linear thermoviscoelastic model in calculating the thermally induced stress of PS and ABS plates and compared with experimental results. Most of the previous theoretical studies on residual stresses focused on amorphous polymers, in order to avoid the effect of crystallization on stresses. The stress distribution in semicrystalline polymers is different from amorphous materials because of factors such as the change of material properties with crystallinity. For an accurate description of the viscoelastic responses of semicrystalline polymers, a good understanding of crystalline kinetics is therefore required. Research of Kamal et al.7 showed a distinctive difference between the residual stress distributions of the HDPE and PC. Specifically, the stress profile of HDPE would display compressive stresses in the core region, leading to higher tensile stress in the skin layer. Chapman31 proposed a mathematical model in calculating the residual stress of semicrystalline polymers. Free quenching was assumed, and it was found that crystallinity would obviously increase the magnitude of residual stresses. Farhoudi and Kamal32 developed a thermoelastic model in calculating the residual stress of HDPE rectangular plates. This model accounted for the effect of crystallinity on the elastic modulus of materials and showed a nice agreement with experimental data by the layer removal method.

161

6.1.2.2 Current Model 6.1.2.2.1 Thermoviscoelastic Model A linear thermoviscoelastic model can describe the stress–strain relation and stress relaxation of materials in injection molding precisely. Assuming a thermorheologically simple viscoelastic behavior of materials during the injection molding, this model can be applied in the description of stress–strain relation covering the melt state above the glass transition temperature T g to the glassy state. The stress of linear thermoviscoelastic models is expressed as the sum of the hydrostatic stress and deviatoric components of stress6,33 : σij = −Ph δij + τij ,

i, j = 1, 2, 3

(6.4)

where P h is the hydrostatic stress or the negative value of stress spherical tensor, τ ij denotes the deviatoric components of stress, and δij is the kronecker delta. For isotropic materials, these two parts can be obtained from: 1 Ph = − Trσ = − 3



t

t

τij (t) =





∂εth ∂εm − ′ G1 (ξ(t) − ξ(t )) ′ ∂t ∂t −∞ ′

G2 (ξ(t) − ξ(t ′ ))

−∞

∂εijd ∂t ′

dt ′



dt ′

(6.5) (6.6)

where G 1 and G 2 are the bulk and shear relaxation modulus functions, respectively, Trσ is the trace of the stress tensor σ ij , t and t ′ are time variables, ξ (t) represents the pseudotime, εm and εijd express the spherical and deviatoric components of strain tensors, and εth is the thermal strain due to temperature changes. The relationship among the Young’s modulus E , shear modulus G, and bulk modulus K can be obtained from the simple tensile test of materials. And the bulk and shear relaxation modulus functions G 1 and G 2 can therefore be written as33 : E φ(t) = 3Kφ(t) 1 − 2u E G2 (t) = φ(t) = 2Gφ(t) 1+u

G1 (t) =

(6.7) (6.8)

where μ is the Poisson’s ratio and φ(t) is the relaxation function, which can be expressed as the sum of weighted exponential functions:

φ(t) =

N  k=1

  t gk exp − λk

(6.9)

where λk is the relaxation time  and g k denotes the relaxation parameters satisfying N k=1 gk = 1.

162

RESIDUAL STRESS AND WARPAGE SIMULATION

Assuming that there is no stress or strain in the material when t < 0, namely, the initial stress and strain value is 0, substituting Equation 6.7 into Equation 6.5, it follows that    t ∂εth ∂εm ′ 3Kφ(ξ(t) − ξ(t )) − ′ dt ′ Ph = ∂t ′ ∂t 0  t ∂εth = 3Kφ(ξ(t) − ξ(t ′ )) ′ dt ′ ∂t 0  t 3∂εm ′ − Kφ(ξ(t) − ξ(t ′ )) dt (6.10) ∂t ′ 0 where the thermal strain εth is given by  t α(t)T (t)dt ′ εth =

(6.11)

0

in which α represents the thermal expansion coefficient. If α is not time dependent, it can be taken as a constant, t resulting in εth = α 0 T (t)dt ′ . Similarly, we can get  t ∂εijd (6.12) Gφ(ξ(t) − ξ(t ′ )) ′ dt ′ τij (t) = 2 ∂t −∞ where εijd = εij − 13 Trεδij . Let K (t) = Kφ(t), β(t) = 3αKφ(t), G(t) = Gφ(t), and on substituting Trε = ε11 + ε22 + ε33 = 3εm into Equation 6.10, a simplified expression reads as follows:   t ′ ∂T ′ ∂Trεm β(ξ(t) − ξ(t )) ′ − K(ξ(t) − ξ(t )) dt ′ Ph = ∂t ∂t ′ 0

(6.13)

τij = 2



0

t

G(ξ(t) − ξ(t ′ ))

∂εijd ∂t ′

dt ′

(6.14)

The pseudotime ξ (t) in the above equations can be defined from the linear Maxwell model as  t 1 ′ ξ(t) = dt (6.15) 0 αT where α T is the time–temperature conversion factor. For amorphous polymers, when the temperature falls between the glass transition temperature and 100 ◦ C above that, the classical WLF equation can be applied to describe the conversion equation of time and temperature6 : log10 αT = −

c1 (T − Tr ) c2 + (T − Tr )

(6.16)

And for temperatures beyond that range or semicrystalline materials, the Arrhenius formula can be adopted6 : ln αT = −c3 (T − Tr )

(6.17)

In the above two equations, c 1 ,c 2 , andc 3 are material parameters and T r represents the reference temperature.

6.1.2.2.2 Thermoelastic Model Although the thermoviscoelastic model is popular in current literature,4,34 its application in commercial software has encountered several problems. First, a thermorheologically simple viscoelastic behavior is assumed, which means that the linear viscoelastic change of materials can be handled with the time–temperature equivalence principle. However, the relaxation spectrum of such materials is difficult to obtain, as most polymers are actually thermorheologically sophisticated. Second, as the relaxation function φ(t) is closely related to the internal structures of materials, especially for the semicrystalline polymer, it is still difficult to master the relationship between them either theoretically or experimentally. A thermoviscoelastic model is, therefore, widely applied in current commercial software when calculating the residual stress. The polymer is taken as a viscous melt above the glass transition temperature, retaining no stress. And the material is assumed as an elastic solid when below the transition temperature T g . This means that the relaxation behavior of materials is neglected and the calculation of relaxation spectrum is therefore taken off. And the thermoelastic model of residual stresses can be derived as ⎧ ⎪ 0 T ≥ Tg ⎪ ⎪    t ⎪ ⎪ α ∂T ∂Trεm 1 ⎨ ′ − dt − Tr σij = κ ∂t ′ κ ∂t ′ 0 ⎪  t ∂εd ⎪ ⎪ ij ⎪ ⎪ dt ′ T ≤ Tg ⎩ + 2G ′ 0 ∂t (6.18) in which T g is the glass transition temperature, G represents the shear modulus of materials, α and κ denote the thermal expansion and isothermal compressive coefficients, respectively, which are defined as     1 ∂ρ 1 ∂ρ , κ= (6.19) α=− ρ ∂T p ρ ∂p T where ρ and p represent local density and pressure of materials, respectively. 6.1.2.2.3 Specific-Volume Model The coefficients in the hydrostatic stress calculation can be obtained from the PVT relation or specific-volume model of materials. For amorphous polymers, the following Tait equation is widely applied35 : ⎧ (a0m ⎪ g )) ⎪   + a1m (T − T ⎪ ⎪ p ⎪ ⎪ , T > Tg 1 − 0.0894 ln 1 + ⎨ Bm v(p, T ) = ⎪ (a0s )) ⎪   + a1s (T − Tg  ⎪ ⎪ ⎪ p ⎪ ⎩ , T ≤ Tg 1 − 0.0894 ln 1 + Bs (6.20)

RESIDUAL STRESS ANALYSIS

Bm (T ) = B0m e−B1m T ,

Bs (T ) = B0s e−B1s T

Tg (p) = Tg (0) + sp

(6.21) (6.22)

where T g is the pressure-dependent glass transition temperature; and a 0m , a 1m , B 0m , B 1m , a 0s , a 1s , B 0s , B 1s , and s are material constants. 6.1.3

Numerical Simulation

6.1.3.1 Basic Assumptions As most of injection-molded products possess a thin shell structure, with in-plane dimensions often 1 order of magnitude larger than the thickness dimension, the major assumptions are as follows6,36 : 1. The polymer is amorphous with a thermorheologically simple behavior. 2. A local orthogonal coordinate system can be built, where the flow direction of polymer melt and the thickness direction of products act as the first and third principle directions, respectively. 3. Owing to various structure constraints such as floors and veins in the mold, the product may experience complete planar mold constraints. Planar strains are therefore not considered before demolding, that is, ε11 = ε22 = 0, which only allows the nonzero strain component ε33 in the thickness direction. 4. Lateral deformation is also neglected, that is, εxz = εyz = 0. 5. The principle stress σ 33 is normal to the flow plane. It is uniform in the thickness direction (z -direction) and does not vary with positions. 6. The product sticks to the mold surface closely once there exists a cavity pressure (or σ 33 < 0). 7. The deformation of mold is ignored. 8. Deformation of products within the mold is neglected. 9. As the flow residual stress is generally 1–2 orders of magnitudes lower than the thermal residual stress, it can be neglected as long as the accuracy of calculation is not so rigid, and the initial stresses and strains are therefore assumed to be zero. 10. The anisotropy of materials is not considered. The viscoelastic relaxation function of isotropic materials can be written as Gijkl =

3[G2 (t) − G1 (t)]δij δkl 3 [G1 (t)(δik δil + δil δik )] + 2

163

1. Stresses and strains at the interface of materials are consistent (this is not valid for the thermoelastic model). 2. Friction between the part and the mold is neglected. 3. The in-plane stress profile follows hˆ σ (z , t)dz = 0.

6.1.3.2 Boundary Conditions Boundary conditions vary during different injection molding stages. According to the pressure on products, multiple boundary conditions can be given as follows: 1. Once the filling stage is completed and packing stage begins, surface materials start to solidify, and there are two frozen skin layers with a molten core region in the thickness direction of products. Packing pressure is therefore forced onto the product through the molten core region. Pressurized polymer melt is injected into the cavity to compensate for the shrinkage caused by cooling and solidification. The cavity pressure P of this stage is controlled by the injection machine and can be calculated from packing analysis. P can be regarded as a known parameter, resulting in σ 33 = − P . 2. With the solidification of gates or end of packing, no compensation from new polymer melts is available, and the product is free from the packing pressure. With the cooling of materials, the cavity pressure would gradually lower down. The product would stick to the mold surface closely under the cavity pressure, and there is no variation in the thickness, that is,  l/2

ε33 (z)dz = 0

(6.24)

−l/2

where l is the thickness of products 3. The product starts to detach from the mold surface, and shrinkage in the thickness direction begins until it is ejected from the mold. 4. After the ejection, postdemolding thermal stresses will be induced when the product cools down to the ambient temperature. It is free from any mold constraints, and free deformation is available. Release of residual stresses at this stage will eventually result in warpage and deformation. It can be seen that the above multiple boundary conditions can account for impacts of the packing stage and solidification of gates on residual stresses.

(6.23)

where G ijkl is the relaxation modulus, δ is the Kronecher delta, and G 1 and G 2 represent the shear and bulk relaxation modulus, respectively.

6.1.3.3 Space-Time Discretization When calculating the residual stresses, the finite difference method can be applied in the discretization of equations in the time. For example, the linear viscoelastic constitutive equation can

164

RESIDUAL STRESS AND WARPAGE SIMULATION

be transformed into a discretized iterative form in this way. And the stress value at current time step can be obtained from the analysis results of last time step combined with current boundary conditions. Employing the time discretization of hydrostatic stress formula, it becomes37 : Ph (tn+1 ) = Ph (tn ) + βT − KTrε

k=1

in which 

t 0

  ξ(t) − ξ(t ′ ) exp − dεiid (t ′ ) λ

(6.27)

The above formula can be transformed into the partial differential equation as dεd 1 dξ dsii + sii = 2Ggk ii dt λ dt dt

(6.28)

The finite difference format can be written as εd sii (tn+1 ) − sii (tn ) 1 ξ + sii (tn+1 ) = 2Ggk ii t λ t t

(6.29)

which leads to   sii(k) (tn+1 ) = ςk sii(k) (tn ) + 2Ggk εiid

where λk is the relaxation time, and   ξ −1 ςk = 1 + λk

(6.30)

(6.31)

On the basis of the above basic assumptions, ε11 = ε22 = 0, deviatoric components of strains can therefore be expressed as Tr(ε) = ε11 + ε22 + ε33 = ε33 1 1 εm = (ε11 + ε22 + ε33 ) = ε33 3 3 1 εiid = εii − εm = − εii , i = 1, 2, 3 3

(6.32)

(6.34)

N 

(k) ςk s33 (tn )

(6.36)

k=1

And the normal strain variation is as follows: ε33 =

∗ σ33 (tn+1 ) − σ33  N 4 k=1 ςk gk + K 3G

(6.37)

6.1.3.4 Numerical Simulation In the numerical simulation, the finite element mesh adopted by the postfilling stage analysis can still be employed in the residual stress analysis. For example, the product can be discretized into numerous layers in the thickness direction. On the basis of the above discretized iterative formulas, the stresses at time step t n + 1 can be calculated from the stresses of the last time step t n . Final residual stresses would therefore be dependent on the heat and pressure evolution within the cavity. At each time step, numerical simulations can be employed on every layer of elements as follows: 1. Obtain the temperature change at time step t n + 1 : T = T (t n + 1 ) − T (t n ), based on the calculated thermal field. 2. With current temperature, calculate the thermal expansion coefficient α T from Equation 6.16 or 6.17. Then get the pseudotime ξ from Equation 6.15, and thus ς k (k = 1, 2, . . . , N ) from Equation 6.31. 3. Substitute the last hydrostatic stress and current T ∗ into Equation 6.36 to get σ33 . 4. Obtain the normal stress σ 33 according to the specific boundary condition at each molding stage: Stage 1: Packing pressure is taking effect on the product and the normal stress is equivalent to the negative pressure, that is, σ33 (tn+1 ) = −P (tn+1 ) (6.38) Stage 2: Packing pressure no longer works, and the thickness variation is zero. According to Equation 6.24, l0 =

N 

ε33 l0(i) = 0

(6.39)

i=1

(6.33)

Substituting Equations 6.25 and 6.34 into Equation 6.4, the normal stress reads as   N  4 ∗ G σ33 (tn+1 ) = σ33 + ςk gk + K ε33 (6.35) 3 k=1

∗ = −(Ph (tn ) + βT ) + σ33

(6.25)

where T and ε represent changes of the temperature and strain during the period of t = t n + 1 − t n ,respectively, and β and K are defined the same as in Equation 6.13. The deviatoric components of stress can be discretized as37 : N  sii(k) (t), i = 1, 2, 3 (6.26) τii (t) =

sii(k) (t) = 2Ggk

where

where l0(i) is the initial thickness of the i th layer. Combining Equation 6.37 with Equation 6.39 yields σ33 (tn+1 ) = in which V = 43 G

N

k=1 ςk gk

∗

σ33 (i) i=1 V l0 N 1 (i) i=1 V l0

N

+ K.

(6.40)

165

RESIDUAL STRESS ANALYSIS

Stage 3: Cavity pressure reduces to zero, and the product starts to detach from the wall in the thickness direction, resulting in (6.41) σ33 (tn+1 ) = 0 1. Substitute current σ 33 into Equation 6.37 to obtain ε33 of current layer. 2. Calculate the deviatoric components of strains from Equations 6.33 and 6.34. And compute the current hydrostatic stress P h (t n + 1 ) with Equation 6.25. 3. Obtain the deviatoric components of stress τ 11 , τ 22 , and τ 33 from Equation 6.26. 4. Get final in-plane stresses σ 11 and σ 22 from Equation 6.4. 6.1.4

TABLE 6.1 Specific-Volume Model Constants for the PS Used in Numerical Calculationsa Symbol

Value

a 0m , m3 /kg a 1m , m3 /(kg K) B 0m , Pa B 1m , K−1 T g (0), K a 0s , m3 /kg a 1s , m3 /(kg K) B 0s , Pa B 1s , K−1 s, K/Pa a Ref.

