Fundamentals of Enriched Finite Element Methods 9780323855150

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Fundamentals of Enriched Finite Element Methods

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Fundamentals of Enriched Finite Element Methods Alejandro M. Aragón Department of Precision and Microsystems Engineering Delft University of Technology Delft, the Netherlands

C. Armando Duarte Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Urbana, IL, United States of America

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-85515-0 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Joshua Mearns Production Project Manager: Sujithkumar Chandran Cover Designer: Alejandro M. Aragón (cover design), Armando C. Duarte (mode I fracture image), and Miles Hitchen (text layout) Typeset by VTeX

To Sofía and Leonardo, for enriching my life. —Alejandro M. Aragón To my beloved wife Carolina and daughters Camila and Clara. —C. Armando Duarte

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Contents

Preface

xi

1

1 2 3 6

Introduction 1.1 Enriched finite element methods 1.2 Origins and milestones of e-FEMs References

Part One Fundamentals

11

2

The finite element method 2.1 Linear elastostatics in 1-D 2.1.1 The strong form 2.1.2 The weak (or variational) form 2.1.3 The Galerkin formulation 2.1.4 The finite element discrete equations 2.1.5 The isoparametric mapping 2.1.6 A priori error estimates 2.1.7 A posteriori error estimate 2.2 The elastostatics problem in higher dimensions 2.2.1 Strong form 2.2.2 Weak form 2.2.3 Principle of virtual work 2.2.4 Discrete formulation 2.2.5 Voigt notation 2.2.6 Isoparametric formulation in higher dimensions 2.3 Heat conduction 2.4 Problems References

13 13 16 20 25 28 33 35 37 39 39 41 43 43 44 46 49 52 55

3

The p-version of the finite element method 3.1 p-FEM in 1-D 3.1.1 A priori error estimates 3.2 p-FEM in 2-D 3.2.1 Basis functions for quadrangles 3.2.2 Basis functions for triangles 3.3 Non-homogeneous essential boundary conditions 3.3.1 Interpolation at Gauss–Lobatto quadrature points 3.3.2 Projection on the space of edge functions

57 57 62 66 66 68 71 72 73

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4

5

Contents

3.4 Problems References

74 74

The Generalized Finite Element Method 4.1 Finite element approximations 4.2 Generalized FEM approximations in 1-D 4.2.1 Selection of enrichment functions 4.2.2 What makes the GFEM work 4.3 Applications of the GFEM 4.4 Shifted and scaled enrichments 4.5 The p-version of the GFEM 4.5.1 High-order GFEM approximations for a strong discontinuity 4.6 GFEM approximation spaces 4.7 Exercises References

77 78 79 80 81 82 91 93 97 99 101 101

Discontinuity-enriched finite element formulations 5.1 A weak discontinuity in 1-D 5.2 A strong discontinuity in 1-D 5.3 Relationship to GFEM 5.4 The discontinuity-enriched FEM in multiple dimensions 5.4.1 Treatment of nonzero essential boundary conditions 5.4.2 Hierarchical space 5.5 Convergence 5.6 Weak and strong discontinuities 5.7 Recovery of field gradients References

105 106 107 110 114 116 117 121 122 123 127

Part Two Applications 6

GFEM approximations for fractures 6.1 Governing equations: 3-D elasticity 6.1.1 Weak form 6.2 GFEM approximation for fractures 6.2.1 Approximation of uˆ 6.2.2 Approximation of u˜˜ 6.2.3 Cohesive fracture problems 6.2.4 Approximation of u˘ 6.2.5 Topological and geometrical singular enrichment 6.2.6 Discrete equilibrium equations 6.3 Convergence of linear GFEM approximations: 2-D edge crack 6.3.1 Topological enrichment 6.3.2 Comparison with best-practice FEM

129 131 132 133 133 134 135 136 140 148 150 153 155 156

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6.3.3 Geometrical enrichment 6.4 Convergence of linear GFEM approximations: 3-D edge crack References

157 158 163

7

Generalized enrichment functions for weak discontinuities 7.1 Formulation 7.1.1 Linear elastostatics 7.1.2 Heat conduction 7.1.3 Discrete equations 7.1.4 Enrichment functions for weak discontinuities 7.1.5 Enrichment performance 7.2 Discussion and further reading References

167 167 168 170 171 172 174 176 178

8

Immerse boundary (fictitious domain) problems 8.1 Formulation 8.1.1 Treatment of boundary conditions 8.1.2 An immersed popcorn example References

179 180 182 186 187

9

Non-conforming mesh coupling and contact 9.1 Formulation 9.2 Examples 9.2.1 Infinite plate with a circular hole 9.2.2 Hertzian contact 9.3 Further reading References

191 192 195 196 197 198 200

10

Interface-enriched topology optimization 10.1 Formulation 10.1.1 Enriched finite element analysis 10.1.2 Design space 10.1.3 Optimization 10.2 Compliance minimization 10.3 Fracture resistance 10.4 Discussion and further reading References

203 204 204 207 208 208 211 217 219

Part Three Computational aspects

223

11

225 225 226 227

Stability of approximations 11.1 Conditioning control of GFEM matrices 11.1.1 Scaling of global matrices 11.1.2 Conditioning control of Heaviside enrichments

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11.1.3 Conditioning control of singular enrichments 11.1.4 Well-conditioned first-order GFEM approximations 11.1.5 Solution of singular systems of equations 11.1.6 Two-dimensional inclined edge crack 11.2 Stability of interface-enriched generalized finite element formulations 11.2.1 IGFEM scaling in 1-D 11.2.2 IGFEM stability in higher dimensions References

241 241 244 248

12

Computational aspects 12.1 A basic FEM code structure 12.2 e-FEM considerations 12.2.1 Computational geometry 12.2.2 Numerical integration References

251 251 253 254 258 261

13

Approximation theory for partition of unity methods 13.1 Approximation theory for the GFEM 13.2 A priori error estimates for partition of unity approximations References

265 265 268 273

A

Recollections of the origins of the GFEM A.1 The hp-cloud method A.2 The name partition of unity method is coined A.3 A partition of unity method for problems with singularities A.4 Wrapping up my PhD at UT Austin A.5 Work on partition of unity methods at COMCO and Altair Engineering A.5.1 The Element Partition Method (EPM) A.5.2 Collaboration with Ivo Babuška and Tinsley Oden A.6 Early work on partition of unity methods at Texas A&M University A.7 Early work on partition of unity methods at Northwestern University References

275 275 278 278 280

Index

233 236 236 239

281 281 285 286 286 287 293

Preface

This book is a treatise on enriched finite element methods (e-FEMs) for the solution of partial differential equations of continuum mechanics. These methods emerged in the early 1990s with the key idea of using any a priori knowledge of the solution of a problem to improve—or enrich—the finite element approximation space in a continuous Galerkin framework. This knowledge is embedded in the approximation by means of analytical functions (for instance, Westergaard’s solution for cracks) or functions obtained numerically (for instance, the solution of another boundary value problem). The Galerkin projection, in an attempt to minimize the error with respect to the exact solution according to Céa’s lemma, will then assign nonzero coefficients to enrichments if they are indeed useful. e-FEMs, as described in this book, can therefore be seen as a generalization of the finite element method (FEM) for building nonpolynomial functional spaces. We discuss the theory thoroughly, and also provide details on the implementation of these techniques in standard displacement-based finite element codes. The methods are then demonstrated on several applications, including multiphase materials, fracture, immerse boundary (fictitious domain) problems, the coupling of nonconforming discretizations and contact, and topology optimization. Although general, the formulations herein are actually described in the context of elliptic boundary value problems, and particularly linear elasticity. We look at enriched formulations of the primal (nodal) field, and thus we do not discuss other methods that deal with element enrichments (for instance, enhanced strain fields or embedded discontinuity approaches at the element level). The authors have worked most of their academic careers in the development of the techniques described in this treatise, and in fact various subjects in this book are still ongoing development. The content presented herein has been taught for many years in graduate courses at the University of Illinois at Urbana–Champaign and at Delft University of Technology. Our primary aim is to fill a gap in the existing literature regarding e-FEMs. After more than two decades of development, the literature on this class of methods is vast, but there is still no introductory treatise on the subject. Other books on e-FEMs are mostly concerned with fracture problems, which is but a subset of the types of problems that can effectively be addressed by e-FEMs, as demonstrated herein. e-FEMs have not seen yet a wide adoption due to intrinsic limitations, many of which have only been lifted very recently. For instance, recent work on stable formulations has solved a long-standing issue of ill-conditioned system matrices. Also, interface- and discontinuity-enriched FEMs have been shown to solve issues with oscillatory tractions in boundaries with prescribed essential boundary conditions. These methods have now achieved the required maturity, evidenced by the emergence of first implementations in mainstream finite element analysis (FEA) software like Abaqus. As a result, these advanced discretization techniques are used not only by academics

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Preface

but also by practitioners in several industries, including among others automotive, aerospace, and medical. This book is intended for a graduate audience, as a graduate course in advanced finite element methods, or as a reference source for computational mechanicians in science and industry who have faced the limitations of the standard FEA. There are many problems where e-FEMs have an edge over the standard FEM, including those where complex meshing is required, or those where the final computational model is obtained by meshing independent components, problems with evolving discontinuities (fracture propagation or topology optimization), to name a few. In e-FEMs the problems are usually solved on a simple (usually structured) finite element mesh. This flexibility regarding the choice of mesh is perhaps the most appealing feature of these methods, for it has been recognized that creating the computational model for finite element analysis can take most of the analysis time. Therefore, much emphasis is placed throughout this treatise in removing some of the requirements on finite element meshes, like the need for meshes that conform to crack surfaces or material interfaces. e-FEMs are therefore preferred over the standard FEM not only because they can use simple structured finite element meshes, but also because there is no remeshing required for problems where discontinuities evolve. We note, however, that mesh refinement is still needed for many applications since enrichments based on a priori knowledge about the exact solution are in general not able to deliver acceptable accuracy on coarse meshes. Furthermore, remeshing is in general also needed for problems with severe mesh distortion. Mesh refinement can be robustly implemented in e-FEMs since the mesh does not need to fit crack surfaces or material interfaces. Therefore, as shown in this work, it can be performed by simply bisecting elements in a neighborhood of these localized features. Throughout the book we discuss e-FEMs that have their roots in the so-called Partition of Unity Method (PoUM). In particular, the Generalized Finite Element Method (GFEM) showed first incredible flexibility in the construction of (nonpolynomial) finite element spaces. The application of the method to the analysis of problems with discontinuities (e.g., fracture) was later coined as the eXtended Finite Element Method (XFEM) and provided a means to decouple the problem’s discontinuities from the finite element discretization. In GFEM/XFEM enriched degrees of freedom are associated to nodes of the original finite element discretization, which inevitably showed some intrinsic limitations of the method. Consequently, another family of discontinuity-enriched methods, whereby enrichments are associated to nodes placed directly along discontinuities, surged to circumvent some of those limitations. The pros and cons of both families of methods are discussed in depth throughout the book, primarily in the context of problems with discontinuities (e.g., material interfaces or cracks). The book is divided into three parts: Part I: Fundamentals, Part II: Applications, and Part III: Computational aspects. Although the reader is expected to have prior knowledge on FEA, we provide in Part I concise reviews of the h-version (Chapter 2) and p-version (Chapter 3) of FEM. These chapters give the reader sufficient knowledge to understand the more advanced enriched formulations of subsequent chapters. Partition of unity methods, i.e., PUFEM and GFEM/XFEM, are introduced in Chap-

Preface

xiii

ter 4. Interface- and discontinuity-enriched formulations are presented in Chapter 5. In Part III we showcase the different formulations in a series of applications dealing with discontinuities, including fracture, material interfaces, and immersed boundaries. We also present newer developments on the use of e-FEMs in other realms of mechanics, including topology optimization, and the coupling of nonconforming discretizations and contact. Finally, in Part III we deal with more advanced (implementation) aspects of e-FEMs. Issues that can arise regarding the stability of the formulations—insofar as the condition number of system matrices is concerned—are discussed in Chapter 11. Details for implementing e-FEMs in standard displacement-based FEM packages are then thoroughly discussed in Chapter 12, for which prior programming knowledge is desired for reaping the knowledge presented herein. We are indebted to the many people who have contributed in one way or another to the writing of this book. We thank students who provided feedback throughout the teaching of our courses. Most importantly, we thank the many contributions of our graduate students, many of whom today occupy positions in academia, industry, and national laboratories, and who continue to develop and transmit this knowledge. In particular, we hold dear memories of the late Dr. Jeronymo Peixoto Pereira. We thank Luziana Reno, Dr. Dae-Jin Kim, Dr. Patrick O’Hara, Dr. Julia Plews, Dr. Hasan Ozer, Aditya Krishnan, Dr. Jorge Garzon, Dr. Varun Gupta, Dr. Jongheon Kim, Dr. Nathan Shauer, Dr. Phillipe Daniel Alves, Elena De Lazzari, David Douwes, Dongyu Liu, Dr. Sanne van den Boom, Dr. Jian Zhang, and Dr. Elena Zhebel. We also thank our colleagues and friends, who have over the years challenged our mind with puzzles and motivated us, including Prof. Ivo Babuška, Prof. Angelo Simone, Prof. Tinsley Oden, Prof. Uday Banerjee, Prof. Philippe H. Geubelle, Prof. Jean-François Molinari, Prof. Marc Alexander Schweitzer, and Prof. Fred van Keulen. Finally, this book would not be possible without the support of our families (Yulia, Teresa, Fernando, Camila, Clara, Carolina), so our warmest gratitude goes to them. Alejandro M. Aragón Delft, the Netherlands C. Armando Duarte Urbana, IL, United States of America August 2023

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Introduction

1

Finite element analysis (FEA) is nowadays ubiquitous in academia and industry alike. For instance, every structural engineering studio uses some kind of finite element software for structural design, and it is common practice for companies to use FEA for prototyping—and even optimizing—their products. This has largely been facilitated by a wide availability of commercial software. However, like any discretization method, there are problems that are impractical or at best challenging when solved by FEA. More sophisticated procedures are needed in these cases. The total FEA simulation time is inversely proportional to the size of finite elements used (denoted h throughout this book for meshes with approximately uniformly sized elements); therefore, it is not uncommon to have commercial software running for weeks when using extremely fine finite element meshes with current computer hardware. Yet, small elements may be needed to properly resolve stress fields near reentrant corners or even more abrupt singularities as those that arise near crack fronts, or for properly resolving wave propagation at high frequencies. The vast computational demands required by standard FEA are in part due to the fact that low-order polynomials (usually linear or quadratic at best) are used within each element, and as it is explained in Chapter 2, these approximations converge only algebraically—i.e., the rate of convergence remains constant as the mesh size h is reduced. Furthermore, for fracture problems, for instance, the rate of convergence can be far from optimal since it is controlled by the strength of the singularity. One strategy for obtaining better convergence is to simply increase the order p of the polynomial interpolant within each element. This is the rationale behind the p-version of the finite element method (p-FEM) [1,2], which is described in detail in Chapter 3. Contrary to h-FEM, where the error in the approximation is reduced by decreasing the mesh size h, in p-FEM the error is reduced by increasing the degree of the polynomial approximation over a fixed mesh. The p-FEM approximation improves the finite element space hierarchically, i.e., for a given finite element, the shape functions of order up to p − 1 do not change when increasing the polynomial order from p − 1 to p. In other words, a quadratic shape function can be seen as improving over the linear interpolant, the cubic over the quadratic, and so forth. This hierarchical nature of the procedure can also be identified in the resulting system matrices. For sufficiently smooth problems, p-FEM converges exponentially—i.e., the rate of convergence increases with increasing polynomial order. However, the rate of convergence is only algebraic—like in h-FEM—in problems with nonsmooth solutions, which is often the case in practice. Exponential convergence can be recovered in these cases if strongly graded meshes are used but the design of such meshes, in particular for 3-D problems, can require significant user effort. Computational time may not be the most worrying drawback of standard FEA. It has been estimated that, for complex domains, about 80% of the overall analysis time is spent in creating the computational model [3]; this includes the geometry of the domain, which is usually obtained from a computer-aided design (CAD), and its disFundamentals of Enriched Finite Element Methods. https://doi.org/10.1016/B978-0-32-385515-0.00007-6 Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved.

2

Fundamentals of Enriched Finite Element Methods

cretization by means of finite elements. As a result, a plethora of more sophisticated techniques have been proposed to address this problem, including isogeometric analysis, immersed FEMs, unfitted FEMs, and the procedures outlined in this book, as explained in the next section.

1.1 Enriched finite element methods Enriched finite element methods (e-FEMs) can address the aforementioned limitations of standard FEM. These methods can reduce the error in the solution by enhancing or enriching the function space used to build the FE approximation, e.g., by adding functions to the standard FEM approximation that take into account known features of the solution. Similarly to p-FEM, e-FEMs can also be seen as procedures that improve the approximation over the standard finite element space hierarchically. However, this hierarchy, which can also be identified in the system matrices, goes beyond simply increasing the polynomial approximation order. For instance, singular functions can be used to enrich the approximation and thus capture stress singularities near crack tips on coarse FE meshes. Similarly, especial “spike” enrichments can be used to capture the effect of point heat sources within a single finite element. Enrichments can also be added to capture the kinematics of a perfectly bonded material interface or the displacement jump across a crack, regardless of how such discontinuities cut the finite elements in the mesh. Most importantly, for several classes of problems, e-FEMs can eliminate or significantly reduce the aforementioned 80% of total analysis time by automating the creation of accurate computational models via the use of structured finite element meshes! This advantage becomes even more pronounced in problems where the geometry evolves over time, or even when the actual geometry is the sought solution as is the case in topology optimization. But since “there ain’t no such thing as a free lunch,” where is the catch? The complexity of generating a geometry-conforming mesh in standard FEA, where edges and faces of finite elements align with boundaries and other discontinuities, is transferred in e-FEMs to the formulation and implementation. This necessitates complex and robust computational geometry operations that automatically identify elements cut by discontinuities, splitting such elements in socalled integration elements, adding new enriched degrees of freedom (DOFs) that are associated with enrichments, among others. Although properly implementing a computational geometry engine for all these operations is far from trivial (as discussed in detail in Chapter 12), the effort is well worth it because the process of creating computational models is then fully automated or considerably simplified for the end user. e-FEMs have shown to outperform standard FEM in problems with discontinuities. These include the cases of discontinuous data (e.g., material properties), and discontinuous primal fields (e.g., displacements). Some problems with discontinuities include (see examples in Fig. 1.1): (i) fracture; (ii) multiphase materials (e.g., composites); (iii) material phase transformations due to changes in temperature; (iv) multiphase flows; (v) immersed boundaries (fictitious domains); (vi) dislocations; and (vii) topology optimization. It is worth noting that all these problems can be solved with e-FEMs

Introduction

3

Figure 1.1 Examples of problems where e-FEMs have an edge over standard FEM, including multiphase materials (a), fracture (b), and topology optimization (c).

using simple uniform structured finite element meshes defined over Cartesian grids. e-FEMs can therefore readily be used for addressing one of the main bottlenecks of standard FEM discussed in the preceding section. In addition to these problems, eFEMs have successfully been applied to various other problems, including multiscale analysis, in capturing high field gradients, in contact and coupling non-conforming meshes, in wave propagation, to name a few.

1.2 Origins and milestones of e-FEMs The standard FEM, with the limitations described above, has also been used as a building block to construct more sophisticated approximations. In fact, the methods described in this book can be seen as a generalization of the finite element method for constructing non-polynomial approximations. The development of e-FEMs can be traced back to the work of Babuška et al. on “special finite element methods for a class of second order elliptic problems with rough coefficients” [4]. Some of the methods proposed in [4] use a partition of unity (PoU) to define basis functions that enhance the standard FEM approximation. A PoU is simply a set of functions whose sum at any point in an analysis domain is equal to unity. Any set of Lagrange FE shape functions has this property and can therefore be used to define shape functions able to exactly represent virtually any function—polynomial or not. Another noteworthy property of shape functions defined using a PoU is that they inherit the compact support of standard FEM shape functions—they are nonzero only over a small set of elements, thus leading to sparse matrices like in FEM. The PoU property of standard FE shape functions was used by several authors in the 1990s to hierarchically enhance the standard FEM (see timeline of important contributions to enriched FEMs in Fig. 1.2). Examples are the partition of unity FEM [5,14], the cloud-based hp FEM [6,7], and FEM for crack growth with minimal or without remeshing [8,9]. This new paradigm in constructing finite element approximation spaces was later called Generalized or eXtended Finite Element Method (GFEM/XFEM). It is noted that the GFEM terminology had been used earlier in different contexts [10–12]. GFEM, and more generally PoU methods, were developed concurrently by Babuška and colleagues at the University of Maryland [4,13,14], and by Duarte and

Fundamentals of Enriched Finite Element Methods

Figure 1.2 Important contributions to e-FEMs.

4

Introduction

5

Oden at the University of Texas at Austin [6,7,15–17]. Towards the end of the 1990s, Professor Belytschko and coworkers at Northwestern University published a series of papers on the development and application of PoU methods for 2-D fracture mechanics problems [8,9,18–20]. The method was first denoted as an FEM for crack growth with minimal or without remeshing in [8,9] and later coined the eXtended Finite Element Method (XFEM) [18,20,21] by the Northwestern group. Nowadays, XFEM and GFEM are understood as essentially the same method [22]. The XFEM can also be interpreted as a GFEM for the analysis of problems whose fields are discontinuous, and as such XFEM can be seen as a subset of GFEM. In this treatise both terminologies are used interchangeably. Appendix A presents Armando Duarte’s recollections on his early work on partition of unity methods and on the origins of what today is known as the Generalized/eXtended Finite Element Method. Inspired by GFEM, another family of enriched FE formulations seek to solve problems with discontinuities by assigning enriched DOFs to nodes placed directly along discontinuities. The interface-enriched generalized finite element method (IGFEM) [23] (see Soghrati et al. in Fig. 1.2) was first proposed to solve 2-D problems with weak discontinuities. Since its introduction, IGFEM was developed in many directions to solve 3-D problems [24], resolve multiple discontinuities (even junctions) within a single finite element with linear [25] and quadratic [26] approximations, curved material interfaces [27], NURBS discontinuities [28–31], immerse boundary (fictitious domain) problems with weak enforcement of nonzero essential boundary conditions [32], shape optimization [33–36], and topology optimization [37,38]. In particular, because of the ability to construct an enriched approximation hierarchically for multiple discontinuities within a single element, IGFEM was renamed to the Hierarchical Interface-enriched Finite Element Method (HIFEM) [25]. IGFEM and its successor HIFEM were later generalized to the discontinuity-enriched finite element method (DE-FEM) [39] (see Aragón and Simone in Fig. 1.2), whereby the kinematics of both weak and strong discontinuities are resolved with a unified three-term approximation. DE-FEM, which was also first introduced in 2-D for fracture problems, was later developed to solve three-dimensional problems [40], discontinuities represented by NURBS [41], immersed boundary problems with strong enforcement of nonzero essential boundary conditions [42], and to resolve multiple interacting weak and strong discontinuities [43]. In addition to numerical developments, these techniques have been applied to study a large range of problems, including the analysis of microvascular cooling [44–50] for batteries [48,51] and nanosatellites [52], the thermal and mechanical behavior of porous media [26], pitting corrosion [26], fiber composites [26], the damage response of heterogeneous adhesives [53,54], band structure analysis of phononic crystals [55], the electromagnetic analysis of periodic composite materials with complex microstructures [34,56], the biomechanics of bone fracture [40], and even the optimization of chocolate metamaterial unit cells with tuned fracture anisotropy [57]! As it will be shown in Chapter 5, interface- and discontinuity-enriched formulations can actually be derived from GFEM. They can be seen as methods laying in between standard FEM and GFEM: Though they benefit from the mesh-decoupling property of GFEM, they also share many of the advantages of standard FEM, including a simpler computer implementation, a straightfor-

6

Fundamentals of Enriched Finite Element Methods

ward handling of nonzero essential boundary conditions (these can also be enforced strongly), no loss of accuracy in blending elements (neighboring to cut elements), a stable condition number of system matrices, and smooth reactive tractions recovered from Dirichlet boundaries. Some of these benefits are a direct consequence of placing enriched DOFs directly along discontinuities. Other fundamental contributions to e-FEMs include works devoted to the control of the condition number of stiffness matrices [40,42,58–60]. e-FEMs can address the shortcomings of the standard FEM discussed earlier, but enrichments can lead to illconditioned, and sometimes singular, matrices. This issue, well-known since the early days of these methods, was only properly addressed much later. The goal of conditioning control strategies for e-FEMs is to modify the basis functions of these methods such that the condition number of the stiffness matrix is of the same order as that of FEM defined on a fitted mesh (with comparable mesh size). The modified basis should deliver at least the same accuracy of the original basis, be simple to implement, and of a low computational cost. Two fundamental works on this topic are illustrated in the timeline shown in Fig. 1.2: The 2012 paper by Babuška and Banerjee and the 2019 paper by Sanchez-Rivadeneira and Duarte. Finally, an important milestone in e-FEMs is the development of strategies to compute enrichment functions on the fly, instead of relying on functions available in closed form [61,62]. These works bring the benefits of e-FEMs to problems with limited or no analytical a priori information that can be used to enrich the basis functions of the standard FEM—this was one shortcoming of GFEM and in fact of any partition of unity method. Many nonlinear and multiscale problems fall into this category and can now be solved by adopting numerically computed enrichments.

References [1] I. Babuska, B.A. Szabo, I.N. Katz, The p-version of the finite element method, SIAM Journal on Numerical Analysis 18 (3) (1981) 515–545, https://doi.org/10.1137/0718033. [2] B.A. Szabó, I. Babuška, Finite Element Analysis, John Wiley and Sons, New York, 1991. [3] J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, 1st ed., Wiley Publishing, 2009. [4] I. Babuška, G. Caloz, J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM Journal on Numerical Analysis 31 (4) (1994) 945–981, https://doi.org/10.1137/0731051. [5] I. Babuška, J. Melenk, The partition of unity method, International Journal for Numerical Methods in Engineering 40 (1997) 727–758, https://doi.org/10.1002/(SICI)10970207(19970228)40:43.0.CO;2-N. [6] J. Oden, C. Duarte, Chapter: Clouds, cracks and FEMs, in: B. Reddy (Ed.), Recent Developments in Computational and Applied Mechanics, International Center for Numerical Methods in Engineering, CIMNE, Barcelona, Spain, 1997, pp. 302–321, http:// gfem.cee.illinois.edu/jmartincolor/. [7] J. Oden, C. Duarte, O. Zienkiewicz, A new cloud-based hp finite element method, Computer Methods in Applied Mechanics and Engineering 153 (1998) 117–126, https:// doi.org/10.1016/S0045-7825(97)00039-X. [8] T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (1999) 601–620.

Introduction

7

[9] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) (1999) 131–150. [10] S. Nordholm, G. Bacskay, Generalized finite element method applied to bound state calculation, Chemical Physics Letters 42 (2) (1976) 253–258, https://doi.org/10.1016/00092614(76)80358-2. [11] O.C. Zienkiewicz, The generalized finite element method—state of the art and future directions, Journal of Applied Mechanics 50 (4b) (Dec. 1983) 1210–1217, https:// doi.org/10.1115/1.3167203. [12] I. Babuška, J.E. Osborn, Generalized finite element methods: Their performance and their relation to mixed methods, SIAM Journal on Numerical Analysis 20 (3) (1983) 510–536, https://doi.org/10.1137/0720034. [13] J. Melenk, On generalized finite element methods, PhD thesis, University of Maryland, 1995. [14] J. Melenk, I. Babuška, The partition of unity finite element method: Basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 (1–4) (1996) 289–314, https://doi.org/10.1016/S0045-7825(96)01087-0. [15] C. Duarte, J. Oden, Hp clouds—an hp meshless method, Numerical Methods for Partial Differential Equations 12 (1996) 673–705, https://doi.org/10.1002/(SICI)10982426(199611)12:63.0.CO;2-P. [16] C. Duarte, J. Oden, An h–p adaptive method using clouds, Computer Methods in Applied Mechanics and Engineering 139 (1) (1996) 237–262, https://doi.org/10.1016/S00457825(96)01085-7. [17] C. Duarte, The hp Cloud Method, PhD thesis, The University of Texas at Austin, Austin, TX, USA, Dec. 1996. [18] N. Sukumar, N. Moës, B. Moran, T. Belytschko, Extended finite element method for three-dimensional crack modelling, International Journal for Numerical Methods in Engineering 48 (11) (2000) 1549–1570, https://doi.org/10.1002/1097-0207(20000820)48: 113.0.CO;2-A. [19] J. Dolbow, N. Moës, T. Belytschko, Modeling fracture in Mindlin–Reissner plates with the extended finite element method, International Journal of Solids and Structures 37.48–50 (2000) 7161–7183, https://doi.org/10.1016/S0020-7683(00)00194-3. [20] C. Daux, N. Moës, J. Dolbow, N. Sukumar, T. Belytschko, Arbitrary branched and intersecting cracks with the extended finite element method, International Journal for Numerical Methods in Engineering 48 (12) (2000) 1741–1760, https://doi.org/10.1002/10970207(20000830)48:123.0.CO;2-L. [21] J. Dolbow, An extended finite element method with discontinuous enrichment for applied mechanics, PhD thesis, Northwestern University, Evanston, IL, Dec. 1999. [22] T. Belytschko, R. Gracie, G. Ventura, A review of extended/generalized finite element methods for material modeling, Modelling and Simulation in Materials Science and Engineering 17 (043001) (2009) 1–24, https://doi.org/10.1088/0965-0393/17/4/043001. [23] S. Soghrati, A. Aragón, C. Duarte, P. Geubelle, An interface-enriched generalized finite element method for problems with discontinuous gradient fields, International Journal for Numerical Methods in Engineering 89 (8) (2012) 991–1008. [24] S. Soghrati, P.H. Geubelle, A 3D interface-enriched generalized finite element method for weakly discontinuous problems with complex internal geometries, Computer Methods in Applied Mechanics and Engineering 217–220 (2012) 46–57, https://doi.org/10.1016/j. cma.2011.12.010.