9.72e−4 5.44e−7 2.53e8 4.08e−3 373 9.72e−4 2.24e−7 3.53e8 3.00e−3 5.1e−7

6.

TABLE 6.2 Material Data for the PS Used in Numerical Calculations

Case Study

6.1.4.1 Case Study 1: Validation by Experimental Data The layer removal method is widely employed in measuring the residual stresses of injection-molded products. It is convenient to perform with slabs or plaque-like products, and the gapwise distribution of residual stresses can be obtained. Treuting and Read38 initially measured the residual stresses in steel plates with this method. Isayev11 reviewed its successful application in injection-molded products. After the removal of a thin layer from the surface, the specimen will warp to a circular arc and reestablish equilibrium of residual stresses. The gapwise residual stresses in the specimen before the layer removal can be calculated by measuring the resultant curvature as a function of removed materials, using the general biaxial stress relation. Zoetelief6 provided a detailed introduction to this method. To validate the mathematical model and numerical simulation, residual stresses of a simple PS plate, as shown in Figure 6.4, was calculated, and the numerical results were compared with existing experimental data by Zoetelief.6 The trade name of material PS Styron 678E is provided by Dow Chemical. The melt and mold temperature were 200 ◦ C and ◦ 55 C, respectively; the injection rate, 7.6 × 10−6 m3 /s; the

FIGURE 6.4

Elastic Modulus, MPa 2700

Poisson’s Ratio

Shear Modulus, MPa

Thermal Expansion Coefficient, K−1

0.35

906

8.31e−005

packing pressure, 55 MPa; and the packing and cooling time, 10.0 s and 30.0 s, respectively. The specific-volume model constants, mechanical data, stress relaxation data, and time–temperature conversion data employed in the numerical calculations are outlined in Tables 6.1–6.4, respectively, supplied by Reference 6. The layout of runner and cooling systems applied in the numerical simulation is shown in Figure 6.5, in which the product is located between two symmetrical cooling circuits. Residual stresses of the midpoint P in the specimen were calculated with the proposed thermoelastic and thermoviscoelastic models, respectively. And simulation results were compared with experimental data measured by Zoetelief et al.6 as shown in Figure 6.6. As can be seen, all the curves collect close to the measured data, that is, a tensile stress in the surface, followed by a compressive stress valley and a tensile stress

Schematic graph of the specimen (unit: mm).

166

RESIDUAL STRESS AND WARPAGE SIMULATION

TABLE 6.3 Stress Relaxation Data for the PS Used in Numerical Calculations i

θi , s

G i , Pa

1 2 3 4 5 6

5.0 3.0e−1 6.66e−3 1.18e−4 2.5e−6 4.77e−8

9.14e7 3.43e8 2.22e8 1.20e8 5.96e7 6.92e7

TABLE 6.4 Time–Temperature Conversion Data for the PS Used in Numerical Calculations C1

C 2, K

C 3 , K−1

13.37

51.06

0.6242

Cooler

FIGURE 6.6 Comparison of in-plane residual stress distribution between predicted and measured data (the Zoetelief viscoelastic and Zoetelief viscous elastic analysis are the predicted results of Zoetelief).

Runner

FIGURE 6.5 Layout of runner and cooling systems in the numerical calculations.

in the core region. This follows the trend of most relevant literature. It is clear that the present viscoelastic result seems to be in better agreement with the experimental data than the viscous elastic analysis, especially in the prediction of surface tensile stress value and the position of the maximum compressive stress, but the value of the maximum compressive stress seems overpredicted. Results show that the thermoelastic model stands up to the thermoviscoelastic model in terms of validity and accuracy and could also be employed in the calculation of residual stresses for commercial packages, especially when the relaxation data are difficult to obtain. 6.1.4.2 Case Study 2: Influence of Process Parameters on Residual Stresses To investigate the relationship between residual stresses and key processing parameters, another simple plate with a dimension of 300 × 120 × 2.5 mm was chosen as the specimen, as the effects of processing conditions can be clearly demonstrated with this simple geometry. Three key processing parameters were studied: packing pressure, melt temperature, and mold temperature. The concept of deliberately varying one variable and maintaining all the other parameters constant was used. For example, in one set of numerical simulation, all other parameters were maintained constant at the standard level,

FIGURE 6.7 sures.

Variation of residual stresses with packing pres-

while packing pressure varied from 35 to 75 MPa to study its effect on residual stresses. The standard level of processing conditions is the same with previous case study, and the material data can be referred from Tables 6.1–6.4. Variations of simulated residual stresses in injectionmolded parts with processing parameters are displayed in Figures 6.7–6.9. 6.1.4.2.1 Effect of Packing Pressure It can be seen from Figure 6.7 that with the increase in packing pressure, the compressive stress valley in the subskin layer gets deeper,

RESIDUAL STRESS ANALYSIS

167

Flow direction

150 mm P1 Central position 300 mm 2.5 mm

FIGURE 6.8 tures.

Variation of residual stresses with melt tempera75 mm

FIGURE 6.10

Rectangular strip mold.

of packing pressure on residual stresses is offset by other factors. 6.1.4.2.2 Effect of Melt Temperature Residual stresses at various melt temperatures are shown in Figure 6.8. With the increase of melt temperature, solidification of the subskin layer will be delayed with a reduced cavity pressure, which results in a lower compressive stress at this region, as shown in Figure 6.8. And a lower tensile stress can also be seen in Figure 6.8, because of the reestablishment of stress equilibrium with the lower compressive stress in the subskin layer.

FIGURE 6.9 atures.

Variation of residual stresses with mould temper-

indicating a more severe maximum compressive stress. This may be contributed to the fact that this layer may be solidified under a higher cavity pressure, and the stress state may be retained even after a long period of relaxation. After the reestablishment of stress equilibrium after demolding, a more severe maximum compressive stress may therefore be attained. This will in turn result in a higher tensile stress in the skin layer and the core region, as shown in the figure. Another interesting thing about Figure 6.7 is that the impact of packing pressure on residual stresses is more obvious under a lower packing pressure. For example, the stress curve at 35 MPa is clearly deviated from the 55 MPa one, whereas this deviation seems much less pronounced between the curve of 55 and 75 MPa. This may imply that above a certain level of packing pressure, the effect

6.1.4.2.3 Effect of Mold Temperature The relationship between residual stresses and mould temperatures is shown in Figure 6.9. It can be seen that the effect of mould temperature on residual stresses is similar to that of melt temperature. This may also be due to a longer solidification process with an increased mould temperature. A lower tensile stress in the core region is also observed with a higher mould temperature. 6.1.4.3 Case Study 3: Evolution of Residual Stresses To investigate the evolution and distribution of the residual stresses, numerical simulation was performed on a fan-gated specimen (300 × 75 × 2.5 mm), as shown in Figure 6.10. The material used was ABS (Novodur P2X of Bayer). The triangular element was applied in the analysis, which discretized the product into 2912 nodes and 5820 elements. The mechanical data, stress relaxation data, time–temperature conversion data, and processing conditions employed in the numerical calculations are outlined in Table 6.5–6.8, respectively.39

168

RESIDUAL STRESS AND WARPAGE SIMULATION

TABLE 6.5 Material Properties for the ABS Used in Numerical Calculations

TABLE 6.7 Parameters in Time–Temperature Shift Functions for the ABS Used in Numerical Calculations

Solidification Thermal Young’s Shear Temperature, Expansion Modulus, Modulus, Poisson’s MPa MPa Ratio K Coefficient, K−1

c1

c2, K

c 3 , K−1

Tr, K

14.22

47.01

0.3291

373

368

8 × 10−5

2240

805

0.392 TABLE 6.8 Simulation

6.1.4.3.1 Evolution of Cavity Pressure Cavity pressure has a significant effect on the residual stresses. As the normal stress equals to the negative packing pressure in most of the time in the simulation, the evolution of the normal stress will therefore will be demonstrated by the development of cavity pressure. The cavity pressure history of the central position P1 is shown in Figure 6.11. During the filling stage the injection pressure increases all the time to overcome the increasing resistance, with cavity pressure growing from 0 to 27.64 MPa. Once the packing stage starts, the cavity pressure rises sharply to the highest value 56.56 MPa (point A in Fig. 6.11) at t = 1.8 s. With the compensation of molten polymers by the packing pressure, the cavity pressure holds constant at first (segment AB in Fig. 6.11). With the accumulation of the frozen layer toward the core region, compensation for the shrinkage reduces gradually, causing the decay of the cavity pressure (segment BC in Fig. 6.11). When the packing process ends at t = 7.5 s, no more packing pressure is applied to the part, and the cavity pressure collapses (segment CD in Fig. 6.11). When t = 25.58 s, the cavity pressure of the central position drops to zero (point D), after which the part is detached from the mold in the thickness direction and shrinks freely. In injection molding, the history of the cavity pressure is affected by the injection pressure, packing pressure, and part cooling comprehensively. Before the end of packing stage or the complete solidification of products in the thickness direction, the cavity pressure is mainly determined by the packing pressure. After that, the cavity pressure decreases with the cooling of products. As the temperature history varies at different positions of products, the evolution of cavity pressure will also show difference. Figure 6.12 depicts the predicted evolution of cavity pressure at different locations along the flowpath, including the pressure profiles near the gate, a quarter to the gate, TABLE 6.6

λk , s gk

Processing Conditions for Injection Molding

Injection Packing Cooling Melt Mold Time, s Time, s Time, s Temperature, K Temperature, K 1.2

6.3

21.7

503

323

halfway down to the end, a quarter to the end, and near the end of the flowpath. 6.1.4.3.2 Evolution and Distribution of Residual Stresses The stresses in the flow plane are called in-plane stresses. For transversely isotropic materials, the first and second principle stresses are assumed equal. To begin with, the evolution of the in-plane stresses is considered. Figure 6.13 shows the predicted in-plane stresses distribution of central position P1 in the gapwise direction. And the distribution at five typical instants are plotted in Figure 6.14, including (i) at the beginning of the filling stage, (ii) at the end of the filling stage, (iii) at the end of the packing stage, (iv) when the pressure of the central position drops to zero, and (v) just before ejection. During the filling stage, the molten material is forced into the mold and forms a very thin frozen layer quickly once contacting the cold mold surface, leading to a high stress. There exists a large flat region of stresses in the core region as shown in Figure 6.14, where the stress remains uniform and the material is essentially in a rubbery state with low elastic modulus and equals to the negative packing pressure. As the packing stage starts, in-plane stress goes into a steady decline and quickly reaches the lowest value with the rise of the cavity pressure. At the end of packing, the shape of the profile changes and shows that relatively high stresses have been built up in the surface layer, followed by a mild decline in the region more inward. When no more packing pressure is applied to the part, the cavity pressure drops quickly, causing the compressive stress to decrease throughout the product. Just before ejection, the residual stress distribution profile

Relaxation Spectrum Data for the ABS Used in Numerical Calculations 1

2

3

4

5

6

4.706 × 10−9 5.014 × 10−2

4.410 × 10−6 8.585 × 10−2

2.082 × 10−3 2.869 × 10−1

6.198 × 10−1 4.043 × 10−1

3.035 × 106 3.531 × 10−4

2.749 × 108 1.171 × 10−4

RESIDUAL STRESS ANALYSIS

FIGURE 6.11

FIGURE 6.12

169

Pressure history at central position P1.

Calculated pressure histories at different locations along the flowpath.

consists of three distinct regions made up of two skins and a core. The stress exhibits a high surface tensile value and changes to a compressive peak value close to the surface, with the core region experiencing a parabolic peak. Clearly, the existence of the relatively high stresses in the skin and core regions is associated with the relatively low pressure under which the skin and the core have frozen, while the existence of the minimum stresses in the intermediate region is due to the fact that the material in this region is frozen at the maximum pressure.

Then we will consider the effect of flowpath on the distribution of the in-plane stresses. In injection molding, the pressure and temperature history vary throughout the part, leading to different in-plane stress distributions. Figures 6.15, 6.16, and 6.17 show the stress gapwise distribution along the flowpath at (i) the end of filling, (ii) the end of packing, and (iii) just before ejection, respectively. Results show that the gapwise profiles of stresses have the same shape along the flowpath.

170

RESIDUAL STRESS AND WARPAGE SIMULATION

FIGURE 6.13

FIGURE 6.14

Predicted stress evolution of central position P1 in the gapwise direction.

Predicted stress distribution in the thickness direction at some typical instants.

At the beginning, the stress near the gate is much lower than the stress at other locations, as shown in Figure 6.15. And the difference in the stress values at different locations vanishes gradually. Figure 6.16 shows that the stress distribution tends to be unique along the flowpath, probably because of a more uniform distribution of the cavity pressure at the end of packing. Just before ejection, profiles of in-plane stresses along the flowpath seem to be the uniform, consisting of three distinct regions as mentioned before. The stresses near the gate are lower than the stresses farther away, probably because the material near the gate has frozen at a higher pressure.

From the above-mentioned results and analysis, it can be concluded that the stress distribution and evolution are closely related to the development of the cavity pressure.

6.2

WARPAGE SIMULATION

The final dimensions and geometry of molded parts may be a primary concern in evaluating the manufacturing quality.40 During the injection molding process, a reduction of up to 35% specific volume may be induced, as polymers are cooled from molten to a solid state.41 Once the part in

WARPAGE SIMULATION

FIGURE 6.15

FIGURE 6.16

171

Predicted stress distribution at the end of filling stage (t = 1.2 s).

Predicted stress distribution at the end of packing stage (t = 7.5 s).

ejected, it may generally undergo a continuous cooling and shrinkage for up to 30 days. In a practical view, shrinkage and warpage can be defined as the deviations from the mold geometry.40 Shrinkage regards with dimensional differences or a geometrical reduction in the size of the part,42 and warpage concerns more with form deviations. It is generally assumed that warpage is closely related with shrinkage. Typically, warpage will only be induced with a nonuniform shrinkage, especially in the regions with unequal shrinkage where stresses are generated. In other words, if the shrinkage is uniform, the part will not deform or change its shape, thus no warpage.42 In the following section, shrinkage and warpage may be regarded as an integrated term as S&W most of the time.

In determining the quality of injection-molded parts, especially the thin-wall products, S&W are usually concerned as the major observed quality factors. During the plastic injection molding, one of the biggest challenges may be the reduction of S&W, which will deteriorate the quality of products.43 In other words, reducing S&W of injectionmolded products has become one of the top priorities in the injection molding industry in order to improve the quality44 or probably the most dominant underlying problems in the successful injection molding. 6.2.1

Development of Warpage

Warpage in injection-molded products results from the differential shrinkage or variations in shrinkage42 throughout

172

RESIDUAL STRESS AND WARPAGE SIMULATION

FIGURE 6.17

Predicted stress distribution just before the demolding (t = 29.2 s).

the part and is dependent on many processing and material factors, including the part geometry effect, the mechanical stiffness of polymer material, orientation of polymer molecules, differential crystallinity, and temperature gradients along a part thickness.45 Shrinkage alone does not lead to warpage, as a highly uniform shrinkage will only contribute to a perfectly shaped part that is smaller.42 And the shrinkage change in injection-molded parts is mainly determined by shrinkage because of the cooling effect or the pressure effect.46 As the temperature and pressure of polymers vary across the thickness and throughout the product, the part will shrink differently at different locations, which leads to warpage. A clear understanding of the origin of shrinkage is therefore required for the warpage study. In analyzing how the shrinkage causes warpage, types of shrinkage effects are discussed in the following sections. 6.2.1.1 Differential Cooling Effects Variations in cooling can also affect shrinkage differences. A dramatic example would be the temperature difference on opposite mold surfaces.42 Once removed from the mold, the part may undergo different contraction because of different cooling rates, and this shrinkage difference on different sides will build a bending moment that results in warpage. In general, differential cooling effect may be caused by the effect on orientation, regional volumetric shrinkage (caused by thermal contraction or crystallization), side-to-side cooling variations (this will affect orientation and volumetric shrinkage from one side of a part to the other side), pressure, etc.47 Take the pressure effect, for example, regions under different pressures will have shrinkage at different rates, with a lower shrinkage under a higher pressure. A typical example would be the box-shaped products as shown in Figure 6.18. In this case, the inside wall

(a)

(b)

FIGURE 6.18 Warpage caused by differential cooling of a boxshaped product: (a) original shape and (b) after warpage.

of the box will be more difficult to cool down than the outside, and so there exists a temperature and shrinkage difference. Sections with nonuniform thicknesses may also create differential cooling effect, especially with crystalline materials. The thicker part may be hotter because of a more difficult cooling, and different shrinkage may occur because of temperature differences, which will set up internal stresses. 6.2.1.2 Area Shrinkage Effects The shrinkage difference between different regions of injection-molded parts can be described by the area shrinkage, most notably, the change in the area because of parallel and perpendicular shrinkage.42 6.2.1.3 Orientation Effects Orientation effects arise from the molecular or fiber orientation in materials. This may cause the difference between parallel and perpendicular shrinkage, and therefore the warpage42 . It can be defined by comparing variations in different directions within a region. Impact factors concerning orientation effects cover the magnitude of polymer and polymer-additive orientation,

WARPAGE SIMULATION

direction of orientation (variations in flow direction may induce conflicting shrinkage vectors), type of flow, transient flow (where the flow direction may change through the thickness of the flow channel due to unbalanced filling), etc.47 As has been mentioned, the driving force for warpage is the shrinkage difference. And the variations in shrinkage are resisted only by the geometric constraints or stiffness of the mold. Even a part with a high variation in shrinkage can be free from warpage as long as it is stiff enough, but a high internal stress will be created in this situation. Shrinkage of polymers can often be classified into two groups from a practical point of view: volumetric and linearized shrinkage.47 Volumetric shrinkage is caused by the well-known thermal contraction of materials when it is cooled. This can be assumed isotropic if no external forces are applied. The linearized shrinkage originates from the orientation effects caused by shear and extensional flows, which has a direct impact on the way the polymer shrinks. Shrinkage of materials can be explained in terms of the internal microstructure, such as the molecular motion, the cohesive bond energies of both the primary atomic and secondary molecular bonds, and change in the specific volume.47 The relationship between pressure, specific volume, and temperature can be evaluated through the commonly used PVT graph. As shown in Figure 6.19, by tracing the shrinkage curve of the material at 0.1 MPa, one can see that it contracts by almost 20%, as it is cooled from its processing temperature of 210 ◦ C to ambient temperature of 30 ◦ C.48 Owing to the significant amount of free space when a plastic is heated, materials can also be compressed by a applied pressure. Potential variations in volume are created by the combination of thermal

FIGURE 6.19

173

contraction and pressure during molding in this particular situation. The inflection in the curve indicates the glass transition temperature of the polymer. In order to produce a warp-free part, variations in shrinkage need to be eliminated. This is almost impossible in a conventional injection molding process, as there are numerous impact factors coupled together. Let us consider the mechanism of a typical injection molding process: due to variations in shear through the cross sections of the cavity, the shear distribution, shear direction, and cooling rate vary across the thickness during the mold filling and shrinkage would vary in both volume and linear. The differential shrinkage will also develop residual stresses. When these stresses are sufficient enough to overcome the resistance of the mold, warpage will be induced. With the warpage of parts, some of residual stresses will be relaxed. 6.2.2

Model Prediction

With the development of plastic injection molding techniques, mechanisms and prediction of warpage have been widely investigated. Two major principles have been applied in the numerical study of warpage. 1. Residual Stress Method . During the injection molding process, residual stresses (including the flow-induced and thermally induced stresses) obtained from an elastic or viscoelastic model, or the experiments are taken as the initial stress and input into the structure analysis package for a further analysis of S&W. Molecular and fiber orientation, along with other impact factors such as the crystallization effect will be included in the stress model. With the modification of

A PVT graph of POE (provided by Dupont-Dow company, trade name: 8150).

174

RESIDUAL STRESS AND WARPAGE SIMULATION

the stress model, more effects can be accounted for, such as the processing parameters during the packing and cooling stage. It has become the most popular method in the warpage analysis. 2. Residual Strain Method . Shrinkage in this process is calculated from an empirical or experimental model and adopted as the initial strain in the structure analysis package for the analysis of warpage and deformation. The calculation of shrinkage considers the effects of molecular and fiber orientation, as well as the crystallization effect, etc. It is superior in its flexibility, but an accurate coefficient in the model requires numerous experiments, as an empirical formula is employed in the shrinkage calculation. And its application has been limited because of great property differences among various polymers. Yet it has been adopted in the first commercial software calculating S&W, SWIS by Moldflow Corporation. A typical formula for the strain calculation can be written as49 S || =

5  i=1

bi Mi , S ⊥ =

10 

bi Mi

(6.42)

i=6

where S || and S ⊥ represent the shrinkage strain parallel and perpendicular to the flow direction, respectively; b i is the material constant; and M i denotes the effect of processing parameters, which can be obtained from filling and packing analysis. With b i (i = 1, . . . , 10) gained from experiments and M i (i = 1, . . . , 10) from simulation results, shrinkage in the two directions can be calculated from the above equation. And in order to calculate warpage of products, the bending moment applied on the product needs to be calculated first, by modifying the strains on the two sides of parts based on the temperature differences between the cavity and core. Later, a corrected residual in-mold stress (CRIMS) model was proposed by Moldflow. It combines the abovementioned residual stress and residual strain methods, and a more accurate residual stress result can be obtained by modifying the calculated stress with the residual strain method. This may be the most accurate method in calculating residual stresses so far, as it shores up the deficiencies of ignoring crystallization and molecular orientation effects in the conventional residual stress method by experiments. However, it also experiences the similar inadequacy of requiring a large quantity of experiments in the residual strain method As mentioned above, earlier researchers generally input the residual stress or strain as the load into a specific finite element structure analysis package for analysis of warpage and deformation. Since the 1990s, the integrated simulation of deformation with residual stress has become the trend,

and the flat shell element is widely employed in calculating the postmolded warpage. 6.2.2.1 Review of Warpage Models 6.2.2.1.1 Review of Theoretical Study of Warpage Study of warpage based on the residual stress has begun since the 1980s and early attempts have been focused on the warpage induced by thermal stresses. Jacques50 initiated the numerical simulation of warpage for amorphous plastic plates caused by differential cooling. He analyzed the heat transfer of injection molding with the one-dimensional finite difference method and calculated the thermal stresses of each layer after the glassy transition, and then simulated the warpage with pure bending theory that was suitable for simple products. Chiang et al.51 conducted a coupled analysis of filling and cooling simulation and later calculated the flow and thermal stresses in order to predict the S&W. Results showed that flow stresses during the molding process might have a large effect on the optical property of part surface, but little effect on warpage was detected. With the development of injection molding simulation, the integrated whole-process simulation has become a trend. And the effects of processing conditions on the warpage are included in the analysis of stress and warpage during each stage. Porsch and Michacli52 discretized a product into several layers and predicted the warpage after calculating the thermal stresses of each layer, which are caused by uneven temperature distribution and contraction. Their model excluded the effects of packing pressure, orientation, anisotropy, and stress relaxation. Matsuoka et al.53 predicted warpage of fiber-strengthened products using a threedimensional thin-wall model, after integrating the analysis of mold cooling, polymer behaviors, fiber orientation, material properties, and stress. However, relaxation behaviors of flow stresses and crystal property of materials were not included in the model. Many researchers on the warpage simulation have shown that the liquid–solid phase transformation during the cooling stage and the stress relaxation behavior has a great impact on the prediction of residual stresses and warpage. Numerous attempts have been made on a more accurate and reasonable mathematical model in describing the viscoelastic and phase transformation behaviors, in order to enhance the accuracy of warpage simulation through a more accurate residual stress result.54 – 56 Meanwhile, the impact of crystallinity, orientation, and processing conditions on stresses and warpage has aroused increased attention. For example, Chang and Tsaur57 adopted a modified Tait equation for describing the pressure-volume-temperature relationship for crystalline materials and the Malkin crystalline kinetics for the crystallization behavior of polymers. They calculated the residual stresses with a linear thermoviscoelastic model, and input the stress result as the initial conditions for

WARPAGE SIMULATION

the simulation of S&W with a three-dimensional finite element method. Attention was shown on the effect of fiber orientation on the stress and warpage of fiberreinforced plastics. Kikuchi and Koyama58 considered the fiber orientation as the major factor in inducing the warpage of fiber-reinforced plastic products and the differential distribution of temperature and pressure as the critical factor for the warpage of nonreinforced plastic parts. They calculated the warpage with the nonlinear structure software MARC, using the predetermined results of flow field, fiber orientation, and thermal stresses. Bushko and Stokes12,28,29 focused on the warpage of amorphous polymers and built the warpage mechanism of injection-molded products with the solidification of polymer melts between parallel cooled plates. Temperature difference and packing pressure effects were accounted for, but the flow effect was excluded. Simulation of warpage based on the shrinkage result is performed through the analysis of PVT relationship during molding and combined with experiments. Early attempts were centered on the relation of various processing parameters with shrinkage. Egbers and Johnson59 measured the shrinkage of different HDPE under different cooling times, mold and melt temperatures, injection pressure, and gate sizes. It was concluded that about 80% of shrinkage was related with the part thickness and gate size and the other 20% would go to processing conditions. Hebert et al.60 measured the effect of temperature variation on shrinkage for the 30% filled PS molded products and summarized their relation with a statistical method. Simple empirical models have recently been developed in the quantitative study of shrinkage. Nievelstein and Menges61 investigated the effects of packing pressure, mold temperature, part thickness, and flow direction on shrinkage and proposed an empirical model of predicting shrinkage with a linear superposition principle. Shoemaker et al.62 studied the shrinkage of products with uniform and nonuniform thickness and tried to reduce the shrinkage by optimizing the packing stage. Sanschagrin et al.63 focused on the shrinkage of fiber-reinforced plastic products and developed a corresponding shrinkage model. Their experiments showed that the impact factors on the axial and transverse shrinkage of fiber-reinforced products include the packing pressure, melt temperature, mold temperature, and fiber-reinforced ratio, but especially the fiber-reinforced ratio. Bernhardt64 introduced a shrinkage evaluation package made by TMconcept company, which accounted for the effects of major factors on shrinkage, such as processing conditions, flow orientation, and mold geometry. It claimed that the simplified shrinkage model, which solely based on the PVT data, was not accurate for the cases of complex products concerning the impact factors such as orientation and anisotropy. On the basis of the shrinkage result from the above quantitative empirical models, a number of researchers have

175

started the warpage study of injection-molded products in this way. Thomas and Mccffery65 initiated a warpage predictive model based on the analysis of previous filling, packing, and cooling stages. The volumetric shrinkage, stress relaxation, and orientation were considered in the model, and the relationship between these factors with shrinkage was obtained with experiments combined with the linear regression method. And warpage was calculated by the structure analysis package based on the prediction of shrinkage result. In the 1990s, the Moldflow Corporation measured the shrinkage of numerous materials under different flow velocities, packing pressures, packing times, mold temperatures, filling times, thicknesses, etc. The results formed the basis of impact factors on shrinkage, including the volumetric shrinkage, crystallinity, stress relaxation, and orientation effect. 6.2.2.1.2 Review of Flat Shell Elements Injectionmolded products generally possess a thin-wall structure, and the in-plane dimensions are often 1 order of magnitude larger than the thickness dimension, so they can be normally considered as perfect examples of thin shells. In the formulations of shell elements, there are four distinct choices: flat shell elements, curved shell elements, axisymmetric shell elements, and Mindlin type degenerated solid elements.66 Among these elements, the flat shell element is one of the most popular approaches in the finite element analysis, as it is simplest, computationally efficient, and free from complex shell equations. In general, flat shell elements should be able to endure the bending and stretching deformation. In case of small deformation, the stretching deformation and bending deformation of a flat element can be assumed to be independent of each other and, therefore, considered as the superposition of a membrane problem and a plate bending problem. Accordingly, flat shell elements can be developed by including the membrane and bending properties through the combination of a membrane element and a plate bending element. The constant strain triangle (CST) element is the simplest and the most widely used membrane element. However, the CST element is too stiff, and the stiffness of the assembled CST elements increases, as the aspect ratio of the elements becomes larger. Owing to the lack of a drilling degree of freedom, the CST element will cause a rotational singularity in the stiffness matrix, when all the elements sharing one node are coplanar and the local coordinate systems of the elements coincide with the global coordinate system. In this case, the global stiffness matrix becomes singular. To overcome this problem, a small fictitious stiffness with rotational degrees of freedom or membrane elements with rotational degrees of freedom can be employed.

176

RESIDUAL STRESS AND WARPAGE SIMULATION

The first successful membrane triangular elements with drilling freedoms were presented by Allman,67 followed by Bergan and Felippa.68 These elements were nonconforming and could pass displacement-specified patch tests. Allman later proposed a more sophisticated formulation69 . Since then, other elements with drilling degrees of freedom have been derived. But most of them suffer from the aspect ratio locking, which may occur when the response of the element is highly dependent on its aspect ratio. And during the modeling of a thin-walled structure, some elements frequently have very high or low aspect ratios such as in modeling stiffened plates. Felippa70 developed an optimal membrane triangular (OPT) element with a drilling degree of freedom. This element was an LST3/9R membrane element (linear strain triangular membrane element with three corner nodes and nine degrees of freedom, three per node, including the rotational degrees of freedom). The formulation of this element was based on the assumed natural deviatoric strain (ANDES) template, and its strain energy was accurate for any arbitrary aspect ratio. It has been shown that this element can achieve acceptable results even with relatively coarse meshes.71 As for bending elements, many types of triangular plate bending elements have been developed. There are normally two major principle theories for plate bending elements: (i) classical Kirchhoff thin-plate theory, which neglects the transverse shear strain and requires C 1 continuity; and (ii) Mindlin–Reissner plate theory,72 which accounts for the transverse shear strain and requires C 0 continuity. Most of the early attempts focused on the classical Kirchhoff thin-plate theory and tried to build perfect compatible elements satisfying C 1 continuity. However, owing to the stiffness of such elements and difficulty of building them, attention was turned to building noncompatible elements such as the BCIZ element. The BCIZ element73 may be one of the simplest Kirchhoff plate bending elements. It was developed by Bazeley et al. and was named after the authors’ initials. Research into the thicker plate element has begun since the 1970s and most of them were based on the Mindlin–Reissner plate theory. As only the C 0 continuity is required, various elements could be easily applied on the thick plate. However, the application of thick-plate theory into the thin-plate analysis commonly encounters the shear locking problem, resulting in an underestimation of deformation prediction. Numerous studies have been published to address this issue of shear locking. Batoz74,75 proposed the triangular (DKT, discrete Kirchoff triangular) and quadrilateral (DKQ) plate bending element, based on the discrete Kirchhoff theory. Results showed that these two elements possess excellent properties and could overcome the shear locking. But they could only be employed in the thin-plate analysis. Later, Batoz

modified them and constructed the DST-BL element76 and DST-BK element,77 which were suited for thick plates. The DKT element is probably the most influential bending element so far. A number of efficient 9-DOF (degree of freedom) triangular elements based on the discrete Kirchhoff constraint were developed later, such as the DKMT element.78 These elements can converge toward the discrete Kirchhoff plate bending elements when the thickness of the plate is very small. A number of effective elements that are free from shear locking have also been proposed.79 – 82 Brunet and Sabourin83 presented a rotation-free triangular element, which required only three translation degrees of freedom per node to describe membrane and bending effects. This element displays high effectiveness and sufficient accuracy, especially in springback prediction of sheet metal forming. But it will fail when there is more than one neighbor on the same boundary or when there are significant changes in thickness between neighboring elements, which is common in the plastic injection-molded parts.84 Recently, a refined nonconforming element method (RNEM) was presented by Chen and Cheung.85,86 It was based on the Mindlin–Reissner plate theory so that the model can be applied to both thin- and thick-plate analysis. And based on this theory, a new quadrilateral plate element RDKQM with the reconstitution of the shear strain has been established.87 Later, based on this method, they introduced the displacement function of the Timoshenko’s beam theory into the formulation and derived new triangular thin-/thickplate (RDKTM) elements.88 It was shown that the RDKTM element possesses high accuracy for both thin and thick plates and is capable of passing the patch test required for Kirchhoff thin-plate elements, without exhibiting the extra zero energy modes. In addition, the RDKTM is free of shear locking for very thin plate analysis (where the thickness/span ratio is B , it follows  L = 0.5γ + 0.5 4 + γ 2 R0 −0.5   B = 0.5γ + 0.5 4 + γ 2 R0 −1   tan θ = 0.5γ + 0.5 4 + γ 2

(7.39) (7.40) (7.41)

where γ is the accumulated shear history experienced.