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[25] S. Soghrati, Hierarchical interface-enriched finite element method: An automated technique for mesh-independent simulations, Journal of Computational Physics 275 (2014) 41–52, https://doi.org/10.1016/j.jcp.2014.06.016. [26] S. Soghrati, J.L. Barrera, On the application of higher-order elements in the hierarchical interface-enriched finite element method, International Journal for Numerical Methods in Engineering 105 (6) (2016) 403–415, https://doi.org/10.1002/nme.4973. [27] S. Soghrati, C.A. Duarte, P.H. Geubelle, An adaptive interface-enriched generalized FEM for the treatment of problems with curved interfaces, International Journal for Numerical Methods in Engineering 102 (6) (May 2015) 1352–1370, https://doi.org/10.1002/nme. 4860. [28] M. Safdari, A.R. Najafi, N.R. Sottos, P.H. Geubelle, A NURBS-based interface-enriched generalized finite element method for problems with complex discontinuous gradient fields, International Journal for Numerical Methods in Engineering 101 (12) (2015) 950–964. [29] M. Safdari, A.R. Najafi, N.R. Sottos, P.H. Geubelle, A NURBS-based generalized finite element scheme for 3D simulation of heterogeneous materials, Journal of Computational Physics 318 (2016) 373–390. [30] S. Soghrati, R.A. Merel, NURBS enhanced HIFEM: A fully mesh-independent method with zero geometric discretization error, Finite Elements in Analysis and Design 120 (2016) 68–79. [31] M.H.Y. Tan, M. Safdari, A.R. Najafi, P.H. Geubelle, A NURBS-based interface-enriched generalized finite element scheme for the thermal analysis and design of microvascular composites, Computer Methods in Applied Mechanics and Engineering 283 (2015) 1382–1400. [32] A.C. Ramos, A.M. Aragón, S. Soghrati, P.H. Geubelle, J.-F. Molinari, A new formulation for imposing Dirichlet boundary conditions on non-matching meshes, International Journal for Numerical Methods in Engineering 103 (6) (2015) 430–444. [33] A.R. Najafi, M. Safdari, D.A. Tortorelli, P.H. Geubelle, A gradient-based shape optimization scheme using an interface-enriched generalized FEM, Computer Methods in Applied Mechanics and Engineering 296 (2015) 1–17, https://doi.org/10.1016/j.cma.2015.07.024. [34] K. Zhang, A.R. Najafi, J.-M. Jin, P.H. Geubelle, An interface-enriched generalized finite element analysis for electromagnetic problems with non-conformal discretizations, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 29 (2) (2016) 265–279, https://doi.org/10.1002/jnm.2073. [35] A.R. Najafi, M. Safdari, D.A. Tortorelli, P.H. Geubelle, Shape optimization using a NURBS-based interface-enriched generalized FEM, International Journal for Numerical Methods in Engineering 111 (10) (2017) 927–954, https://doi.org/10.1002/nme.5482. [36] X. Zhang, D.R. Brandyberry, P.H. Geubelle, IGFEM-based shape sensitivity analysis of the transverse failure of a composite laminate, Computational Mechanics (May 2019) 1–18. [37] J. Zhang, F. van Keulen, A.M. Aragón, On tailoring fracture resistance of brittle structures: A level set interface-enriched topology optimization approach, Computer Methods in Applied Mechanics and Engineering 388 (2022) 114189, https://doi.org/10.1016/j.cma. 2021.114189. [38] S.J. van den Boom, J. Zhang, F. van Keulen, A.M. Aragón, An interface-enriched generalized finite element method for level set-based topology optimization, Structural and Multidisciplinary Optimization 63 (1) (2021) 1–20, https://doi.org/10.1007/s00158-02002682-5.

Introduction

9

[39] A.M. Aragón, A. Simone, The discontinuity-enriched finite element method, International Journal for Numerical Methods in Engineering 112 (11) (2017) 1589–1613, https:// doi.org/10.1002/nme.5570. [40] J. Zhang, S.J. van den Boom, F. van Keulen, A.M. Aragón, A stable discontinuity-enriched finite element method for 3-D problems containing weak and strong discontinuities, Computer Methods in Applied Mechanics and Engineering 355 (2019) 1097–1123, https:// doi.org/10.1016/j.cma.2019.05.018. [41] E. De Lazzari, S.J. van den Boom, J. Zhang, F. van Keulen, A.M. Aragón, A critical view on the use of non-uniform rational B-splines to improve geometry representation in enriched finite element methods, International Journal for Numerical Methods in Engineering 122 (5) (March 2021) 1195–1216, https://doi.org/10.1002/nme.6532. [42] S.J. van den Boom, J. Zhang, F. van Keulen, A.M. Aragón, A stable interface-enriched formulation for immersed domains with strong enforcement of essential boundary conditions, International Journal for Numerical Methods in Engineering 120 (10) (2019) 1163–1183, https://doi.org/10.1002/nme.6139. [43] D. Liu, J. Zhang, A. Simone, A.M. Aragón, Discontinuity-enriched finite element method for intersecting discontinuities, International Journal for Numerical Methods in Engineering (2019). [44] S. Soghrati, P.R. Thakre, S.R. White, N.R. Sottos, P.H. Geubelle, Computational modeling and design of actively-cooled microvascular materials, International Journal of Heat and Mass Transfer 55 (19) (2012) 5309–5321. [45] S. Soghrati, A.R. Najafi, J.H. Lin, K.M. Hughes, S.R. White, N.R. Sottos, P.H. Geubelle, Computational analysis of actively-cooled 3D woven microvascular composites using a stabilized interface-enriched generalized finite element method, International Journal of Heat and Mass Transfer 65 (2013) 153–164. [46] M.H.Y. Tan, A.R. Najafi, S.J. Pety, S.R. White, P.H. Geubelle, Gradient-based design of actively-cooled microvascular composite panels, International Journal of Heat and Mass Transfer 103 (2016) 594–606, https://doi.org/10.1016/j.ijheatmasstransfer.2016.07.092. [47] M.H.Y. Tan, P.H. Geubelle, 3D dimensionally reduced modeling and gradient-based optimization of microchannel cooling networks, Computer Methods in Applied Mechanics and Engineering 323 (2017) 230–249, https://doi.org/10.1016/j.cma.2017.05.024. [48] S.J. Pety, M.H.Y. Tan, A.R. Najafi, P.R. Barnett, P.H. Geubelle, S.R. White, Carbon fiber composites with 2D microvascular networks for battery cooling, International Journal of Heat and Mass Transfer 115 (2017) 513–522, https://doi.org/10.1016/j.ijheatmasstransfer. 2017.07.047. [49] S.J. Pety, M.H.Y. Tan, A.R. Najafi, A.C. Gendusa, P.R. Barnett, P.H. Geubelle, S.R. White, Design of redundant microvascular cooling networks for blockage tolerance, Applied Thermal Engineering 131 (2018) 965–976, https://doi.org/10.1016/j.applthermaleng. 2017.10.094. [50] R. Pejman, S.H. Aboubakr, W.H. Martin, U. Devi, M.H.Y. Tan, J.F. Patrick, A.R. Najafi, Gradient-based hybrid topology/shape optimization of bioinspired microvascular composites, International Journal of Heat and Mass Transfer 144 (2019) 118606, https:// doi.org/10.1016/j.ijheatmasstransfer.2019.118606. [51] M.H.Y. Tan, A.R. Najafi, S.J. Pety, S.R. White, P.H. Geubelle, Multi-objective design of microvascular panels for battery cooling applications, Applied Thermal Engineering 135 (2018) 145–157, https://doi.org/10.1016/j.applthermaleng.2018.02.028. [52] M.H.Y. Tan, D. Bunce, A.R.M. Ghosh, P.H. Geubelle, Computational design of microvascular radiative cooling panels for nanosatellites, Journal of Thermophysics and Heat Transfer 32 (3) (2018) 605–616, https://doi.org/10.2514/1.T5381.

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[53] A.M. Aragón, S. Soghrati, P.H. Geubelle, Effect of in-plane deformation on the cohesive failure of heterogeneous adhesives, Journal of the Mechanics and Physics of Solids 61 (7) (2013) 1600–1611, https://doi.org/10.1016/j.jmps.2013.03.003. [54] S. Soghrati, B. Liang, Automated analysis of microstructural effects on the failure response of heterogeneous adhesives, International Journal of Solids and Structures 81 (2016) 250–261, https://doi.org/10.1016/j.ijsolstr.2015.12.002. [55] S.J. van den Boom, F. van Keulen, A.M. Aragón, Fully decoupling geometry from discretization in the Bloch–Floquet finite element analysis of phononic crystals, Computer Methods in Applied Mechanics and Engineering 382 (2021) 113848, https://doi.org/10. 1016/j.cma.2021.113848. [56] K. Zhang, J. Jin, P.H. Geubelle, A 3-D interface-enriched generalized FEM for electromagnetic problems with nonconformal discretizations, IEEE Transactions on Antennas and Propagation 63 (12) (Dec. 2015) 5637–5649. [57] A. Souto, J. Zhang, A.M. Aragón, K.P. Velikov, C. Coulais, Edible mechanical metamaterials with designed fracture for mouthfeel control, Soft Matter 18 (2022) 2910–2919, https://doi.org/10.1039/d1sm01761f. [58] I. Babuška, U. Banerjee, Stable generalized finite element method (SGFEM), Computer Methods in Applied Mechanics and Engineering 201–204 (2012) 91–111. [59] A. Sanchez-Rivadeneira, C. Duarte, A stable generalized/extended FEM with discontinuous interpolants for fracture mechanics, Computer Methods in Applied Mechanics and Engineering 345 (2019) 876–918, https://doi.org/10.1016/j.cma.2018.11.018. [60] A.M. Aragón, B. Liang, H. Ahmadian, S. Soghrati, On the stability and interpolating properties of the hierarchical interface-enriched finite element method, Computer Methods in Applied Mechanics and Engineering 362 (2020) 112671, https://doi.org/10.1016/j.cma. 2019.112671. [61] T. Strouboulis, I. Babuška, K. Copps, The design and analysis of the generalized finite element method, Computer Methods in Applied Mechanics and Engineering 81 (1–3) (2000) 43–69. [62] C. Duarte, D.-J. Kim, Analysis and applications of a generalized finite element method with global-local enrichment functions, Computer Methods in Applied Mechanics and Engineering 197 (6–8) (2008) 487–504, https://doi.org/10.1016/j.cma.2007.08.017.

Part One Fundamentals

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The finite element method

2

The finite element method is considered today the standard numerical procedure in solid mechanics for solving boundary and initial value problems. Although initially proposed for structural analysis, today FEM is used in virtually every realm of physics. Simply put, the governing equations that describe a physical phenomenon are solved by discretizing functional spaces (or sets) that describe the field(s) of interest. A domain of interest is divided into nonoverlapping subdomains—called finite elements—where the unknown field is described by means of polynomial approximations. The basis (or shape) functions in each finite element, which are used to span the aforementioned discrete spaces, are usually low-order polynomials. After prescribing boundary conditions, an approximate solution to the unknown field in the whole domain is then obtained by solving a discrete algebraic system of equations. Although simple within each element, the captured behavior of the whole element assembly can be fairly complex because highly nonlinear functions can be approximated by loworder piecewise-polynomial functions as long as their supports (elements where the corresponding function is nonzero) are small enough. This intuitively gives the notion of convergence: The approximate solution of a boundary value or initial value problem should become increasingly more accurate as the mesh size is reduced. This is the basis of the h-version of the finite element method, h-FEM, or simply FEM henceforth. This chapter gives a concise introduction to FEM. We start by looking at the elastostatic boundary value problem (BVP) for a one-dimensional bar, i.e., a structural element that can only sustain axial loads. Although simplistic, this 1-D problem provides most of the insight required to derive the formulation in higher dimensions. The strong and weak (variational) forms of the BVP are stated first. Then the Galerkin projection is used to arrive at the finite element discrete equations. The elastostatics BVP in higher dimensions follows, and towards the end of the chapter we briefly look at heat conduction problems. We also try to understand the error incurred by the finite element approximation by calculating rates of convergence, which are compared with a priori error estimates. An a posteriori error estimate is also used to estimate an approximation of the exact strain energy, which is useful for problems whose exact solution is simply unavailable. The material here should by no means be taken as a thorough treatment on the subject, for which the reader is referred to classical textbooks [1–3].

2.1 Linear elastostatics in 1-D Fig. 2.1 shows a 1-D bar, which can be represented mathematically as an open set  ⊂ R (a subset of the real 1-D coordinate space). A point in the bar can be located by its coordinate x ∈ R. We denote by  the closure of the open set, i.e., the bar plus its boundary (here the left and right ends). This set closure is often used to deFundamentals of Enriched Finite Element Methods. https://doi.org/10.1016/B978-0-32-385515-0.00009-X Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved.

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Fundamentals of Enriched Finite Element Methods

fine the boundary ∂ ≡  simply as  =  \ . The bar has a total length L and cross-sectional area A that is not necessarily constant (A = A(x)). We are now interested in obtaining the static equilibrium of the bar when subjected to appropriate boundary conditions (BCs), e.g., fixing one of its ends and applying a tensile axial force at the other end. The state of deformation is characterized by the displacement field u(x). To determine the governing equation that describes static equilibrium, we consider a bar segment of length x. We see in Fig. 2.1 the forces acting on it, which include the axial forces at the segment’s ends N(x) and N (x + x), an axial load per unit length b(x), and a displacement-dependent force per unit length s(x)u(x) acting in opposite direction of the displacement, with s denoting stiffness per unit length. The latter may represent, for instance, the resistance to remove a pile from the soil (an example is given later in the chapter). The bar segment is in static equilibrium when all forces balance each other: N (x + x) − N (x) − s(x)u(x)x + b(x)x = 0. Dividing by x and taking the limit as x → 0 yields lim

x→0

N (x + x) − N (x) − s(x)u(x) + b(x) = 0, x dN (x) − s(x)u(x) + b(x) = 0. dx

The dependence on the spatial coordinate x is dropped henceforth for simplicity. We assume that the material of the bar behaves linearly elastic according to Hooke’s du are related through σx = Ex , where E is law, i.e., the stress σx and strain x = dx du . ThereYoung’s modulus. The axial force can thus be written as N = σx A = EA dx fore, the equilibrium of every point along the bar is described by   du d EA − su + b = 0 ∀ x ∈ . (2.1) dx dx This is a second-order ordinary differential equation, and thus its solution requires two integration constants that are determined through boundary conditions. Three types of BCs can be applied to our 1-D bar (see Fig. 2.1): • Essential or Dirichlet BC on the primal variable, which in this case is the displacement u. Denoting by  D the part of the boundary with prescribed displacement, this condition is stated as ¯ u| D = u,

(2.2)

i.e., the restriction of the displacement field at  D is equal to the prescribed displacement u. ¯ Notice that a displacement could be prescribed at either or both bar ends, and in either case (2.2) is valid since u¯ may be a function of position. • Natural or Neumann BC imposed on the dual variable, which in this case is the axial force N . With  N denoting the part of the boundary with prescribed axial

The finite element method

15

Figure 2.1 (Top) Schematic of a 1-D bar used to derive the governing differential equation; (bottom) Types of boundary conditions.

force, this BC is usually expressed as  du  EA  n = N¯ , dx  N

(2.3)

where N¯ is the prescribed axial force value and n is either −1 for the left end or 1 for the right end, and it is used as a consequence of our sign convention: A force in the positive direction applied to the left end would produce a negative strain (compression in the bar). Similarly, a force in the positive direction at the right end would produce tension and thus a positive strain. • Mixed or Robin BC, which can be seen as a combination of the previous two BCs. Denoting by  R this region, the mixed BC is    du  EA  n = H u¯ − u| R , dx  R

(2.4)

where H represents the stiffness value of a spring that connects the bar to a point with prescribed value u. ¯ 1 The same sign convention for n applies here, since u¯ > u| R at the left end of the bar produces a force in the positive direction that compresses the bar. It is not possible to prescribe more than one type of BC at the same end of the bar. Boundary conditions are therefore disjoint, a condition usually expressed as  D ∩  N = ∅, or  D ∩  R = ∅, or  N ∩  R = ∅. 1 The symbol H was chosen because this type of boundary condition is more common in heat transfer

problems, where H refers to the heat transfer coefficient (or film coefficient), as described later in § 2.3.

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Fundamentals of Enriched Finite Element Methods

2.1.1 The strong form At this point we have everything we need to formally state the strong form of the 1-D elastostatics boundary value problem: Given the Young’s modulus E :  → R, the cross-sectional area A :  → R, the spring per unit length s :  → R, the axial load per unit length b :  → R, and prescribed quantities for BCs, find the displacement field u ∈ C 2 such that2   du d EA − su + b = 0 ∀ x ∈ , dx dx satisfying boundary conditions u = u¯ on  D u = u¯ on  D EA

du n = N¯ on  N dx

du n = N¯ on  N , dx   du and EA n = H u¯ − u| R on  R , dx   du and EA n = H u¯ − u| R on  R , dx and EA

or or

where u¯ :  D → R is the given prescribed displacement, N¯ :  N → R the prescribed force, and H :  R → R the mixed BC spring constant. Notice that if  =  \  =  D we are dealing with a pure Dirichlet problem where the displacement is prescribed at both ends of the bar. Following a similar reasoning, we could also be dealing with a pure Neumann or pure Robin problem. For the pure Neumann problem, since only the field derivatives are known, the solution can only be determined up to a constant. In other words, there are an infinite number of configurations that have the same deformation (any two of which are separated by a translation of the bar, i.e., a rigid body motion). Example 2.1: Bimaterial interface. Fig. 2.2 shows a bar of length L clamped at the left end and subjected to a load N¯ at the right end. A material interface at x = x subdivides the bar in two parts i , i = 1, 2, with corresponding axial rigidity ki = Ei Ai (constant in each component). State the strong form for this problem and obtain the exact displacement field. Solution: The strong form for this problem is: Given the subdomains’ axial rigidities ki : i → R, i = 1, 2,  du  = N¯ , find the displacethe prescribed displacement u1 |x=0 = 0, and the prescribed axial load k2 dx2  ment field u such that   d dui ki =0 dx dx

x=L

in i , i = 1, 2,

u1 = 0

at x = 0,

du k2 2 = N¯ dx

at x = L,

2 Here C 2 denotes the space of continuous functions with continuous derivatives up to second order. We

therefore require that the displacement field be differentiable, with well-defined derivatives up to second order.

The finite element method

17

Figure 2.2 1-D bar composed of two materials, with interface located at x = x .

with interface conditions (i.e., continuity of displacement and axial forces) u1 = u2

at x = x ,

du du k1 1 = k2 2 dx dx

at x = x .

The solution to this boundary value problem is obtained by integrating the equilibrium equation twice: ki

dui = Ci , dx

ui =

Ci x + Di , ki

i = 1, 2,

(2.5)

where the four integration constants (two for each subdomain) are determined after applying boundary and interface conditions: D1 = 0,

C1 = C2 = N¯ ,

and D2 =

x k N¯ , k1 k2

where k = k2 − k1 represents the jump in axial rigidity at x . After replacing these constants in (2.5), the final displacement field is given by ⎧ ⎨ N¯ x for x ≤ x , u(x) = Nk¯1x (2.6) ⎩  + N¯ (x−x ) for x ≥ x . k k 1

2

The displacement field is shown in Fig. 2.3 for k1 /k2 = 10.

Figure 2.3 Displacement field given by Eq. (2.6) for k1 /k2 = 10 and x = L/2. As long as the axial rigidity in each subdomain is different, the gradient field has a discontinuity at the interface, i.e., the displacement field is C 0 -continuous (although we still require that ui ∈ C 2 ). 

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Fundamentals of Enriched Finite Element Methods

Example 2.2: One-dimensional cracked bar. Consider the 1-D bar of length L shown in Fig. 2.4, composed of two parts joined at x = x by a spring of constant k > 0. Each part has the same axial rigidity k = EA. State the strong form of the problem and obtain its exact solution.

Figure 2.4 One-dimensional cracked bar connected with a spring with constant k . The bar is fixed at the left end and subjected to a prescribed axial load N¯ at the right end. Solution: For this problem the strong form is: Given the axial rigidity k, the spring constant k , the predu scribed displacement u1 (0) = 0, and the prescribed force k dx2 = N¯ , find the displacement field u such that k

d2 ui =0 dx 2 u1 = 0

in i , i = 1, 2, at x = 0,

du k 2 = N¯ dx

at x = L,

with interface condition (force continuity) du du1 = k u(x ) = k 2 at x = x , dx dx     where u(x ) = u x+ − u x− = u2 (x ) − u1 (x ) denotes the displacement jump across the discontinuity at x = x . Similarly to Example 2.1, the solution is obtained by integrating the governing equation twice: k

k

dui = Ci , dx

ui =

Ci x + Di , k

i = 1, 2.

Integration constants are again found by means of boundary and interface conditions: D1 = 0,

C1 = C2 = N¯ ,

and D2 =

N¯ , k

where the last constant is obtained using u(x ) = N¯ /k . The exact solution to this problem, which is shown in Fig. 2.5 for x = L/2, is

¯ Nx for x < x , (2.7) u(x) = Nk¯ ¯ Nx for x > x . k + k 

The finite element method

19

Figure 2.5 Displacement field given by (2.7) for x = L/2. The strong discontinuity at x produces a ∈ C −1 -continuous field, although in each subcomponent ui = C 2 as in the previous example. It is worth noting that, since the axial rigidity at each side of the discontinuity is the same, the slope of the field at either side of the discontinuity is the same. 

Example 2.3: Constrained bar pullout. Fig. 2.6 shows a bar of constant axial rigidity k = EA constrained by a semi-infinite elastic foundation with stiffness per unit length s. Since there is no axial load per unit length, (2.1) simplifies to k

d2 u − su = 0. dx 2

(2.8)

du The boundary conditions are u(∞) = 0 and −k dx = −N¯ (so the bar is in tension). Find the exact solution and use it to compute the strain energy given by

  2 1 ∞ du 2 U = + su dx . k 2 0 dx

Figure 2.6 Pullout of an infinite 1-D bar constrained laterally via a displacement-dependent force per unit length su. The bar is subjected to a prescribed axial load −N¯ at x = 0. Solution: Eq. (2.8) can be rewritten as d2 u = ς 2 u, dx 2

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Fundamentals of Enriched Finite Element Methods

with ς 2 = s/k. This equation has solution u = C1 eς x + C2 e−ς x , where the constants C1 and C2 are again determined by the boundary conditions. For the Dirichlet boundary condition to hold, C1 = 0 since only the second term decays as x → ∞. The constant C2 is obtained with the Neumann boundary condition:  N¯ du  k n = kC2 ς = −N¯ ⇒ C2 = − , dx x=0 kς so the final solution is u=−

N¯ −ς x . e kς

(2.9)

The solution can be easily verified by applying the derivative twice: du N¯ = e−ς x dx k

⇒

d2 u N¯ = − ςe−ς x = ς 2 u. k dx 2

The exact solution is shown in Fig. 2.7 for different values of ς. As expected, as the stiffness of the foundation relative to that of the bar decreases (lower values of ς), the displacement along the bar is felt at deeper coordinates (increasing values of x). Also, the effect of a stiffer foundation relative to the bar translates into a sharper gradient in the displacement field close to x = 0.

Figure 2.7 uk/N¯ for different values of ς . With u given by (2.9), the energy in the system is  

  2 e−2ς x e2ς x − 1 ς N¯ 2 1 x ς N¯ 2 du U = lim + su2 dx = lim = . k x→∞ 2 0 x→∞ dx 2s 2s

(2.10) 

2.1.2 The weak (or variational) form Obtaining the solution of the strong form, that is, satisfying the equilibrium equation at every point along the bar and the boundary conditions, may not be straightforward even for simple load configurations. Thus we seek a weak formulation where equilibrium is instead satisfied on average. The weak or variational statement of the problem can be obtained in different ways, for instance, by minimizing potential energy or by looking at the virtual work on our mechanical system caused by the prescribed loads. Herein we use the method of weighted residuals because of its generality—for instance, not every weak form may be derived from a potential, or the principle of virtual work may not make sense for non-mechanical systems.

The finite element method

21

To satisfy equilibrium on average (weakly) along our 1-D bar, we multiply (2.1) by a weight or test function w and we set the integral over the domain equal to zero. To wit,    L   du d EA − su + b dx = 0 ∀ w ∈ E (), (2.11) w dx dx 0 where E () is the so-called energy space, which is defined by the set of functions that have finite strain energy U (u):

    1 L dw 2 2 E () := w(x), ∀x ∈  : U (w) = + sw dx < ∞ . (2.12) EA 2 0 dx We use E () ≡ E for simplicity henceforth, but we keep in mind that functional spaces are defined on the domain of interest. The space given by (2.12) induces the energy norm 

w E () := U (w), (2.13) which gives us a means to quantitatively measure functions. Notice the analog √ with Euclidean spaces since the norm of a vector v ∈ R3 is simply obtained as v = v · v. The weighted residual given by (2.11) can be written as  w, D(u) = wD(u) dx = 0 ∀w ∈ E ,

(2.14)



where we use ·, · to denote the dot or inner product between two functions, and D(u) is our differential equation. Note that the expression w, D(u) = 0, ∀w ∈E is a statement of orthogonality between our differential equation D(u) and the entire space E (since the expression needs to be satisfied for all functions w ∈ E ). However, because u does not necessarily satisfy the differential equation, then D(u) = 0 is a residual; therefore the statement w, D(u) = 0, ∀w ∈E essentially states that the residual be orthogonal to the space E . The weighted residual is therefore a projection where the solution u to (2.14) is the best at satisfying the orthogonality condition and therefore at minimizing the error with respect to the exact solution. This is demonstrated more formally later in the chapter.

In (2.11) u may not necessarily be the “exact solution” anymore so it is called the trial solution. In addition to requiring that the strain energy remains finite, the trial solution needs to be taken from a set of functions that satisfy the essential boundary conditions a priori. This set of kinematically admissible functions is then defined as    (2.15) Eˆ := v  v ∈ E , v| D = u¯ , where u¯ is the prescribed primal field as defined in (2.2). Choosing w ∈ E and u ∈ Eˆ ensures that (2.11) remains bounded from above (no mechanical system can sustain infinite strain anyways).

22

Fundamentals of Enriched Finite Element Methods

Notice that the first term in Eq. (2.11) requires that the trial solution be differentiable twice, whereas there is no differentiability requirement on the test function. By performing integration by parts, we not only balance the derivative orders of both functions, but also relax the differentiability requirement on the trial solution:     L  L du du L d dw du EA dx = wEA  − dx . (2.16) w EA dx dx dx 0 dx dx 0 0    boundary terms

The boundary terms on the right-hand side are replaced by considering the actual du would be replaced boundary conditions for the problem at hand. For instance, EA dx   by N¯ if the BC at the corresponding end were of Neumann type, or by H u¯ − u| R du if it were of Robin type. If the BC were of Dirichlet type, EA dx would be replaced by the reaction force R, but in such case we normally choose the weight function to vanish at the Dirichlet boundary to prescind of this term (more on this later). After replacing (2.16) into (2.11) and collecting terms, the weak or variational form is finally stated as: Find u ∈ Eˆ such that    L  L dw du du L EA + swu dx = wb dx + wEA  ∀ w ∈ E . (2.17) dx dx dx 0 0 0 Notice that the left-hand side of the equation depends on both the test function and the trial solution, whereas the right-hand side depends only on the test function. Although the boundary terms seem to depend on u, they will have to be replaced by the known boundary conditions (an exception to this is the Robin boundary condition, where the unknown part that depends on u will have to be moved to the left-hand side). Eq. (2.17) can therefore be written in abstract form as: Find u ∈ Eˆ such that B(u, w) = L (w)

∀w ∈ E ,

where the bilinear form3 is given by   L dw du EA + swu dx , B(u, w) = dx dx 0

(2.18)

(2.19)

and the linear form by  L (w) = 0

L

 du L wb dx + wEA  . dx 0

3 For functions u, v, w and C ∈ R, the bilinear form has the following properties:

B(u, u) ≥ 0, B(u, v) = B(v, u), B(u + v, w) = B(u, w) + B(v, w), B(Cu, v) = CB(u, v).

(2.20)

The finite element method

23

By inspecting (2.12), (2.13), and (2.19), it can be seen that 1 U (u) = B(u, u) and u E () = 2



1 2 B(u, u).

(2.21)

2.1.2.1 Sobolev spaces More formally, for solving partial differential equations we choose functions from the Hilbertian Sobolev space H κ (), which comprises square-integrable functions in  that have square-integrable derivatives up to order κ. For our 1-D bar, our space is H 1 (), which is mathematically described as   dv (2.22) ∈ L 2 () . H 1 () ≡ v| v ∈ L 2 () : dx A function v is square integrable on , i.e., it belongs to the Lebesgue space L 2 () if  v, v = v 2 dx < ∞. 

2.1.2.2 Non-homogeneous Dirichlet boundary conditions We now consider the case when Dirichlet boundary conditions are nonzero, i.e., u| ¯  D = 0. It was mentioned above that in (2.16) the boundary term associated with du = R. the prescribed displacement had to be replaced by the reaction force, i.e., EA dx However, since (2.18) has to be valid for any function w, certain choices of test functions simplify the formulation. For instance, by defining   E0 () := w| w ∈ E (), w| D = 0 , choosing a weight function w ∈ E0 causes the term containing the reaction force to drop out from the formulation. More rigorously, when dealing with non-homogeneous Dirichlet boundary conditions, we take special consideration regarding the choice of function spaces (and function sets). By considering the function space   V0 () = v| v ∈ H 1 () : v| D = 0 , (2.23) where H 1 () is the first-order Hilbertian Sobolev function space as defined in (2.22), ˜  D = u. ¯ In other words, V

we now define a linear variety V := u˜ + V0 , where u| is a translation of the space V0 by u˜ so that every element in V satisfies the nonhomogeneous Dirichlet boundary condition. The weak form of the boundary value problem can then be expressed as: Find the displacement field u ∈ V such that B(u, w) = L (w)

∀w ∈ V0 .

(2.24)

24

Fundamentals of Enriched Finite Element Methods

Since u˜ is known, (2.24) is equivalent to finding the “unknown” part of the solution in the space V0 and adding it to u; ˜ therefore, an equivalent statement is: Given u = v + u, ˜ find v ∈ V0 such that B(v, w) = L (w) − B(u, ˜ w)

∀ w ∈ V0 .

(2.25)

This last expression explicitly states that the known nonzero essential boundary condition modifies the linear form, which will eventually result in a force vector once (2.25) is discretized (as explained later in § 2.1.4). Notice that in (2.25) both the test function and the trial solution are chosen from the same space. We will use this last expression henceforth for the weak form when dealing with non-homogeneous essential boundary conditions. Example 2.4: Weak formulation. The previous section outlined the variational formulation for our 1-D bar. However, because no concrete boundary conditions were given, the formulation was kept general per Eqs. (2.18)–(2.20). Consider a prescribed displacement u(0) = u¯ 0 at the left end and a mixed boundary condition du  EA dx  = H (u¯ L − u(L)) at the right end (notice that possibly u¯ 0 = u¯ L ). State the L weak or variational form. Solution: For this problem, (2.17) is written as   L  L dw du EA + swu dx = wb dx + w(L)H (u¯ L − u(L)) − w(0)R dx dx 0 0

∀w ∈ E ,

where we recognize that the axial force at the left end corresponds to the reaction force R. However, we eliminate the reaction term by choosing E = V0 so that w(0) = 0. Notice that, because of the Robin boundary condition, two terms are added to the linear form; while one of these terms is known, the other involves the yet unknown displacement field u so it becomes part of the bilinear form. The final problem statement is: Given u = v + u, ˜ u(0) ˜ = u¯ 0 , find v ∈ V0 such that B(v, w) = L (w) − B(u, ˜ w) where

∀ w ∈ V0 ,

  L dw dv + swv dx + w(L)H v(L), EA dx dx 0  L L (w) = wb dx + w(L)H u¯ L .