 B = −0.5γ + 0.5 4 + γ 2 R0

(7.42)

7.3.2.2.6 Cox Theory The prediction of deformation parameter and the orientation angle is given as D = Ca

θ=

1 19p + 16  16p + 16 (19pCa/20)2 + 1

1 1 π + tan−1 4 2



19pCa 20



(7.43)

(7.44)

In the following sections, we apply the above models to different systems under various flow conditions. The predictions of these models will be compared with the available experimental data in published literature. Some comparisons results may have been published elsewhere and may not be listed in this chapter. Steady Deformations The predictions of the various models are compared with the experimental results of steady droplet deformation in simple shear flow, as shown in Figures 7.13 and 7.14. Several simulation results can be found elsewhere.72 It can be seen that most models agree well with the experimental results for both the deformation and the orientation angle, except for the failure of the Cox model in predicting the orientation angle at p = 3.6. The experimental results show large scattering at large capillary numbers, indicating that the shape of the droplet may not exactly follow the ellipsoidal form. Transient Deformation Transient deformation depends on the flow field and the characteristic time of the blends. Any nonsteady deformation can be regarded as a transient process. The capability of Yu model, MM model, and JT model in describing such a transient process have been examined before.72 And here we focus on the validation of two empirical models: affine deformation and shear deformation. For transient deformation at start-up with small capillary numbers, both models describe well the initial experimental results as clearly shown in Figure 7.15, but the large deviation from experimental data at subsequent stages indicates that the affine and shear deformations may not be adequate for describing the transient deformation in weak flows. For capillary numbers larger than the critical one, the droplet will deform and may eventually breakup. Predictions of transient deformation before breakup are

PHASE MORPHOLOGICAL EVOLUTION IN POLYMER BLENDS

0.5

0.5

Experimental

Experimental MM1 Model

0.4

209

MM1 Model

0.4

MM2 Model

MM2 Model 0.3

D

D

0.3

p = 3.6

p = 0.08 0.2

0.2

COX Model

Yu Model

Yu Model

0.1

JT Model

0.1

COX Model

JT Model 0.0

0.0 0.0

0.1

0.2 Ca

0.3

0.4

0.0

0.2

0.4 Ca

0.6

0.8

40 Experimental

Experimental

20

MM1 Model 35

MM2 Model

MM2 Model

10

COX Model

45 – θ

45 – θ

15 20

Yu Model

p = 0.08 COX Model 10

Yu Model

5

JT Model p = 3.6 Experimental

JT Model 0 0.0

0.1

0.2 Ca

0.3

0.4

0 0.0

0.1

0.2 Ca

0.3

0.4

FIGURE 7.13 Steady deformation in simple shear flow for p = 0.08. Source: The experimental results are from Torza et al.76

FIGURE 7.14 Steady deformation in simple shear flow for p = 3.6. Source: The experimental results are from Torza et al.76

shown in Figure 7.16. As can be seen, affine deformation seems to describe the slender deformation well, as it assumes an equal length of two shorter axes, while shear deformation appears better for the deformation of flat sheet at extremely large capillary numbers. Results of reversing the direction of shear after the droplet reaches its steady shape are displayed in Figure 7.17. Owing to the equal length assumption of two shorter axes, results from affine deformation are not listed. It is clear that shear deformation model fails to describe this phenomenon well.

When the capillary number is slightly greater than the critical value, the mode of breakup depends on the viscosity ratio. For example, when p is far lower than 1, the drop assumes a pointed sigmoidal shape and small droplets are released from the ends (tip streaming). For p ≈ 1, the slightly extended drop will breakup via a dumbbell shape, that is, the central portion of the droplet gradually necks until the droplet breaks into two daughter droplets, with small satellite droplets scattered in-between.47 When Ca is much greater than Cacrit , the original liquid drops are extended into long slender filaments because of affine stretching and local radii are decreased so that the interfacial tension becomes dominant, which tends to minimize the interfacial area between the two phases. The filament will subsequently break up by a capillary-wave instability (Rayleigh instability). Basic parameters governing droplet breakup are outlined as follows:

7.3.2.3 Break When a droplet is subjected to flow, it will deform, orient, and possibly break up. The droplet behavior is dependent on its viscosity ratio, the capillary number, the flow type (shear vs elongation ratio), etc. For capillary numbers Ca less than a critical value Cacrit , the droplet will arrive at a steady shape and orientation.

210

MICROSTRUCTURE AND MORPHOLOGY SIMULATION

1.4 L/R

B/R W/R

10

Ca = 0.24, p = 1.4 L/R B/R W/R

1.2

L/R, B/R, W/R

L/R, B/R, W/R

1.3

Experimental Affine Shear

1.1 1.0

P = 1, Ca = 5.0

MM1 Model Affine Shear

L/R B/R W/R Experimental

1

0.9 0.1

0.8 0

1

2 Strain

3

0

4

FIGURE 7.15 Transient droplet deformation in simple shear flow. Source: The experimental results are from Guido and Villone.48

4

L/R

B/R W/R

Cacrit —The critical capillary number is defined as the minimum capillary number required in case of breakup of a deformed drop. It depends on the type of flow and is customarily plotted as a function of p. For simple shear, breakup of droplet is impossible if the viscosity ratio exceeds 4, whereas in elongation, it becomes possible whatever the viscosity ratio is. The following de Bruijn empirical equation of Cacrit is widely adopted for shear flows78 : log10 Cacrit = −0.506 − 0.0995 log10 p + 0.124(log10 p)2 0.115 − log10 p − log10 4.08

(7.45)

Although the critical capillary number will be related to the elasticity ratio in viscoelastic systems, incorporation of this parameter is impossible because of the lack of elasticity effects. The conditions of breakup are limited to isolated drops. In more concentrated systems, however, this relation requires modification. As droplets impinge against each other, drops will be destabilized and breakup becomes easier, indicating a lower final critical capillary number. It has been shown that breakup predictions using a critical capillary number based on the emulsion viscosity instead of the matrix viscosity agree well with the experimental results, which confirms that breakup is caused by the average emulsion stress.79 Assuming that the forces on a single droplet as being proportional to the viscosity of the surrounding emulsion, the breakup relation can be modified as79 

p (7.46) ηr,em Cacrit = fGrace ηr,em

L/R, B/R, W/R

10

8 Strain

MM1 Model Affine Shear

12

16

P = 1, Ca = 70

L/R B/R W/R Experimental

1

0.1 0

4

8

12

16

Strain

FIGURE 7.16 Transient droplet deformation in simple shear flow for large capillary number. Source: The experimental results are from Almusallam et al.51

where ηr,em = ηem /ηm is the relative emulsion viscosity and ηem is the emulsion viscosity. k *—A reduced capillary number is defined as k * ≡ Ca/Cacrit . There are four regions both in shear and elongation, which define the general rule for droplet deformation or breakup80,81 : If k * < 0.1, droplets do not deform. If 0.1 < k * < 1, droplets deform, but do not break. If 1 < k * < 4, droplets deform and then split into two primary daughter droplets. If k * > 4, droplets extend into steady filaments, following the affine deformation of the matrix and may eventually breakup under two particular conditions: if the diameter of the fiber falls under a critical value d *, or a capillary-wave instability occurs after the cessation of flow.

PHASE MORPHOLOGICAL EVOLUTION IN POLYMER BLENDS

L/R

2.0

B/R

W/R Experimental

1.8

Experimental Fitting result

100

Shear

211

1.4 P = 0.11, Ca = 0.3 7

1.2

t*b

L\R,B/R,W/R

1.6

10 1.0 0.8 0.6 0.4

1 –2

0

2

6

4

10–4

10–3

10–2 10–1 Viscosity ratio

8

t/T

10–0

101

FIGURE 7.18 Effect of viscosity ratio on the dimensionless time to break. Source: The experimental data are from Elemans81 and Grace.78

40 30 P = 0.11 Ca = 0.37

20 10

Experimental

0

Shear deformation –10 –20 –30 –40 –2

0

2

4

6

8

t/T

FIGURE 7.17 Shear reversal. Source: Scatter symbols are experimental results from Guido et al.77

t∗b —It should be noted that all the breakup mechanisms described above (drop deformation, affine fibrillation, drop splitting, or thread breakup) are not instantaneous but necessitate a required time,82 which can be determined from initial drop diameter, interfacial tension, and viscosity ratio.78,83,84 A dimensionless time for breakup tb∗ can be defined as tb∗ ≡ tb γ˙ /(2Ca), which can be plotted as a function of p 78 : tb∗ = 84p0.345 k ∗−0.559

(7.47)

The fitting result to the experimental data is shown in Figure 7.18. Droplet breakup normally requires that the local residence time in the flow field should be greater than the necessitated breakup time. D 0 ,d *—Apart from the choice of the mechanisms of morphological changes, two parameters play an important

role in the droplet simulation: the initial diameter D 0 and the critical fiber diameter d *. Researchers have shown that the computation results are affected by the choice of d * only under a given value.82 Many researchers started their computation with the initial pellet size while forgetting the melting zone, which often resulted in an unrealistic prediction. It has been demonstrated that a real predictive model could only be attained by including the melting mechanism in the actual calculation.82 For drop splitting process that leads to two identical smaller drops, without considering the effect of neighboring droplets, the diameter of resulting daughter droplet after splitting can be obtained from constant volume principle, reading d ≈ 0.794D’. Using the definition of time to breakup, the rate of change of the total number of drops N d can be expressed as dNd γ˙ Nd = (7.48) dt Cacrit tb∗ where Nd = π6φV , V is the total volume of emulsion. The D3 drop splitting rate in terms of drop diameter can therefore be derived as85 

dD dt



break

=

−γ˙ D 3Cacrit tb∗

(7.49)

The above equation is valid as long as the reduced capillary number k * satisfies 1 < k * < 4. As has been mentioned earlier, the drop deforms into a long fiber for k * > 4 and may subsequently disintegrate under the capillary instability or interfacial tension.83,84 For the typical sinusoidal Rayleigh disturbance, the liquid

212

MICROSTRUCTURE AND MORPHOLOGY SIMULATION

cylinder radius follows80 R (z) = R¯ + α sin



2π z λ



(7.50)

where the average radius  R¯ is expressed in terms of the 2 initial radius R 0 as R¯ = R02 − α2 and the disturbance amplitude α grows exponentially with time as α = α0 exp (qt)

(7.51)

where α 0 is the distortion at t = 0 and q is the growth rate parameter. The thread breakup occurs once α = R ∼ = 0.81R 0 . For incompressible liquids, the resulting droplets after thread breakup can also be obtained from conservation of volume:  3π (7.52) Rdrops = R0 3 2Xm where R 0 is the initial fiber radius and X m is the dominant wave number. The major difficulty of the above computation comes from the estimation of the initial distortion α 0 . As it is obvious that breakup of fibers in quiescent state will become almost instantaneous as the fiber diameter falls into the submicron range, one can alternatively assume that the break of fiber will become independent of k * once the filament diameter decreases below a critical value d *.85 For a spherical droplet subjected to a large shear strain, the resulting fiber length L and diameter B can be approximated by the affine deformation model as85 B = bD0 γ −1/2

L = D0 γ ,

(7.53)

where the constant b depends on the shape of the deformed drop, b = 1 for a spheroid, and b = (2/3)1/2 for a cylinder. When a cylindrical fiber disintegrates, the resulting droplets will have a diameter about twice of the fiber diameter. The diameter of resulting droplets from the cylindrical breakup can therefore be given by 

D1 = 2

2 D0 γ −1/2 3

and breakup produces the phenomenon of morphological hysteresis.47 Tokita86 suggested that the drop diameter in polymer blends originates from the two competing processes: continuous breakup and coalescence of the dispersed particles. Experimental results87 have shown that when the volume fraction of dispersed phase exceeds 0.5%, the final drop size is usually larger than predicted if no coalescence is considered. The critical event in coalescence is a collision between the two droplets. Colliding droplets develop a thin film of matrix fluid separating them. The matrix fluid drains from the film when hydrodynamic forces push the drops together as shown in Figure 7.19. If the film thickness falls below a critical value h c , then van der Waals forces become dominant, which cause the film to rupture, leading to the coalescence of droplets. Alternatively, the hydrodynamic forces may reverse before the film ruptures, and in this case, the droplets separate without coalescing.47 In the modeling of coalescence, two events have to be considered: drops collide within a given process time and the film between the drops drain sufficiently during the available interaction time.80 For equal spheres following the basic external flow, the collision frequency can be estimated. For example, in simple shear flow, the collision frequency of an individual drop reads C = 8γ˙ φtloc (7.55) And the probability of collision reads80 Pcoll



−π = exp 8γ˙ φtloc



(7.56)

where t loc is the local residence time. The probability for expulsing the liquid film depends on the mobility of the interface, and the mobility is usually determined by the viscosity ratio p: a greater p indicates

(7.54)

where D 1 is the final drop diameter after complete disintegration. 7.3.2.4 Coalescence The droplet size will decrease because of the breakup and increase because of coalescence. Coalescence represents a limit on our ability to obtain fine microstructures, and the interaction between coalescence

FIGURE 7.19

Coalescence process of two deformable drops.

213

PHASE MORPHOLOGICAL EVOLUTION IN POLYMER BLENDS

pexp

for partially immobile interfaces (7.58)    R 3 = exp − ln for mobile interfaces k∗ 2 hc (7.59)

The probability of coalescence p coal can thus be defined as the product of p coll and p exp . Although many previous investigations have been devoted to two droplets coalescence, there still remain a lot of unsolved problems. The major difficulties come from the shear flow field, properties of fluids, interfacial properties, etc. Taking the interfacial properties, for instance, the effects of interfacial thickness and mobility have seldom been studied in the coalescence process. For simplification, theoretical analysis of coalescence often makes certain assumptions, which limit the application of the model. To account for the complete effect of all the basic parameters, direct numerical simulation seems a good choice. A straightforward approach is using a mesh that has grid points on the interface and deforms under the flow field on both sides of the interface. Numerical approaches include the boundary-integral and boundary element method,88 finite element method (FEM),89 and finite difference method.90 However, the track of moving interface in these methods is at the cost of cumbersome computations. Alternatively, one can make use of fixed grid in handling moving interfaces, which include volume-offluid method,91 level-set method,92 and diffuse interface method.93 The coalescence rate of droplet diameter usually follows the theoretical relation as85

 dD = CD −1 φ 8/3 γ˙ (7.60) dt coal where C is the coalescence constant. The final drop diameter can be assumed as the sum of the breakup and coalescence, yielding85 

dD dt



=



dD dt



break

+



dD dt



(7.61)

coal

Diameter evolution can thus be tracked by solving the above equation over time with a finite difference method.

2.0 Deq = 0.116 + 3.259 × Φ4/3

R = 0.983

1.6 Experimental Deq Deq (um)

that the interface is more immobile. It can be expressed as80   

9 R 2 ∗2 pexp = exp − k for immobile interfaces 8 hc (7.57)  √ 

 3 R pk ∗3/2 pexp = exp − 4 hc

Linear fitteed Deq

1.2

0.8

0.4

0.0 0.0

0.1

0.2 Φ4/3

0.3

0.4

FIGURE 7.20 Average equilibrium diameter of POE dispersed in PP as a function of POE volume fraction.

As the evolution rate becomes 0 at equilibrium leading to85  1/2 0 Deq = Deq + 6CCacrit tb∗ φ 8/3 (7.62)

0 is its value where D eq is the equilibrium diameter and Deq extrapolated to zero concentration. The coalescence constant C for a given polymer blend can thus be obtained by preparing various compositions of the blend and calculated from the slope of the equilibrium diameter D eq versus composition φ 4/3 plot. As shown in Figure 7.20, the coalescence constant C for the given blend is computed as C = 2.77 × 10− 15 . The resulting droplet size after coalescence can also be computed from volume conservation, simply as82

R∗ = R 7.3.3



2 2 − pcoal

1/3

(7.63)

Case Study

A computation of the droplet morphology development for polymer blends along a twin-screw extruder was conducted by Delamare and Vergnes,82 based on the basic mechanisms of breakup and coalescence. Depending on the value of a local capillary number and on local flow conditions, different morphological changes were accounted for, covering affine deformation, drop splitting, break up by capillary instability, and coalescence. Although the computation was performed in the twin-screw extruder molding, it can easily be applied to other molding cases such as the injection molding. A blend with a polypropylene matrix (Hoechst PPU 1780) in which a terpolymer ethylene-ethyl acrylate-maleic anhydride (Elf Atochem Lotader 3700) is dispersed (10%

214

MICROSTRUCTURE AND MORPHOLOGY SIMULATION

(a)

(b)

FIGURE 7.22 Influence of initial diameter D 0 on morphology evolution calculation (blend 90/10, 200 rpm, 20 kg/h, D 0 = 0.53 µm, d * = 0.3 µm).82

(c)

FIGURE 7.23 Comparison of theoretical and experimental mean diameter (blend 85/15, 200 rpm, 20 kg/h, D 0 = 0.53 µm, d * = 0.3 µm).82 (d)

FIGURE 7.21 Example of theoretical result (blend 90/10, 200 rpm, 20 kg/h, D 0 = 0.53 µm, d * = 0.1 µm): (a) average diameter, (b) ratio of local to critical capillary number, (c) ratio of breakup time to local time, and (d) probability of coalescence.

in weight) is presented. Predicted changes along the screw profile of mean diameter D, of the ratio of local to critical capillary number, of the ratio of breakup time to local residence time, and of the probability of coalescence are presented in Figure 7.21,85 respectively. It is clear that the change in the mean diameter is the coupled result of breakup and coalescence. For example, when entering the zone under pressure preceding the mixing section (point B), conditions for fibrillation are fulfilled (k /k * ≈ 10) and breakup occurs rapidly because the limiting value (d * = 0.1 µm) is reached, leading to droplets of 0.2 µm after breakup. As has been said, the initial diameter D 0 would impact on the computational result, as shown in Figure 7.22.82 It appears that the choice of D 0 is not crucial as soon as

D 0 is less than 10 µm. But with D 0 = 1 mm, a fibrillar morphology will remain all along the screw profile, as the fibers are too big to be broken in the flow field (breakup time of capillary instability is too high).82 And a real predictive model should consider the melting mechanism in the actual computation. Simulation results were compared with experimental data, in which the experimental dispersion was characterized by an average diameter (Fig. 7.23) and a distribution (Fig. 7.24). It can be seen from Figure 7.23 that the magnitude and slight evolution of the morphology along the screws are well predicted by the model.82 In addition, the model captures the drastic change between initial and final distributions as shown in Figure 7.24.82 Therefore, the proposed theoretical model may thus be applied to define the best conditions for obtaining a desired morphology.82 7.4

ORIENTATION

The study of orientation is closely related to the fact that the orientation in molecules or fibers will inevitably lead to the anisotropy in polymer properties.

ORIENTATION

215

7.4.1.1 Uniaxial As observed in fibers and uniaxially stretched films, the uniaxial orientation of molecular chains in polymers can be statistically expressed as a function of the angle θ between the symmetry axis and the chains segments as95 3cos2 θ − 1 (7.64) fH = 2

(a)

where f H is the Hermans orientation factor that represents the value of second moment of uniaxial molecular orientation and ranges from −1/2 to 1. For the cubic orientation systems, the value of Hermans orientation factor is 0. For crystalline materials, the Hermans orientation factor is defined by the angles between the symmetry axis and crystallographic axes j as96 fj =

3cos2 θj − 1 2

(7.65)

The cos2 θj terms are not independent of each other but are interconnected through trigonometric relationships. For an orthorhombic unit cell, the Pythagorean theorem requires that96 (b)

FIGURE 7.24 Comparison of theoretical and experimental size distribution. (a) End of melting zone (initial choice for the computation) and (b) die exit.

7.4.1

Molecular Orientation

Molecular orientation refers to the directional alignment of molecular chains when subjected to either a shear or an elongational flow. In a quiescent polymer melt, molecular segments are in a random state, that is, in a state of a maximum disorder. When external flow is applied, for example, during molding processes, molecular chains will orient along preferential directions by the flow. Part of this orientation freezes during solidification, resulting in anisotropy of the solid polymer properties.94 For instance, orientation of macromolecules can cause different properties of crystals in different directions. Therefore, one of the biggest challenges in present polymer science concerns the reliable prediction of molecular orientation during processing. On the basis of the number of principal refractive indexes, a polymer can be classified into three fundamental symmetry systems in an optical point view: cubic (isotropic), uniaxial, and biaxial. The molecular orientation in fabricated polymers can be characterized in terms of the orientation factor in the following sections.

cos2 θa + cos2 θb + cos2 θc = 1,

fa + fb + fc = 0 (7.66) It should be noted that the orientation factor of the crystalline phase f Hc can be distinguished from the orientation factor of the amorphous phase f Ha for a semicrystalline polymer. Alternatively, one can define the molecular orientation based on the anisotropy of polarizability tensor α ij : fH =

α11 − α22 α11 − α33 = α0 α0

(7.67)

where α 0 is the value of the difference in polarizability along and perpendicular to polymer chains; α 11 , α 22 , and α 33 are the components of polarizability tensor along “1” and perpendicular “2,3” to the symmetry axis. 7.4.1.2 Biaxial Biaxial orientation is common in polymeric films and injection-molded products. It can be represented by various sets of Euler’s angles, as a function of the angle θ 1 of the molecular chain axis with the primary symmetry axis 1 and the angle θ 2 of the molecular chain axis with the second latitude 2. For example, White and Spruiell defined the orientation factor as97 f1B = 2cos2 θ1 + cos2 θ2 − 1

(7.68)

f2B = 2cos2 θ2 + cos2 θ1 − 1

(7.69)

216

MICROSTRUCTURE AND MORPHOLOGY SIMULATION

The above equations are symmetric with regard to angles θ 1 and θ 2 . As for crystalline materials, the above equations for biaxial molecular orientation can be derived by introducing angles between appropriate crystallographic axes j and defined laboratory axes 1 and 2, reading97 B f1j = 2cos2 θ1j + cos2 θ2j − 1

(7.70)

B f2j

(7.71)

= 2cos2 θ2j + cos2 θ1j − 1

where θ 1j represents the angle between 1 laboratory axis and the j -crystallographic axes, and θ 2j is the angle between the 2 laboratory axis and j -crystallographic axes. The birefringence measurement is one of the simplest methods of studying molecular orientation in polymers, which measures double refraction, namely, the existence of two different refractive indexes in two directions. For an anisotropic polymer, its molecular polarizability becomes a second-order tensor as98 ni − nj =

 (n¯ + 2)2 •  N αi − αj 18n¯

n1 − n2 

◦

=

n1 − n3 

◦

A=



3 < RR > − < RR>0 2 < R0 >

(7.75)

And the constitutive equation for the subchain population can be written as94 1 D A − ∇ν T • A − A • ∇ν = − A + ∇ν + ∇ν T Dt λ

(7.76)

The relaxation time, λ, is considered constant, this means that the model is not able to predict shear-thinning behavior of polymer melts. To remedy situation, λ should be allowed to vary with shear rate temperature, which was proposed as follows94 :

(7.72)

where N represents the number of molecules per unit volume, α i and α j are the polarizabilities along the principle axes, n i and n j are the principle refractive indexes, and n¯ is a mean refractive index of the polymer. The definition of the Hermans orientation factor for uniaxial orientation can thus be written as fH =

where < RR>0 is the value of < RR > at rest and the endto-end distance of the molecular chain follows < R02 >= tr < RR>0 ; λ is the relaxation time. Defining fractional “deformation” of the population of dumbbell subchains with respect to the equilibrium conformation as

  λ T , P , γ ′, χ =

λα ′ (T , P , χ )

1−b 1 + E λα ′ (T , P , χ ) γ ′

and the this and

(7.77)

α ′ (T , P , χ ) = 10−(F1 (T −B1 −B2 P ))/(F2 +T −B1 ) h′ (χ ) (7.78)

 e2 ′ h (χ ) = 1 + e1 exp − p χ

(7.79)

(7.73)

◦

where  is the intrinsic birefringence of the polymer. Qualitatively, the orientation of macromolecules along a preferred direction is the result of a competition between the characteristic relaxation time λ, which is a function of thermomechanical and crystallinity histories, and the flow characteristic time, t f , which is the reciprocal of the deformation rate.1 A high ratio of t f /λ corresponds to a high orientation level. The most appreciated model would be the Leonov constitutive equation, upper convected Maxwell models, and, more recently, Pom–Pom model.99,100 A nonlinear formulation of the elastic dumbbell model101 has been widely adopted in describing the evolution of molecular orientation by the effect of kinematics obtained using a viscous approach. If R is the end-to-end vector of a molecular chain and the symbol < > is the average over the configuration space, the second-order conformation tensor < RR > can be linked to the velocity gradient, ∇ ν by the elastic dumbbells as102 D < RR > −∇ ν • < RR > − < RR > • ∇ ν T Dt

1 < RR>0 − < RR> = (7.74) λ

7.4.2

Fiber Orientation

During the injection molding process of fiber-reinforced composites, the molten polymer and the fibers flow as a suspension into the mold cavity when injected through a gate under high pressure. When the fibers are transported along with the resin, they will change their orientation during the flow. The complicated flow phenomena normally create variations in the orientation throughout the part. Deformation of the suspension can influence fiber orientation state and conversely the flow will also be influenced by fiber orientation. The frozen-in orientation of fibers will greatly affect the microstructure and final properties of the molded product. One of the key issues is to find the orientation of the fibers, as they flow into the mold cavity.6 Qualitatively, fibers align in the direction of shear and stretching. And a skin/core effect has been observed in many injection-molded parts. Specifically, fibers are aligned in the flow direction near the surface, and they can be oriented transverse to the flow direction, which can be explained from the flow kinematics. Any motion of the fiber must necessarily involve a cooperative motion of neighboring fibers.

ORIENTATION

According to the fiber volume fraction φ and the fiber aspect ratio a R (defined by the length-to-diameter ratio L/d ), fiber suspensions can be classified into three concentration categories: dilute, rodlike, and concentrated fibers. 1. Dilute Suspensions: the volume fraction satisfies φaR2 < 1, and each fiber rotates freely. 2. Rodlike Fibers: the volume fraction satisfies φ = nπ d 2 L/4, where n is the number density of the fibers. 3. Concentrated Suspensions: the volume fraction satisfies φa R > 1. As the average distance between fibers is less than the fiber diameter, fibers cannot rotate independently except around their symmetry axes. And the suspension with 1 < φaR2 < aR is called semiconcentrated , where each fiber has only two rotating degrees of freedom. Most commercial composites commonly used in injection molding fall into the semi- or highly concentrated regimes.103 Isotropic constitutive models are no longer valid for injection-molded fiber-reinforced composites. Unless the embedded fibers are randomly oriented, fiber orientation in the fiber-reinforced composites should be accounted for and anisotropy in the thermomechanical properties of the injection-molded products should be introduced. A lot of research has been carried out in predicting the fiber orientation. A typical example would be Folgar and Tucker’s work104 on modeling the motion o f a single fiber in concentrated suspensions. Their approach, based on Jeffery’s equation105 , which is only valid for dilute fiber suspensions without considering fiber–fiber interaction, added the interaction that occurs between fibers through a diffusion term. The governing equation for fiber orientation in planar flows can be derived as θ˙ =

∂vx ∂vx −CI γ˙ ∂ψ − cos θ sin θ − sin2 θ ψ ∂θ ∂x ∂y ∂v ∂v y y + cos2 θ + sin θ cos θ (7.80) ∂x ∂y

where θ is the angle at which the fiber is oriented with respect to the flow field, ψ is the orientation distribution function, v x and v y are the derivatives of velocity, and C I is an empirical constant called the interaction coefficient, which can be considered as a measure of the intensity of fiber interactions in the suspension. C I is assumed to be isotropic and independent of the orientation state, as a first approximation. Instead of representing the orientation of a fiber by a single angle θ , Advani and Tucker77 developed a more efficient method in representing the fiber orientation, with the use of the comments of a unit vector along the fiber axis and various second-order tensors. This approach

217

dramatically reduced the computational requirements when solving the equations using the Folgar–Tucker method. Adopting a unit vector p to specify the fiber orientation in suspensions, the probability distribution function of the fiber orientation can be described by the second- and fourthorder orientation tensors < pp > and < pppp > (or, in Cartesian tensor notation, a ij and a ijkl ), where the angular brackets denote the average with respect to the probability density function. The orientation p of each fiber depends on its initial configuration, aspect ratio, the number density of fibers in the suspension, and the shear deformation.103 The Folgar–Tucker model in terms of the orientation tensors can be formulated as follows106 :   Daij = ωik akj − aik ωkj + λ Dik akj + aik Dkj − 2Dkl aijkl Dt   +2CI δij − αaij (7.81)

where δ ij defines the unit tensor; ωij the local vorticity, defined as ωij = (v i , j − v j , i )/2; D ij is the rate of strain tensors; α is a constant that equals 3 for three-dimensional orientation and 2 for planar orientation to satisfy the condition tr < pp > = 1; and λ is a function of the fiber  aspect ratio defined as aR2 − 1 / aR2 + 1 . In order to express the fourth-order tensor in terms of the second-order tensor, a closure approximation is needed. Different forms of closure approximations have been proposed, including the quadratic, hybrid, and composite approximations. And the validity of the closure schemes depends on the type of flow and the degree of alignment of the fibers. The major obstacle with the use of the Folgar–Tucker model is the determination of the C I value. A direct numerical simulation of fiber–fiber interaction for simple shear flow was recently developed by Fan et al.107 Shortrange interaction is modeled by lubrication forces and longrange interaction was calculated using the boundary element method. The components of both the second- and fourthorder tensors a ij and a ijkl can be obtained by an appropriate averaging procedure, and their numerical values are then used in an anisotropic version of Folgar–Tucker’s rotary diffusion equation to determine the interaction coefficient in the defined shear flow as follows103 :   Daij = ωik akj − aik ωkj + λ Dik akj + aik Dkj − 2Dkl aijkl Dt   − 3γ˙ Cjk akj + Cjk aik + 6γ˙ Ckl aijkl   + 2γ˙ Cij − Ckk aij (7.82)

where the symmetrical second-order tensor C ij accounts for the anisotropic behavior and is extended from the isotropic diffusivity C I δ ij . Once the tensor C ij has been determined, the evolution of fiber orientation in injection molding can be simulated.