B(v, w) =

0

Notice again that in the right-hand side the linear form is modified by the known part of the solution u˜ that satisfies the non-homogeneous Dirichlet BC at x = 0. 

Example 2.5: One-dimensional cracked bar (revisited). In Exercise 2.2 we looked at obtaining the exact solution for a 1-D cracked bar that was connected with a linear spring (see Fig. 2.4). Derive the weak formulation for this problem. Solution: Since the weak formulation is equivalent to the principle of virtual work, here we use the latter. The virtual work done by the external force is simply We = w(L)N¯ , where we denote w = δu the virtual displacement. The virtual work of internal forces in the bar is  L  L du dw Wi = σ δ dx = EA dx , dx dx 0 0

The finite element method

25

where we note that the virtual strain is δ = dw dx . To derive the virtual work of the spring, Fig. 2.4 shows the free body diagram of the spring, and the jump in displacement and virtual displacement at x = x . The force on the linear spring is given by fs+ = fs− = k u(x ) ,     where u(x ) = u x+ − u x− . The virtual work of fs+ and fs− is respectively   Wi+ = fs+ w x+ ,   Wi− = −fs− w x− , so the virtual work of internal forces on the spring is given by     Wi+ + Wi− = fs+ w x+ − fs+ w x− = fs+ w(x ) = k u(x ) w(x ) . The weak formulation of the problem can then be stated as: Find u ∈ V0 such that  L EA 0

du dw dx + w(x ) k u(x ) = w(l)N¯ dx dx

∀w ∈ V0 ,

with the function space V0 defined as

  2  L 1 1 dv 2 V0 = v : v(0) = 0, U (v) = dx + k v(x ) < ∞ . EA dx 2 0 2 

2.1.3 The Galerkin formulation So far we have made no choice regarding the space of functions used to solve (2.17). In fact, if E is infinite-dimensional—i.e., any arbitrary function can be represented as long as it has finite strain energy—then both weak and strong statements are equivalent (this can be proved mathematically [1, § 1.4]). An infinite-dimensional function can be written as u(x) =

∞ 

ϕi (x)Ui ,

(2.26)

i=1

where ϕi :  → R form a basis that spans E (), and Ui ∈ R are the coefficients of those functions—usually referred to as degrees of freedom (DOFs). One can immediately foresee that finding a solution in an infinite-dimensional space is not practical, so instead we search for a solution in a finite-dimensional space E h () ⊂ E (). In other words, by truncating (2.26) to a finite number of terms, we accept the possibility that our trial solution uh ∈ E h () may not be exact. Nevertheless, this poses no problem as long as we have information about the magnitude of the error made and we know how to reduce it, which we will investigate later in this chapter. Our finite-dimensional problem statement now reads: Find uh ∈ E h such that     ∀ wh ∈ E h , (2.27) B uh , w h = L w h

26

Fundamentals of Enriched Finite Element Methods

where the linear and bilinear forms now use trial and weight functions taken from the finite-dimensional space E h . Eq. (2.27) is usually referred to as the Galerkin projection. If uh and w h are taken from the same space the procedure is called Bubnov– Galerkin, and otherwise it is called Petrov–Galerkin.

2.1.3.1 Orthogonality of Galerkin error The variational form (2.18) and its discrete counterpart (2.27) can be used to give some insight into the solution obtained by the Galerkin projection. In Eq. (2.18) u denotes the exact solution, and since such equation is valid for all w ∈ E , it is certainly valid for a subset of test functions w h ∈ E h ⊂ E :     B u, w h = L w h ∀ wh ∈ E h . (2.28) Then subtracting (2.27) from (2.28) yields   B u − uh , w h = 0 ∀ wh ∈ E h .

(2.29)

Eq. (2.29) states that the error in the approximation ε ≡ u − uh is orthogonal (in the sense of the bilinear form) to the subspace E h . Two functions f and g are orthogonal with respect to the bilinear form if B(f, g) = 0. In other words, uh is the projection (in the sense of the bilinear operator) of the exact solution onto the subspace E h . This is tantamount to saying that from all w h ∈ E h , the approximated solution uh obtained by the Galerkin projection is the one that minimizes the error with respect to the exact solution. This is demonstrated in the following theorems. Theorem 2.1.1. Function uh ∈ E h minimizes the error with respect to the exact solution, i.e., uh = arg min u − v h v h ∈E h

Eh

.

(2.30)

Proof. We do a proof by contradiction; assume that v h has a lower error than uh , then 2

ε 2E h > u − v h h E   B(ε, ε) > B u−uh + uh − v h , u−uh + uh − v h         B(ε, ε) >  B(ε, ε) + B ε, uh − v h + B uh − v h , ε + B uh − v h , uh − v h       v h− h + B uh − v h , uh − v h . 0 > 2Bε, u



We first added zero and then used the distributive andsymmetry properties  of the bilinear h h h h form (footnote 3); then we used Eq. (2.29). Since B u − v , u − v ≥ 0, v h cannot have a smaller error than uh .



The finite element method

27

A generalization of Eq. (2.30) for asymmetric bilinear forms is provided by Céa’s approximation theorem [4]. Theorem 2.1.2 (Céa’s approximation theorem). Let V be a closed Hilbert space with norm · V . Let B : V × V → R be a bilinear form, not necessarily symmetric with the following properties: • B is continuous, and therefore bounded: |B(u, w)| ≤ C1 u V v V

C1 > 0, ∀u, v ∈ V .

(2.31)

• B is coercive (bounded below) on V , i.e., there exists a constant C2 > 0 such that B(u, u) ≥ C2 u 2V

∀u ∈ V ,

(2.32)

then u − uh

V

≤

C1 u − vh V C2

vh ∈ V .

(2.33)

Proof. C2 u − uh

    ≤ B u − uh , u − uh = B u − uh , u − uh +v h − v h V      h , = B u − uh , u − v h + B u  −u v h − uh  2



≤ C1

u − uh

V

u − vh

V

∀v h ∈ V .

The first inequality is a restatement of the coercive relation (2.32) and the last inequality  results from (2.31). The canceled term results from (2.29) since v h − uh ∈ V . Theorem 2.1.3. Given the energy space E h , the error ε = u − uh in the energy norm can be determined as !  # 1"

u − uh E = B(u, u) − B uh , uh (2.34) 2 Proof. 2 u − uh

     h ,  = B ε, u − uh = B(ε, u) − B  u −u uh  E     h h = B u − u , u +B u − u , uh         = B(u, u) − B  uh , u + B u, uh − B uh , uh    = B(u, u) − B uh , uh . 2

In this proof the first cancellation is due to (2.29) (for the same reason note that the term added in orange (gray in print version) is equally zero) and two terms cancel one another because of the symmetry of the bilinear form (footnote 3). This result tells us that

28

Fundamentals of Enriched Finite Element Methods

we can compute the error in the approximation by simply computing the strain energy of our approximate solution and compare it to the strain energy of the exact solution. By inspecting (2.21), we can then rewrite (2.34) as   

u − uh E = U (u) − U uh . (2.35) 

2.1.4 The finite element discrete equations The discrete variational form (2.27) could already be solved à la Rayleigh–Ritz by choosing trial and test functions that act globally on the entire bar and that are kinematically admissible (i.e., that satisfy a priori the essential boundary conditions). For the trial solution to be a truncated Fourier series $ instance, one could$take N a cos πx) + u(x) = N (j j =1 j j =1 bj sin (j πx), aj , bj ∈ R. However, this approach is not practical for large because their resulting algebraic system of equations are dense. One major accomplishment of the finite element method was to circumvent this problem by choosing basis functions that have local support—i.e., they are nonzero only in their corresponding elements and zero elsewhere—resulting in sparse algebraic systems. We divide our 1-D bar in % Fig. 2.8 into nE finite elements at nN node locations E ei , ei ∩ ej = ∅ for i = j . With a Bubnov–Galerkin (nE = nN − 1) such that  = ni=1

Figure 2.8 1-D bar discretized using 4 finite elements and their corresponding shape functions.

projection, consider the test function and the trial solution of the form w = h

nN 

ϕi wi = W and u = h

i=1

nN 

ϕi ui = U ,

(2.36)

i=1

# # " " and U = u1 u2 . . . unN are coefficient where W = w1 w2 . . . wnN vectors (the latter collects all DOFs and is called the global DOF vector), and # "  = ϕ1 ϕ2 . . . ϕnN is a vector that collects all shape functions.4 The ith shape 4 In this book we will also use a different notation for (2.36). By defining as ι the index set of all finite h

element nodes, the test function and the trial solution can also be written as   ϕi wi and uh = ϕi ui . wh = i∈ιh

i∈ιh

The finite element method

29

function in , which is associated with node xi , is given by ⎧ x−xi−1 ⎪ ⎨ xi −xi−1 for ei−1 = [xi−1 , xi ] , −x ϕi (x) = xxi+1−x for ei = [xi , xi+1 ] , i i+1 ⎪ ⎩ 0 elsewhere.

(2.37)

The shape function ϕi is a piecewise linear polynomial that has a maximum value ϕi (xi ) = 1, it ramps down linearly to zero at contiguous nodes i − 1 and i + 1, and it is equally zero elsewhere. Obviously (2.37) is truncated for the first and last shape functions. Although the behavior in each element is represented with a linear polynomial, we note that we may still be able to reproduce fairly complex solutions because the finite elements can be created as small as needed.   These functions satisfy the so-called Kronecker-δ property since ϕi xj = δij . Because of this property, the degree of freedom ui physically represents the displacement of the bar at xi , i.e., ui = u(xi )—and as a consequence $nN U is commonly referred to as ϕi (x) = 1, ∀x ∈  and therethe global displacement vector. Notice also that i=1 fore the chosen basis forms a partition of unity. This property is widely exploited in this book when describing the more advanced enriched formulations. By using the symmetry property of the bilinear operator, and by inserting (2.36) into (2.27), we see that B

nN 

ϕi wi ,

wi

i=1 nN 

nN 

j =1 nN 

∀ wi ∈ R, i = 1, . . . , nN ,

i=1

nN      B ϕi , ϕj uj = wi L ϕi

wi

nN    ϕj uj = L ϕi wi

j =1

i=1 nN 

nN 

∀ wi ∈ R, i = 1, . . . , nN ,

i=1

    B ϕi , ϕj uj − L ϕi = 0

∀ wi ∈ R, i = 1, . . . , nN ,

j =1

i=1

and since the ith equation has to be equal to zero for an arbitrary value of wi , for wi = 0 then we must have nN 

    B ϕi , ϕj uj − L ϕi = 0

i = 1, . . . , nN .

j =1

This last expression, which is a set of nN equations with unknown coefficients uj , j = 1, . . . , nN , can be written in matrix form5 as KU = F ,

(2.38)

5 Note that the ith equation can be expanded to

B(ϕi , ϕ1 ) u1 + B(ϕi , ϕ2 ) u2 + · · · + B(ϕi , ϕN ) unN = L (ϕi ) .             Ki1

Ki2

KinN

Fi

30

Fundamentals of Enriched Finite Element Methods

  where Kij = B ϕi , ϕj and Fi = L (ϕi ); here K and F are the global stiffness matrix and global force vector, respectively. We now consider an alternative approach to derive (2.38) by looking at the local equilibrium of a single finite element of the discretization. Consider element # " ei = xi , xj , whose equilibrium in the weak sense is described by (2.27): 

xj



xi

dw h duh + sw h uh EA dx dx



 dx =

xj

  w h b dx + w h (xj )N xj

xi

− w h (xi )N (xi )

(2.39)

∀ wh ∈ E h ,

  where N(xi ) ≡ Ni and N xj ≡ Nj denote the axial forces at the left and right ends of the finite element, respectively. In this element the only nonzero shape functions are ϕi , ϕj , and therefore the displacement field can be written as uh = ϕi ui + ϕj uj = ϕu, " # " # where ϕ = ϕi ϕj is the element shape function vector, and u = ui uj is the element degree of freedom vector. Within the element the weight function can be written analogously as w h = ϕw = # " w ϕ  , where w = wi wj is now an arbitrary vector, i.e., the statement ∀ w h in Eq. (2.27) implies ∀ w ∈ R2 since ϕ is known. Then (2.39) becomes w



xj



   EAB  B + sϕ  ϕ dx u = w

xi

xj

xi

ϕ  b dx +



−Ni Nj

 ∀ w ∈ R2 , (2.40)

where coefficient vectors have been taken out of the integrals because they are constant and B = dϕ dx is the so-called strain–displacement matrix, i.e., the matrix that relates du strains to displacements at element level since u = ϕu, and therefore  = dx = dϕ dx u. Because (2.40) must be valid for any w ∈ R2 , we must have    xj  xj   −Ni EAB  B + sϕ  ϕ dx u = ϕ  b dx + . (2.41) Nj xi xi       k

f

Denoting by k the local element stiffness matrix and by f the local element force vector, the system ku = f

(2.42)

describes the static equilibrium of our finite element. Clearly, the solution over the entire bar can only be obtained after considering the contributions of all finite elements in the discretization. This process, which is called assembly, results in the global stiffness matrix and global force vector. Denoting by k e and f e the local stiffness matrix

The finite element method

31

and local force vector corresponding to the eth element, respectively, the global arrays are obtained as nE

K=

A

nE

ke ,

e=1

F=

Af , e

(2.43)

e=1

where A is the standard finite element assembly operator; this operator ensures that components in the local arrays are properly added in the global arrays at their correct global degree of freedom locations. Noteworthy, the assembly process produces the cancellation of many of the boundary forces in (2.41) since for a given node, the force Nj from the preceding element would balance Ni from the subsequent one (otherwise the node would not be in static equilibrium). In practice these are therefore not considered and only the ones at the ends of the entire bar (if any) are included according to the boundary conditions of the problem (see the discussion after (2.16)). However, notice that this means of interpreting the discrete equations allows us to also consider concentrated forces at nodes. As before, the resulting discrete equation that describes the static equilibrium for the entire bar is KU = F . This equation can now be solved by a direct or an iterative solver. Since the trial and weight functions are of the same form (Bubnov–Galerkin), the resulting stiffness matrix is symmetric positive semidefinite. The only zero eigenvalue physically corresponds to the rigid body translation of the bar, a situation that produces no deformation, i.e., the strain energy is zero. After prescribing Dirichlet boundary conditions then the system is positive definite and there exists a unique solution. Once the global displacement vector U is obtained, the strain energy of the approximation can be calculated—for linear problems—simply as 1 U h = U  KU . 2

(2.44)

If non-homogeneous Dirichlet boundary conditions are present, the formulation in the element that contains the prescribed node is not modified for simplicity, and the prescribed essential boundary condition is enforced during the solution process: The system of equations can be subdivided in free (f ) and prescribed (p) parts as      Kff Kfp Uf F = f . (2.45)  Kfp Kpp Up Fp Since the prescribed displacement Up is known, we are interested mainly in the unknown portion of the displacement vector U f , which can be obtained by solving the  −1  first equation, i.e., Uf = Kff F f − Kfp Up . In the latter expression it can be seen that the known prescribed displacement modifies the global force vector when accounting for nonzero essential BCs (U p = 0). Although it is possible to subdivide the global stiffness matrix in components as per (2.45), an amenable computer implementation solves the original nN × nN system by keeping the DOF ordering of the global arrays intact. To that end, the force vector is modified (subtracting Kfp Up ) and the value of the prescribed displacement is placed at the corresponding prescribed DOF;

32

Fundamentals of Enriched Finite Element Methods

in the global stiffness matrix we zero-out rows and columns of prescribed DOFs and assign 1’s in their diagonal location. This procedure, which essentially replaces the equations of prescribed DOFs with trivial equations of the form 1 · U i = u(x ¯ i ), also removes the singularity of the stiffness matrix caused by zero energy modes (rigid body motions). Example 2.6: h-FEM solution to the bimaterial interface problem. For the bimaterial interface problem in Example 2.1, the exact solution was found in Eq. (2.6). Given the bilinear and linear forms for this problem, respectively  B(u, w) =

L

k(x) 0

dw du dx and L (w) = w(L)N¯ , dx dx

where k(x) = E(x)A, obtain the solution with two finite elements (whose common node is placed at the material interface at x = x ). Solution: Finite element approximations studied so far are piecewise linear. In other words, they are C 0 continuous since the field gradient is discontinuous at element boundaries. Therefore, as long as a node is placed at x = x , the finite element solution would capture the jump in the gradient at that location. Moreover, because the exact solution is piecewise linear, a linear finite element approximation should recover (2.6) exactly. The discretization for the problem is shown in Fig. 2.9, this time for x = L/3.

Figure 2.9 Finite element discretization for the bimaterial interface problem, composed of two finite elements (and three nodes). The center node is placed exactly at the material discontinuity at x = L/3. Taking our trial and weight functions as uh (x) =

3  i=1

ϕi (x)ui and wh (x) =

3 

ϕi (x)wi ,

i=1

respectively, where ϕi (x) are the linear shape functions given by (2.37), we can now compute the local stiffness matrix and the local force vector for each element: Element e1 : The first element is connected to nodes 1 and 2, and therefore its element freedom table (EFT) " # is 1 2 . EFTs help us in adding the elements of the local stiffness and force arrays to the right locations in their global counterparts. With our choice of trial and weight functions above, the approximation in the first element can be written as   " # u1 uh1 = ϕ1 ϕ2 = ϕ 1 u1 , u2

The finite element method

33

and the strain–displacement matrix as B1 =

dϕ 1 ' 1 = −l 1 dx

(

1 l1 .

With the latter, we compute the local element stiffness matrix as ⎡

 k1 =

l1

k1 ⎣

− l1

⎤

1⎦

'

1 l1

− l1 1

1 l1

(

dx = k¯1



1

1 −1

2  −1 1

1 2

k with k¯1 = 1 . l1

We superposed the DOFs of the EFT to know where each component will be added into the global stiffness " # matrix. The element force vector is simply f 1 = −R 0 . # " # " Element e2 : Similarly to the first element, the second element’s EFT is 2 3 . With ϕ 2 = ϕ2 ϕ3 , the strain–displacement matrix for the second element is B2 =

(

dϕ 2 ' 1 = −l 2 dx

1 l2 ,

and the stiffness matrix ⎡

 k2 =

l2

k2 ⎣

− l1

⎤

2⎦

1 l2

'

− l1 2

1 l2

(

dx = k¯2

" The force vector for this element is f 2 = 0



2

1 −1

3  −1 1

2 3

k with k¯2 = 2 . l2

# N¯ .

Assembly and solution: The global stiffness matrix and force vector are now assembled, respectively as K = A2e=1 k e and F = A2e=1 f e , yielding the system

where we show already struck the row and column corresponding to prescribed DOF 1. The modified system to solve is

1 2 3

⎡1 1 ⎣0 0

2

0 k¯1 + k¯2 −k¯2

3 ⎤ ⎡ ⎤ 0 0 u1 ¯ ⎦ −k2 u2 = ⎣ 0 ⎦ N¯ u3 k¯2

1 2 3

⎡ and its solution U = ⎣

⎤ 0 ¯ ¯ ⎦, N /k1 N¯ /k¯1 + N¯ /k¯2

and therefore the DOFs exactly correspond to the displacement field at the nodes. By replacing the DOFs  in U in our trial solution uh adopted earlier, it can be verified that (2.6) is recovered.

2.1.5 The isoparametric mapping

# " Let us consider a single finite element of the bar e = xi , xj ; a bijective function Q can be used to map a reference or master element eˆ = [−1, 1] into the actual finite

34

Fundamentals of Enriched Finite Element Methods

# " Figure 2.10 Mapping from ξ to x coordinate for a finite element e = xi , xj and shape functions on the master element eˆ = [−1, 1].

element e via the coordinate ξ (see schematic in Fig. 2.10), i.e., Q(ξ ) : eˆ → e. Let us take the simple linear mapping given by  1+ξ 1−ξ xi + xj = ϕi (ξ ) xi , 2  2      2

x (ξ ) =

(2.46)

i=1

ϕi

ϕj

where we note that linear shape functions are now a function of master coordinate ξ . This choice of mapping is by no means arbitrary since, so far, the trial solution is interpolated in the same way via (2.36); we therefore adopt an isoparametric formulation, whereby both the primal field and the geometry are interpolated with the same polynomial order. We could have also opted for a sub- or superparametric mapping, where the geometry interpolation is of lower or higher order, respectively, compared to the trial solution. In this isoparametric case, the inverse mapping Q−1 (x) : e → eˆ is given by ξ (x) =

2x − xi − xj . xj − xi

(2.47)

This mapping is mostly used for integration, since standard Gauss quadrature rules are defined on a master element #(here on eˆ = [−1, 1]). Integrating a function f (x) in the " physical element e = xi , xj can be rewritten as 

 f (x) dx = e

+1

−1

f (x(ξ ))j dξ ∼ =

nGP 

f (x(ξi ))γi j,

(2.48)

i=1

where j ≡ dx / dξ is the Jacobian of the coordinate transformation, and the last term evaluates numerically the integral by sampling the integrand f (x) at nGP Gauss quadrature points with coordinates ξi and corresponding weights γi . For this mapping, j represents the ratio between the lengths between physical and master elements, i.e.,

The finite element method

35

j = le /2, with le = xj − xi denoting the length of the element. Notice that in the last equation the equality was changed to approximately equal since most likely the numerical quadrature of f (x) will not be exact. The summation will only be exact in the case of integrating polynomials with a sufficient number of integration points—nGP quadrature points integrate exactly a polynomial of order 2nGP − 1. The benefits of transformation (2.48), which are not obvious for the 1-D case, will become evident in higher dimensions. Example 2.7: Stiffness matrix calculation with isoparametric formulation. Using the isoparametric mapping above, obtain the stiffness matrices in Example 2.6. Solution: The stiffness matrix for the ith element is  ki B  B dx , ki =

(2.49)

li

where the strain–displacement matrix is B = dϕ dx . We now want to transform this integral so that the isoparametric formulation is used. In the master element eˆ = [−1, 1], which is the same regardless of the element considered, the linear shape functions are ( ' 1+ξ . ϕ = 1−ξ 2 2 Because these are a function of the master coordinate ξ but the strain-displacement matrix in (2.49) requires derivatives with respect to the global coordinate x, we apply the chain rule:   1  dϕ dξ  dϕ dξ ki = ki j dξ . dξ dx dξ dx −1 Finally, noting that dϕ ' 1 = −2 dξ

(

1 , 2

2 dξ = , dx li

the integral becomes  1 1 ' −2 ki = ki − 12 1 −1

2

1 2

and

j=

(  2 2 l

 1 dξ = k¯i −1 2 i

li

l dx = i, dξ 2

which is the same matrix obtained earlier in Exercise 2.6.

 k −1 , with k¯i = i , 1 li 

2.1.6 A priori error estimates For the h-version of FEM using uniform meshes—i.e., the mesh size h is distributed roughly uniformly throughout the discretization—the error of the finite element approximation with respect to the exact solution is bounded by

u − uh E ≤ C1 hβh u E ,

(2.50)

where C1 is a constant independent of mesh size h, and βh the rate of convergence. With the relative error in energy ε¯ ≡ u − uh E / u E , and taking the natural logarithm, we get ln ε¯ = ln C1 + βh ln h,

(2.51)

36

Fundamentals of Enriched Finite Element Methods

Table 2.1 Algebraic convergence rates for h-FEM based on problem category. Category

Convergence rate

A

B

βh = p †

min (p, κ − 1)‡ βh = p



C for uniform h for optimal mesh

βh > 0

† p ≡ polynomial order. ‡ κ ≡ regularity of the solution. For u ∈ H κ , κ is very large or κ > 1 for categories A and B, respectively.

which is the equation of a line with slope βh in ln h × ln ε¯ space. Therefore, hFEM exhibits “algebraic convergence,” whereby the convergence rate (the slope of the line (2.51)) remains constant as the mesh size h decreases. The convergence rate is, however, dependent on the smoothness (also called regularity) of the exact solution, the latter which can belong to either of the following three categories [2]: • Category A. The field u is an analytic function in , i.e., it can be expanded in Taylor series; • Category B. The field u is analytic in , except for a finite number of sets of zero measure, i.e., points in 1-D, points and curves in 2-D, or points, curves, and surfaces in 3-D; • Category C. The field u is neither in Category A nor in B. Depending on the problem category, the convergence rates for h-FEM are summarized in Table 2.1. Note that the rate of convergence is limited either by the polynomial interpolation order or by the regularity of the exact solution, so for problems exhibiting singularities the latter is likely to control the convergence rate. The error in the energy norm can also be expressed as a function of the total number of degrees of freedom nD . Since for uniform meshes h ≈ 1/nD 1/d , where d is the dimensionality of the problem, we rewrite (2.51) as ln ε¯ = ln C1 −

βh ln nD , d

(2.52)

which is again the equation of a line—with negative slope βh /d this time—in ln nD × ln ε¯ space. Plotting the relative error as a function of the total number of DOFs makes more sense when the mesh size is kept fixed and the polynomial order of the approximate solution increases, which is the topic of Chapter 3. Example 2.8: h-FEM solution of the constrained bar pullout example. Obtain the finite element solution for the boundary value problem given in Example 2.3 for N¯ = −1 N, ς = 1 m−2 , and 4, 8, 16, and 32 elements of uniform size. Use a domain  = [0, 10 m] and prescribe the displacement at x = 10 m (use Eq. (2.9) to calculate its value). In addition, create a convergence plot of the relative error in the energy norm, defined as

u − uh E U −U h , = ε¯ =

u E U

The finite element method

37

where U h is the energy obtained by the finite element analysis computed using (2.44), and U is the exact energy obtained via (2.10). Finally, calculate the convergence rate βh . Solution: Noting that for this problem ς 2 = s/k, the 2 × 2 local stiffness matrix for the eth element can be written as ke =

  e

2     B  B + ς 2 ϕ  ϕ dx = B  B + ς 2 ϕ  ϕ γi j,

(2.53)

i=1

where the continuous integral is transformed into a 2-point Gauss quadrature rule for numerical integration following (2.48). Notice the equality sign (as opposed to approximately equal in Eq. (2.48)) since numerical quadrature for this problem is actually exact. After assembling the global stiffness and force vectors, the rigid body translation is eliminated by prescribing the exact value of the displacement at the last node obtained via (2.9), i.e., u¯ = unN = −0.000 045 399 9 m. A unit load is also prescribed to the right-hand side force vector as F1 = N¯ = −1 N. The problem is then solved numerically with 4, 8, 16, and 32 finite elements of uniform size. The displacement field as a function of coordinate is shown in Fig. 2.11a, where as expected, the numerical solution improves (it is closer to the exact solution) as the number of elements is increased.

Figure 2.11 (a) uk/N¯ for different mesh sizes; (b) Relative error in energy norm as a function of mesh size h. In order to quantify the error with respect to the exact solution, we need the reference value for the exact strain energy in the system. The result of the energy given by (2.10) was obtained for a bar of infinite length. Thus, by chancing the upper integration limit to x = 10 m, the reference energy is computed as U = 0.499 999 998 969 423 2 J. Per Eq. (2.44), the energy obtained from the finite element solution is computed as U h = 12 U  KU . The relative error in the energy norm as a function of mesh size is illustrated in Fig. 2.11b, showing a constant convergence rate βh = 1 (rounded from βh = 0.9956). Note that indeed h-FEM shows algebraic convergence since the rate of convergence (the slope of the line) roughly remains constant as the mesh size h decreases. 

2.1.7 A posteriori error estimate In the previous example we computed the convergence rate of a linear approximation. This algebraic convergence, of the form given by (2.50), was obtained with knowledge of the exact solution. However, there are cases—most of them—where the exact solution is simply unavailable. How can we tell, without an exact solution, whether

38

Fundamentals of Enriched Finite Element Methods

our finite element approximation is good enough? Fortunately, it is possible to obtain an approximate value of the exact strain energy by means of three finite element solutions. Let us write (2.35) as

u − uh 2E = U − U h , which is the left-hand side of (2.50) squared; thus we can write U − U h = C12 h2βh U ,

(2.54)

where we change the inequality for an equality since the bound (2.50) is increasingly tighter as h → 0. Eq. (2.54) has three unknowns, namely C1 , βh , and U , for which we need three equations. By solving the problem with three different mesh discretizations with increasingly smaller mesh sizes h0 , h1 , and h2 —with their corresponding approximate strain energies U h0 , U h1 , and U h2 —we write three different expressions for (2.54): 2β

U − U h0 = C12 h0 h U , 2β

U − U h1 = C12 h1 h U , 2β

U − U h2 = C12 h2 h U . We now manipulate these three equations to eliminate constants C1 and βh : Divide the first equation by the second and then take the natural logarithm to get     U − U h0 h0 = 2β , ln ln h U − U h1 h1 and similarly with the second and third equations:     h1 U − U h1 = 2βh ln . ln U − U h2 h2 Divide the last two equations to get     h0 −U h0 ln U ln h h1 U −U 1   = Q.   = U −U h1 ln U −U h2 ln hh12

(2.55)

(h0 / h1 ) After computing Q = ln ln (h1 / h2 ) , the approximation to the exact strain energy U can be obtained by solving the nonlinear equation

U − U h0 = U − U h1



U − U h1 U − U h2

Q (2.56)

.

Also, since for uniform meshes we can relate the mesh size h to the total number of DOFs nD and the problem dimensionality d as h ≈ n 11/d , it is also possible to use DOFs to compute Q =

ln (nD 1 /nD 0 ) ln (nD 2 /nD 1 ) .

D

The finite element method

39

Example 2.9: A posteriori error estimate for the constrained bar pullout example. Revisit Example 2.8 and use the strain energy values of the finest three discretizations to compute an approximation to the exact strain energy. Solution: For mesh sizes h0 = L/8, h1 = L/16, and h1 = L/32 (with L = 10 m), the strain energy values are, respectively, U h0 = 0.470 317 081 698 943 76 J, U h1 = 0.491 960 380 220 035 9 J, U h2 = 0.497 977 827 952 915 23 J, and the value Q = 1. Eq. (2.55) is now solved with a root-finding algorithm to yield U = 0.500 295 121 117 217 9 J, which has a 0.06% relative error with respect to the exact strain energy obtained in Example 2.8. 

2.2 The elastostatics problem in higher dimensions In the previous section we studied in detail the 1-D elastostatics boundary value problem. Starting from the differential equation that describes static equilibrium, we posed the problem in weak form and discretized it using finite element spaces. Along the way, we gained considerable insight that now helps us devise the finite element formulation in higher dimensions. As the reader will realize, the processes in 2-D and 3-D are remarkably similar although we give a slightly more formal presentation. Our aim is not to overcomplicate matters but to expose the reader to an alternative description that is normally encountered in published works. As in the previous section, we start with the strong form of the boundary value problem.

2.2.1 Strong form Consider the d-dimensional Euclidean vector space Rd (d = {2, 3}), spanned by an orthonormal basis {ei }. A point in Rd is represented by its coordinates in such basis as x = xi ei (summation convention).6 √ This space is equipped with the inner product structure that induces the norm v = v · v, ∀v ∈ Rd . Now assume a body  ⊂ Rd (see Fig. 2.12), which represents a continuum solid. The closure of the body is  and its boundary ∂ ≡  =  \  has outward unit normal vector field n = n(x). The boundary is divided into regions  ≡  D ∪  N ∪  R , where Dirichlet, Neumann, and Robin boundary conditions are prescribed, respectively. These regions are also disjoint, i.e.,  D ∩  N ∩  R = ∅. As we did earlier in § 2.1, we now determine the governing static equilibrium equation by simply looking at the forces acting in an infinitesimal volume, as illustrated in Fig. 2.12. For instance, the balance of forces in the x direction is σxx (x + dx) dy dz − σxx (x) dy dz + σxy (y + dy) dx dz − σxy (y) dx dz + σxz (z + dz) dx dy − σxz (z) dx dy + bx dx dy dz = 0, 6 In this book x ≡ x, x ≡ y, and x ≡ z; both notations will be used interchangeably. 1 2 3

40

Fundamentals of Enriched Finite Element Methods

Figure 2.12 Schematic of a 3-D domain  showing regions of the boundary with prescribed essential ( D ), natural ( N ), and mixed ( R ) boundary conditions. Also, forces in the x direction acting on an infinitesimal volume dx dy dz are also shown.

where σij is the (i, j )th component of Cauchy’s stress tensor σ :  → Rd × Rd , and bx the x component of the body force vector b :  → Rd , i.e., bx = b · e1 . Dividing by the infinitesimal volume dx dy dz and taking the limit as the volume vanishes, we obtain ∂σyx ∂σzx ∂σxx + + + bx = 0. ∂x ∂y ∂z Force balance equations in the y and z directions are obtained analogously, and the three equations that describe static equilibrium in d-dimensional space can be written compactly as ∇ ·σ + b = 0

∀ x ∈ ,

(2.57)

where ∇· is the divergence operator.7 The elastostatics boundary value problem in strong form consists of (2.57), together with boundary conditions u = u¯

on  D ,

σ · n = t¯

(2.58)

N

on  ,

(2.59)

σ · n = H · (u¯ − u) on  ,

(2.60)

R

7 At this point it seems appropriate to remind the divergence and gradient operations. Given a scalar s, a

vector v, and a tensor T , we have ∇ · v := ∇s :=

∂vi , ∂xi ∂s ei , ∂xi

∇ · T := ∇v :=

∂Tij ei ∂xj

(divergence),

∂vi ei ⊗ ej ∂xj

(gradient).

The finite element method

41

where u¯ and t¯ denote the prescribed displacement and tractions, respectively, and H :  R → Rd × Rd is a tensor that characterizes the elastic support. Eq. (2.59), known as Cauchy’s relation, relates the stress field to the tractions on the boundary. Contrary to the one-dimensional bar of § 2.1, in higher dimensions we can prescribe the three types of boundary conditions simultaneously. In fact, the actual boundary value problem can include any combination of boundary conditions (2.58)–(2.60). To complete the formulation, we adopt a linear constitutive relationship (generalized Hooke’s law) between stress σ and strain , i.e., σ = C, where C is a fourthorder tensor. We also assumesmall deformation theory so that the linearized strain  tensor can be written as  = 12 ∇u + ∇u .

2.2.2 Weak form As before we start by writing the weighted residual  w · (∇ · σ + b) d = 0

∀w ∈  .

(2.61)



For elastostatics in 2-D or 3-D we turn to vector-valued function spaces. We therefore define ' (d    () ≡ H 1 () = v : v(x) ∈ Rd ∀x ∈ ; vi ∈ H 1 () , i = 1 . . . d , (d  '  0 () ≡ H01 () = v ∈  () : v| D = 0 ,

(2.62) (2.63)

where H 1 is the first-order Sobolev function space on , defined earlier in § 2.1.2.1. Eq. (2.62) states that every component of the vector-valued function v ∈  belongs to H 1 . The subscript 0 in (2.63) emphasizes the fact that vector-valued functions from the space vanish on the region with prescribed essential boundary conditions. As with the 1-D bar studied earlier, we can lower the differentiability requirement on u by  balancing derivatives. To that effect, we use the identity w · (∇ · σ ) = ∇ · σ  · w − ∇w : σ ,8 where : denotes the contraction between two tensors, i.e., 8 Recalling the definition of the divergence of a tensor (footnote 7), we can write

σ  · w = σij ej ⊗ ei wk ek = σij wk δik ej = σij wi ej , and therefore   ∂ σ w  ∂σ ∂wi ij j ij  ∇ · σ ·w = = wi + σij . ∂xj ∂xj ∂xj       (∇·σ )·w

σ :∇w

42

Fundamentals of Enriched Finite Element Methods

S : T = Sij Tij (summation convention implied). Eq. (2.61) can be rewritten as9      ∇w : σ d = w · b d + ∇ · σ  · w d ∀w ∈  . 





By using the divergence theorem, the last integral over the volume is converted to an integral over the boundary,         ∇ · σ · w d = n · σ · w d = w · (σ · n) d ,

 





where the last equality was obtained using the transpose  of a tensor, i.e., for vectors v 1 , v 2 and a tensor T , then v 1 · (T · v 2 ) = v 2 · T  · v 1 . Since  ≡  D ∪  N ∪  R , the last integral is then split into the different boundary regions: 

 w ·(σ · n) d = 

N

w · t¯ d +





D

w ·R d +

R

w ·H ·(u¯ − u) d , (2.64)

with R denoting the reaction force on the Dirichlet boundary. By conveniently choosing w ∈ V0 the integral containing the reaction term is eliminated. Notice that, as in the 1-D case, the boundary conditions arise naturally after balancing the derivatives between trial and test functions. The weak form is finally expressed as: Find u ∈ 0 such that B(u, w) = L (w)

∀ w ∈ 0 () ,

(2.65)

with bilinear and linear forms given, respectively, by  B(u, w) =

 ∇w : σ d +

R



w · H · u d

(2.66)

and 

 w · b d +

L (w) = 

N

w · t¯ d +

 R

w · H · u¯ d .

(2.67)

In the case that the elastostatics BVP has non-homogeneous essential BCs, similarly to the 1-D case, we define the linear variety  := u˜ + 0 as a translation of ˜  D = u. ¯ Then the weak form reads: Given 0 by a function u˜ ∈  that satisfies u| ˜ find v ∈ 0 such that u = v + u, ˜ w) B(v, w) = L (w) − B(u,

∀ w ∈ 0 ().

(2.68)

9 Note that we have chosen to keep σ  though the stress tensor is symmetric, i.e., σ = σ  , which is the

result of the balance of angular momentum.

The finite element method

43

2.2.3 Principle of virtual work For simplicity, assume there is no Robin BC. Since the stress tensor is symmetric,  ∇w : σ = sym (∇w) : σ , where sym (∇w) := 12 ∇w + ∇w (notice the similarity with linearized strain). Then by taking sym (∇w) = δ and w = δu, (2.65) can be rewritten as    t¯ · δu d , σ : δ d = b · δu d + 

N



which is the well-known expression for the principle of virtual work.

2.2.4 Discrete formulation The Galerkin formulation is obtained by choosing finite-dimensional vector-valued function sets  h ⊂  and 0h ⊂ 0 so that the weak formulation reads: Find uh ∈  h such that B(uh , w h ) = L (wh )

∀ wh ∈ 0h ().

(2.69)

Selecting trial and test functions from the same space 0h —the Bubnov–Galerkin approach—yields a symmetric stiffness matrix. We now focus on the finite element discrete equations. The domain  is dis% E  ei and ei ∩ ej = ∅, ∀i = j cretized into nE finite elements so that h = int ni=1 (see Fig. 2.13). Contrary to the 1-D case and unless the geometry is relatively simple, in higher dimensions the discretized domain h ≈  due to geometry discretization error—this also extends to the domain boundary, so  h ≈ .

Figure 2.13 Discretization of domain h into finite elements showing the original geometry (dashed).

44

Fundamentals of Enriched Finite Element Methods

After choosing vector-valued finite-dimensional function spaces, typically using low-order polynomials, the trial and test functions can be written as uh = U and w h = W ,

(2.70)

" # where U = u1x u1y u1z u2x u2y u2z . . . unN x unN y unN z , i.e., coefficient vectors "stack up the x, y, and z #degrees of freedom of all nodes in a single vector, and  = ϕ1 I ϕ2 I . . . ϕnN I with I denoting the d × d identity matrix; in 3-D the latter is ⎡ ⎤ ϕ1 0 0 ϕ2 0 0 ϕnN 0 0 0 ϕnN 0 ⎦.  = ⎣ 0 ϕ1 0 0 ϕ2 0 · · · (2.71) 0 0 ϕnN 0 0 ϕ1 0 0 ϕ2 At element level, (2.70) is written as w h = ϕw, (2.72) uh = ϕu, # " where ϕ = ϕ1 I ϕ2 I . . . ϕn I is now a d × nd matrix (n being the number of nodes in the element). For a particular element e (with boundary ∂e), the static equilibrium is expressed as    h h ∇w : σ de = w · b de + w h · (σ · ne ) d∂e ∀ wh ∈ 0h , (2.73) e

e

∂e

where ne is the unit normal to the element boundary and therefore σ · ne = t e is the traction field on the element boundary. By considering  = Bu, with B denoting again the strain–displacement matrix (whose shape is yet to be determined), (2.73) can be rewritten as

 

 w

B  CB de u = w ϕ  b de + ϕ  t e d∂e e ∂e      

∀ w ∈ Rn×d . (2.74)

e

k

f

And since this equation has to hold for any w ∈ Rn×d , the local equilibrium at element level is ku = f . Following the assembly procedure discussed at the end of § 2.1.4, the final discrete equation that describes the static equilibrium of the entire finite element assembly is again KU = F where K and F are obtained via (2.43). As was the case for the 1-D discretization, element tractions t e between neighboring elements will be canceled, leaving only those at the boundary of the discretized domain. As a result, the last term in (2.74) will be nonzero only when ∂e ∩  N = ∅ and/or ∂e ∩  R = ∅.

2.2.5 Voigt notation The only remaining piece of the puzzle is to determine how to compute local element arrays k and f . Notice that in (2.73) we deal with the contraction of two second-order

The finite element method

45

tensors, and that stress is related to strain via a fourth-order tensor C. To simplify the formulation, we use Voigt notation to take advantage of symmetry of tensors so we can reduce their rank. This allows us to use a second-order tensor (written as a matrix) to represent the fourth-order elasticity tensor, and first-order tensors (written as vectors) to represent stress and strain. This transformation is made such that the energy σ :  remains the same. In Voigt notation, stress and strain are " # σ = σxx σyy σxy , ( " # ' ∂uy ∂uy ∂ux x  = xx yy γxy = ∂u + ∂x ∂y ∂x ∂y for 2-D, and # " σ = σxx σyy σzz σyz σxz σxy , " #  = xx yy zz γyz γxz γxy ' ∂uy ∂uy ∂uz ∂uz ∂uz x = ∂u ∂x ∂y ∂z ∂y + ∂z ∂x +

∂ux ∂z

∂uy ∂x

+

∂ux ∂y

(

for 3-D. It can be easily verified that σ :  = σij ij yields the same result as σ ·  when the latter are expressed in Voigt notation as vectors. At element level, the strain is  = Bu where now the strain–displacement matrix is " # B =  ϕ1  ϕ2 . . .  ϕn ≡  ϕ, (2.75) where the differential operator  is given by ⎡∂ ⎡

∂ ∂x

⎢ ≡⎣ 0

∂ ∂y

0

∂x

⎢0 ⎢ ⎢0 ⎢ ≡⎢ ⎢0 ⎢∂ ⎣ ∂z

⎤

∂ ⎥ ∂y ⎦ ∂ ∂x

and

∂ ∂y

⎤ 0 0⎥ ⎥ ∂ ⎥ ∂z ⎥ ∂ ⎥ ∂y ⎥ ∂ ⎥ ⎦ ∂x 0

0 ∂ ∂y

0 ∂ ∂z

0 ∂ ∂x

(2.76)

for 2-D and 3-D, respectively.10 The relationship between stress and strain, assumed linearly isotropic elastic herein, is therefore given by σ = C. Since we now use Voigt notation, the fourth-order constitutive tensor is replaced by   E 0 (1 − ν)I + ν1 C= (2-D plane stress), 1−ν 0 1 − ν2 2 10 For an n-node element in 2-D, for instance, the strain–displacement matrix is

⎡

∂ϕ1

0

∂ϕ1 ∂y

∂ϕ1 ∂y ∂ϕ1 ∂x

⎢ ∂x B =⎢ ⎣ 0

which is a 3 × 2n matrix.

∂ϕ2 ∂x

0 ∂ϕ2 ∂y

0 ∂ϕ2 ∂y ∂ϕ2 ∂x

···

∂ϕn ∂x

0 ∂ϕn ∂y

⎤

0

∂ϕn ⎥ ⎥ ∂y ⎦ , ∂ϕn ∂x

46

Fundamentals of Enriched Finite Element Methods

C= C=

 E (1 − 2ν)I + ν1 0 (1 + ν)(1 − 2ν) (1 − 2ν)I + ν1 E (1 + ν)(1 − 2ν)

0



1−2ν 2



0

0  1−2ν I 2

(2-D plane strain),

(3-D),

where I and 1 are the d × d identity and unity matrices, respectively. Noting the Lamé Eν E and μ = G = 2(1+ν) (shear modulus), the latter can be writconstants λ = (1+ν)(1−2ν) ten as   λ1 + 2μI 0 C= . (2.77) 0 μI

2.2.6 Isoparametric formulation in higher dimensions Similarly to the one-dimensional case, in higher dimensions an isoparametric formulation is also used since it eases the formulation of shape functions and also numerical integration. We look at shape functions of widely-used elements in 2-D, noting that extrapolating to 3-D is straightforward.   For the triangle = ξ = (ξ, η) ∈ R2 : ξ ≥ 0 ∧ η ≥ 0 ∧ ξ + η ≤ 1 , the linear shape functions are (see left column in Fig. 2.14) ϕ1 = 1 − ξ − η,

ϕ2 = ξ,

ϕ3 = η.

(2.78)

  For the bilinear quadrangle = ξ = (ξ, η) ∈ [−1, 1] × [−1, 1] ⊂ R2 , the shape functions are obtained as the product of the 1-D linear functions in (2.46) in two orthogonal directions, i.e., ϕi (ξ ) ϕj (η) , i, j = {1, 2}: 1 ϕ1 = (1 − ξ )(1 − η), 4 1 ϕ3 = (1 + ξ )(1 + η), 4

1 ϕ2 = (1 + ξ )(1 − η), 4 1 ϕ4 = (1 − ξ )(1 + η). 4

(2.79)

These bilinear shape functions are also shown in Fig. 2.14. Functions in (2.78) and (2.79) (and those in (2.37) for 1-D) have two important properties: 1. Kronecker-δ property. The function ϕi is equal to unity at the location of its associated node  and  zero at the location of every other node—mathematically expressed as ϕi x j = δij . A direct consequence of this property is that the DOF associated with shape function ϕi physically represents the displacement at x i . 2. Partition of unity property. $ At every coordinate x in the domain, all shape functions add up to unity, i.e., i ϕi (x) = 1, ∀x ∈ h . Example 2.10: 2-D shape functions. Noting the Kronecker-δ property above, derive the 2-D shape functions in Eqs. (2.78) and (2.79).

The finite element method

47

Figure 2.14 Master element used for a 2-D isoparametric formulation. Both the triangular (left) and quadrangular (right) elements are shown.

Solution: Noting that the distance to a line is zero for every point on it, we then use lines passing through nodes to derive the shape functions. In other words, for a node ξ i , its associated shape function is derived by looking at lines that pass through other element nodes. Constant strain triangle: For the first node ξ 1 , we use a line that passes through nodes ξ 2 and ξ 3 . This line is shown in Fig. 2.15 as ξ + η = 1. Therefore the first shape function for the triangle is ϕ1 (ξ, η) = 1 − ξ − η. For the second node ξ 2 , the function ξ = 0 passes through nodes x 1 and x 3 . Therefore the second shape function is ϕ2 (ξ, η) = ξ.

Figure 2.15 Master element showing lines passing through nodes.

48

Fundamentals of Enriched Finite Element Methods

The last shape function is found analogously by using the line η = 0, so ϕ3 (ξ, η) = η. Noteworthy, these functions satisfy the Kronecker-δ property without the need for scaling to ensure a unity value at the location of their associated nodes. Bilinear quadrangle: Taking the first node ξ 1 = (−1, −1), the two lines that pass through all other nodes are ξ = 1 and η = 1. Using these, the first shape function is ϕ1 = C(1 − ξ )(1 − η),

  where the constant C = 1/4 satisfies ϕ1 ξ 1 = 1. The other shape functions are obtained similarly. All quadrangle shape functions can be written compactly as ϕi (ξ, η) =

1 (1 + ξi ξ )(1 + ηi η), 4

i = {1 . . . 4} ,

where (ξi , ηi ) is the coordinate of the ith node on the master element.



With the shape functions in parent coordinates, an isoparametric formulation is then used to represent the trial solution, the test function, and the geometry: uh = ϕu

w h = ϕw,

x = ϕX,

(2.80)

# " where X = x1 y1 x2 y2 . . . xn yn , and ϕ is a d × nd matrix of shape functions (see discussion after (2.72)). The derivatives of functions that depend on the master coordinate ξ , with respect to global coordinates, are obtained by applying the chain rule. For a function g = g(ξ, η), for instance: ∂g ∂g ∂x ∂g ∂y = + , ∂ξ ∂x ∂ξ ∂y ∂ξ ∂g ∂g ∂x ∂g ∂y = + , ∂η ∂x ∂η ∂y ∂η

(2.81) (2.82)

which can be written in matrix form as ∂g

∂y ∇ξ g =

∂ξ ∂g ∂η

= 

∂x ∂ξ ∂x ∂η



∂ξ ∂y ∂η



∂g ∂x ∂g ∂y

= J ∇ x g,

(2.83)

J

where J is the Jacobian matrix of the geometry mapping. Notice that with this definition J = ∇ ξ x  (see footnote 7). The derivatives of this function with respect to global coordinates are then simply determined as ∇ x g = J −1 ∇ ξ g. Noteworthy, the ratio between the area of the physical element to that of the master element is given by j = det (J ), which is exactly the last missing piece to transform the integrals for numerical quadrature. Given a function f (x), its integral over master element , for instance, is transformed as 

 f (x) de = e

f (x(ξ ))j dξ =

nGP     f x ξ i j γi . i=1

(2.84)

The finite element method

49

Local arrays in (2.74) are then computed as  k e = B  CBj dξ ,    f e = ϕ bj dξ + ϕ  t¯ j d∂ξ ,

(2.85)

∩ N

where the derivatives in the strain–displacement matrix are computed considering the mapping.11 A more thorough discussion on the computer implementation of FEM is given in Chapter 12.

2.3 Heat conduction We turn our attention to heat conduction, which is another elliptic differential equation whose discrete formulation is in many ways simpler than that of elastostatics. Fourier’s law of heat conduction states that the rate of heat transfer through a solid body is proportional to the negative gradient in the temperature. This is expressed mathematically as q = −κ

du , dx

(2.87)

where q is the heat flux, κ the thermal conductivity, and u the temperature.12 We can obtain the differential equation for the steady state heat conduction problem in 1-D by following a similar approach to that of § 2.1. Considering a bar of constant cross-sectional area A, the heat that flows into an infinitesimal bar volume Ax is q(x)A. Similarly, the heat that flows out is q(x + x)A. Given a heat source f per unit volume, conservation of heat implies that q(x + x)A − q(x)A = f Ax. Dividing by Ax and taking the limit yields dq q(x + x) − q(x) −f = − f = 0. x→0 x dx lim

11 Considering a 2-D problem, for instance, the strain–displacement matrix is computed as

⎡ "

B =  ϕ1

...

#

⎤

∂ϕ1

0

∂ϕ1 ∂y

∂ϕ1 ∂y ∂ϕ1 ∂x

⎢ ∂x =⎢ ⎣ 0

⎡ ∇ x ϕ1 · e x ⎥ ⎥ · · ·⎦ = ⎣ 0 ∇ x ϕ1 · e y

0 ∇ x ϕ1 · e y ∇ x ϕ1 · e x

⎤ · · ·⎦

(2.86)

" # " # where ex = 1 0 , ey = 0 1 , ∇ x ϕ1 = J −1 ∇ ξ ϕ1 , and J −1 is the inverse of the Jacobian matrix. 12 We use the same symbol used earlier for displacements. Thus, u must be understood as the primal variable of the boundary value problem.

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Fundamentals of Enriched Finite Element Methods

Therefore, steady-state heat conduction in 1-D is governed by   d du κ +f =0 ∀ x ∈ , dx dx

(2.88)

which is known as Poisson’s equation. In the case of no source term and constant conductivity, (2.88) simplifies to Laplace’s equation, i.e., ∇ 2 u = 0. The boundary value problem is fully determined after prescribing two boundary conditions out of the following three: • Given the prescribed temperature u¯ :  D → R, the essential or Dirichlet BC is ¯ u| D = u. • With prescribed heat flux q¯ :  N → R, the natural or Neumann BC is ¯ −q| N n = q, where n is again either −1 or 1 for the left and right end, respectively. Our sign convention then assumes that heat flux is positive when heat enters the bar and negative otherwise. • Finally, the mixed, Robin or in this case convective boundary condition is −q| R n = H (u∞ − u), where H is the heat transfer coefficient and u∞ the ambient temperature. Example 2.11: Heat conduction weak formulation in 1-D. Use the method of weighted residuals to obtain the weak formulation for a heat conduction problem in 1-D. Consider a prescribed temperature u = u¯ at x = 0 and a convective BC −q| R = H (u∞ − u) at x = L. Solution: As done previously for elastostatics, the weighted residual is obtained by multiplying (2.88) with a weight function and integrating over the domain of interest  = [0, L]:    L   du d κ + f dx = 0 ∀ w ∈ V0 . w dx dx 0 Integration by parts on the first term balances the derivatives:     L  L d dw du du du L w κ κ dx = wκ  − dx dx dx dx 0 dx dx 0 0  L dw du κ dx . = w(L)H (u∞ − u(L)) − dx dx 0 Note that since w ∈ V0 , the reactive flux term drops out. Then the weak form is: Given a function u˜ that satisfies the Dirichlet BC u| ˜  D = u, ¯ find v ∈ V0 , v = u − u˜ such that B(v, w) = L (w) − B(u, ˜ w) where B(v, w) =

 L 0

κ

∀ w ∈ V0 ,

dw dv dx + w(L)H v(L), dx dx

(2.89)

(2.90)

The finite element method

L (w) =

 L

51

wf dx + w(L)H u∞ .

0

(2.91)

Notice that the Robin boundary condition arises naturally after integrating by parts. In Euclidean space Rd , we consider an open domain  ⊂ Rd , with



closure . The boundary of the domain is ∂ ≡  and has outward unit normal n; the boundary is comprised of disjoint regions  D ,  N , and  R . Given the thermal conductivity tensor κ :  → Rd × Rd , the heat source f :  → R, prescribed temperature u¯ :  D → R, prescribed heat flux q¯ :  N → R, ambient temperature u∞ , and heat transfer coefficient H :  R → R, find the temperature field u ∈ C 2 such that κ ∇u) + f = 0 ∇ · (κ

in ,

(2.92)

with boundary conditions u = u¯

on  D ,

κ ∇u · n = q¯

(2.93)

N

(2.94)

R

(2.95)

on  ,

κ ∇u · n = H (u∞ − u)

on  .

To obtain the weak form of (2.92), we consider the linear variety V := u˜ + V0 so that u = u˜ + v satisfies a priori the essential BC (2.93). The weak form of the heat conduction BVP is: Given u˜ ∈ V , find v ∈ V0 , v = u − u˜ such that B(v, w) = L (w) − B(u, ˜ w)

∀ w ∈ V0 ,

with bilinear and linear forms given by   κ ∇v) d + B(v, w) = ∇w · (κ  L (w) =

(2.97)

R



and

H wv d

(2.96)

 wf d +



 w q¯ d +

N

R

H wu∞ d ,

(2.98)

respectively. The Galerkin statement follows by choosing finite-dimensional function sets V h ⊂ V and V0h ⊂ V0 : Find uh ∈ V h := u˜ + V0h such that B(uh , w h ) = L (w h )

∀ w h ∈ V0h .

(2.99)

A Bubnov–Galerkin formulation is then obtained by choosing trial and test functions from the same space V0h . To obtain the finite element discrete equations, we apply the discrete formulation to elements of the domain discretization h . Let the trial and test functions within the eth element be given by uh = ϕu and w h = ϕw, respectively. Inserting these into (2.99) leads to the local arrays:    k e = B κ B de + H ϕ  ϕ d∂e , e  R ∩∂e    (2.100) ϕ  q¯ d∂e + ϕ  H u∞ d∂e , f e = ϕ  f de + e

 N ∩∂e

 R ∩∂e

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Fundamentals of Enriched Finite Element Methods

" where B =  ϕ1 tor is simply ≡

'

∂ ∂x

...

∂ ∂y

(

#  ϕn ≡  ϕ and for heat conduction the differential opera-

and  ≡

'

∂ ∂x

∂ ∂y

∂ ∂z

(

,

(2.101)

for 2-D and 3-D, respectively. Notice that, because" of the structure of the # differential operator, this matrix can also be written as B = ∇ x ϕ1 . . . ∇ x ϕn ≡ ∇ x ϕ. The equations in (2.100) also contain the contributions of the boundary conditions, though it is understood such integrals will be nonzero only if the element has a side that belongs to the respective domain boundary, i.e., if  N ∩ ∂e = ∅ or  R ∩ ∂e = ∅. Since an isoparametric formulation is commonly used to compute element local arrays, (2.100) is calculated as   H ϕ  ϕj d∂ξ , k e = B  κ Bj dξ + R ∩    (2.102)  f e = ϕ fj dξ + ϕ  qj ¯ d∂ξ + ϕ  H u∞ j d∂ξ , ∩ N

∩ R

where " B = J −1 ∇ ξ ϕ1

J −1 ∇ ξ ϕ2

...

# J −1 ∇ ξ ϕn ≡ J −1 ∇ ξ ϕ,

(2.103)

j = det (J ), and J −1 is the inverse of the Jacobian matrix given by (2.83). After assembling the contribution of all elements, i.e., K = Ae k e and F = Ae f e , the global degree of freedom vector U is obtained by solving the discrete system of linear equations KU = F (see computational aspects in Chapter 12).

2.4 Problems PROBLEM 2.1.— BAR WITH VARYING CROSS SECTION Consider a 1-D bar with varying cross section as shown in Fig. 2.16. The cross-sectional area varies linearly from A = A0 at x = L to A = 7A0 at x = 0. The bar has a prescribed displacement u = 0 at x = 0 and a prescribed force N¯ at x = L.

Figure 2.16 Bar with varying cross-section. 1. 2. 3.

Determine the exact solution. With the exact solution, compute the exact value of the strain energy. Solve the problem using h-FEM with an increasing number of elements (for instance, nE = {2, 4, 8, 16}) and compare the solution with respect to the analytical solution.

The finite element method

4.

53

Using the exact value of the strain energy, compute the approximated strain energy and create a convergence plot: Make a log–log plot of the relative error in energy norm ε¯ as a function of mesh size h.

FIRST-ORDER SHEAR DEFORMATION THEORY BEAM PROBLEM 2.2.— First-order shear deformation theory (FSDT), contrary to classical beam theory, accounts for shear deformation. While in classical beam theory the beam angle of rotation θ is related to the transverse displacement w as θ = dw / dx, in FSDT the rotation angle is approximated independently from the transverse displacement. However, FSDT is not usually used to model thin beams since the formulation suffers from shear locking: As the thickness of the beam reduces, the formulation introduces inaccurate shear strain energy that results in spurious stiffness and thus in overly stiff models. Consider in Fig. 2.17 a cantilever beam of length L subjected to a uniformly distributed load q. The beam has cross-sectional area A = BH , moment of inertia I , and Young’s and shear moduli E and G, respectively. The strong form of this problem reads: Given the beam’s geometric and material properties, and the distributed load q, find the transverse displacement w and the rotation angle θ such that   d2 dθ EI =q in , dx dx 2     dθ d dw EI + Ks GA − θ = 0 in , dx dx dx

(2.104)

with boundary conditions on the primal variables w(0) = 0, θ(0) = 0. With M = −EI κ denoting the bending moment and Q = −Ks GAγ the shear force, the boundary conditions on the dual variables at the end of the beam are Q(L) = 0 and M(L) = 0; κ = dθ / dx denotes the curvature, γ = −θ + dw / dx the shear strain, and Ks the shear correction factor (5/6 for rectangular cross-section).

Figure 2.17 Cantilever beam of length L and cross sectional area A = BH . The beam is loaded with a uniformly distributed load q. Given the strong form above, 1. 2.

3.

Determine the exact solution of (2.104); ˆ the trial (test) functions for the transverse displacement and the angle of Considering w, θ (w, ˆ θ) ˆ their corresponding spaces—state the weak form for this rotation, respectively—and W ,  (Wˆ , ) problem and the expressions for bilinear and linear forms. Show that by setting wˆ = δw, θˆ = δθ, the weak form of equilibrium is equivalent to the principle of virtual work, given by  L 0

(Mδκ + Qδγ ) dx −

 L 0

qδw dx = 0.