218

MICROSTRUCTURE AND MORPHOLOGY SIMULATION

z/b (a)

z/b (b)

FIGURE 7.25 Gapwise distribution of birefringence n at various radial locations at the end of cooling (packing pressure = 0 MPa): (a) Experimental data and (b) numerical analysis results.108

7.4.3

Case Study

7.4.3.1 Molecular Orientation Molecular orientation in injection-molded products was analyzed by Kim et al.,108 through a numerical simulation and experimental measurement of birefringence. A center-gated disk was presented as the specimen. The relationship between the birefringence and residual stress for polymeric materials under stress is well established in terms of the stress-optical law. Numerical analysis results have been compared with the corresponding experimental data for the case of no packing stage (i.e., zero packing pressure) in Figure 7.25108 and for the case of a packing pressure of 16.5 MPa and a packing time of 6.0 s in Figure 7.26.108 As can be seen, the most important effect of the packing stage on the birefringence distribution is the appearance of an

(a)

(b)

FIGURE 7.26 Gapwise distribution of birefringence n at various radial locations at the end of cooling (packing pressure = 16.5 MPa): (a) Experimental data and (b) numerical analysis results. The meaning of symbols is the same as explained in Figure 7.25.108

inner peak in addition to the outer peak appearing during the filling stage, as one can find when Figure 7.26 is compared with Figure 7.25. The inner peak appearing with packing is due to the deformation, and thus more molecular orientation is obtained when additional flow is allowed into the cavity, while the outer peak, which appears during the filling stage, remains unchanged in postfilling stages, as the birefringence is already frozen near the cavity wall.108 It is clear that the overall shapes of the birefringence distribution of the numerical analysis results are in good agreement with experimental measurements for both cases. However, numerical analysis of birefringence values appears smaller than the experimental data, possibly because the stress-optical coefficient might not be a constant

ORIENTATION

219

(a)

(a)

(b)

FIGURE 7.28 Filling simulation results for a tension test specimen: (a) filling pattern (step increment = 0.0464 s, filling time = 0.8807 s) and (b) velocity field.

(b)

FIGURE 7.27 Gapwise distribution of birefringence n at the end of the cooling at (a) melt temperature = 200 ◦ C (t = 8.090 s) and (b) melt temperature = 240 ◦ C (t=8.412 s).108

as used in this numerical analysis, or the overall modeling via the Leonov model is not accurate enough for this kind of prediction.108 Significant effect of melt temperature on the simulated birefringence distribution is shown in Figure 7.27.108 (The meaning of symbols is the same as explained in Fig. 7.25.) As melt temperature increases, the birefringence decreases particularly near the cavity wall, because of the fact that the frozen layer gets thinner and the duration time, when the elastic stress can be relaxed, gets longer as melt temperature increases. 7.4.3.2 Fiber Orientation A numerical simulation program to predict the transient behavior of fiber orientations together with a mold filling simulation for

short-fiber-reinforced TPs in arbitrary three-dimensional injection mold cavities was developed by Chung and Kwon.109 The Folgar–Tucker model introduced above was employed, with the hybrid closure approximation given for the three-dimensional orientation for the fourth-order tensor a ijkl . And the fiber orientation tensor is determined at every layer of each element across the thickness of molded parts with appropriate tensor transformations for arbitrary three-dimensional cavity space. Numerical simulations have been carried out for cavity geometry of a tension test specimen with a pin-end gate. The successive flow fronts and the velocity fields are shown in Figure 7.28109 after the cavity gets filled. The diverging flow around the gate is followed by parallel flow, converging-parallel-diverging and parallel flow in sequence. In order to study the effect of the fiber–fiber interaction coefficient C I , orientation state under two conditions was simulated, as shown in Figure 7.29 (C I = 0.001)109 and Figure 7.30 (C I = 0.05).109 The orientation fields at the end of filling at four different layers are represented by three eigenvectors with the magnitude of each eigenvalue of the second-order orientation tensor plotted. It is clear that the fiber orientation state approaches the randomized orientation state as the magnitude of C I increases.109

220

MICROSTRUCTURE AND MORPHOLOGY SIMULATION

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

FIGURE 7.29 Fiber orientation state when the cavity is filled for a tension test specimen at (a) z = 0, (b) z = 0.2b, (c) z = 0.4b, and (d) z = 0.9b.

FIGURE 7.30 Fiber orientation state with C I = 0.05 when the cavity is filled for a tension test specimen at (a) z = 0, (b) z = 0.2b, (c) z = 0.4b, and (d) z = 0.9b.

To investigate the effect of the initial orientation state at the gate, the transient fiber orientation states at the center layer is shown in Figure 7.32,109 with the aligned initial condition at the gate as injection mold filling proceeds. At the initial stage, fibers around the gate are aligned to be perpendicular to the flow direction because of the strong diverging flow, as shown in Figure 7.32a. As the melt front advances, fibers near the gate tend to follow the initial orientation, as indicated in Figure 7.32b–d, probably because the diverging flow around the gate becomes weaker because of the geometrical effect of the specimen, and the convective transport from the gate becomes relatively dominant.109 Meanwhile, fiber orientation away from the gate is not affected much by the initial orientation.

7.5.1

7.5

NUMERICAL IMPLEMENTATION

The modeling and simulation results of crystallization, phase morphology, and orientation have been discussed earlier. This section briefly introduces their numerical implementation.

Coupled Procedure

In general, the numerical implementation is particularly challenging because relevant physical processes must be accounted for at two vastly different length scales.110 On the macroscale, the flow in the injection machine must be modeled. And it has been well established that the flow in the mold cavity during the injection molding process is a transient process. On the micro scale, the morphological behavior of dispersed phase, including deformation, reorientation, and advection of the microstructure, must be calculated. Although each situation can be addressed individually with existing models and methods, little work has been made on the combined problem. The primary concern could be developing a strategy that allows simultaneous representation of microstructural detail and global variations in microstructure, while accurately modeling both macroscopic complex flows and microstructural evolution.110 For simplification, a semicoupled method can be used to consider the interactive effects of flow field and microstructure. The calculation in each modeling is done separately. And the result of flow field simulation provides the input

NUMERICAL IMPLEMENTATION

221

At every moment, the following steps are obeyed: 1. the variable distribution (velocity, temperature, pressure, and viscosity) is obtained by solving the constitutive equation of the flow field; 2. the local flow field details are substituted into the microstructure constitutive equation and the microstructure distribution at that moment is obtained; 3. the rheological parameters are updated based on the microstructure distribution; 4. the above steps should be followed again and again until the iteration ends, and the final morphology distribution results. 7.5.2

(a)

(b)

(c)

(d)

FIGURE 7.31 Transient orientation state at core layer (z = 0) with aligned initial condition at the gate at (a) t = 0.081 s, (b) t = 0.255 s, (c) t = 0.557 s, and (d) t= 0.881 s.

parameters for the calculation of the microstructure evolution, and in turn, the result of microstructure simulation affects the flow field by changing the rheological properties of the blends. A sketch for the semicoupled modeling of polymer blends in injection molding has been illustrated in Figure 7.32.

FIGURE 7.32

Stable Scheme of the FEM

The FEM is most widely used in the simulation of injection molding for its better applicability to the complex geometry of the thin-walled injection-molded product. When integrating the governing equations, assumption should be made on how the physical quantities vary between nodes, because different assumption will lead to different discrete scheme. The standard Galerkin method, which has been discussed in Chapter 4, uses the shape function N β as the weighting function, but false oscillation will appear once there are convection terms, and the numerical instability will grow as the convection strengthens. To address this problem, Steamline-Upwind/Petrov-Galerkin (SUPG) method is used.111 This method introduces artificial dissipation terms (the derivatives of the shape functions) in the streamline direction to improve the shape functions. Besides, while solving the NS equation by the standard Galerkin method, the pressure and velocity are solved cooperatively by equal-order interpolations, which often produces singular stiffness matrix, so the pressure results are not quite ideal or even loses physical sense. The pressurestabilizating/Petrov-Galerkin (PSPG) method, with stable

Semicoupled modeling of polymer blends in injection molding.

222

MICROSTRUCTURE AND MORPHOLOGY SIMULATION

discrete scheme, overcomes this problem and is able to eliminate the numerical oscillation and guarantee the compatibility of the interpolation space.112 The injection molding process is typically convection dominant (the Reynolds number is high). To get rid of the numerical oscillation problem and get satisfactory results, both the SUPG and PSPG methods should be used to solve the governing equations (NS equations, temperature and microstructure equation). 7.5.3 Formulations of the Velocity and Pressure Equations The spatial region  is divided into e , e = 1, 2, . . . , n el , where n el is the total number of the elements. Within an element region e , assume that on the boundary Ŵ h , the stress σ (x , y, z )ij n j = h Ŵ , where n j is the outer normal unit vector on the boundary and h Ŵ is a known load function, the standard Galerkin method discrete scheme of the incompressible NS equation reads

equation; the third term on the right-hand side of the mass conservation equation represents the standard Galerkin scheme, which is also called the incompressible restrictive condition; the fourth term represents the PSPG stabilizing term, which is a product of the perturbing term τ PSPG ∇ q and the residual of the momentum conservation equation, 2 where τPSPG = ρh 4η is the PSPG stabilizing factor, h representing a certain characteristic length of the element, and h Ŵ representing the load on the boundary Ŵ. Obviously, when τ PSPG = 0, the above equation is reduced to the classic Galerkin scheme. Assuming that q = 0 and w = 0, it is obtained that



ηwj (ui,j + uj,i )d =

 e

= (w, hŴ )Ŵh   q, ui,i = 0

(7.83)

(7.84)

where the symbol (·, ·) represents the inner product, while w and q represent weighting functions, which equal the shape functions of the element. The incompressible NS equation is simplified as ∇ · σ = f (momentum conservation equation) and ∇ · u = 0 (mass conservation equation), then the PSPG discrete scheme within the whole solved region can be simplified as • • w (∇ s − f)d + q • (∇ • u)d   + τPSPG ∇q • (∇ • s − f)d = 0 (7.85) 

Substituting the concrete forms of every term of the equation, the pressure FEM equation is obtained based on the PSPG scheme:   hw,j ui,j + uj,i d − w,j P δij d   nel + quj,j d − τPSPG • ρ −1 q,j δij • e=1 e   

η ui,j + uj,i − Pj δij d = whŴ dŴ (7.86) Ŵ

The first two terms on the left-hand side and the terms on the right-hand side of the equation represent the standard Galerkin scheme of the momentum conservation



wj P δij d +



wh dŴ Ŵ

τPSPG q,i p,i d = − ρqui,i d e   + τPSPG q,i η(ui,j + uj,i ),j d e

    w, rui,t + ruj ui,j + (D (w) , 2hD (u)) + w,i , P,i δ ij



(7.87)

(7.88)

e

Assuming that w = N α and q = N α , where α = 1, 2, 3, 4 is the index of the tetrahedron element node, from the above equation, the FEM schemes of the velocity and pressure equation can be obtained as  e

e

  ηNα,j Nβ,j ui α + Nβ,i uj α d

 − e

e

Nα,j Nβ Pβ δij d =

 e

e

 e

Nα Nβ,i ui β d +

Nα,i Nβ,i Pβ d = 0

Ŵe

Nα hŴ dŴ

nel  e

e

(7.89)

τGLS ρ −1 (7.90)

where N β is the above-mentioned interpolating function. The velocity and pressure can be solved separately. As the equations include both the pressure and velocity parameters, an iterative procedure should be used, as shown in Figure 7.33. In the solving process, it is necessary to amend the results when the velocity and pressure are renewed, to ensure the stability of the computational process. For example, the low relaxation strategy can work P k+1 = ς P k + (1 − ς ) P k+1 , uk+1 = ς uki + (1 − ς ) uk+1 i i

(7.91)

where ζ is the relation factor, with the value ranging between 0 and 1.

NUMERICAL IMPLEMENTATION

223

where C V and K represent the specific heat capacity and the coefficient of heat conductivity of the material, respectively, and τ SUPG is the SUPG stabilizing factor, determining the value of the artificial dissipation term introduced in the streamline direction. By comparing the value of t 2 (it represents the transient character of the fluid, where t is the time step adopted h in the computation), 2 u (it represents the convection 2

FIGURE 7.33

Solving steps of velocity–pressure equations.

7.5.4 Formulations of Temperature and Microstructure Equations

 + e

Ŵ

 e

e

  w + τSUPG uj • ∇w e

Ŵe

wq0 dŴ (7.93)

Assuming that w = N a , the first-order backward difference T , t = (T n − T n − 1 )/t is used as the derivative of temperature with regard to time, then the stable scheme of the energy equation is obtained as    ρCV Nα + τSUPG uj Nα,j e

e

Nβ + ui Nβ,i Tβn+1 d t  + KNα,i Nβ,i Tβn+1 d 

e

e

= (7.92)

e

w,i KT,i d =

    2 0.5η ui,j + uj,i + H˙ C d +

e

 2  d − 0.5η ui,j + uj,i  2   d + wq0 dŴ w 0.5η ui,j + uj,i = 

e

e

In general, the governing equations of the evolution of temperature and microstructure are always characterized with apparent convection. As discussed above, the traditional standard Galerkin method, whose weighting function and shape function are the same and numerical oscillation is inevitable, will fail. Therefore, the SUPG method proposed by Brooks and Hughes111 is thought to be the first stable scheme that can manage, eliminating the numerical oscillation in the convection-dominant problem. Assuming that there is a second boundary condition KT , i n i = q 0 on the boundary Ŵ of the energy equation during the filling process, the SUPG scheme goes as   w,i KT,i d ρwCV T,t + ui T,i d +      τSUPG uj • ∇w ρCV T,t + ui T,i − KT,ii + e

h (it represents the diffusion character of the fluid), and 4V character of the fluid), the value of τ SUPG can be optimized and usually approaches the smallest of the above three h h2 t values. For instance, when t 2 0

(10.9)

where φ ij = φ(x j ) − φ(x i ), φ(x i ), and φ(x j ) are objective function values of candidate and current designs, φ is a normalization constant, which is the running average of φ ij , and T k is the strategy temperature. Parameter φ is updated as follows when φ ij > 0 before computing the acceptance probability9 : φ¯ =

M × φ¯ + φij M +1

(10.10)

where M is the number of terms in the running average. The initial values for φ¯ and M are taken as 1 and 0, respectively. The second half of the acceptance probability criterion is carried out by generating a random number, rn, uniformly distributed over the interval [0,1]. The candidate design is accepted as the current design if rn < Aij , otherwise the iterations continue with the previous design. The strategy temperature T k is gradually decreased, while the annealing process proceeds according to a cooling schedule. This requires the values for T s and T f , the initial and final temperature values. Moreover, the values of ¯ the final starting acceptance probability P s for φij = φ, ¯ and the number acceptance probability P f for φij = φ, of temperature reduction cycles N are needed. Once P s is given, the starting temperature T s can be found as Ts = −

1 ln Ps

(10.11)

The strategy temperature T k is reduced to Tk+1 = αTk

(10.12)

where α is the cooling factor and less than 1 and T k + 1 is the temperature of the next cycle. While T approaches 0, for design transitions with φ ij > 0, Aij also approaches 0. Given P f , T f is obtained as Tf = −

1 ln Pf

(10.13)

After N cycles, the final temperature value T f can be expressed as Tf = Ts α N−1 (10.14) and α is obtained by α=



ln Ps ln Pf

1/(N−1)

(10.15)

As to the optimization problem with n design variables, an iteration around the current design i comprises n designs generated by randomly perturbing one of the n variables, provided that each variable is changed only once and the temperature is kept constant during these neighborhood searches. The variables are selected randomly for perturbation. As the acceptance probability of a candidate design is greater at high temperatures, the algorithm can escape from the local minima easily, and thermal equilibrium is rapidly reached in a fewer number of iterations. As the aforementioned probability is smaller at low temperatures, the algorithm needs a larger number of iterations to escape from the local minima and attain equilibrium. For that reason, it is necessary to increase the number of iterations while the temperature is reduced to

PARTICLE SWARM ALGORITHMS

reach the thermal equilibrium. The number of iterations are carried out during each temperature reduction cycle, IPC(T ), and are given as 

T − Tf IPC(T ) = IPCf + (IPCf − IPCs ) T f − Ts



(10.16)

where IPCs is the number of iterations per cycle at the initial temperature T s , whereas IPCf is the number of iterations per cycle needed at the final temperature T f .

10.2.2 Optimum Design Algorithm for Simulated Annealing The optimum design algorithm using SA consists of the following steps: Step 1. Assign the values for P s , P f , and N , and calculate the cooling schedule parameters T s , T f , and α from Equations 10.11, 10.13, and 10.15. Initialize the cycle counter ic = 1. Set the variable counter iv and iterations per cycle counter il to 0. Step 2. Construct the initial design x 0 randomly. Assign this design as current design x i . Carry out the nonlinear analysis and obtain the response of the frame. Calculate the value of the objective function φ(x 0 ). Step 3. Determine the iterations required per cycle IPC from Equation 10.16. Step 4. Randomly select a variable (the section number of the frame’s member group) k a ∈ [1, . . . , n] to be changed and set iv = iv + 1. Give a random perturbation to the variable k a to generate a candidate design x j in the neighborhood of the current design x i . Step 5. Calculate the related objective function value φ(x j ) and φ ij . Step 6. If φ ij ≤ 0, accept the candidate design x j as the current design. Meanwhile, if the new current design is a feasible one and better than the previous optimum, assign it temporarily as the optimum design. If iv > n, go to step 9; otherwise, return to step 4. Step 7. If φ ij > 0, update φ¯ using Equation 10.10. Calculate the acceptance probability Aij (T k ) from the second half of Equation 10.9. Generate a uniformly distributed random number rn over the interval [0,1]. If rn < Aij , go to step 8, otherwise check if iv > n. If it is, go to step 9; otherwise, go to step 4. Step 8. Accept the candidate design as the current design; that is, set x i = x j and φ(x i ) = φ(x j ). If the new current design is a feasible one and better than the previous optimum, assign it temporarily as the optimum design. If iv > n, go to step 9; otherwise, go to step 4. Step 9. If il ≤ IPC, set iv = 0, il = il + 1, and go to step 4. Otherwise, go to step 10.

287

Step 10. Update the temperature T k using Equation 10.12 and set the cycle counter ic = ic + 1. If ic > n, terminate the algorithm and define the last temporary optimum as the final optimum design. Otherwise, set il = iv = 0 and return to step 3. 10.3

PARTICLE SWARM ALGORITHMS

PSAs were first introduced by Dr. James Kennedy and Dr. Russell Eberhart in 1995.10 They were inspired by the biological and social behavior of birds flocking and fish schooling. This behavior was collectively referred to as swarming. When a swarm looks for food, its individuals will spread out in the environment and move around independently. Each individual has a degree of freedom or randomness in its movements, which enables it to find food. Sooner or later, one of them will find something digestible and, being social, will announce this to its neighbors.11 In the PSA, the potential solutions, called particles, are analogous to birds or fish. These particles fly according to their own flying experiences and their companions’ flying experiences. PSAs share many similarities with other evolutionary algorithms such as GAs. They both initialize a population in a similar manner and search for an optimum solution by updating generations. However, unlike GAs, the PSA has no evolution operators such as crossover and mutation operators.12 Owing to easy implementation and fewer parameters for adjustments, the PSA has been successfully applied in many research and application areas. In general, the PSA can be used where the other evolutionary algorithms can be applied. A more complete description about PSAs can be found in the books by Kennedy,13 Engelbrecht,14 Nedjah,15 and Clerc.16 10.3.1

General Procedures

In a PSA, each potential solution is regarded as a particle in a multidimensional space, which adjusts its position according to its own experience and the experiences of neighboring particles. Particles have two primary operators: velocity update and position update. During each generation, each particle is accelerated toward the particle’s previous best position and global best position. That is, a new velocity value for each particle is calculated based on its current velocity, the distance from its previous best position, and the distance from the global best position. The new velocity value is then used to calculate the next position of the particle in the search space. This process is repeated until the stopping criterion is satisfied. Suppose that the dimension of the feasible solution space is D and the size of the population is n. The i th

288

INTELLIGENT OPTIMIZATION ALGORITHMS

particle in the particle swarm at generation k is composed of three vectors: the current position

k D-dimensional  k k = s , . . . , s , the previous best position Pik = S i,1  i,D 

i k k k k . , . . . , vi,D pi,1 , . . . , pi,D , and the velocity Vik = vi,1 k The current position Si can be considered as a set of coordinates describing a point in the D-dimensional search space. At each generation, the position is regarded as a problem solution and evaluated by a fitness function. If this position is better than any other position found so far, the position is stored in the second vector, Pik . In addition, the best position found so far at generation k in

 k the whole swarm is stored in a vector Pgk = g1k , . . . , gD . During evolution, the new velocity and position of the i th particle are calculated by using Equations 10.17 and 10.18, respectively, Vik+1 = w × Vik + c1 × r1 × (Pik − Sik ) + c2 × r2 × (Pgk − Sik ) Sik+1 = Sik + Vik+1

(10.17)

Start algorithm PSO for each particle Initialize velocity V i and position S i for particle i Evaluate particle i and set P i = S i end for P g = max(P i ) repeat for i = 1 to n Update the velocity and position of particle i Evaluate particle i if fit(S i ) > fit(P i ) Pi = S i ; if fit(P i ) > fit(P g ) Pg = Pi ; end for until stopping criteria is satisfied print P g end algorithm

(10.18)

where w is the inertia weight, c 1 and c 2 are constants called acceleration coefficients, and r 1 and r 2 represent two independent random numbers in the range [0,1]. Moreover, in order to control the excessive flying of particles outside of the feasible solution space, the values of each component in Vi are kept within the range [−V max , V max ]. The pseudocode and the flowchart of the implementation procedure for a standard PSA are illustrated in Table 10.1 and Figure 10.2, respectively. First, an initial population is generated with random positions and velocities. Next, the performance of each particle is evaluated according to a predefined fitness function. The fitness function is related to the problem to be solved. Then, all particles adjust their positions via Equations 10.17 and 10.18. This process is repeated until the stopping criterion is satisfied. In general, the stopping criterion is that either a given number of generations has been reached or that the population has become uniform.

10.3.2

TABLE 10.1 The Pseudocode of a Standard Particle Swarm Algorithm

Determination of Parameters

The PSA described earlier has a few parameters that need to be determined. Suitable parameters are conducive to find the optimum solution with fewer iterations on average. These parameters include the size of the population n, the inertia weight w, the maximum velocity V max , and the acceleration coefficients c 1 and c 2 . 1. The size of the population n. This parameter is often set empirically on the basis of the dimensionality and perceived difficulty of a problem; its recommended range is 20–50.

2. The acceleration coefficients c 1 and c 2 . These parameters determine the magnitude of the random forces in the direction of the previous best P i and the global best P g . Typically, the value of c 1 = c 2 = 2.0 is adopted. 3. The inertia weight w . This parameter is used to control the impact of the previous history of velocities on the current velocity, thereby influencing the balance between global (wide ranging) and local (nearby) exploration abilities for the flying particles. A larger inertia weight tends to facilitate global exploration (searching new areas), while a smaller inertia weight tends to facilitate local exploration to fine tune the current search area. Therefore, a time decreasing inertia weight is better than a fixed inertia weight. That is, larger inertia weights at the beginning help to find good regions, while lesser inertia weights facilitate fine searches. In addition, some other strategies, such as fuzzy logic and random component, were adopted to adjust the inertia weight.17 4. The maximum velocity V max . As seen from Equation 10.17, this parameter acts as a constraint that controls the maximum global exploration abilities of the PSA. A small maximum velocity results in a limited global exploration ability, no matter what the inertia weight is. In contrast, if the maximum velocity is high, then there is a large range of global exploration abilities available for determining the value of the inertia weight. In practice, it is difficult to select the best maximum velocity without trial and error. As the maximum velocity affects the global exploration

ANT COLONY ALGORITHMS

289

Start

Initialize all particles with random positions and velocities

Evaluate each particle with fitness function

Yes

Stopping criterion is satisfied? No

Update velocities and positions for all particles by Equations 10.17 and 10.18

Evaluate each particle with fitness function

Update previous best value Pi for particle i

Update global best value Pg

Output global best value Pg

End

FIGURE 10.2 algorithm.

Flowchart of the implementation procedure for the standard particle swarm

ability indirectly while the inertia weight influences it directly, in most cases, the maximum velocity is not taken into account in the PSO (particle swarm optimization) algorithm, thus allowing the inertia weight to control the PSO algorithm’s global exploration ability. That is, the maximum velocity is set to the maximum initialization range of S i ; that is, V max = S max .

10.4

ANT COLONY ALGORITHMS

In the early 1990s, the ACA was first proposed by M. Dorigo and his colleagues, as inspired by the foraging behavior of real ants.18 Nearly blind, ants employ their secretions (called pheromones) as a communication medium to establish the shortest path from their nest to a food source. When searching for food, ants initially explore the area surrounding their nest in a random direction, secreting pheromones on the ground as they go. The deposited pheromones dissipate after some time. If a lucky ant finds

a food source, it will track its deposited pheromone trail back to the nest. As such, the lucky ant secretes another layer of pheromones on the path. Other ants sense these pheromones and follow the path with a certain probability. In general, the higher the pheromone densities in a path, the more likely the ants are to follow. After a period of time, the pheromone density of the shortest path from the nest to the food source will be enhanced by more and more ants moving along it; and inversely, the pheromones deposited on the other paths dissipate without reinforcement.11 This collective behavior emerging from the interaction of the different search threads, inspired this computational paradigm for solving combinatorial optimization problems, such as train scheduling, time tabling, shape optimization, and telecommunication network design.19 ACAs are a kind of parallel optimization algorithm. In the ACA, each ant acts as a simple computational agent as it searches good solutions over several constructive computational threads according to heuristic and pheromone trail information. ACAs simply iterate a main loop where an army of ants construct their good solutions.20 There are

290

INTELLIGENT OPTIMIZATION ALGORITHMS

TABLE 10.2 Algorithm

The Pseudocode of a Basic Ant Colony

Start algorithm ACA Initialize pheromone values repeat for each ant (currently in state i ) choose next state j with the probability calculation equation end for for each ant movement update the pheromone values end for until stopping criteria is satisfied print result end algorithm

where m is the number of ants and τijk is the amount of pheromone laid down on the movement (i → j ) by ant k , which can be computed by ant density model Equation 10.23 or ant quantity model Equation 10.24.  Q, if ant k moves from i to j at iteration t k τij = 0, otherwise (10.23) ⎧ ⎨ Q , if ant k moves from i to j at iteration t τijk = dij ⎩ 0, otherwise (10.24) where Q is a constant parameter. 10.5

many variants of ACAs, and the pseudocode for a basic ACA is shown in Table 10.2. For the basic ACA, problem solutions are seen as states. At each iteration, each ant computes a set of feasible expansions to its current state and moves to one of these in probability. For an ant k , the probability Pijk (t) of moving from state i to state j at iteration t is defined as β

Pijk (t)

τijα (t) + ηij = β (τiα (t) + ηi )

Pijk (t) = 0

if i → j is allowed

if i → j is not allowed

(10.19) (10.20)

where τ ij and ηij represent the pheromone and attractiveness values from state i to state j , respectively. The pheromone value indicates the degree of proficiency in the past to make this movement. The attractiveness value denotes the prior desirability of this movement. For example, in the case of the traveling salesman problem (TSP), the attractiveness value can be defined as 1/d ij ; that is, ηij = 1/d ij , where d ij is the distance between city i and city j . Parameters α and ß are user defined (0 ≤ α, ß ≤ 1) and are used to control the impact of pheromones and attractiveness, respectively. After all ants have completed their movements, the pheromone values are updated by means of Equation 10.21 τij (t + 1) = ρτij (t) + τij

(10.21)

where ρ is a user-defined parameter (0 ≤ ρ ≤ 1) called the evaporation coefficient. It represents the decay of pheromones between iterations t and t + 1. The parameter τ ij denotes the sum of the contributions of all of the ants that followed i → j movements to construct their solutions, which can be defined as τij =

m k=1

τijk

(10.22)

10.5.1

HILL CLIMBING ALGORITHMS General Procedure

HCAs are a relatively simple optimization algorithm based on the metaphor of a human climber who cannot see the final destination but can see one step ahead.21 Although more advanced algorithms—such as the GA, SA algorithm, and PSA—may give better results, the HCA works well in some situations, especially for some real time systems, where HCA can give a relatively better solution in a limited time. The pseudocode of an HCA is illustrated in Table 10.3. The HCA starts with an initial solution and iteratively makes small changes to the solution for generating a neighborhood solution. If the neighborhood solution is better than the current solution, then the neighborhood solution is substituted for the current solution. When the current solution can no longer be improved or other stopping criteria are satisfied, the process terminates. In brief, the HCA follows a path of solution candidates that improves the fitness function(s) without keeping any history of prior events. HCAs are easy to implement and can achieve good results in a short time. However, HCAs are always heading toward a solution that is better than the current solution. As such, a major problem for HCAs is that they can easily get stuck on a local optimum, which, while better than any surrounding solution, is not the best overall solution. Unless the problem to be solved is convex, a global maximum may not be reached.11 In this form, the HCA is a local search algorithm rather than a global optimization algorithm. For overcoming this problem, some researchers have made modifications to the HCA, such as the dynamic HCA22 and the stochastic HCA23 . 10.5.2 Flow Path Generation with Hill Climbing Algorithms For further explanation, a typical injection molding application of an HCA is described in this section. This application,

REFERENCES

TABLE 10.3

291

The Pseudocode of a Hill Climbing Algorithm

Start algorithm HCA Initialize all solutions in search space Select an initial solution as current solution repeat neighbors = getNeighbors(current solution) next solution = getBestNeighbors(neighbors) if fit(next solution) > fit(current solution) current solution = next solution until stopping criterion is satisfied print current solution end algorithm

Boundary node: Nb

Generated flow path (X2, Y2 ) (X1, Y1)

r (Xc , Yc)

Gate node: Ng

(Xj, Yj)

FIGURE 10.3 Schematic diagram of the implementation procedure for generating the flow path.

first proposed by Lam and Seow, describes a flow path generation method from a fill pattern’s time contours.24 The implementation procedure is summarized in the following steps: Step 1. Store all boundary nodes N b in a list LIST(N b ). Step 2. Choose a boundary node N b (X i , Y i ) from the list LIST (N b ), and set (X c , Y c ) as the coordinate of N b (X i , Y i ); that is, X c = X i , Y c = Y i . Step 3. Determine n and r for the next calculation, where n is the number of points on the circumference of a circle of radius r with (X c , Y c ) as the center. For example, if searching at 5◦ interval, n is equal to 72 (360/5 = 72). The fill time will be calculated at these points, (X j , Y j ), where(j = 1, . . . , n). The determination of n and r depends on the accuracy required for the flow path. A higher n and a lower r will increase not only the resolution of the flow path but also the computational cost. Step 4. Calculate the fill time f j for each point (X j , Y j ). These fill times are calculated using the interpolation method from the fill time at the corresponding nodes. Step 5. Find the minimum fill time, min (f j ), from j = 1, . . . , n, and set (X c , Y c ) as the coordinate of the point with the minimum filling time. Step 6. If (X c , Y c ) is not the gate node N g , repeat the calculation from step 3. Step 7. If all of the boundary nodes in the list LIST (N b ) have not been processed, repeat the calculation from step 2. As an example, Figure 10.3 shows a schematic diagram of generating a flow path for a quarter section of a uniform thickness plate by using the above-mentioned calculation procedure. The initial location of the algorithm was at a boundary node N g . From there, the HCA was used to trace a path back to the injection node N g at the bottom left. As discussed previously, a major problem of the HCA is premature convergence; that is, it is only able to find a global maximum if the problem to be solved is

convex. Fortunately, filling patterns are well-defined convex problems, with the global minimum of the filling time equal to 0, which is the filling time for the injection node. Therefore, the flow path generation problem is an ideal application for an HCA. Furthermore, Lam and Seow combined the above-mentioned flow paths generation method and an optimization routine to balance mold cavities. Numerical experiments show that the proposed cavity balancing method is robust and effective.24

REFERENCES 1. Holland J., Adaptation in Natural and Artificial Systems. 1975, Ann Arbo, WI: The University of Michigan Press. 2. Davis L., The Handbook of Genetic Algorithms. 1991, New York: Van Nostrand Reingold. 3. Goldberg D., Genetic Algorithms in Search, Optimization, and Machine Learning. 1989, MA: Addison-Wesley. 4. Michalewicz Z., Genetic Algorithms + Data Structures = Evolution Programs. 1994, New York: Springer. 5. Blanco A., Delgado M., Pegalajar M.C., A real-coded genetic algorithm for training recurrent neural networks. Neural Networks, 2001. 14(1): 93–105. 6. Christopher R.H., Jeffery A.J., Michael G.K., A Genetic Algorithm for Function Optimization: A MATLAB implementation. 1996. 7. Kirkpatrick S., Gelatt C.D., Vecchi M.P., Optimization by simulated annealing. Science, 1983. 220(4598): 671–680. 8. Laarhoven P.J.M., Aarts E.H.L., Simulated Annealing: Theory and Applications. 1987, Dordrecht: Kluwer Academic publishers. 9. Balling R.J., Optimal steel frame design by simulated annealing. Journal of Structural Engineering-ASCE, 1991. 117(6): 1780–1795. 10. Kennedy J., Eberhart R., Particle swarm optimization. in Neural Networks. 1995. Perth, WA, Australia: IEEE International Conference on.