(2.105)

54

Fundamentals of Enriched Finite Element Methods

PROBLEM 2.3.— BOUNDARY VALUE PROBLEM WITH u ∈ C ∞ Consider the differential equation   3 d2 u 2 a + a b(b − x + 1) − 2 = , 2  dx a 2 b2 + 1

(2.106)

with boundary conditions u (0) = u (1) = 0, where a is constant and b = x − xb . The solution to this boundary value problem is given by u(x) = (1 − x)(arctan (ab) + arctan (axb )). This solution has a peak centered near x = xb whose sharpness depends on the value of a. For large a, the solution has a “near discontinuity” while for small a the solution is very smooth. Taking xb = 0.2, Fig. 2.18 shows the exact solution and its derivative for a = {0.5, 50}.

Figure 2.18 Exact solution and its derivative for problem (2.106) considering a = {0.5, 50}. 1.

2. 3.

Solve problem (2.106) for xb = 0.2 and a = 0.5. Use 2, 4, 8, 16, and 32 evenly spaced linear and quadratic Lagrangian elements (10 runs total). Make a log–log plot of the relative error in the energy norm in ordinates, defined as ! U − Uh ε¯ = , U versus the mesh size h and another one versus the number of degrees of freedom nD (2 plots and 4 curves total). The exact strain energy for this problem is U = 0.040 877 754 793 807 99. Calculate the rates of convergence in the energy norm for both element types and indicate them in the convergence plots. How do the computed convergence rates compare with theoretical values, knowing that the solution is very smooth? Using the data points (¯ε and nD ) from the h-version and the a posteriori error estimate, evaluate the value of the exact strain energy. Compare this with the exact value. Solve (2.106) for xb = 0.2 and a = 50 and repeat the above h-version convergence study for only the quadratic element case. This time use 5, 10, 20, and 40 elements evenly spaced. Make a log–log plot of ε¯ as a function of h and compute the rate of convergence. If the plot is not linear, use the last two data points to compute the (asymptotic) rate of convergence. The exact strain energy, U , for this problem is U = 25.138 142 068 251 32.

PROBLEM 2.4.— MATERIAL DISCONTINUITY Consider the boundary value problem   d du − EA + Cu = T (x), 0 < x < L, dx dx

(2.107)

The finite element method

55

with BCs u(0) = 0 and u(L) = 1. The bar consists of two materials with elastic moduli E1 = 10 000 and E2 = 1000. The length of the bar is L = 10 and the cross-section is A = 1. The material interface is located at x = L/2. The bar is subjected to a distributed force per unit length given by T (x) = 25x −

15 2 1 3 x + x . 2 2

The solution to this problem is given by [5]

1 (E2 Bx + g(x)) , u(x) = E1 B(x − L) + 1 + E1 (g(x) − g(L)), 2

(2.108)

x ≤ x , x ≥ x ,

(2.109)

with the constant B and the function g(x) given by B=

E1 E2 − g (x ) (E2 − E1 ) − g(L)E1 25 5 1 and g(x) = − x 3 + x 4 − x 5 , E2 ((E2 − E1 )x + LE1 ) 6 8 40

respectively. Fig. 2.19 shows the solution (2.109) and its derivative, respectively.

Figure 2.19 Exact solution and its derivative for problem (2.107) subjected to load (2.108). 1.

Solve the boundary value problem using the h-version of FEM and uniform meshes. Use (i) a sequence of meshes with an even number of elements, and (ii) a sequence of elements with an odd number of elements. In case (i) the material discontinuity is at a node of the mesh, while in case (ii) it is at the center of an element. Note that in the latter, the stiffness matrix of the element containing the material interface must be integrated with the right material properties. Use linear and quadratic elements. For each element type used, create a convergence plot of the relative error in the energy norm versus the number of DOFs. The exact energy can be computed from Eq. (2.109) as      L  x  E1 du 2 E2 du 2 dx + dx , U (u(x)) = 2 dx dx x 2 0 or alternatively, you may use a posteriori error estimation. Calculate the rate of convergence for both element types and sequences of meshes in (i) and (ii).

3-D SHAPE FUNCTIONS FOR LOW-ORDER ELEMENTS PROBLEM 2.5.— Following Example 2.10, derive the shape functions for the 4-node linear tetrahedron and the 8-node trilinear hexahedron.

References [1] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall International, Englewood Cliffs, NJ, 1987. [2] B.A. Szabó, I. Babuška, Finite Element Analysis, John Wiley and Sons, New York, 1991.

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Fundamentals of Enriched Finite Element Methods

[3] O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, Oxford, 2013. [4] S. Brenner, R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, Springer, New York, 2007. [5] D.W. Kim, W.K. Liu, Y.-C. Yoon, T. Belytschko, S.-H. Lee, Meshfree point collocation method with intrinsic enrichment for interface problems, Computational Mechanics 40 (6) (2007) 1037–1052, https://doi.org/10.1007/s00466-007-0162-1.

The p-version of the finite element method

3

The previous chapter introduced the h-version of the finite element method, where the interpolation in each finite element is a low-order polynomial (linear or quadratic at best) and thus the approximation is improved by refining the mesh. This is the method of choice adopted by the industry not only because it is widely available in commercial software, but also because having an accurate meshed representation of the geometry often necessitates fine finite element meshes. For a given finite element mesh, the approximation can also be improved by increasing the polynomial order p of the interpolant in each finite element. This is the idea behind the p-version of the finite element method [1,2], which is the subject of this chapter. p-FEM exhibits exponential rate of convergence for smooth problems, so the method can be more cost-effective when compared to h-FEM for certain classes of problems, i.e., for a target accuracy p-FEM may require substantially less DOFs than h-FEM. In p-FEM, the linear shape functions studied in Chapter 2—which form a partition of unity (PoU)—are augmented hierarchically with nonlinear polynomial shape functions to build the approximation space. As such, p-FEM can be seen as an enriched FEM, whereby the linear h-FEM space is augmented with nonlinear polynomial enrichments. Because e-FEM can in turn be seen as a generalization of p-FEM for non-polynomial hierarchical finite element spaces, a deep understanding of p-FEM can thus serve as a strong foundation for learning and developing the methods described in later chapters. The second author was in fact inspired by p-FEM during his early work on PoU methods, as discussed in Appendix A. As in the previous chapter, we start describing the method in 1-D to gain insight before exploring it in higher dimensions. Because p-FEM relies on building polynomial approximations of increasing polynomial order, much emphasis is placed on the choice of basis functions. We will see that orthogonal polynomials are the preferred choice since they yield more stable system matrices as the polynomial order increases. A priori error estimates are also discussed. As in Chapter 2, we showcase the method on several examples.

3.1 p-FEM in 1-D We start by recalling the weak form of our 1-D bar elastostatics boundary value problem: Find u ∈ E such that B(u, w) = L (w)

∀w ∈ E ,

Fundamentals of Enriched Finite Element Methods. https://doi.org/10.1016/B978-0-32-385515-0.00010-6 Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved.

(3.1)

58

Fundamentals of Enriched Finite Element Methods

where L

 B(u, w) =

EA 0

 dw du + swu dx dx dx

and 

L

L (w) = 0

 du L wb dx + wEA  , dx 0

and where the last term of the linear form L (w) is yet to be defined according to the actual boundary conditions of the problem (see discussion in § 2.1.2 after Eq. (2.17)). In order to solve the finite-dimensional form of (3.1), for which we will follow a Bubnov–Galerkin procedure as outlined in § 2.1.4, we now focus our attention to building an appropriate finite element space of arbitrary polynomial order, i.e., a basis for E p ⊂ E . The choice of polynomial basis plays a fundamental role in the approximating properties of the method. Consider in Fig. 3.1 two polynomial bases in the interval ξ ∈ [−1, +1]. For the basis ξ p , p = {1, . . . , 9}, the functions become increasingly similar for the higher polynomial degrees—this also holds for only even or only odd orders when ξ < 0. The inability of the basis to introduce new features as the polynomial order is increased translates in practice into stiffness matrices that become increasingly ill-conditioned. Therefore, such basis is of little to no use. A suitable basis function space for p-FEM can be built using orthogonal polynomials,1 since these produce better conditioned systems as the polynomial order of the approximation increases when compared to other bases. One such basis, also shown in Fig. 3.1, consists of Legendre polynomials Pp (ξ ). These polynomials can be calculated using the following recursive formula: Pp (ξ ) =

(2p − 1)ξ Pp−1 (ξ ) + (1 − p)Pp−2 (ξ ) , p

(3.2)

with P0 (ξ ) = 1 and P1 (ξ ) = ξ . However, before using the Legendre polynomials to build our basis, we note that the stiffness matrix involves the shape functions’ derivatives; therefore, it is more appealing to have Legendre polynomials as the derivatives of the shape functions instead, in order to produce stiffness matrices that are better conditioned. In other words, it is desired that the shape functions satisfy  +1 ∂ϕi ∂ϕj dξ = δij . −1 ∂ξ ∂ξ We build our basis by augmenting the linear hat functions used in h-FEM. To wit, ϕ1 =

1−ξ , 2

ϕ2 =

1+ξ , 2

1 Two polynomials u, v are orthogonal on  if



u, v =



uv dx = 0.

ϕi = ϕˆ i−1 ,

i = 3, 4, . . . , p + 1,

(3.3)

The p-version of the finite element method

59

Figure 3.1 Polynomial bases on the interval ξ ∈ [−1, +1] for degrees up to p = 9. Monomial basis ξ p is shown on the left, whereas Legendre polynomials Pp (ξ ) are shown on the right.

Figure 3.2 Hierarchical shape functions based on Legendre polynomials for 1 ≤ p ≤ 8 (left) and their derivatives (right).

where   1 ϕˆj (ξ ) = √ Pj (ξ ) − Pj −2 (ξ ) . 4j − 2

(3.4)

These shape functions, together with their derivatives, are shown in Fig. 3.2. Then p+1 our finite-dimensional high-order polynomial space is E p = span {ϕi }i=1 , with ϕi given by (3.3). For an approximation of order p, the space is spanned by p + 1 basis functions—the two linear shape functions used in h-FEM are kept intact. Therefore, we could see this way of building the finite element space as improving a lower-order polynomial interpolant hierarchically. With the space E p defined, we now choose both the trial solution and the test functions from it, following a Bubnov–Galerkin

60

Fundamentals of Enriched Finite Element Methods

approach; uh , w h ∈ E p can be written as   uh = ϕi ui = U and w h = ϕi wi = W , i∈N

(3.5)

i∈N

where N denotes the index set of nodes associated not only with linear shape functions, but also with generalized nodes associated with the higher-order nonlinear shape functions. Notice that the same matrix notation is used as in h-FEM (see § 2.1.4), with the difference that now the array  contains linear and nonlinear shape functions (and thus U , W have generalized coefficients associated with nonlinear shape functions as well). Inserting (3.5) into (3.1) yields      B ∀ wi ∈ R, i = 1, . . . , |N|, ϕi wi , ϕj uj = L ϕi wi j ∈N

i∈N

i∈N

which can be written in matrix form as ∀ W ∈ R|N| ,

W (KU − F ) = 0

where K and F are the global stiffness matrix and force vector, respectively, and U is the global degree of freedom vector. For W = 0, we therefore must have KU = F , which is the discrete system of linear equations that describes static equilibrium. As in h-FEM, global arrays are calculated by assembling the contribution of local

E E element arrays, i.e., K = Ane=1 k e , F = Ane=1 f e . For a given element e = xi , xj , let ϕ be the array that only contains the nonzero shape functions acting in the element and B = dϕ dx the strain–displacement matrix. Then the local arrays are computed as  ke =

xj

xi

 fe =

xj

xi





EAB B + sϕ ϕ dx ∼ =



nGP  

EAB B + sϕ ϕ γi j,

(3.6)

i=1

ϕ b dx ∼ =

nGP   ϕ b γi j,

(3.7)

i=1

where we prescinded from terms related to boundary conditions for simplicity. We also note that j = dx / dξ is still the Jacobian from the linear mapping of the geometry given by Eq. (2.46). Therefore, here we adopt a subparametric formulation for p ≥ 2. Finally, we evaluate the integrals numerically by sampling the integrands at nGP Gauss quadrature points ξi (with corresponding weights γi ) on the master element eˆ = [−1, 1]. Noteworthy, the number of Gauss points nGP (which is not necessarily the same for both (3.6) and (3.7)), needs to be chosen with care to ensure exact integration of all terms. If only the first term in the stiffness matrix (3.6) is present, then for a given polynomial interpolation of order p and assuming constant EA, the

The p-version of the finite element method

61

highest-order term in the matrix has order p as well. Therefore, nGP = p would integrate exactly the highest-order term (and thus every other term). If both terms in the stiffness matrix are present, then the highest-order term is of order 2p and then nGP = p + 1. For the computation of the force vector (3.7), a similar reasoning yields nGP = p. The recursive nature of (3.2) makes the chosen basis appealing for computer implementation since increasing polynomial orders can reuse already computed quantities. Algorithm 3.1 gives pseudocode to compute the shape functions given by (3.3) and their derivatives. The function HIERARCHICAL receives an input coordinate ξ and a polynomial order p, and uses the LEGENDRE function as a building block. The latter computes the Legendre polynomial for a given master coordinate and polynomial order recursively. Noteworthy, this algorithm works for arbitrary order. Algorithm 3.1 Legendre polynomials and hierarchical shape functions. function LEGENDRE(ξ, p) if p = 0 then return 1 else if p = 1 then return ξ else return p1 [(2p − 1) ξ LEGENDRE(ξ, p − 1) + (1 − p) LEGENDRE (ξ, p − 2)] function HIERARCHICAL (ξ, p) 

 1+ξ , 1−ξ ϕ, dϕ ← − 12 dξ 2 2

1 2



if p ≥ 2 then for j ∈ {2. . . p} do  1 ϕ ← ϕ √4j −2 (LEGENDRE(ξ, j ) − LEGENDRE(ξ, j − 2))    dϕ 2j −1 dϕ LEGENDRE(ξ, j − 1) dξ ← dξ 2

 return ϕ, dϕ dξ

Example 3.1: p-FEM solution to constrained bar pullout problem. Use p-FEM to solve Example 2.8 of Chapter 2. With a fixed mesh of four equally-sized finite elements (h/L = 0.25, with L = 10), compute finite element solutions with polynomial orders p = {1, 2, 4, 8, 16}. Create a convergence plot of the relative error in energy norm ε¯ ≡ u − uh E /uE , as a function of the total number of degrees of freedom nD . Compare these results with those obtained by h-FEM in Example 2.8 (notice they need to be expressed as a function of DOFs instead of mesh size). In addition, compare the condition number for both methods, computed as κ(K) ≡ cond (K) :=

λmax , λmin

(3.8)

where λmax and λmin are the maximum and minimum (nonzero) eigenvalues of the stiffness matrix, respectively.

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Solution: The stiffness matrix for the eth element is   nGP   ke = B B + ς 2 ϕ ϕ dx = B B + ς 2 ϕ ϕ γi j, e

(3.9)

i=1

where nGP = p + 1 since the highest-order term in the matrix has order 2p. Notice that, putting aside the number of integration points, Eqs. (2.53) and (3.9) are identical. As in Example 2.8 we prescribe boundary conditions u (L) = −0.000 045 399 9 = U5 and F1 = −1. With this DOF numbering, then linear DOFs take precedence over quadratic DOFs, quadratic over cubic, and so on, giving the linear system KU = F a hierarchical structure. Results are summarized in Fig. 3.3, where the relative error in energy norm ε¯ as a function of DOFs for both h-FEM and p-FEM is first shown on the left. Incidentally, the mesh sizes h = {4, 8, 16, 32, 64} used in Example 2.8 result in the same number of total DOFs as the polynomial order sequence p = {1, 2, 4, 8, 16} used here. The figure shows a remarkable gain in accuracy by using p-FEM. In fact, notice that the rate of convergence increases with the interpolant’s polynomial order p, indicating exponential convergence—as opposed to the algebraic convergence of h-FEM where the rate remains constant with increasing DOFs. Notice also that in the last segment (shown dotted in the figure) the rate actually decreases. This has to do with the fact that machine precision has been reached (taking the square root of a floating point number actually chops off accuracy by half).

Figure 3.3 Comparison between h-FEM and p-FEM for the constrained bar pullout problem. Results show the relative error in energy norm (left) and the condition number of the stiffness matrix (right) as a function of the total number of DOFs nD . The figure also shows the condition number of the global stiffness matrix as a function of DOFs for   both methods. For this 1-D problem, the condition number of h-FEM is κ(K) = O nD 2 (or κ(K) = O h−2 with respect to mesh size). Conversely, the condition number for p-FEM—and this particular choice of orthogonalbasis—remains roughly constant for p ≥ 2. This is in accordance with theory, since in p-FEM  κ(K) = O p4(d−1) [3], and thus κ(K) = O(1) for dimensionality d = 1. 

3.1.1 A priori error estimates In Chapter 2 we looked at error estimates for h-FEM. We discussed that, depending on the regularity of the exact solution, problems can be classified in one of three categories. Problems in Category A are smooth, and while h-FEM exhibits algebraic convergence, p-FEM has exponential convergence (see Example 3.1). This type of convergence, mathematically expressed by [2]   C   uE () , ≤ (3.10) u − uh  θ γ E () e nD

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Table 3.1 Convergence types and rates for p-FEM based on problem category.

Convergence type Convergence rate

A exponential θ ≥ 1/d †

Category B algebraic βp = 2 (κ − 1)‡ for uniform h

C algebraic βp > 0

† d ≡ dimensionality of the problem. ‡ κ ≡ regularity of the solution. For u ∈ H κ , κ is very large or κ > 1 for categories A and B, respectively.

where γ and θ are constants, is appealing because the error reduces very rapidly with increasing number of DOFs nD . We can actually determine  of change in con the rate vergence rate by obtaining the constant θ . Writing ε¯ ≡ u − uh E () /uE () , and applying the natural logarithm twice yields ln ε¯ ≤ ln C − γ nD θ , ln (− ln ε¯ ) ≥ ln γ + θ ln nD , where the last inequality, which holds for 0 < C ≤ 1, represents a line with slope θ in ln nD × ln (ln 1/¯ε ) space. As shown in Table 3.1, theory predicts θ ≥ 1/d, where d is the dimensionality of the problem. For non-smooth problems (e.g., problems with singularities), p-FEM exhibits algebraic convergence, described by   1   ≤ C2 β uE () , (3.11) u − uh  E () p p where βp is the convergence rate and p is the minimum interpolation order in the mesh. Theory predicts, per Table 3.1, that for problems in Category B using uniform meshes, the algebraic convergence rate is limited by the regularity of the solution, i.e., βp = 2 (κ − 1). For non-smooth problems in Category C, we only know they converge, i.e., βp > 0. Example 3.2: Constrained bar pullout problem (revisited). Analyze the convergence behavior of p-FEM in Example 3.1 and determine the coefficient of exponential convergence θ . Solution: In Example 3.1 it was shown that the convergence of p-FEM for this problem is exponential, as the rate of convergence increased with order p. Exponential convergence follows (3.10), and we can therefore recover the exponent θ by taking the natural logarithm of the reciprocal of the relative error (see Fig. 3.4). For this problem, whose exact solution is smooth, theory predicts a value θ ≥ 1 and a value θ = 1.49 is found numerically. Notice that the last data point (p = 16) was not considered since, as discussed in Example 3.1, machine precision had been reached and thus this data point is meaningless. 

Example 3.3: p-FEM solution for first-order shear deformation beam. Fig. 3.5 shows a cantilever beam of length L, cross-sectional area A = BH , and moment of in3 ertia I = BH 12 . The beam has Young’s modulus E, shear modulus G, and is subjected

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Figure 3.4 Natural logarithm of the reciprocal of the relative error in energy norm, as a function of the total number of DOFs nD .

Figure 3.5 Cantilever beam of length L and cross-sectional area A = BH . The beam is loaded with a uniformly distributed load q.

to a uniformly distributed load q. Consider steel as the base material with E = 200 GPa and thus the ratio E/G = 2.6 (as in [4]). The exact solution of this boundary value problem, which was first enunciated in strong form in Problem 2.2, is     q 2Lx − x 2 q 6L2 x 2 − 4Lx 3 + x 4 + , w(x) = 24EI 2Ks GA (3.12)   q 12L2 x − 12Lx 2 + 4x 3 , θ (x) = 24EI where w(x) is the deflection, θ (x) the angle of rotation, and Ks the shear correction factor (5/6 for rectangular cross-section). The exact solution will be compared with finite element approximations, which are obtained from the weak form of equilibrium: Find w, θ ∈ W ×  such that   ˆ B w, θ ; w, ˆ θˆ = L w, ˆ θˆ ∀w, ˆ θˆ ∈ Wˆ × , (3.13) where 



B w, θ ; w, ˆ θˆ =

 0

L



  dwˆ dθˆ +Q − θˆ M dx, dx dx

(3.14)

The p-version of the finite element method

  L w, ˆ θˆ =

L

65

q wˆ dx .

(3.15)

0

In (3.14) M is the bending moment and Q the shear force (for more details see Problem 2.2). Solve the problem with finite elements of increasing polynomial order (up to fourth order since that is exact according to (3.12)) and show that, while linear approximations exhibit locking, this problem can be alleviated with higher-order approximations. Using a single finite element, study the effect of beam slenderness in the range L/H = [4, 1000]. Thereafter, for the most slender beam, study the effect of increasing the number of elements (from 1 element to 512 elements). Finally, study convergence and the condition number of the stiffness matrix. Noteworthy, similarly to Eq. (2.21), 

the energy norm (w, θ )E () =

1 2 B(w, θ ; w, θ )

can be used to quantify the error.

Solution: For solving the discrete counterpart of (3.13), the trial functions for the deflection and rotation are wh (x) =

 i∈N

ϕi (x)wi and θ h (x) =



ϕi (x)θi ,

(3.16)

i∈N

respectively—similarly to (3.5); the same form is adopted for the test functions. The beam is discretized first with a single finite element and the problem is solved for polynomial orders p = {1, . . . , 4}. Fig. 3.6 h and the exact deflection shows the ratio between the finite element deflection at the tip of the beam wL given by (3.12) w(L) = wL , as a function of the ratio H /L. As apparent from the figure, the linear formulation shows locking, but this can be alleviated by increasing the polynomial order. We also see the results obtained for L/H = 1000 by discretizing the beam in nE = {1, 2, 4, . . . , 512} finite elements. For the linear approximation and for 512 elements, the deflection is only about 45% of the exact value, showing that locking is not mitigated by increasing the number of elements.

h , normalized by Figure 3.6 Ratio between the computed transverse displacement at the tip wL the exact value wL given by Eq. (3.12), as a function of the ratio H /L (left) and of the number of elements nE (right).

In Fig. 3.7 we plot the relative error in energy norm squared and the condition number of the stiffness matrix, as a function of the total number of degrees of freedom nD . The curve labeled p = {1, . . . , 4} shows the result for a single finite element. Contrary to Fig. 3.6, in this figure we see the difference between p = 3 and p = 4. Note that the latter is exact regardless of the number of elements. Even if the same polynomial order was adopted for both approximations in (3.16), the Galerkin projection—as per Céa’s approximation theorem (Theorem 2.1.2)—assigns negligible values to the quartic DOFs associated with the rotation. Note also that we do not reach machine precision accuracy for p = 4 because of the condition number—we obtain the solution vector with 16 − log10 κ(K) digits of accuracy when using double precision floating-point

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Figure 3.7 Relative error in energy norm squared (left) and condition number (right) as a function of the total number of DOFs nD . In addition to the results obtained for each polynomial order p, the curve p = {1, . . . , 4} shows the results for a single beam element. arithmetic [5]. Note also how accuracy for p = 4 is actually reduced as we increase the number of total DOFs, mirroring the increase in condition number. This problem is ill-conditioned for even a single element for nonlinear approximations since κ(K) ≈ 106 , although the rate of growth for these remain stable  as O nD 2 . 

3.2 p-FEM in 2-D While in 1-D constructing a p-FEM finite element space entails simply selecting the set of orthogonal polynomials and the approximation order, in higher dimensions it is not so straightforward. We may ask how many basis functions are necessary to solve a particular problem? The answer to this question, of course, depends on the problem, since there may be cases where a higher-order interpolation is preferred along a particular direction. Intuitively, however, the more functions are considered, the richer the finite element space becomes. We can also foresee that it may be a bad idea to skip lower-order functions—particularly the linear ones! In the following we describe p-FEM spaces for 2-D quadrangles and triangles.

3.2.1 Basis functions for quadrangles Two spaces that have been proposed for quadrangular elements are [2,6]: p

• Trunk space. This space, denoted Sts , where p is the polynomial order, is spanned by the following set of monomials:  i, j = {0, 1, . . . , p} , i j ξ η i + j = {0, 1, . . . , p} , ξη ξ p η, ξ ηp

if p = 1, for p ≥ 2.

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67 p ,p

• Tensor product space. This space, denoted Spsξ η with pξ and pη indicating the polynomial order associated to its corresponding direction, is spanned by  i j

ξ η

  i = 0, 1, . . . , pξ ,   j = 0, 1, . . . , pη .

Polynomial orders do not have to be the same in both directions, so it is straightforward to create p-orthotropic tensor product spaces. The monomials that span these spaces are visualized in the Pascal triangle of Fig. 3.8. The actual finite element spaces used for analysis are, however, not constructed using monomials (see discussion on monomial bases in § 3.1). Similarly to the 1-D case, it is desired to use orthogonal polynomials, and therefore we make use of the shape functions described in the previous section to construct p-FEM spaces hierarchically in higher dimensions.

p

Figure 3.8 Pascal triangle showing the trunk spaces Sts , p = {1 . . . 5}, and the tensor product p ,p space Spsξ η , pξ = pη = {1 . . . 3}, used for quadrangular elements.

Consider a quadrangular element, parameterized by the master coordinate ξ , i.e.,   = ξ = (ξ, η) ∈ [−1, 1] × [−1, 1] ⊂ R2 . One approach for constructing 2-D basis functions over this element considers the product of two functions, each defined with respect to one of the orthogonal coordinates, i.e., ϕi (ξ )ϕj (η), where ϕi (ξ ) and ϕj (η) are expressed as per (3.3) and (3.4). In this procedure, which yields the functions shown in Fig. 3.9, we can distinguish three types of functions: • Nodal functions. These are obtained by the product of linear functions, i.e., ϕi (ξ ) ϕj (η) , i, j = {1, 2}, and are equivalent to the shape functions of the 4-node bilinear standard finite element. • Edge functions. Taking a linear function along one direction and a nonlinear function along the other, i.e., ϕi ϕˆ j , i = {1, 2}, j = {2, . . .}, gives rise to edge functions.

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Figure 3.9 Nodal, edge, and bubble shape functions for a 2-D quadrangular element.

Edge functions, which are shown in the figure up to p = 5, are constructed as ϕ1 (η) ϕˆ i (ξ ), ϕ2 (ξ )ϕˆ i (η) , ϕ2 (η) ϕˆ i (ξ ), ϕ1 (ξ )ϕˆ i (η) ,

i = 2, . . . i = 2, . . . i = 2, . . . i = 2, . . .

first edge, second edge, third edge, fourth edge.

• Bubble functions. These are obtained as ϕˆ i ϕˆj , i, j = {2, . . .}. Noteworthy, we can see that the functions in the first column of Fig. 3.9 span Sps1,1 ≡ Sts1 , the functions in the first and second columns span Sts2 , adding the third column we span Sts3 , and so on.

3.2.2 Basis functions for triangles As in the previous section, shape functions for triangles can be constructed with the aid of orthogonal polynomials. Chebyshev polynomials of the first kind are used here as an alternative to the Legendre polynomials used previously [7]. These can also be

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69

computed recursively via Cp (ξ ) = 2ξ Cp−1 (ξ ) − Cp−2 (ξ ),

(3.17)

with C0 (ξ ) = 1 and C1 (ξ ) = ξ . Chebyshev polynomials are illustrated in Fig. 3.10, and contrary to Legendre polynomials shown in Fig. 3.1, all of the extrema are either −1 or 1.

Figure 3.10 Chebyshev polynomials Cp (ξ ).

  Consider the triangle = ξ = (ξ, η) ∈ R2 : ξ ≥ 0 ∧ η ≥ 0 ∧ ξ + η ≤ 1 . Similarly to the quadrangular case, we define three types of shape functions (see Fig. 3.11):

Figure 3.11 Nodal, edge, and bubble shape functions for a 2-D triangular element.

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Fundamentals of Enriched Finite Element Methods

• Nodal functions. These are the same as those for the constant-strain triangle given by Eq. (2.78): ϕ1 = 1 − ξ − η,

ϕ2 = ξ,

ϕ3 = η.

• Edge functions. These are constructed by using the nodal functions to ensure highorder functions vanish along two edges of the triangle: ϕ1 ϕ2 Ci (2ξ + η − 1) first edge, second edge, ϕ2 ϕ3 Ci (η − ξ ) ϕ1 ϕ3 Ci (1 − ξ − 2η) third edge, with i = {0, 1, . . . , p − 2}. • Bubble functions. The bubble functions for triangles take the form ϕ1 ϕ2 ϕ3 Ci (2ξ − 1) Cj (2η − 1) , with 0 ≤ i + j ≤ p − 3. As was the case for edge functions, here we also use nodal functions to ensure bubble functions are exactly zero along edges. There is a an important caveat for p-FEM of polynomial order p = 3 and above, since a careless computer implementation can result in polynomial non-conforming approximations. In other words, it is possible that the cubic edge function of an element, for instance, does not match the cubic edge function of the contiguous element, yielding a C −1 -continuous function along the edge. Note that this does not happen with even edge functions since these are symmetric with respect to the edge center point. A simple solution to this issue is to simply match the odd edge functions, for instance, by looking at the edge node ids in lexicographical order and use the normal vector to the edge as a criterion to multiply one of the odd shape functions (and their derivatives) by −1. Example 3.4: p-FEM solution for the Laplace equation. Consider the square plate  of unit side length, as shown in Fig. 3.12a. It can be shown that the function (see Fig. 3.12b) u(x, y) = csch (π) sin (πx) sinh (π − πy) ,

(3.18)

where u represents a vector-valued scalar field (e.g., temperature), satisfies the Laplace equation ∇ 2 u = 0. Solve the problem numerically by using the p-FEM and determine the coefficient of exponential convergence θ . Solution: To solve this problem, an essential boundary condition u = 0 is prescribed in three edges of the plate. For the remaining edge, a natural boundary condition q¯ = π coth(π ) sin(π x) is applied. The problem is then solved with p = {1, . . . , 10}. The results are summarized in Fig. 3.13, where exponential convergence is clearly shown. A value θ = 0.56 is determined for the last data points. According to Table 3.1, for d = 2 theory predicts θ ≥ 1/2. It is shown that for p = 10 the results approach numerical machine precision (a direct solver was used to obtain these results). 

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71

Figure 3.12 (a) Domain  = [0, 1]2 discretized with quadrangular elements; (b) Exact solution (3.18).