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11. Weise T., Global Optimization Algorithms–Theory and Application. 2008. 12. Yiqing L., Xigang Y., Yongjian L., An improved PSO algorithm for solving non-convex NLP/MINLP problems with equality constraints. Computers & Chemical Engineering, 2007. 31(3): 153–162. 13. Kennedy J., Eberhart R.C., Shi Y., Swarm Intelligence. 2001, San Francisco: Morgan Kaufmann Publishers. 14. Engelbrecht A.P., Fundamentals of Computational Swarm Intelligence. 2005, New York: Wiley. 15. Nedjah N., de Macedo Mourelle L., Swarm Intelligent Systems. 2006, London: Springer. 16. Clerc M., Particle Swarm Optimization. 2006, London: ISTE. 17. Poli R., Kennedy J., Blackwell T., Particle swarm optimization. Swarm Intelligence, 2007. 1(1): 33–57. 18. Dorigo M., Optimization, learning and natural algorithms [PhD Thesis]. 1992, Politecnico di Milano: Italy.

19. Blum C., Ant colony optimization: Introduction and recent trends. Physics of Life reviews, 2005. 2(4): 353–373. 20. Onwubolu G.C., Babu B.V., New Optimization Techniques in Engineering. 2004, London: Springer. 21. Pearl J., Heuristics: intelligent search strategies for computer problem solving. 1984, MA: Addison-Wesley Pub. Co., Inc. 22. Yuret D., de la Maza M., Dynamic hill climbing: Overcoming the limitations of optimization techniques, in The Second Turkish Symposium on Artificial Intelligence and Neural Networks. 1993, Citeseer. 208–212. 23. Huang X., Kelkar M., Chopra A., et al., Some practical considerations for application of heuristic combinatorial algorithm to reservoir description, in International Exposition and 65th Annual Meeting. 1995, SEG. 1025–1027. 24. Lam Y.C., Seow L.W., Cavity balance for plastic injection molding. Polymer Engineering & Science, 2000. 40(6): 1273–1280.

11 OPTIMIZATION METHODS BASED ON SURROGATE MODELS Yuehua Gao School of Traffic & Transportation Engineering, Dalian Jiaotong University, Dalian, Liaoning, China

Lih-Sheng Turng Polymer Engineering Center, Department of Mechanical Engineering, University of Wisconsin–Madison, Madison, Wisconsin, USA

Peng Zhao State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, Zhejiang, China

Huamin Zhou State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China

Considering the limitations of conventional categories, these approaches employ modeling methods to establish a surrogate model (also called a metamodel ) based on CAE simulations, and thus substitutes the simulations with the surrogate model in the subsequent optimization procedure. Optimization is usually expensive and time-consuming as it requires either excessive computing time and/or extensive physical experimentation. Methods that can extract useful information from evaluated entries and formulate models to correlate process conditions and objectives provide the maximum benefit. In this chapter the optimization methods based on surrogate model are discussed. This type of model can reduce the cost of optimization by using a lower fidelity model most of the time, with occasional recourse to the high fidelity model. Unlike other optimization methods, this class optimization method makes the optimization algorithm perform on a lower fidelity model (an approximate mathematical function) that replaces expensive simulations. In other words, the approximate function, not the simulation code, is computed through the optimization iterations. The

key to these methods is the construction of the lower fidelity model, which is called a surrogate model . The most well-known surrogate models such as the response surface method (RSM), artificial neural network (ANN), support vector regression (SVR), Kriging model, and Gaussian process (GP) are introduced. For this kind of optimization strategy, the common flowchart involves three parts: sampling, modeling, and optimization. The details are as follows: 1. State the optimization problem, then select a physicalbased simulation program to sample in the design space and achieve the corresponding responses by running simulation codes. Most of the sample points are used to construct the surrogate model, while others are used as validation data to check the accuracy of the surrogate model. 2. If needed, variables can be screened to reduce the number of variables by determining the important factors for improving the optimization efficiency.

Computer Modeling for Injection Molding: Simulation, Optimization, and Control, First Edition. Edited by Huamin Zhou.  2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc.

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3. Assume a surrogate model such as the RSM, ANN, or Kriging model, and construct the surrogate model according to the sample points. 4. Use the validation data to evaluate the predictive performance of the model. On the basis of the error observed with the validation data, a decision is made as to whether the initially assumed regression model is appropriate or not. If it is appropriate, go to the next step. If, however, the evidence shows that the model does not achieve the required predictive performance, the proper way to improve the surrogate model is to assume a new regression model and go through the entire process again. 5. Perform an optimization algorithm based on the acceptable surrogate model to search for the optimal solutions. For any optimization algorithm, the approximation functions (not the time-consuming simulation codes) are analyzed for each iteration. Thus, both conventional gradient-based algorithms and intelligent algorithms can be used easily. Throughout the entire procedure, both sampling and modeling are important. Distribution and the amount of samples determine the accuracy of the surrogate model and the computational efficiency of the optimization. Some existing methods such as the orthogonal array and the Latin hypercube sampling (LHS) model are known to be better choices for surrogate modeling. Readers can find more details on these methods in books by Taguchi and Konishi1 and papers by Mckay et al.2 and Park.3 This chapter focuses on popular surrogate models. In these surrogate models, the Kriging model and the GP are special because they can provide the predicted error that helps to update the surrogate model to realize the sequential approximation optimization. This kind of optimization method not only allows a small sample set but also improves the accuracy of the optimum. The details of the sequential approximation optimization are given in Section 11.4. Different surrogates can be used for different problems as not all of these methods are suitable for a specific objective problem. For example, RSMs are not suitable due to the complex correlations between the objectives and the high dimensional inputs that demand an excessive number of data points.

11.1

RESPONSE SURFACE METHOD

One of the most common methods for building an approximate model is the RSM in which a polynomial function of varying order (usually a quadratic function) is fitted to many sample data points using least square regression. RSM was originally developed for the model

fitting of physical experiments by Box and Draper4 and was later adopted in other fields. It is one of the most widely used methods to solve the optimization problem in manufacturing environments.5,6 The RSM is an empirical modeling approach to determine the relationship between various design parameters and responses with various desired criteria and to search for the significance of these process parameters in coupled responses. It is an experimentation strategy for building and optimizing the empirical model. Therefore, RSM is a collection of mathematical and statistical procedures, and is good for the modeling and analysis of problems in which the desired response is affected by several variables. The mathematical model of the desired response to several independent input variables is gained by using the experimental design and applying regression analysis.

11.1.1

RSM Theory

The polynomial models generated by RSM are often referred to as response surface (RS) models in the literature. RS is a model building technique based on the statistical design of the experiment (DOE) and least square error fitting. The true response can be written in the following form: y(x) = f (x) + ε (11.1) where f (x ) is an unknown response function and ε is the random error. The response surface model of Equation 11.1 can be written in terms of a series of observations yi = β0 +

h  j =1

βj xij + εi

(11.2)

where x ij denotes the i th observation of variable x j . This equation may be written in matrix form as y = Xβ + ε

(11.3)

where y = {y1 , y2 , . . . , yn }T ⎡ 1 x11 x12 · · · ⎢1 x21 x22 · · · ⎢ X = ⎢. .. .. .. ⎣ .. . . . 1 xn1

xn2

···

(11.4) ⎤

x1h x2h ⎥ ⎥ .. ⎥ . ⎦

(11.5)

xnh

where n is the number of samples. The term β is an (h × 1) vector of the regression coefficients, and ε is an (n × 1) vector of random errors.

RESPONSE SURFACE METHOD

The vector of least squares estimators, b, is determined by minimizing the below equation: L=

n  i=1

εi2 = (y − Xβ)T (y − Xβ)

(11.6)

The condition simplifies to XT Xb = XT y

(11.7)

Thus, the least squares estimator of β is b = (XT X)−1 XT y

(11.8)

11.1.3

295

Optimization Process Using RSM

The procedure of the RSM for determining the design parameters with optimal performance characteristics is summarized as follows (Fig. 11.1): Step 1: Define the independent input variables and desired responses with the design constraints. Step 2: Adopt a sampling method to plan the experimental design. Step 3: Determine the situation of the RSM model and decide whether the model of the RSM needs screening variables or not.

The widely used quadratic model can be expressed as f = a0 +

nv  i=1

ai xi +

nv  i=1

aii xi2 +

 i 0     xi − xj 2 RBF: K(xi , xj ) = exp − σ2

(11.29) (11.30) (11.31)

Here, ρ represents the degree of the polynomial kernel and σ represents the bandwidth of the Gaussian kernel. The kernel parameter should be carefully chosen as it implicitly defines the structure of the high dimensional feature space ϕ(x ) and thus controls the complexity of the final solution. In addition, the performance of the SVR model is heavily dependent on the regulation parameter C , the width of the tube ε, and the parameter of the chosen kernel function. From an implementation point of view, training the SVR is equivalent to solving a linearly constrained quadratic programming (QP) with a number of variables twice that of the input data dimension. Some significant features of the SVR are as follows: 1. The SVR can model nonlinear relationships. 2. The SVR training process is equivalent to solving linearly constrained QP problems, and the SVR embedded solution is unique, optimal, and unlikely to generate local minima.

301

KRIGING MODEL

3. It only chooses the necessary data points to solve the regression function, which results in the sparseness of the solution.

11.3.4

Selection of SVR Parameters

Despite its superior features, SVR is limited in academic research and industrial applications because the user must define various parameters appropriately. To construct the SVR model efficiently, the SVR’s parameters must be carefully set. Inappropriate parameters in an SVR lead to overfitting or underfitting. Different parameter settings can cause significant differences in performance. Therefore, selecting the optimal hyperparameter is an important step in SVR design. The parameters include the following. 1. Kernel function: The kernel function is used to construct a nonlinear decision hypersurface on the SVR input space. Generally, using a Gaussian function will yield better prediction performance.12 2. Regularization parameter C : Parameter C determines the trade-off cost between minimizing the training error and minimizing the model’s complexity. 3. Bandwidth of the kernel function (σ 2 ): σ 2 represents the variance of the Gaussian kernel function. 4. The tube size of ε-insensitive loss function (ε): This parameter is equivalent to the approximation accuracy placed on the training data points. SVR generalization performance (estimation accuracy) and efficiency depend on the hyperparameters (C , ε, and kernel parameters σ 2 ) being set correctly. Therefore, the main issue is to locate the optimal hyperparameters for a given data set. However, no general guidelines are available to select these parameters. The problem of optimal parameter selection is further complicated by the fact that the SVR model’s complexity, and hence its generalization performance, depends on all three parameters together, including the interaction of the parameters. Generally, when selecting the parameters, most researchers follow the trialand-error procedure, first building a few SVR models based on different parameter sets, then testing them on a validation set to obtain the optimal parameters.

11.4

KRIGING MODEL

The Kriging surrogate model is a statistics-based interpolated method, which was initially developed by geologists to estimate mineral concentrations over an area of interest given a set of sampled sites from the area.13 It was also introduced at about the same time in the field of statistics to include the correlations that exist in the residuals of linear estimators.14

The Kriging model has a statistical interpretation that allows one to construct an estimate of the potential error in the interpolator. This measure of potential error plays the key role in adaptive approximation optimization. This optimization method has a greater ability in exploring an unknown space by considering the predicted uncertainty. In addition, it allows optimization iterations to start with a small initial sample size and maintains the stability of the Kriging surrogate model while it is being improved adaptively.

11.4.1

Kriging Model Theory

The Kriging model can be postulated as a combination of polynomials and stochastic processing y(x ˆ i) =

 h

βh fh (xi ) + z(xi ) = fT (xi )β + z(xi ) (11.32)

  in which xi = x1i , x2i , . . . , xni v is the i th sample point with n v variables; y(x ˆ i ) is an approximate function fitted to n s sample points; fh (xi ) is a linear or nonlinear function of xi ; β h is the regression coefficient to be estimated; and z(xi ) is the realization of a stochastic process with a mean of zero, a variance σz2 , and a nonzero covariance. The first term in Equation 11.32 provides the “global” approximation for the unknown RS. The second term creates “localized” deviations so that the Kriging surrogate model can interpolate the sample points. The covariance matrix of z(x) is        Cov z xi , z xj = σz2 R θ , xi , xj

(11.33)

where R(θ , xi , xj ) is the correlation function between any two of the sample points xi and xj , and all correlation function values of the sample points comprise the correlation matrix R. The spatial correlation function between any two points can be given by several styles, and are illustrated subsequently (Section 11.4.2). In this model, the parameters β and σz2 are unknown. Given the response  T values of samples Y = y 1 (x1 ), y 2 (x2 ), . . . , y ns (xns ) , these unknown parameters will be estimated by choosing them to maximize the likelihood of the samples. β = FT R−1 Y/FT R−1 F 1 σz2 = (Y − FT β)T R−1 (Y − FT β) ns

(11.34) (11.35)

Here, the important parameter θ can be obtained by 

ns  2  1 max − ln σz − ln (|R|) , θk >0 2 2

k = 1, 2, . . . , nv

(11.36)

302

OPTIMIZATION METHODS BASED ON SURROGATE MODELS

  ˆ ∗ ) at Where F = f1 , f2 , . . . , fns . The function value y(x ∗ a new point x can be approximately estimated as a linear combination of response values Y. y(x ˆ ∗ ) = cT Y

(11.37)

The MSE of this predictor is minimized with unbiased estimation, thus ∗

∗

∗ T

y(x ˆ ) = f(x )β + r(x ) ζ

(11.38)

where ζ = R−1 (Y − Fβ)

∗

1

∗

(11.39) ns

∗

r(x ) = [R(θ , x , x ), . . . , R(θ , x , x )]

(11.40)

With the predictor in Equation 11.38, the MSE of the predictor is   σˆ 2 (x∗ ) = σz2 1 + ρ T (FT R−1 F)−1 ρ − rT R−1 r

(11.41)

where

ρ = FT R−1 r − f

(11.42)

Thus, the function value y(x ˆ ∗ ) and the MSE of predictor ∗ ∗ σˆ (x ) at any new point x are predicted by using Equations 11.38 and 11.41. To determine the θ values, an m-dimension optimization problem needs to be solved according to Equation 11.36. Although it requires significant computational time if the sample data set is large, the computational time can still be neglected when compared to more time-consuming numerical simulation analyses. Simpson15 suggested that this method is the best choice for deterministic and highly nonlinear problems with a moderate number of variables (i.e.,