Figure 3.13 Relative error in energy norm (left) and natural logarithm of its reciprocal (right) as a function of the total number of DOFs nD .

3.3 Non-homogeneous essential boundary conditions An important point that needs attention is how to prescribe nonzero Dirichlet boundary conditions. In 1-D this is straightforward because all shape functions higher than linear vanish at the end nodes of the element. In higher dimensions this is no longer the case. The problem is schematized in Fig. 3.14, where a quadrangular element has an edge along  D , the part of the boundary where nonzero Dirichlet BCs need to be prescribed. ˆ → , the right edge corresponds to Let us assume that, through the map Q (ξ, η) :  ξ = 1. Furthermore, let us consider a scalar problem with a quartic field interpolation. If the quadrangle were a Lagrange element, prescribing nonzero Dirichlet BCs on the right edge would be straightforward because of the Kronecker-δ property of Lagrange shape functions: The nodes on the right edge would be located equidistantly at coordinates (1, ηi ) , ηi = {−1, −0.5, 0, 0.5, 1}. The field at the right edge would thus be written as u(1, η) = ϕ2 (1, η)U2 + ϕ3 (1, η)U3 + ϕ6 (1, η)U6 + ϕ10 (1, η)U10 + ϕ14 (1, η)U14 ,

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Fundamentals of Enriched Finite Element Methods

Figure 3.14 A quadrangular finite element has an entire edge along the Dirichlet boundary D .

where only nonzero shape functions   along the right edge were considered. Because of the Kronecker-δ property ϕi x j = δij , evaluating the equation above is trivial: ¯ −1), u(1, −1) = ϕ2 (1, −1)U2 = U2 = u(1, u(1, −0.5) = ϕ6 (1, −0.5)U6 = U6 = u(1, ¯ −0.5), u(1, 0) = ϕ10 (1, 0)U10 = U10 = u(1, ¯ 0), u(1, 0.5) = ϕ14 (1, 0.5)U14 = U14 = u(1, ¯ 0.5),

(3.19)

u(1, 1) = ϕ3 (1, 1)U3 = U3 = u(1, ¯ 1). Note, however, that the function to be prescribed over the boundary u¯ is usually a function of global coordinates x, thus it is understood above that u(ξ, ¯ η) ≡ u(x(ξ, ¯ η)). Once the DOFs are known by solving (3.19), they can be prescribed strongly. In p-FEM prescribing nonzero BCs is more involved because the Kronecker-δ property only applies to the bilinear functions. Still, generalized DOFs corresponding to higher-order shape functions can be computed by solving a local problem, and once known, they can still be enforced strongly. The field over the right edge, when using p-hierarchical shape functions, can be written as u(1, η) = ϕ2 (1, η)U2 + ϕ3 (1, η)U3 + ϕˆ 2 (η)U6 + ϕˆ 3 (η)U10 + ϕˆ 4 (η)U14 ,       (3.20) nodal functions

edge functions

where the edge functions are written in terms of (3.4)—for instance, ϕˆ 3 is the cubic function. An important attribute of (3.20) is that all nonlinear functions vanish at the end nodes of the edge, and it is this very property what allows us to solve a local problem to obtain the DOFs associated with the nonlinear functions. In the following we explore two ways to accomplish this.

3.3.1 Interpolation at Gauss–Lobatto quadrature points One possibility is to evaluate (3.20) at Gauss–Lobatto quadrature points, which are related to the derivatives of the Legendre polynomials—i.e., nGP Gauss–Lobatto points

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73

comprise the interval endpoints ξ ± 1 and the remaining nGP − 2 points are determined as the roots of the polynomial dPnGP −1 / dξ . Following the example given 5 earlier  for the√Lagrange  element, we would evaluate (3.20) at 1 × {ηi }i=1 = √ quartic 1 × − 1, − 3/7, 0, 3/7, 1 and obtain a 5 × 5 system that can be solved for U2 , U3 , U6 , U10 , and U14 . In fact, we know that U2 and U3 are readily available because the linear functions satisfy the Kronecker-δ property and all edge functions vanish at nodal points; therefore it is only necessary to solve a 3 × 3 system. To wit, U2 = u(1, ¯ η1 ), and

⎡

ϕˆ 2 (η2 ) ⎣ϕˆ 2 (η3 ) ϕˆ 2 (η4 )

ϕˆ 3 (η2 ) ϕˆ 3 (η3 ) ϕˆ 3 (η4 )

¯ η5 ), U3 = u(1, ⎤⎡ ⎤ U6 ϕˆ4 (η2 ) ϕˆ4 (η3 )⎦ ⎣U10 ⎦ U14 ϕˆ4 (η4 ) ⎡ ⎤ u(1, ¯ η2 ) − U2 ϕ2 (1, η2 ) − U3 ϕ3 (1, η2 ) ¯ η3 ) − U2 ϕ2 (1, η3 ) − U3 ϕ3 (1, η3 )⎦ . = ⎣u(1, u(1, ¯ η4 ) − U2 ϕ2 (1, η4 ) − U3 ϕ3 (1, η4 )

Once known, generalized DOFs U2 , U10 , and U14 can be prescribed as in standard hFEM.

3.3.2 Projection on the space of edge functions Consider in Fig. 3.15 the function u, ¯ which has been decomposed into linear and nonlinear components. The linear part is obtained by interpolating the extreme values on the interval [−1, 1] with the linear shape functions. Mathematically, u¯ lin = u(−1)ϕ ¯ ¯ 2 (η) + u(1)ϕ 3 (η), where ϕ2 and ϕ3 are, as earlier, the linear shape functions along the edge.

Figure 3.15 The function u¯ to be prescribed on the boundary can be decomposed into linear u¯ lin and nonlinear u˜ components.

We now consider only the nonlinear part, i.e., u˜ (η) = u¯ (η) − u¯ lin —which is exactly zero at the end nodes—and define the function space (dropping the dependence on η for brevity) ε = u˜ − U6 ϕˆ2 − U10 ϕˆ 3 − U14 ϕˆ 4 ,

U6 , U10 , U14 ∈ R.

(3.21)

For a particular choice of DOFs U6 , U10 , U14 in (3.21), the resulting function measures the error between the nonlinear component of u¯ and what can be interpolated by the nonlinear terms in our quartic approximation.

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Fundamentals of Enriched Finite Element Methods

The objective is then to find U6 , U10 , and U14 so as to minimize the L 2 -norm of the error εL 2 , which is induced by the inner product  ε, ε =

+1

−1

ε 2 dη = ε2L 2 .

(3.22)

The coefficients that minimize (3.22) are obtained by ∂ ε2L 2 ∂Ui

i = {6, 10, 14} ,

= 0,

(3.23)

which again results in a 3 × 3 system of equations that can be solved for the generalized DOFs. The first equation is ∂ ε2L 2 ∂U6

 = −2

+1

−1

 ε ϕˆ 2 dη = −2

+1 

−1

u˜ ϕˆ2 − U6 ϕˆ22 

(3.24)

− U10 ϕˆ2 ϕˆ3 − U14 ϕˆ 2 ϕˆ 4 dη = 0. The other two equations are obtained analogously. The system of equations to solve is given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ ϕˆ 2 , ϕˆ2  ϕˆ 2 , ϕˆ 3  ϕˆ 2 , ϕˆ 4  u, ˜ ϕˆ2  U6 1 ⎣ϕˆ 2 , ϕˆ3  ϕˆ 3 , ϕˆ 3  ϕˆ 3 , ϕˆ 4 ⎦ ⎣U10 ⎦ = ⎣u, ˜ ϕˆ3 ⎦ . (3.25) 2 u, U ϕˆ , ϕˆ  ϕˆ , ϕˆ  ϕˆ , ϕˆ  ˜ ϕˆ  2

4

3

4

4

4

14

4

Noteworthy, this approach is computationally more involved because it requires computing integrals of the symmetric coefficient matrix.

3.4 Problems PROBLEM 3.1.— BAR WITH VARYING CROSS SECTION Consider Problem 2.1 given in Chapter 2. Solve the problem with a single finite element and polynomial orders p = {1, 2, 4, 8}. Create a convergence plot and compare with the results obtained for h-FEM. BOUNDARY VALUE PROBLEM WITH u ∈ C ∞ PROBLEM 3.2.— Consider Problem 2.3 given in Chapter 2. Solve the BVP for xb = 0.2 and a = 0.5, and 2 elements of degree p = {1, . . . , 5}. Thereafter solve for a = 50 and 5 elements of degree p = {1, . . . , 5}. Make a convergence plot in energy norm (relative error) and compare with the curves obtained for h-FEM. MATERIAL DISCONTINUITY PROBLEM 3.3.— Solve Problem 2.4 with a bimaterial interface this time using the p-version of FEM and uniform meshes with elements of degree p = {1, . . . , 5}. Use (i) two elements, and (ii) three elements.

References [1] I. Babuska, B.A. Szabo, I.N. Katz, The p-version of the finite element method, SIAM Journal on Numerical Analysis 18 (3) (1981) 515–545, https://doi.org/10.1137/0718033. [2] B.A. Szabó, I. Babuška, Finite Element Analysis, John Wiley and Sons, New York, 1991.

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[3] J.-F. Maitre, O. Pourquier, Condition number and diagonal preconditioning: comparison of the p-version and the spectral element methods, Numerische Mathematik 74 (1) (1996) 69–84, https://doi.org/10.1007/s002110050208. [4] A. Tessler, S.B. Dong, On a hierarchy of conforming Timoshenko beam elements, Computers & Structures 14 (3) (1981) 335–344, https://doi.org/10.1016/0045-7949(81)90017-1. [5] R. Kannan, S. Hendry, N.J. Higham, F. Tisseur, Detecting the causes of ill-conditioning in structural finite element models, Computers & Structures 133 (2014) 79–89, https:// doi.org/10.1016/j.compstruc.2013.11.014. [6] B. Szabó, A. Düster, E. Rank, The p-version of the finite element method, in: Encyclopedia of Computational Mechanics, John Wiley & Sons, Ltd, 2004. [7] P.R.B. Devloo, C.M.A. Ayala Bravo, E.C. Rylo, Systematic and generic construction of shape functions for p-adaptive meshes of multidimensional finite elements, Computer Methods in Applied Mechanics and Engineering 198 (21) (2009) 1716–1725, https:// doi.org/10.1016/j.cma.2008.12.022.

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The Generalized Finite Element Method

4

In the early 1990s, the work of Babuška et al. [1,15] and Duarte and Oden [2,3] on partition of unity methods paved the way for a family of new approximation methods, among which the Generalized or eXtended Finite Element Method (GFEM/XFEM) [1,4–9] are the most widely known. They have been developed and improved for more than two decades and are available in several mainstream commercial software, such as Abaqus, LS-DYNA, OptiStruct, and ANSYS. The GFEM and XFEM are essentially identical methods: The name Generalized FEM was adopted by Babuška, Duarte, and Oden in 1995–1998, and the name eXtended FEM was coined by Belytschko and colleagues in 1999–2000 [10]. The acronym GFEM is adopted in this book. The second author’s recollections on the origins of the GFEM and XFEM are presented in Appendix A. Contrary to standard FEMs, where a polynomial basis is adopted to obtain an approximate solution, GFEM uses any a priori knowledge about the solution to improve its approximation. In GFEM, a standard FEM polynomial basis can be augmented by any function, called enrichment, which need not be polynomial or even known analytically. The key ideas of the Generalized Finite Element Method are presented in this chapter in a 1-D setting. This allows the adoption of a simple notation, the step-bystep solution of representative problems, and a straightforward implementation of the method. Furthermore, these same ideas carry over to the formulation of the method in higher dimensions, which is covered in application-specific chapters. We first recall h- and p-version FEM approximations in Section 4.1, and generalize them in Section 4.2 using the partition of unity property of Lagrangian finite element shape functions. We show that this property allows the definition of GFEM shape functions that can exactly reproduce any enrichment function. A more rigorous discussion on the approximation properties of the GFEM is presented in Chapter 13. After that, we present in Section 4.3 the application of the method to problems with material interfaces and discontinuities. This class of problems has received considerable attention from the e-FEM community since they pose significant challenges to standard FEMs and are found in many engineering applications. We show that by selecting proper enrichment functions for the GFEM, the simulation of this class of problems can be conducted on FEM meshes that are decoupled (also known as unfitted or nonmatching meshes) from the material interfaces or discontinuities. High-order GFEM approximations for problems with smooth and discontinuous solutions are presented in Section 4.5. As a prelude to the mathematical theory of partition of unity methods presented in Chapter 13, the concept of GFEM approximation spaces is introduced in Section 4.6. Fundamentals of Enriched Finite Element Methods. https://doi.org/10.1016/B978-0-32-385515-0.00011-8 Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved.

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Fundamentals of Enriched Finite Element Methods

4.1 Finite element approximations The error of a finite element approximation can be controlled using the h- or p-version of the method, as reviewed in Chapters 2 and 3. The combination of both strategies is also possible [11]. In the first case, the FEM error is controlled by decreasing the mesh size h while not changing the polynomial order of the approximation. In the p-version, the opposite strategy is adopted—the polynomial order of the FEM approximation is increased while keeping the mesh fixed. An example of a finite element mesh covering an analysis domain  with mesh parameter h is shown in Fig. 4.1. In the figure, ϕα (x) is the standard linear finite element shape function associated with node xα , α ∈ Ih , where Ih = {1, . . . , n} is the set of nodes in the mesh. These functions are defined in (2.37).

Figure 4.1 One-dimensional linear finite element shape functions on an analysis domain . Each function ϕα is associated with node xα and with a subdomain ωα ⊂  over which ϕα = 0.

An FEM approximation is simply a linear combination of finite element shape functions. In the case of the h-version and for a scalar-valued field, this approximation is given by (cf. Section 2.1.4)1 uhFEM =



(4.1)

ϕα Uα ,

α∈Ih

where Uα , α ∈ Ih , are degrees of freedom. In the case of the p-version, an FEM approximation is given by p

uFEM =

 α∈Ih





ϕα Uα + 



std. linear FEM

ϕˆ α Uα ,

α∈N\Ih





(4.2)



enriched high-order

where ϕˆ α is the high-order shape function defined in (3.4) and N the set of all nodes— the union of end nodes (in 1-D) in set Ih and interior nodes associated with 1-D high-order shape functions. 1 In this chapter, uh h FEM and u denote h-version FEM and GFEM approximations, respectively. Similarly, p p uFEM and u denote p-version FEM and GFEM approximations, respectively. The type of approximation

is also clear from the context.

The Generalized Finite Element Method

79

FEM approximations often require strong mesh refinement in order to recover the optimal convergence of the method—the same rate of convergence attained in problems with smooth solutions. In addition, the finite element mesh must fit material interfaces, crack surfaces, and domain boundaries. These limitations of the FEM can be removed by hierarchically enriching its approximation with generalized FEM shape functions, as shown next. Remark 1. Definitions (4.1) and (4.2) involve global FEM shape functions. This is just a convenient way to define FEM approximations. The implementation of the method is, of course, performed at the element level. This same strategy is adopted for the generalized FEM.

4.2 Generalized FEM approximations in 1-D A generalized finite element approximation is a finite element approximation hierarchically enriched with functions derived from a priori knowledge about the solution of a problem. A generalized finite element approximation uh (x) of a scalar field u(x) is given by uh (x) =

 α∈Ih



uˆ α ϕα (x) + 



std. FEM

 α∈Ihe



ϕα (x)

mα 

u˜ αj Eαj (x),

j =1



(4.3)



enriched GFEM

where Eαj is an enrichment function assigned to node α ∈ Ihe , j ∈ {1, . . . , mα } is the index of the enrichment function at that node, and Ihe ⊆ Ih is the set of GFEM enriched nodes. In addition to the standard FE shape function ϕα , there are mα GFEM shape functions associated with a node α ∈ Ihe . Nodes α ∈ Ih \ Ihe have only the FE Lagrangian shape function ϕα . The coefficients uˆ α and u˜ αj are denoted regular/standard and enrichment degrees of freedom, respectively. They are computed by solving a set of discrete equations like in the standard FEM. Details and examples are presented later in this chapter. The GFEM approximation given in (4.3) is an example of an enriched finite element method (e-FEM). The p-FEM approximation given in (4.2) can also be seen as an e-FEM, as discussed Chapter 3. The main difference between these methods is the type of hierarchical enrichment adopted. The enrichment strategy adopted in the GFEM is not limited to polynomial functions, as shown later. This provides a high degree of flexibility in the construction of GFEM approximations. Another difference is that all GFEM shape functions are associated with nodes in the set Ihe ⊆ Ih , while p-FEM assigns high-order shape functions to nodes N \ Ih , which are not in a linear FEM discretization. This has an impact on the implementation of the methods and on the sparse structure of the global matrices [7]. The product of a Lagrangian finite element shape function ϕα (x) and an enrichment function Eαj (x) (cf. second term in (4.3)) defines a GFEM shape function. To wit, φαj (x) = ϕα (x)Eαj (x),

α ∈ Ihe ⊆ Ih ,

j ∈ {1, . . . , mα } .

(4.4)

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Fundamentals of Enriched Finite Element Methods

Examples of GFEM shape functions in 1-D are shown in Figs. 4.9 and 4.10. Fig. 4.2 illustrates the construction of a GFEM shape function in 2-D. These figures show that regardless of the choice of the enrichment function, GFEM shape functions are nonzero only over the support of ϕα , denoted by ωα ⊂ . This is the set of elements that are connected to node xα , i.e., ωα := { x ∈ | ϕα (x) = 0}. Set ωα is illustrated in Fig. 4.1 and is also called a cloud [2] or a patch [1]. This property of GFEM shape functions leads to global matrices that are sparse, like in the FEM.

Figure 4.2 Construction of a generalized FEM shape function using a smooth enrichment functions. The top, middle, and bottom functions are the finite element shape function, the enrichment function, and the resulting generalized FE shape function, respectively. The FEM shape function is defined on a patch of four quadrilateral elements. The grid shown in the figure is for visualization purposes only.

Remark 2. GFEM approximation (4.3) adds GFEM enrichments to the h-version FEM approximation (4.1). It is also possible to add GFEM enrichments to the p-version FEM approximation (4.2) [12]. This strategy can be used to define well-conditioned, high-order GFEM approximations [13,14].

4.2.1 Selection of enrichment functions The enrichment functions Eαj are chosen so that they improve the approximation locally in ωα , the support of the standard FE shape function at node α. They are selected

The Generalized Finite Element Method

81

based on any a priori knowledge about the unknown solution of a target class of problems, e.g., the existence of discontinuities (Example 4.3), singularities, and jumps of solution derivatives across material interfaces (Example 4.2), to name a few. These features of the exact solution are approximated by the enrichment functions instead of a finite element mesh with elements whose edges align with cracks or material interfaces, as is usually done in the FEM. Details on how to select these functions for specific classes of problems are given later in this and other chapters. This flexibility in selecting enrichments is perhaps the most important property of the GFEM, which is also shared with all other partition of unity methods [3,15].

4.2.2 What makes the GFEM work Two properties of Lagrangian FEM shape functions are fundamental for the approximation and computational performance of the GFEM. The first one is that the FE shape functions form a partition of unity, i.e.,  ϕα (x) = 1 ∀x ∈  . (4.5) α∈Ih

This property guarantees that any enrichment function can be represented exactly through linear combinations of GFEM shape functions. For simplicity, consider the case in which Ihe = Ih and the same enrichment Ej (x) is adopted for all nodes. The mesh node index can then be dropped from the enrichment function. The summation of the GFEM shape functions at any point x ∈  gives    φαj (x) = ϕα (x)Ej (x) = Ej (x) ϕα (x) = Ej (x) . (4.6) α∈Ih

α∈Ih

α∈Ih

The above reproducing property implies that if the enrichment Ej (x) can locally approximate the solution of a problem, the GFEM shape functions defined using this enrichment can approximate the solution as well. Thus, the approximation performance of the GFEM is directly connected to the adopted enrichments. The partition of unity property of FE shape functions is also a cornerstone of the a priori error estimates for the GFEM approximations presented in Chapter 13. The second important property of FE shape functions is that they have a compact support—they are nonzero only over the finite elements connected to the node to which they are assigned to. This property guarantees that the support of GFEM shape functions is not larger than the support of the FE shape functions—regardless of the size of the support of enrichment functions. This, in turn, leads to GFEM matrices that are sparse like in the FEM. Remark 3. While we have adopted a 1-D setting for simplicity, the definition of a GFEM approximation given in (4.3) and the definition of a GFEM shape function given in (4.4) are valid for any spatial dimension, any type of Lagrangian finite element shape function, and any type of enrichment. Furthermore, GFEM approximations of vector-valued fields, such as displacement fields in 2-D and 3-D elasticity problems, are defined exactly as in 1-D. These approximations are presented in later chapters.

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Fundamentals of Enriched Finite Element Methods

Remark 4. The definition of GFEM shape functions given in (4.4) can adopt other classes of partition of unity functions—not just finite element shape functions. Examples are Shepard functions and moving least square functions [16], which are adopted in the hp-cloud method [2], the method of finite spheres [17], and the particle-partition of unity method [18] to define meshfree shape functions. All these methods can be cast as special cases of the partition of unity method, which has its origins in the works of Babuška et al. [1,15], Duarte and Oden [2,3], and Taylor, Zienkiewicz, and Oñate [19]. The Finite Cover Method [20] and the Manifold Method [21] are also related to the GFEM. A brief summary of the origins of the GFEM and related methods is presented in Appendix A.

4.3 Applications of the GFEM This section presents several examples of GFEM approximations to problems that pose difficulties to the standard FEM. These include problems with material interfaces and discontinuities. The p-version of the method for these applications is presented in Section 4.5. Example 4.1: GFEM approximation on a single element. Write the GFEM approximation (4.3) for a two-node 1-D element, which can be used to solve, e.g., the linear elastostatic problem formulated in Section 2.1. Solution: The set of nodes of the element is given by Ih = {1, 2}. For illustration, we assume that each node is associated with two enrichment functions, but we leave the selection of specific enrichments for later examples. Therefore, the set of GFEM enriched nodes is Ihe = Ih , with mα = 2, α ∈ Ihe . The GFEM approximation (4.3) over the element is given by 

uh (x) =

α∈{1,2}





uˆ α ϕα (x) + 



ϕα (x)

u˜ αj Eαj (x) .

j =1

α∈{1,2}



std. FEM

2 





enriched GFEM

In matrix notation, the above can be written as ⎡

uh (x) =



ϕ1

ϕ1 E11

ϕ1 E12

ϕ2

ϕ2 E21

⎤ uˆ 1 ⎢u˜ 11 ⎥ ⎢ ⎥  ⎢u˜ 12 ⎥ ⎥ ϕ2 E22 ⎢ ⎢ uˆ 2 ⎥ := N U . ⎢ ⎥ ⎣u˜ 21 ⎦ u˜ 22

It is noted that the shape functions are ordered on a node-by-node basis in matrix N. Another option is to have the FE shape functions first, followed by the GFEM functions. Both conventions are used  in  the literature and in this book. Each node α of the element has three degrees of freedom: one standard uˆ α and two enrichment DOFs (u˜ α1 and u˜ α2 ). This example shows again that GFEM enrichments are hierarchically added to nodes of the FE mesh as needed. This feature of the method is analogous to the p-hierarchical FE approximation (4.2). 

Example 4.2: GFEM approximation for a weak discontinuity. Fig. 4.3 shows a 1-D bimaterial bar of length l fixed at the left end and subjected to a force N¯ at the

The Generalized Finite Element Method

83

right end. Obtain the GFEM solution using a mesh with a single element and an enrichment that exhibits a weak discontinuity at the material interface x .

Figure 4.3 1-D bimaterial bar. Solution: This example illustrates the GFEM solution of problems with a displacement field that has a discontinuous gradient. This is the case in problems with material interfaces like the 1-D bimaterial bar shown in Fig. 4.3. The exact solution for this problem is derived in Example 2.1. Its derivative is discontinuous at the material interface x . This can be readily captured by the FEM provided that the FEM mesh has a node at the material interface x , as shown in Example 2.6. While the creation of these meshes is trivial in 1-D, meshing complex interfaces in higher dimensions can be challenging, in particular in problems for which the location of interfaces evolves in time. In this example, we show that with the proper selection of enrichment functions, the GFEM can adopt meshes that do not fit the interface while still delivering a solution that is as accurate as the one provided by the FEM with a mesh fitting the material interface, i.e., the exact solution for this 1-D problem. To simplify the presentation, in the following, we use the nondimensional coordinate ξ = x/ l so the discontinuity is located at w = x / l. The GFEM mesh has a single element and the GFEM approximation is given by   uˆ α ϕα (ξ ) + ϕα (ξ )u˜ α Eα (ξ ) = N (ξ ) U , (4.7) uh (ξ ) = α∈{1,2}

α∈{1,2}

where N and U are defined in Example 4.1. A single enrichment is adopted at element nodes and thus the second index of the nodal enrichment is dropped for clarity. Furthermore, the ramp function [22] shown in Fig. 4.4 is assigned to both nodes. Therefore, E1 = E2 ≡ Rr which is given by ⎧x − x  ⎨ Rr (x) = l − x ⎩ 0

for x ≥ x ,

(4.8)

for x ≤ x .

The nonzero part of function Rr is simply the distance to the interface (cf. Fig. 4.4). This enrichment function introduces a jump in the gradient of uh at x . This is an example of using a priori knowledge about the solution of a problem—a discontinuous gradient at x in this case—to select enrichment functions for the GFEM. We notice that the ramp function not only has the built-in gradient jump at x , but is also identically zero at the left end of the bar, which facilitates prescribing the essential boundary condition u(0) = 0, as shown later. Next, the GFEM solution is computed using the same steps as in Example 2.6. Here, however, trial and test functions are provided by (4.7). The FEM and GFEM functions for the single element used to solve the problem are given by   N (ξ ) = ϕ1 ϕ2 ϕ1 E ϕ2 E ⎧  ⎨ 1−ξ ξ 0 0 for ξ ≤ w,  =  ξ −w ξ −w ⎩ 1 − ξ ξ (1 − ξ ) for ξ ≥ w. ξ 1−w 1−w

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Fundamentals of Enriched Finite Element Methods

Figure 4.4 Ramp enrichment function.  The corresponding DOF vector is given by U = uˆ 1 tions are given by ⎧  dN dξ 1 ⎨−1 1 0 0 B (ξ ) = = dξ dx l ⎩ −1 1 1+w−2ξ

uˆ 2

w−2ξ w−1

1−w

u˜ 1



u˜ 2



. The derivative of the shape func-

for ξ ≤ w, for ξ ≥ w.

The rest of the computations follow standard procedures with one important difference since the bar stiffness and the derivative of GFEM shape functions are discontinuous at ξ = w. This must be taken into account when integrating the element stiffness matrix k (1) . Here, we break the integral into two parts. To wit, k (1) =

 w

 1 B  k1 Bl dξ + B  k2 Bl dξ , 0 w       (1)

(1)

k2

k1

where ⎡ ⎤ 1−w 0 ⎢ ⎥ k 0⎥ ⎢ (1) and k 2 = 2 ⎢ 0⎦ l ⎣ 0

⎡ 1 −1 0 wk 1 0 (1) 1 ⎢ ⎢ k1 = l ⎣ sym. 0

w−1 1−w sym.

−1 1

0 0

⎤ ⎥ ⎥

w−1 ⎥. ⎦ 3 (w−2)w+4 3−3w

1−w 3

The global unconstrained stiffness matrix K = k (1) , since there is only one element in the mesh, and the global load vector is  F = N  (ξ = 1)N¯ = 0

1

0

 1 N¯ .

The GFEM solution at ξ = 0 is uh (ξ = 0) = uˆ 1 . Therefore, the Dirichlet boundary condition u(0) = 0 can be enforced by simply setting uˆ 1 = 0. This is, again, a consequence of Rr (0) = 0. If this were not the case, uh (ξ = 0) = uˆ 1 + u˜ 1 E(0), which is a linear combination of uˆ 1 and u˜ 1 . This is known as a multipoint constraint (MPC), which can be enforced using the same procedures adopted in the context of the FEM, but they are more involved than simply setting uˆ 1 = 0. The global system of equations KU = F is given by ⎡ 1 2 3 4

1

k2 − w k

⎢ ⎢−k + w k ⎢ 2 ⎢ ⎢ 0 ⎣ −k2

2

3

4

−k2 + w k

0

−k2

k2 − w k

0

0

k2 (1−w) 3

k2

−k2 (1−w) 3

⎤ ⎡

⎥ ⎥ k2 ⎥ ⎥ (1−w) ⎥ −k2 3 ⎦ k2 4+w(w−2) 3(1−w)

uˆ 1

⎤

⎡

−R

⎤

⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢uˆ ⎥ ⎢ 2 ⎥ = l ⎢ N¯ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢u˜ ⎥ ⎦ ⎣ ⎣ 1⎦ u˜ 2 N¯

1 2 3 4

The Generalized Finite Element Method

85

where R is the unknown reaction force at the left end of the bar. After imposition of the Dirichlet boundary condition uˆ 1 = 0, the solution of the modified system of equations is given by ⎡

⎤ 0 ⎢ ⎥ N¯ l ⎢ ⎥ ⎢ ⎥ k1 ⎢ ⎥ ¯ ⎢ ⎥, N l (w−1) k   U= ⎢ ⎥ k k ⎣ ⎦ 1 2 (w−1)k N¯ l k1 k2

where k = k2 − k1 is the jump in axial stiffness. The GFEM solution uh = NU is ⎧ N¯ l ⎪ ⎪ ⎪ for ξ ≤ w, ⎨ξ k 1 uh (ξ ) = ⎪ N¯ l N¯ l ⎪ ⎪ ⎩w + (ξ − w) for ξ ≥ w, k1 k2 which is the exact solution (2.6) derived in Example 2.1. The GFEM solution is illustrated in Fig. 4.5 for k2 /k1 = 2. The GFEM displacement at the right end of the bar is uh (ξ = 1) = w

N¯ l N¯ l + (1 − w) . k1 k2

Figure 4.5 Generalized finite element solution for a bar composed of two different materials with axial stiffness ratio k2 /k1 = 2. It is worth noting that when dealing with a constant axial stiffness, i.e., k1 = k2 = k, the solution vector ¯ . In other words, the GFEM shape functions are not needed because 0 N l/k 0 0

becomes U =

there is no jump in the gradient of the solution at x = x , and, as a result, both generalized coefficients, u˜ 1 and u˜ 2 , are zero.



Example 4.3: GFEM approximation for a strong discontinuity. For the 1-D bar with a strong discontinuity at x and shown in Fig. 4.6, obtain a GFEM approximation using a mesh that does not fit the discontinuity and an enrichment that is discontinuous at x . Solution: This example introduces the concept of GFEM approximations for discontinuous functions in a 1-D setting. The solution of fracture mechanics problems exhibits this behavior, which is nontrivial to

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Fundamentals of Enriched Finite Element Methods

Figure 4.6 One-dimensional cracked bar with a spring of stiffness k at x and subjected to force N¯ at x = l. capture by standard FEMs in 2-D and 3-D, in particular when simulating fracture propagation. GFEM approximations for 2-D and 3-D fracture problems are provided in Chapter 6. We recall that the weak formulation for the problem shown in Fig. 4.6 was derived in Example 2.5. The one-dimensional bar has length l and axial rigidity k = EA. The bar has a discontinuity (crack) at x = l/2. A spring with stiffness k is located at the crack and is attached to the crack “faces” at x = x− and x = x+ . The 1-D spring represents a linear cohesive law at the crack faces. It is noted that the GFEM approximation presented in this example can be used with nonlinear cohesive laws (springs) as well. The solution of this problem using the finite element method requires a mesh that has two nodes at the discontinuity in order to approximate the displacement jump at the crack. Like in the previous example, we show that with the proper selection of enrichment functions, the GFEM can adopt meshes that do not fit the crack while still delivering a solution that is as accurate as the one provided by the FEM with a mesh fitting it, i.e., the exact solution for this 1-D problem, as shown later. We adopt the GFEM mesh with three elements shown in Fig. 4.7. The crack is at the middle of element (2). The axial displacement discontinuity in this element can be approximated by a discontinuous enrichment function. We adopt the Heaviside function Hd [23] defined by  0 if x < x , (4.9) H d = +1 if x ≥ x , and illustrated in Fig. 4.7. Function Hd , which is zero at the left of the crack and has a unity value at the right of the crack, introduces a discontinuity in the GFEM approximation of the axial displacement of the bar.

Figure 4.7 Enrichment function for a crack at x = x and GFEM mesh. Nodes 2 and 3 are enriched. The mesh shown in Fig. 4.7 has four nodes. Function Hd is assigned to the nodes of the cracked element, i.e., nodes 2 and 3. Therefore, standard and enriched node sets are, respectively, Ih = {1, 2, 3, 4} and Ihe = {2, 3}. There is at most one enrichment per node with mα = 1, α ∈ Ihe . Thus, from (4.3) with E21 = E31 ≡ Hd , the GFEM approximation of the axial displacement of the bar is given by   uh (x) = uˆ α ϕα (x) + ϕα (x)u˜ α1 Hd (x), (4.10) α∈Ih





std. FEM



α∈Ihe



 enriched GFEM



The Generalized Finite Element Method

87

which in matrix notation can be written as ⎡

uh (x) = N(x)U =



ϕ1

ϕ2

ϕ2 H d

ϕ3

ϕ3 H d

⎤ uˆ 1 ⎢ uˆ 2 ⎥ ⎢ ⎥  ⎢u˜ 21 ⎥ ⎥ ϕ4 ⎢ ⎢ uˆ 3 ⎥ , ⎢ ⎥ ⎣u˜ 31 ⎦

(4.11)

uˆ 4

where ϕα is a linear FE hat function. There are six DOFs before prescribing the support condition at the left end; these are organized in vector U on a node-by-node basis, with the standard FEM DOFs followed by the GFEM DOFs. The approximation of the displacement jump at x = x is given by     uh (l/2) = N(l/2) U = N (x+ ) − N(x− ) U       0 ϕ3 (l/2) Hd 0 U = 0 0 ϕ2 (l/2) Hd = u˜ 21 ϕ2 (l/2) + u˜ 31 ϕ3 (l/2),

(4.12)

  where we use Hd = Hd (x+ ) − Hd (x− ) = 1 from (4.9). The above equation gives a convenient way of computing the displacement jump at the crack location. The bilinear and linear forms for the problem in Fig. 4.6 are given, respectively, by (cf. Example 2.5)  l dv du k dx + v (x ) k u (x ) , 0 dx dx ¯ L (v) = v(l)N.

B(u, v) =

(4.13) (4.14)

Following a Galerkin approach, the same shape functions are used to approximate the displacement and the virtual displacement, i.e., v h (x) = N(x)V ,  with N defined in (4.11) and the virtual DOFs V = vˆ1 vˆ2 v˜21 vˆ3 v˜31 h h Introducing the approximations u and v into (4.13) and (4.14), we get B(uh , v h ) = V 

 l 0

B  kB dx U + V  N(x ) k N(x ) U ,      Ks

K h   L (v ) = V N (l)N¯ .

vˆ4



.

(4.15)

(4.16)

   F

From the weak form, the equilibrium equation is obtained as V  [(K + K s )U − F ] = 0 ∀ V

→

(K + K s )U = F ,

(4.17)

where the bar stiffness matrix K, the spring stiffness matrix K s , and the load vector F are shown in Eqs. (4.15) and (4.16). In the following, these arrays are computed element-by-element and assembled as in the standard FEM. Stiffness matrix calculation: Element (1): The GFEM approximation over element (1), connected to nodes 1 and 2 in Fig. 4.7, is given by ⎡ ⎤   uˆ 1 (1) (1) (1) (1) uh,(1) = ϕ1 ϕ2 ϕ2 H ⎣ uˆ 2 ⎦ = N (1) U (1) , d u˜ 21

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Fundamentals of Enriched Finite Element Methods

where the FEM shape functions in element (1), which has length l/4, are (1)

ϕ1 = 1 −

4x 4x (1) and ϕ2 = . l l

The Heaviside enrichment Hd over element (1) is identically zero; therefore, the GFEM shape function is (1)

(1)

also zero over this element, i.e., ϕ2 H = 0. d Using the above, the strain–displacement matrix is obtained as B (1) =

dN (1) 4 −1 = dx l

 0 ,

1

and the stiffness matrix is  l/4  k (1) = B (1) kB (1) dx 0

⎡

−1 1 0

1 ⎣−1 = 2 (l/4) 0 k

⎤ ⎡ 0  l/4 1 4k ⎣−1 dx = 0⎦ l 0 0 0

−1 1 0

⎤ 0 0⎦ . 0

Element (2): This element is connected to nodes 2 and 3. The GFEM approximation over the element is ⎡ ⎤ uˆ 2   ⎢u˜ ⎥ (2) (2) (2) (2) (2) (2) ⎢ 21 ⎥ h,(2) u = ϕ2 ϕ 2 H ϕ3 ϕ3 H ⎣ ⎦ = N (2) U (2) , d d uˆ 3 u˜ 31 where the FEM shape functions for element (2), which has length l/2, are (2)

ϕ2 = 1 −

2x − l/2 2x − l/2 (2) and ϕ3 = . l l (2)

The Heaviside enrichment over element (2), H , is given by (4.9). The strain-displacement matrix is given d by  ⎧  2 if x < x , dN (2) ⎨ l −1 0 1 0 (2)   (4.18) = B = ⎩ 2 dx if x > x . −1 −1 1 1 

l

Because B (2) is discontinuous across x = x , the local stiffness matrix is computed by breaking the integral over the element into two parts—one on each side of the crack. To wit,  l/2  3l/4   B (2) kB (2) dx + B (2) kB (2) dx k (2) = l/4

⎡

1 0 −1 4k ⎢ 0 0 0 ⎢ = 2 ⎣ −1 0 1 l 0 0 0 ⎡ 2 1 −2 k⎢ 1 1 −1 = ⎢ 2 l ⎣−2 −1 −1 −1 1

l/2

⎤ ⎡ 0 1 ⎥ 4k ⎢ 0⎥ l 1 ⎢ + 2 ⎣ 0⎦ 4 −1 l 0 −1 ⎤ −1 −1⎥ ⎥. 1⎦ 1

1 1 −1 −1

−1 −1 1 1

⎤ −1 −1⎥ ⎥l 1⎦ 4 1

Element (3): The GFEM approximation over element (3), connected to nodes 3 and 4, is given by ⎡ ⎤   uˆ 3 (3) (3) (3) (3) uh,(3) = ϕ3 ϕ 3 H ϕ4 ⎣u˜ 31 ⎦ = N (3) U (3) , d uˆ 4

The Generalized Finite Element Method

89

where the FEM shape functions for element (3), which has length l/4, are (3)

ϕ3 = 1 −

4x − 3l 4x − 3l (3) and ϕ4 = . l l (3)

The Heaviside enrichment over element (3) is H = 1. The strain–displacement and stiffness matrices d are, respectively, B (3) =

dN (3) 4 −1 = dx l

 1

−1

and k (3) =

 l 3l/4



B (3) kB 3 dx ⎡

1 ⎣ 1 = 2 (l/4) −1 k

⎤ ⎡ −1 1 l 4k ⎣ 1 −1⎦ = 4 l 1 −1

1 1 −1

1 1 −1

⎤ −1 −1⎦ . 1

Linear spring term and load vector: The global spring stiffness matrix and the global load vector are given by K s = N(x ) k N(x ) and F = N  (l)N¯ , respectively. This matrix and vector are computed and assembled element-by-element in the case of 2-D and 3-D problems. For the 1-D problem considered here, the above expressions can be used directly. The matrix N(x) with GFEM shape functions evaluated at x = l is, from (4.11), given by   N (l) = 0 0 0 0 0 1 . Thus, the global load vector is  F = N  (l)N¯ = 0 0

0

0

0

 N¯ .

Recall that a GFEM shape function is given by φαj = ϕα Eαj . Therefore, since the FEM shape functions are continuous, we get     φαj = ϕα Eαj . In the case of the Heaviside enrichment, as we saw earlier,   Hd = Hd (x+ ) − Hd (x− ) = 1. Using the above, and the definition of matrix N (x) given in (4.11), the matrix with the jump of GFEM shape functions is given by       0 ϕ3 (x ) Hd 0 , N(x ) = N (x+ ) − N(x− ) = 0 0 ϕ2 (x ) Hd and since ϕ2 (x ) = ϕ3 (x ) = 1/2, we get  N(x ) = 0 0 1/2 0 1/2

 0 .

The spring stiffness matrix is then given by ⎡

0 ⎢0 ⎢ ⎢0 K s = N(x ) k N(x ) = k ⎢ ⎢0 ⎢ ⎣0 0

0 0 0 0 0 0

0 0 1/4 0 1/4 0

0 0 0 0 0 0

0 0 1/4 0 1/4 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ 0⎦ 0

(4.19)

90

Fundamentals of Enriched Finite Element Methods

Assembly and solution: The element stiffness matrices k (e) , e = {1, 2, 3}, are assembled into the global stiffness matrix K = A3e=1 ke as in the finite element method. The global equilibrium equation (4.17) is given by

(4.20)

where the row and column corresponding to uˆ 1 are struck because of the Dirichlet boundary condition at uh (0) = 0, and R is the unknown reaction force at that node. The solution of the system of equations is   ¯ Nl N¯ N¯ N¯ l 3N¯ l N¯ U= 0 . (4.21) + 4k k 4k k k k It is noted that uˆ 4 is the displacement at the right end of the bar, u(l), and that uˆ 2 is the displacement at node 2. However, uˆ 3 is not the displacement at node 3 since that node is enriched with a function that is not zero at the node (see also the remark at the end of this example). From Eq. (4.10), the GFEM solution, shown also in Fig. 4.8, is given by uh (x) = ϕ1 uˆ 1 + ϕ2 uˆ 2 + ϕ3 uˆ 3 + ϕ4 uˆ 4 +    std. FEM approx. uˆ h (x)

Hd ϕ2 u˜ 21 + Hd ϕ3 u˜ 31   

enriched GFEM approx. Hd u˜ h (x)

= uˆ h (x) + Hd u˜ h (x),

(4.22)

where uˆ h (x) and u˜ h (x) are continuous functions defined using linear FEM shape functions. Functions uˆ h (x) and Hd u˜ h (x) approximate the continuous and discontinuous components of the exact solution, respectively. Chapter 6 generalizes this decomposition concept to fracture problems with singularities and to 2-D and 3-D problems.

Figure 4.8 GFEM/exact displacement of cracked bar with elastic spring and its decomposition into continuous and discontinuous components. GFEM approximation (4.22) is exact for this problem. It is worth noting that a standard FEM solution on a mesh with double nodes at the crack is also exact. Displacement jump at the crack: The axial displacement jump at x = x can be computed using (4.12): 

 1 N¯ 1 N¯ N¯ + = . uh (x ) = u˜ 21 ϕ2 (x ) + u˜ 31 ϕ3 (x ) = 2 k 2 k k

The Generalized Finite Element Method

91

This jump is equal to the displacement of the spring. It can also be computed using the decomposition of uh (x) given in (4.22), i.e.,       uh (x ) = uˆ h (x ) + Hd u˜ h (x ) = Hd (x ) u˜ h (x ) = u˜ h (x ). 

Remark 5. This problem can be solved using a single element (see Exercise 4.1). The displacement boundary condition at the left end of the bar (x = 0) can be enforced without difficulty since the Heaviside enrichment is zero at the node with the boundary condition, as shown in Fig. 4.7. If that were not the case, the GFEM approximation at x = 0 would be a linear combination of the standard and enrichment DOFs of the node at x = 0. The enforcement of the displacement boundary condition cannot be done by simply setting these DOFs to zero. This issue can be avoided by simply shifting the enrichment so that it is zero at the node it is assigned to, as shown in Section 4.4. See also Section 5.2 for the solution of this problem using the discontinuity-enriched FEM with a single finite element.

4.4 Shifted and scaled enrichments The enrichments adopted in Examples 4.2 and 4.3 have the desired approximation properties and allow the use of meshes that are not acceptable for the FEM. However, those enrichments may lead to ill-conditioned stiffness matrices, in particular when the material interface or the discontinuity is close to a mesh node. This is demonstrated in Chapter 11, together with strategies to control the conditioning of GFEM matrices. Another drawback of those enrichments is that depending on the adopted mesh and location of the crack or the material interface, they may be nonzero at mesh nodes with Dirichlet boundary conditions. This, as discussed in Examples 4.2 and 4.3, causes difficulties in the imposition of Dirichlet boundary conditions. One simple strategy to address the above boundary condition issue and to significantly improve the conditioning of GFEM matrices is to shift and scale the enrichment functions. Consider a generic enrichment Eα assigned to node xα of a mesh. The shifted and scaled enrichment function is given by Eα (x) − Eα (xα ) , E˜ α (x) = hα

(4.23)

where the scaling factor hα is given by the size of the largest element connected to node xα . In higher dimensions, the size of an element is taken as its diameter. The value of the enrichment function at the node it is assigned to, Eα (xα ), is used to shift it. As a result, the modified enrichment E˜ α evaluated at node xα is zero. The scaling factor hα leads to nondimensional enrichment functions, i.e., enrichments whose values are independent of the unity system adopted in the definition of the FE mesh. Shifting and scaling of polynomial and nonpolynomial enrichment functions were first proposed in [2,3,7] and [24], respectively. Shifting (only) of a broad class of enrichments was also proposed in [25].

92

Fundamentals of Enriched Finite Element Methods

As an example of the application of transformation (4.23), consider the singular enrichment function given by   3π 0.65 Eα (x) = x + sin x . (4.24) 2 Fig. 4.9 shows (at the top) the shifted and scaled enrichment functions at all nodes of the 1-D mesh shown in the figure. The modified enrichments are computed using (4.23). The FE shape functions ϕα (x), and the GFEM shape functions ϕα (x)E˜ α (x), α ∈ {1, . . . , 7}, are also shown in the figure. The GFEM shape functions based on the modified enrichments are zero at the mesh nodes. This facilitates the imposition of Dirichlet boundary conditions in the GFEM since the only nonzero shape function at a node is the FE shape function, as shown in the figure. Therefore, the coefficient of this function has the same physical meaning as in the standard FEM. By setting the value of this coefficient to a prescribed value, the GFEM approximation will be equal to this value at the mesh node (cf. Examples 4.2 and 4.3). However, it is noted that in 2-D or 3-D problems, this strategy to enforce Dirichlet boundary condi-

Figure 4.9 (Top) One-dimensional shifted and scaled enrichment functions. The enrichments are zero at the node they are assigned to; (Middle) FE shape functions; (Bottom) Generalized FEM shape functions.

The Generalized Finite Element Method

93

tions may not be able to enforce the prescribed values of the boundary condition along element edges or faces at a Dirichlet boundary. This can be addressed by, e.g., penalty [26] or Nitsche’s [18] methods.

4.5 The p-version of the GFEM This section presents high-order polynomial GFEM approximations [5–7,27] for onedimensional problems. One appealing feature of this class of high-order approximations is its implementation simplicity. The implementation in any dimension and for any type of element is conceptually identical and involves only vertex shape functions—in contrast with the p-hierarchical approximations of Chapter 3, which involve edge, face, and interior shape functions whose definitions depend on the shape of the finite element. However, there is a price for this simplicity—high-order GFEM shape functions are not linearly independent, as shown later in Theorem 4.6.2. We show how to deal with this issue in Chapter 11. High-order polynomial GFEM approximations are defined using (4.3) with the enrichment functions given by any complete set of polynomial functions. The p simplest possible polynomial basis of degree p in 1-D is the set of monomials x i i=1 . However, as discussed in Section 3.1, this basis leads to ill-conditioned matrices. One way to lessen this issue is to scale and shift the polynomial functions using (4.23). This leads to the following set of enrichments for a node xα [5–7]:  x − xα (x − xα )2 (x − xα )3 (x − xα )p , (4.25) Eαj (x) = , , , . . . , p hα h2α h3α hα where p is the highest polynomial order of the enrichments and the scaling factor hα is taken as the size of the largest element connected to node xα . We adopt onedimensional monomials in (4.25). Other polynomial bases, such as Legendre polynomials (cf. Section 3.1), can be adopted as well. Using enrichment functions (4.25) in (4.3) leads to a p-hierarchical GFEM akin to the p-hierarchical FEM presented in Chapter 3. This GFEM approximation is given by up (x) =

 α∈Ih



uˆ α ϕα (x) + 

linear FEM



 α∈Ihe



ϕα (x)

mα 

u˜ αj

(x − xα )j

j =1



j

hα

,

(4.26)



p-enriched GFEM

where mα is the highest polynomial order of the enrichments at the node with index α. The corresponding GFEM shape functions are  x − xα (x − xα )2 (x − xα )3 (x − xα )p φαj (x) = ϕα (x) , (4.27) , , , . . . , p hα h2α h3α hα where p = mα . Fig. 4.10 shows 1-D GFEM shape functions of degree 5 or less and their first-order derivatives. If the partition of unity is taken as linear FE shape functions, from (4.27) it is clear that the GFEM shape functions are of degree less than or

94

Fundamentals of Enriched Finite Element Methods

equal to p + 1. Theorem 4.6.1 shows that GFEM shape functions (4.27) can exactly represent any polynomial function of degree less than or equal to p + 1.

Figure 4.10 Hierarchical polynomial GFEM shape functions at a node xα (top) and their derivatives (bottom).

Example 4.4: Bar with varying cross-section. Consider the one-dimensional bar shown in Fig. 4.11. It has a length l = 1 m, is fixed at the left end, and is subjected to an axial force N¯ = 1 kN at the right end. The bar behaves linearly elastic with modulus of elasticity E = 70 GPa. The bar cross-section varies linearly from A0 = 10 mm2 at the right end to A = 7A0 at the fixed support. Find the analytical solution of this problem. Then compute the finite element solution with three linear elements of equal length and also the generalized finite element solution. For the latter, use linear enrichments.

Figure 4.11 Bar with varying cross-section. Solution: Analytical: The cross-section of the bar, as a function of position, can be written as A(x) = 7A0 − 6A0 x/ l. Given that the strain = N¯ /(EA(x)), the exact displacement field can simply be found by integrating the strain along the bar. To wit,    ¯ 6x Nl + C. (4.28) ln 7 − u(x) = (x) dx = − 6EA0 l l The constant of integration is found considering the boundary condition of zero displacement at the fixed support. Then, the exact displacement field is " ! ¯ Nl 1 . (4.29) u(x) = ln 6EA0 1 − 6x 7l

The Generalized Finite Element Method

95

The exact strain energy needed to deform the bar is equal the work of the external forces, namely 1/2 N¯ times the displacement at the end of the bar. Thus, the strain energy is given by U=

1 ¯ Nu(l) = 0.231 656 N m. 2

(4.30)

Finite element approximation: We now seek an approximate solution using three linear finite elements of equal length. The stiffness matrix for the ith element with nodes {xi , xj } is ki =

 x j

#

− l1

$

e

EA(x)

1 le

xi

 − l1

1 le

e



dx =

 E A¯ 1 le −1

 −1 , 1

(4.31)

  where le = xj − xi and A¯ = Ai + Aj /2, with Ai and Aj the cross-section areas at xi and xj , respectively. Therefore, we note that this is the stiffness matrix of an element of uniform cross-section, with an area equal to the average between the cross-sections at the ends of the element. After assembling the contributions of all elements, the global arrays are ⎡ ⎤ ⎡ ⎤ −R 18 −18 0 0 ⎢ ⎥ EA0 ⎢ 30 −12 0⎥ ⎢−18 ⎥ and ⎢ 0 ⎥ . K= ⎣ 0 ⎦ 18 −6⎦ l ⎣ 0 −12 0 0 −6 6 N¯ After prescribing the essential boundary condition at the left end (eliminating the first row and column), the system KU = F has a unique solution given by U=

N¯ l  0 EA0

1 18

5 36

11 36



.

The strain energy for the finite element solution is U=

1 ¯ h Nu (l) = 0.218 254 N m, 2

(4.32)

which has a relative error of −5.79%. Notice that this value is negative, so the energy of the approximation is lower than the exact energy value. In other words, for this problem, the strain energy converges from below. Generalized finite element approximation: The linear FEM shape functions provide a partition of unity for the GFEM. Enrichment functions are usually chosen based on any known behavior of the exact solution. Instead of using a logarithmic function, however, here we approximate the solution using a polynomial enrichment, namely the function x/ l. It is noted that no shifting or scaling is adopted. With this choice, the vector of FEM and GFEM shape functions for an element e with nodes {xi , xj } is given by    x −x x−xi x (xj −x) x x−xi . N (e) = j (4.33) le

le

l

le

l

le

The element stiffness matrix is then   k ss k sg , k (e) =  k sg k gg where k ss =

 E A¯ 1 le −1

 −1 1

is the stiffness matrix produced by standard FEM shape functions, and           E 2 2Ai + Aj xi + Aj − Ai xj Aj − Ai xi − 2 Ai + 2Aj xj     , k sg = ¯ i 2 Ai + 2Aj le + 6Ax 6le Ai − Aj xj − 2 2Ai + Aj xi

96

Fundamentals of Enriched Finite Element Methods

k ss =

#   ¯ 2 E 2 3Ai + Aj xi2 − 4Ai xi xj + 2Ax j ¯ i xj + 2Aj x 2 −2Ai xi2 + 2Ax 6le j

$ ¯ i xj + 2Aj x 2 −2Ai xi2 + 2Ax j  ¯ 2 − 4Aj xi xj + 2Ai + 6Aj x 2 . 2Ax i j

The matrices for each element are then given by Element (1): ⎡

18 ⎢ ⎢ EA0 ⎢ −18 k (1) = 1 l ⎢ ⎣ −3 − 17 3

−18

− 13

18

1 3 2 3 − 59

1 3 17 3

− 17 3

⎤

17 ⎥ ⎥ 3 ⎥, 5 −9⎥ ⎦ 22 9

Element (2): ⎡

12 ⎢ ⎢ EA0 ⎢ −12 k (2) = 11 l ⎢ ⎣ 3 23 −3

−12 12 − 11 3 23 3

11 3 − 11 3 14 9 − 25 9

− 23 3

11 3 − 11 3 22 9 − 11 3

− 17 3

⎤

23 ⎥ ⎥ 3 ⎥, 25 −9⎥ ⎦ 16 3

Element (3): ⎡

6 ⎢ ⎢ −6 EA0 ⎢ k (3) = 11 l ⎢ ⎣ 3 17 −3

−6 6 − 11 3 17 3

⎤

17 ⎥ ⎥ 3 ⎥. 11 −3⎥ ⎦ 50 9

The assembled global stiffness matrix is ⎡ 18 −18 0 0 ⎢ ⎢ −18 30 −12 0 ⎢ ⎢ 18 −6 ⎢ 0 −12 ⎢ ⎢ 0 0 −6 6 EA0 ⎢ K= ⎢ 1 l ⎢ −1 0 0 3 ⎢ 3 ⎢ 17 11 28 ⎢− 3 − 0 3 3 ⎢ ⎢ 0 − 23 34 − 11 ⎣ 3 3 3 17 0 0 − 17 3 3

− 13

− 17 3

0

1 3

28 3 11 −3

− 23 3

0

0

34 3 − 11 3

2 3 5 −9

− 59

0

4

− 25 9

0

− 25 9

0

0

70 9 − 11 3

0

0

⎤

⎥ 0⎥ ⎥

⎥ − 17 3 ⎥ ⎥ 17 ⎥ 3 ⎥, ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 11 −3⎥ ⎦ 50 9

where we recognize the standard finite element portion on the top left. The global force vector is given by   F = −R 0 0 N¯ 0 0 0 l N¯ . The global stiffness matrix is singular even after the imposition of the displacement boundary condition since the shape functions are linearly dependent, as shown later in Theorem 4.6.2. One solution of such system can be obtained by perturbing the stiffness matrix to make it positive definite and then apply the iterative algorithm presented in Chapter 11, Section 11.1.5, to find a solution of the unperturbed system. Alternatively, in the case of a small problem like this, a pseudoinverse of K can be calculated, for instance, by means of a singular value decomposition so that K = U V  , where  is a diagonal matrix that stores the singular values [28]. The solution vector can then be expressed as U=

 u y i σi =0

σi

vi ,

where y ∈ span K (which can be obtained, for instance, by multiplying the stiffness matrix by a unitary vector). For this problem, one solution is obtained as

The Generalized Finite Element Method

U=

N¯ l 10−4 EA0  0 1.05947

1.92577

97

0.58151

−1.17573

−0.775192

0.136662

4.03277



,

which can be verified since KU − F = 0. The displacement at the right end of the bar is uh (l) =  N¯ l/EA0 0.000461428, which gives an approximation of the strain energy U = 1/2N¯ uh (l) = 0.230714 N m. This value has a relative error of −0.41%.

Figure 4.12 Normalized displacement (left) and stress (right) obtained using the FEM and the GFEM, compared to the exact solution given by (4.29). Fig. 4.12 shows the FEM and GFEM approximations compared to the analytical solution. The left figure shows the normalized displacement field so that it does not depend on Young’s modulus, the area at the right end, A0 , and the magnitude of the prescribed load. Similarly, the right figure shows the stress field. For the finite element solution, the stress field (derivative of the displacement) is piecewise constant. Notice that while the displacement approximation is quite accurate, the stress field is accurate only at element midpoints. Noteworthy, the GFEM piecewise linear stress field is a drastic improvement over the standard FEM field. 

4.5.1 High-order GFEM approximations for a strong discontinuity This section presents high-order GFEM approximations for problems with a strong discontinuity in a 1-D setting. Extensions to higher-dimensional problems are presented in Chapter 6. The starting point is the GFEM approximation (4.22) derived in Example 4.3 and given by uh (x) = uˆ h (x) + Hd (x)u˜ h (x)   uˆ α ϕα (x) + Hd (x) u˜ α ϕα (x), = α∈Ih

(4.34) (4.35)

α∈Ihe

where uˆ h (x) and u˜ h (x) are continuous functions, Hd is the Heaviside function defined in (4.9), Ihe is the set of nodes enriched with this function, and uˆ α and u˜ α are degrees of freedom. Set Ihe is taken as the set of nodes whose support intersects the discontinuity (cf. Example 4.3). The support of a node α is simply the support ωα of ϕα . Functions uˆ h (x) and u˜ h (x) are taken as linear FEM approximations in (4.35). In this section, high-order GFEM approximations, as defined in Section 4.5, are used instead.

98

Fundamentals of Enriched Finite Element Methods

The p-hierarchical GFEM approximation (4.26) can be written as up (x) =



ϕα (x)

α∈Ih

=



ϕα (x)

mα  j =0 mα 

u˜ αj

(x − xα )j j

hα

u˜ αj Eαj (x),

(4.36)

j =0

α∈Ih

with u˜ α0 ≡ uˆ α and Eα0 (x) = 1. Thus, if mα = 0, the only shape function at node α is ϕα (x). Using (4.36) in the decomposition of the GFEM approximation given in (4.34), a high-order GFEM approximation of a function with a strong discontinuity is defined as up (x) = uˆ p (x) + Hd (x)u˜ p (x) =



ϕα (x)

mα 

uˆ αj Eαj (x) + Hd (x)

j =0

α∈Ih



ϕα (x)

α∈Ihe

mα 

u˜ αj Eαj (x), (4.37)

j =0

where Ihe is the set of nodes enriched with a discontinuous function, uˆ αj and u˜ αj are GFEM degrees of freedom, and Eαj are the same polynomial enrichments used in (4.36). Function Hd (x)u˜ p (x) can also be written as Hd (x)u˜ p (x) = Hd (x)



ϕα (x)



ϕα (x)

 α∈Ihe

u˜ αj Hd (x)Eαj (x)

j =0

α∈Ihe

=

mα 

u˜ αj Eαj (x)

j =0

α∈Ihe

=

mα 

ϕα (x)

mα 

H u˜ αj Eαj (x),

j =0

where  x − xα (x − xα )2 (x − xα )p H Eαj (x) = Hd , Hd , Hd , . . . , H  p d hα h2α hα

(4.38)

are high-order discontinuous enrichment functions [29,30]. Using these functions, the GFEM approximation (4.37) can be written in terms of high-order continuous and discontinuous GFEM shape functions. To wit, up (x) =

mα  α∈Ih j =0

uˆ αj ϕα (x)Eαj (x) +    cont. sh. fn.

mα  α∈Ihe

H u˜ αj ϕα (x)Eαj (x) .    j =0 discont. sh. fn.

(4.39)

The Generalized Finite Element Method

99

This equation shows that the GFEM approximation up (x) is p-hierarchical, like in the p-version of the GFEM presented in Section 4.5. If mα = 0, α ∈ Ih , Eq. (4.39) reduces to the linear GFEM approximation (4.35) with uˆ α0 ≡ uˆ α and u˜ α0 ≡ u˜ α since H Eα0 (x) = 1 and Eα0 (x) = Hd (x). The GFEM approximation introduced in this section brings the benefits of highorder approximations, like high convergence rates and accuracy, to problems with a discontinuous solution. This type of approximation can be used to solve fracture mechanics problems, as shown in Chapter 6. Its extension to 2-D and 3-D fracture problems is presented in Section 6.2.3.1.

4.6 GFEM approximation spaces This section introduces the concept of GFEM approximation spaces and presents two theorems that are useful for the understanding of high-order approximations. A more in-depth mathematical analysis of the method is presented in Chapter 13. The GFEM approximation defined in (4.3) belongs to the GFEM space SGFEM = ShFEM + SeGFEM ,

(4.40)

where ShFEM is the space defined using FEM approximation (4.1), i.e.,  ShFEM

=

%  u % uh = ϕα Uα , h%

& Uα ∈ R ,

(4.41)

α∈Ih

with R denoting the set of real numbers. The GFEM enrichment space is given by  mα %   % e SGFEM = uh % uh = ϕα (x) u˜ αj Eαj (x) j =1

α∈Ihe

=



mα 

α∈Ihe

j =1

u˜ αj φαj (x),

& u˜ αj ∈ R .

(4.42)

Definition (4.40) shows again that a GFEM space SeGFEM is obtained by hierarchically enriching a standard FEM space, ShFEM , with GFEM shape functions. Theorem 4.6.1. Let SGFEM = ShFEM + SeGFEM be the GFEM approximation space over a domain  with ShFEM spanned by 1-D linear FE shape functions and SeGFEM spanned by GFEM shape functions defined in (4.27). Then SGFEM = Pp+1 where p is the highest polynomial order of the enrichments and Pp+1 denotes the space of 1-D polynomials of degree less than or equal to p + 1.

100

Fundamentals of Enriched Finite Element Methods

Proof. In order to simplify the notation, we consider here the case in which no shifting or scaling is applied. The enrichment functions (used at all mesh nodes) are therefore given by ( ' Ej (x) = x, x 2 , x 3 , . . . , x p . (4.43) From the definition of linear FE shape functions, the following holds:  ϕα (x) = 1, α∈Ih



xα ϕα (x) = x.

(4.44)

α∈Ih

Therefore {1, x} ⊂ SGFEM . Furthermore, for any q ≤ p, using the definition of GFEM shape functions with enrichments (4.43):     xα ϕα (x)x q = x q xα ϕα (x) = x q+1 . α∈Ih

α∈Ih

Therefore, x q+1 ∈ SGFEM for any q ≤ p. The same approach presented above to define high-order GFEM shape functions can be used in two- and three-dimensional spaces and also for any type of finite element: triangles, quadrilaterals, tetrahedrons, etc. [5–7]. This makes the implementation of this class of high-order approximation much simpler than in the case of p-hierarchical FEM. The polynomial enrichments at the nodes can be nonisotropic (different polynomial orders in different directions), regardless of the choice of finite element partition of unity (hexahedron, tetrahedron, etc.). Examples of p-orthotropic approximations built on a tetrahedron mesh are given in [7,31]. Significant reduction in the number of degrees-of-freedom relative to isotropic approximations can be achieved in problems exhibiting strong gradients in only one direction [31]. On the other hand, the use of polynomial enrichments and an FE partition of unity may lead to GFEM shape functions that are linearly dependent. The proof of the linear dependence for 1-D GFEM shape functions is given in the next theorem (the threedimensional case is treated in [7]). Theorem 4.6.2. The basis functions of the GFEM space defined in Theorem 4.6.1 are linearly dependent. Proof. Adopting enrichments (4.43), the linear combination of GFEM shape functions φα1 (x) = ϕα (x)x gives   ϕα (x) = x. (ϕα (x)x) = x α∈Ih

α∈Ih

Using the above and (4.44) gives   xα ϕα (x) − (ϕα (x)x) = 0. α∈Ih

α∈Ih

.

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101

4.7 Exercises PROBLEM 4.1.— GFEM SOLUTION FOR A ONE-DIMENSIONAL CRACKED BAR Solve the problem described in Example 4.3 using a single element. Compute the displacement at the right end of the bar and the displacement jump at the crack. Compare your solution with the one shown in Fig. 4.8. Show the decomposition of the solution as in Fig. 4.8. PROBLEM 4.2.— GFEM SOLUTION FOR A ONE-DIMENSIONAL BAR WITH A PRESSURIZED CRACK Solve the problem described in Example 4.3, but with the following modifications: Both ends of the bar are fixed, the spring at the discontinuity is removed, and a pressure p is applied at the discontinuity. HIGH-ORDER GFEM APPROXIMATIONS FOR A WEAK DISCONTINUITY PROBLEM 4.3.— Derive a high-order GFEM approximation for problems with a weak discontinuity in a 1-D setting. Solution: The derivation follows the same steps as in Section 4.5.1. To wit, up (x) =

mα   α∈Ih j =0

uˆ αj ϕα (x)Eαj (x) +    cont. sh. fn.

mα   α∈Ihe j =0

u˜ αj

R (x) , ϕα (x)Eαj   

(4.45)

weak discont. sh. fn.

where Eα0 (x) = 1, Eαj (x) is defined in (4.25) for j > 0, and  & x − xα (x − xα )2 (x − xα )p R , Rr , . . . , Rr Eαj (x) = Rr , Rr p hα h2α hα

(4.46)

with p = mα and Rr (x) defined in (4.8). Other enrichments for weak discontinuities can be used [22].

References [1] J. Melenk, I. Babuška, The partition of unity finite element method: Basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 (1996) 289–314. [2] C. Duarte, J. Oden, Hp clouds—an hp meshless method, Numerical Methods for Partial Differential Equations 12 (1996) 673–705, https://doi.org/10.1002/(SICI)10982426(199611)12:63.0.CO;2-P. [3] C. Duarte, J. Oden, An h–p adaptive method using clouds, Computer Methods in Applied Mechanics and Engineering 139 (1) (1996) 237–262, https://doi.org/10.1016/S00457825(96)01085-7. [4] T. Strouboulis, I. Babuška, K. Copps, The design and analysis of the generalized finite element method, Computer Methods in Applied Mechanics and Engineering 81 (1–3) (2000) 43–69. [5] J. Oden, C. Duarte, O. Zienkiewicz, A new cloud-based hp finite element method, Computer Methods in Applied Mechanics and Engineering 153 (1998) 117–126, https:// doi.org/10.1016/S0045-7825(97)00039-X. [6] J. Oden, C. Duarte, Chapter: Clouds, cracks and FEMs, in: B. Reddy (Ed.), Recent Developments in Computational and Applied Mechanics, International Center for Numerical Methods in Engineering, CIMNE, Barcelona, Spain, 1997, pp. 302–321, http:// gfem.cee.illinois.edu/jmartincolor/. [7] C. Duarte, I. Babuška, J. Oden, Generalized finite element methods for three dimensional structural mechanics problems, Computers & Structures 77 (2000) 215–232, https:// doi.org/10.1016/S0045-7949(99)00211-4. [8] T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (1999) 601–620.

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[9] J. Dolbow, N. Moës, T. Belytschko, Discontinuous enrichment in finite elements with a partition of unity method, Finite Elements in Analysis and Design 36 (2000) 235–260. [10] T. Belytschko, R. Gracie, G. Ventura, A review of extended/generalized finite element methods for material modeling, Modelling and Simulation in Materials Science and Engineering 17 (2009) 043001, https://doi.org/10.1088/0965-0393/17/4/043001. [11] B.A. Szabó, I. Babuška, Finite Element Analysis, John Wiley and Sons, New York, 1991. [12] A. Byfut, A. Schroder, hp-adaptive extended finite element method, International Journal for Numerical Methods in Engineering 89 (2012) 1392–1418. [13] A. Sanchez-Rivadeneira, C. Duarte, A stable generalized/extended FEM with discontinuous interpolants for fracture mechanics, Computer Methods in Applied Mechanics and Engineering 345 (2019) 876–918, https://doi.org/10.1016/j.cma.2018.11.018. [14] A. Sanchez-Rivadeneira, N. Shauer, B. Mazurowski, C. Duarte, A stable generalized/extended p-hierarchical FEM for three-dimensional linear elastic fracture mechanics, Computer Methods in Applied Mechanics and Engineering 364 (2020) 112970, https:// doi.org/10.1016/j.cma.2020.112970. [15] I. Babuška, G. Caloz, J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM Journal on Numerical Analysis 31 (4) (1994) 945–981, https://doi.org/10.1137/0731051. [16] P. Lancaster, K. Salkauskas, Surfaces generated by moving least squares methods, Mathematics of Computation 37 (155) (1981) 141–158. [17] S. De, K. Bathe, The method of finite spheres, Computational Mechanics 25 (2000) 329–345. [18] M. Griebel, M. Schweitzer, A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic PDEs, SIAM Journal on Scientific Computing 22 (3) (2000) 853–890. [19] R. Taylor, O. Zienkiewicz, E. Onate, A hierarchical finite element method based on the partition of unity, Computer Methods in Applied Mechanics and Engineering 152 (1998) 73–84. [20] K. Terada, M. Asai, M. Yamagishi, Finite cover method for linear and nonlinear analyses of heterogeneous solids, International Journal for Numerical Methods in Engineering 58 (2008) 1321–1346. [21] G. Shi, Manifold method of material analysis, in: Transactions of the 9th Army Conference on Applied Mathematics and Computing, US Army Research Office, 1991, Report No. 92-1. [22] A.M. Aragón, C.A. Duarte, P.H. Geubelle, Generalized finite element enrichment functions for discontinuous gradient fields, International Journal for Numerical Methods in Engineering 82 (2) (2010) 242–268, https://doi.org/10.1002/nme.2772. [23] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) (1999) 131–150. [24] F. Barros, S. Proenca, C. Barcellos, On error estimator and p adaptivity in the generalized finite element method, International Journal for Numerical Methods in Engineering 60 (14) (2004) 2373–2398. [25] G. Zi, T. Belytschko, New crack-tip elements for XFEM and applications to cohesive cracks, International Journal for Numerical Methods in Engineering 57 (2003) 2221–2240. [26] J. Chessa, P. Smolinski, T. Belytschko, The extended finite element method (XFEM) for solidification problems, International Journal for Numerical Methods in Engineering 53 (2002) 1959–1977.

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[27] C. Duarte, I. Babuška, J. Oden, Generalized finite element methods for three dimensional structural mechanics problems, in: S. Atluri, P. O’Donoghue (Eds.), Proceedings of the International Conference on Computational Engineering Science. Volume I. Modeling and Simulation Based Engineering, Atlanta, GA, October 5–9, 1998, Tech Science Press, Oct. 1998, pp. 53–58. [28] G. Golub, C. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, 1996. [29] J. Pereira, C. Duarte, D. Guoy, X. Jiao, Hp-generalized FEM and crack surface representation for non-planar 3-D cracks, International Journal for Numerical Methods in Engineering 77 (5) (2009) 601–633, https://doi.org/10.1002/nme.2419. [30] C. Duarte, L. Reno, A. Simone, A high-order generalized FEM for through-the-thickness branched cracks, International Journal for Numerical Methods in Engineering 72 (3) (2007) 325–351, https://doi.org/10.1002/nme.2012. [31] C. Duarte, I. Babuška, Mesh-independent p-orthotropic enrichment using the generalized finite element method, International Journal for Numerical Methods in Engineering 55 (12) (2002) 1477–1492, https://doi.org/10.1002/nme.557.

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Discontinuity-enriched finite element formulations

5

The previous chapter introduced the approximation theory of GFEM. While giving tremendous flexibility, many properties that are taken for granted in standard FEM are often lost in GFEM. Prescribing non-homogeneous essential boundary conditions, for example, requires shifting so that enrichments vanish at the location of enriched nodes or mandates for more sophisticated procedures (e.g., penalty or Lagrange multiplier formulations). Another issue pertains  to the condition number of the system matrix, which grows in h-FEM as O h−2 , but which can grow unbounded in GFEM with certain choices of enrichment functions. Ongoing efforts therefore pursue GFEM formulations—and enrichment functions in particular—that result in a stable GFEM (SGFEM) [1–3] as discussed in Chapter 11. Another major issue of GFEM is the fact that, even with stabilization, it is currently not possible to recover smooth reactive tractions in Dirichlet boundaries [4,5]. It was shown that in GFEM the partition of unity shape functions localize the behavior of enrichments, thereby keeping the sparsity of the discretized system of equations. However, this requirement can be relaxed if enrichment functions are local by construction. For instance, in the analysis of problems with discontinuities, the required behavior is confined to lower-dimensional manifolds—e.g., lines in 2D or surfaces in 3-D. For these, jumps in the field or its gradient, i.e., strong or weak discontinuities, respectively, can be introduced in the formulation by means of enrichments with the corresponding discontinuity kinematics. And because such kinematics are confined to discontinuities, enrichments localized to cut elements can be constructed and associated to enriched nodes that are placed directly along discontinuities (as opposed to GFEM, where enrichments are associated to nodes of the original discretization). This is the rationale behind discontinuity-enriched finite element methods. In this chapter we introduce the Interface-enriched Generalized Finite Element Method (IGFEM) [6] and the Discontinuity-Enriched Finite Element Method (DEFEM) [7] as alternative methodologies to GFEM for modeling weak and strong discontinuities, respectively. We show that both methods—which can actually be derived from GFEM as shown later in this chapter—not only keep the most salient feature of GFEM, i.e., decoupling discontinuities from the finite element discretization, but also all appealing features of standard FEM. As done in previous chapters, we start the exposition in 1-D before moving to the formulation in higher dimensions. A hierarchical enrichment procedure is also described, whereby several discontinuities that interact within a single finite element can be resolved properly. Finally, we discuss the issue of stress overestimation when discontinuities approach standard nodes, and it is shown that accurate stresses can be obtained by post-processing the displacement field. Fundamentals of Enriched Finite Element Methods. https://doi.org/10.1016/B978-0-32-385515-0.00012-X Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved.

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5.1 A weak discontinuity in 1-D We revisit the problem in Exercise 2.1, later solved using GFEM in Exercise 4.2. Here we solve the problem with a weak discontinuity using a single finite element and an enriched node α associated to an enrichment function ψ that introduces the kinematics of a weak discontinuity, i.e., a jump in the field gradient. Consider a trial solution of the form uh (x) =

2  i=1

ϕi (x)uˆ i + ψ(x)α, 

  

standard FEM

enrichment





(5.1)

where the standard FEM component consists of standard DOFs uˆ i associated with Lagrange shape functions ϕi ; the enrichment component has a single enriched DOF α associated with the enrichment function ψ=

(1)

ϕi (x)

for x ≤ x ,

(2) ϕi (x)

for x ≥ x ,

(1)

(5.2)

(2)

where ϕi (x) = x/x and ϕi (x) = (L − x)/(L − x ) are the Lagrange shape functions of subdomains (also integration elements) to the left and right of the discontinuity, respectively (see Fig. 5.1).

Figure 5.1 Bimaterial 1-D bar (top), and shape (middle) and enrichment (bottom) functions acting on a single finite element.

We define  = ϕ1 are

ϕ2

 ψ and B =

d dx ,

so the stiffness matrix and force vector

Discontinuity-enriched finite element formulations

K= 0

=

x

⎡

k1 B  B dx +



L

107

k2 B  B dx

x

k2 − w k

1⎢ ⎢ ⎢−k2 + w k L⎣ k

−k2 + w k

k

k2 − w k

− k

− k



k1 w

+

k2 1−w

⎤ ⎥ ⎥ ⎥ ⎦

  and F = R N¯ 0 , respectively, where k ≡ k2 − k1 , R is the reaction force at the clamped end, and dashed lines subdivide arrays in free and prescribed components. The system KU = F has solution U=



¯

¯

w Nk1L + (1 − w) Nk2L

0

¯

w(1 − w) k kN1 kL2



,

(5.3)

and the exact displacement field obtained earlier in Exercises 2.1 and 4.2 is once again recovered: ¯ Nx for x ≤ x , (5.4) uh (x) = Nk¯ 1x N¯ (x−x )  for x ≥ x . k1 + k2

5.2 A strong discontinuity in 1-D Here we solve the problem in Example 2.2 with a discontinuity-enriched formulation. The problem consists of a 1-D bar of length L clamped at the end and subjected to a load N¯ at the right end. The bar has constant axial stiffness k = EA, and is split at x = L/2 in two parts that are reattached via a spring of constant k . In Example 2.2, the exact displacement found was ¯ N x for x < x , h u (x) = EA (5.5) N¯ N¯ for x > x , EA x + k which has a jump u ≡ u(x+ ) − u(x ) = N¯ /k . We now solve the problem with a single finite element employing a solution of the form u (x) = h

2  i=1



ϕi (x)uˆ i + χ(x)β. 



standard FEM

(5.6)

   enrichment

Eq. (5.6) shows a structure similar to (5.1)—the standard FEM component remains the same—but the weak enrichment has been replaced by one that has the kinematics of a displacement jump (a strong discontinuity). This enrichment function χ (see Fig. 5.2), which is associated with enriched DOF β, can simply be constructued by

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Fundamentals of Enriched Finite Element Methods

Figure 5.2 Split bar showing standard (middle) and enrichment (bottom) functions over a single element.

using the standard FEM shape functions as −ϕ2 (x) for x < x , χ= ϕ1 (x) for x > x .

(5.7)

Due to the partitionof unity property, χ has a displacement jump of unit magnitude    at x , i.e., χ = χ x+ − χ x− = 1. This is important because the enriched DOF represents the displacement jump at x = x —i.e., the enriched DOF has a physical meaning. Because integrating discontinuous enrichment functions is usually done in EFEA by means of integration elements, the enrichment function χ can also be expressed in terms of Lagrange shape functions in these integration subdomains (see Fig. 5.1): (1) −wϕi (x) for x < x , χ= (5.8) (2) (1 − w) ϕi (x) for x > x , where w = x /L (the position of the discontinuity relative to the length). Notice that (5.7) and (5.8) are equivalent. With the discontinuous enrichment defined we can now proceed to solve the prob



 lem. Thus, defining M = 0 0 1 ,  = ϕ1 ϕ2 χ , and B = d/dx, the stiffness matrix and the force vector are given by ⎤ ⎡ 1 −1 1 L ⎥ k⎢ ⎥ ⎢ (5.9) K= kB  B dx + k M  M = ⎢−1 1 −1 ⎥ , ⎦ ⎣ L 0 k L 1 −1 k +1

Discontinuity-enriched finite element formulations

109

  and F = R N¯ 0 , respectively. Notice that the strong enrichment function χ is equal to zero at the left end; therefore, prescribing the essential boundary condition is as straightforward as in standard FEM. The system KU = F has solution U=



N¯ L k

0

+

N¯ k

N¯ k



(5.10)

,

and thus the approximation recovers the exact displacement given by (5.5). Approximations (5.1) and (5.6) are notably simpler than those obtained by the GFEM approximation (4.3). In these simple examples, the partition of unity is not used to localize the effect of enrichment functions since these are local by construction. Furthermore, both weak and strong enrichment functions ramp linearly to zero at mesh nodes so standard FEM shape functions keep the Kronecker delta property. This makes it straightforward to prescribe non-homogeneous essential boundary conditions. Eq. (5.1) is the simplest instance of the Interface-enriched Generalized Finite Element Method (IGFEM), which was proposed by Soghrati et al. [6] for solving problems with discontinuous gradient fields. The approximation (5.6) then introduced the enrichment function used in the Discontinuity-Enriched Finite Element Method (DEFEM) [7], which generalizes IGFEM for solving problems that contain both weak and strong discontinuities—and with a unified formulation. Notice, however, that because the axial stiffness of the bar in the strong discontinuity example was assumed to be constant, there was no jump in the gradient (in the strain) at x . In the general case, an approximation of the form (5.6) would not recover two independent kinematic fields because there are simply not enough DOFs—four degrees of freedom would be required. The general discontinuity-enriched finite element approximation for 1-D can be written as

uh (x) =

 i∈ιh



strong

weak

      ϕi (x)uˆ i + ψi (x)αi + χi (x)βi , 



standard FEM

i∈ιw





i∈ιs

(5.11)



enrichment

where we now use ιh , ιw , and ιs to denote index sets associated with standard, weak, and strong nodes, respectively. The enrichment term has both weak and strong enrichments; while the former is able to describe a jump in the field gradient, adding the latter enhances the approximation so it can capture jumps in the field itself. A 1-D example that combines both weak and strong discontinuities was solved in the article that introduced DE-FEM [7].

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Fundamentals of Enriched Finite Element Methods

5.3 Relationship to GFEM The one-dimensional GFEM approximation, from § 4.2, was given by Eq. (4.3): u (x) = h



uˆ i ϕi (x) +



ϕi (x)

u˜ ij Eij (x).

(5.12)

j =1

i∈Ihe

i∈Ih

mi 

For a single finite element discretization containing a strong discontinuity, we can write the GFEM approximation with two enrichment functions as uh (x) =

2 

uˆ i ϕi (x) +

i=1

2 

ϕi (x)

2 

u˜ ij Eij (x),

(5.13)

j =1

i=1

where we have chosen mi = 2 and thus we enrich each standard node with two functions. From this equation, it is possible to recover (5.11) by an appropriate choice of enrichments Eij and by clustering DOFs. The enrichment term in (5.13) can be written as 2 

ϕi (x)

i=1

2  j =1

u˜ ij Eij = (ϕ1 E11 + ϕ2 E21 ) α + (ϕ1 E12 + ϕ2 E22 ) β,       ψ

(5.14)

χ

where enriched DOFs are clustered, i.e., u˜ 11 = u˜ 21 = α and u˜ 12 = u˜ 22 = β. The enrichment functions ψ and χ in the DE-FEM approximation (5.11) can be recovered by means of properly scaled Heaviside GFEM enrichments. Referring to Fig. 5.3, where the local coordinate along the element is ξ = x/L, ψ = ϕ1 c11 H (ξ − w) + ϕ2 c21 H (w − ξ ) ,

(5.15)

χ = ϕ1 c12 H (ξ − w) + c22 ϕ2 H (w − ξ ) ,

(5.16)

and the scaling parameters are determined as a function of the discontinuity location w = x /L: c11 =

1 , 1−w

c21 =

1 , w

c12 = −1, and c22 = 1.

(5.17)

Eq. (5.11) is then retrieved from GFEM. However, notice that since the scaling parameters are (almost) arbitrary, (5.15) and (5.16) represent a family of enrichment functions—and thus a family of discontinuity-enriched formulations. Example 5.1: p-IGFEM solution for bimaterial beam. Recalling Example 3.3, consider in Fig. 5.4 a cantilever beam subjected to a uniformly distributed load q. The beam is composed of two materials with an interface located at x = L/2. While the materials have different Young’s moduli, geometric properties remain the same throughout the beam, i.e., A1 = A2 = A and I1 = I2 = I . Consider E1 = 200 GPa, E1 /E2 = 10, q = 10 N/m2 , L = 1 m.

Discontinuity-enriched finite element formulations

111

Figure 5.3 The enrichment functions ψ and χ used in DE-FEM are obtained by clustering DOFs and properly scaling of Heaviside enrichment functions.

The formulation is based on first-order shear deformation theory, so the solution is expected to exhibit shear locking for slender beams. Obtain first the exact solution to this problem. Then solve it using a p-IGFEM approximation, whereby the linear IGFEM is augmented with smooth and nonsmooth higher-order polynomials that resolve properly the material discontinuity. Solve the problem with a single element with the interface in the middle, and study the numerical behavior for the slenderness range L/H = [4, 1000] (the same as that of Example 3.3). Finally, conduct a convergence analysis and show the error in energy norm and the condition number as a function of the total number of degrees of freedom. Solution: The governing differential equations for this beam are   d2 dθi E = q, (x)I i dx dx 2     d dθ dwi Ei (x)I i + Gi Ks A − θ = 0, dx dx dx

in i , (5.18) in i ,

112

Fundamentals of Enriched Finite Element Methods

Figure 5.4 Cantilever beam of length L, area A = BH , and moment of inertia I , loaded with a uniform distributed load q. The beam is composed of two materials with different Young’s moduli such that E1 /E2 = 10. where wi = w|i is the transverse displacement and θi = θ|i the rotation in the ith subdomain (with Young’s modulus Ei ). The boundary value problem includes boundary conditions w1 (0) = 0, θ1 (0) = 0, M2 (L) = 0, Q2 (L) = 0, and material interface conditions w1 (x ) = w2 (x ) and θ1 (x ) = θ2 (x ). The exact solution is obtained by integrating the first governing differential equation twice and applying boundary and interface conditions to determine integration constants. This procedure yields     x4 q x2 q Lx 3 L2 x 2 w1 = + Lx − , − + E1 I 24 6 4 G1 K s A 2 (5.19)     x4 q Lx 3 L2 x 2 q x2 C1 x C2 w2 = − + + Lx − + + , E2 I 24 6 4 G2 K s A 2 E2 I G2 K s A for the transverse displacement, and   x3 x2 q x −L + L2 , θ1 = E1 I 6 2 2   x3 C q x x2 + 1 , θ2 = −L + L2 E2 I 6 2 2 E2 I

(5.20)

for the rotation; constants C1 and C2 are, respectively, given by    x2 x3 1 x 1 , − C1 = qE2  − L  + L2 6 2 2 E1 E2      Lx3 L2 x2 qLx 1 q x4 C2 1 1 1 + = − + − − G2 K s A I 24 6 4 E1 E2 K s A G1 G2   qx2 1 1 C x − − − 1 . 2Ks A G1 G2 E2 I To solve this problem numerically, we adopt p-hierarchical interface-enriched finite element approximations. These can be decomposed into smooth and non-smooth components—the latter with discontinuous field gradient. For the smooth component we use the hierarchical p-FEM approximations (3.16)

Discontinuity-enriched finite element formulations

113

in Example 3.3. The discontinuous enrichment is constructed by multiplying the IGFEM bilinear enrichment (5.1) by the Legendre polynomials defined on the canonical element as per Eq. (3.2), i.e.,  4  3 ψˆ i i=1 = ψ × Pj j =0 (note that P0 = 1).

Figure 5.5 Shape and enrichment functions up to quartic order used for the approximations (5.21). The approximations for the transverse displacement and for the rotation are, respectively, wh (x) =

 i∈N

ϕi (x)wi +

4  i=1

ψˆ i (x)w˜ i ,

θ h (x) =

 i∈N

ϕi (x)θi +

4 

ψˆ i (x)θ˜i ,

(5.21)

i=1

where we denote by w˜ j and θ˜j their corresponding enriched DOFs. Shape and enrichment functions acting on a beam discretized with a single finite element are shown in Fig. 5.5. Note that (5.21) is valid regardless of the number of finite elements used to describe the beam because there is always a single element containing the interface (all others use the hierarchical p-FEM smooth approximation). We adopt the same polynomial order for both approximations and we rely on the Galerkin projection to assign negligible coefficients to the quartic polynomials associated with the rotation field, since the quartic term is not needed as per Eq. (5.20). Fig. 5.6 shows the ratio between approximate and exact transverse displacements at the right end of the beam, as a function of the ratio H /L, showing once again the issue of shear locking for linear interpolations as the beam becomes increasingly more slender, i.e., as H /L → 0.

h , normalized by the exact Figure 5.6 Ratio between the transverse displacement at the tip wL value wL given by (5.19), as a function of the ratio H /L.

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Fundamentals of Enriched Finite Element Methods

Figure 5.7 Convergence plot showing the relative error in energy norm squared (left) and condition number of the global stiffness matrix (right) as a function of the total number of DOFs nD . For the convergence analysis, we compute the exact strain energy as  2    1 dθi 2 dwi U (w, θ) = + Gi K s A − θi Ei I dx . 2 i dx dx

(5.22)

i={1,2}

Notice that U (w, θ) = 12 B(w, θ; w, θ ), with the bilinear form given in (3.14) for a beam without a material interface. Using (5.19) and (5.20) in (5.22) yields U = 0.192 188 052 5 J. The results of the convergence analysis are shown in Fig. 5.7 (left), for which we used an element set E = {1, 3, 5, 9, 17, 33, 65, 129, 257, 513} (the interface is always at the middle of the center element). Similar results as those in Chapter 3 are obtained, where exponential convergence is recovered for approximations of increasing polynomial order on a single finite element (see curve p = {1 . . . 4}). The quartic interpolation recovers the exact solution, so the curve labeled p = 4 reaches the highest accuracy for the floating-point computations. As in Example 3.3, the apparent loss of accuracy for p = 4 as the number of DOFs increase is due to the poor condition number. Notice that the problem is poorly conditioned even for a single element, and that enriched approximations have higher condition number that the non-enriched ones. Chapter 11 discusses strategies to deal with ill-conditioned enriched stiffness matrices. 

5.4 The discontinuity-enriched FEM in multiple dimensions

Without loss of generality, we consider in Fig. 5.8 a cracked domain  ⊂ Rd composed of two material phases 1 and 2 such that  = 1 ∪2 . The domain boundary ∂ =  \  ≡ , with unit outward normal n, is comprised of mutually exclusive regions with prescribed Dirichlet and Neumann boundary conditions,  D and  N , respectively. The ith material phase has its own boundary ∂i = i \ i ≡ i . The interface between material phases is 12 = 1 ∩ 2 , and the outward unit normal vector field to this interface is denoted by n12 . The schematic figure shows a crack  C that originates on both material phases. While this crack produces a jump in the displacement, the material interface 12 is still responsible for a jump in the displacement gradient. The domain  is discretized  E into finite elements that completely disregard ei . In order to account for this mismatch, the discontinuities, i.e.,  ≈ h = ni=1

Discontinuity-enriched finite element formulations

115

Figure 5.8 Mathematical representation of a cracked d-dimensional solid domain  composed of material phases 1 and 2 , which here represent the matrix and an inclusion, respectively. The schematic thus shows a weak discontinuity 12 and a strong discontinuity  C .

standard FEM approximation space will be enriched or augmented by functions that capture the jump in the displacement field, and the gradient thereof. The discrete form of the elastostatics boundary value problem is: Find uh ∈  h such that       ˜ wh ∀ wh ∈ 0h , (5.23) B v h , w h = L w h − B u, where u˜ = uh − v h is the part of the approximation uh that satisfies essential boundary ¯ and the linear and bilinear forms are given by ˜  D = u, conditions, i.e., u|     h h h ¯ w hi · t C d , L w = w i · bi d + w i · t d + N

i

i∈{1,2}

C

(5.24) and

   B v h , wh = i∈{1,2}

i

     w hi : σ i wi d ,

(5.25)

respectively. The traction field along the crack is denoted t C . Both v h and w h are now taken from the discontinuity-enriched finite element space in d dimensions, which can be written as eh = uh (x) =

 i∈ιh



strong

weak

       ϕi (x)uˆ i + si ψi (x)α i + χi (x)β i , 



standard FEM

i∈ιw





i∈ιs

ˆ α i , β i ∈ Rd . u,



enrichment

(5.26) As before, the first term of the approximation is the standard FEM component and the other terms enrich the approximation to reproduce the kinematics of weak and strong

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discontinuities. In (5.26), ιw and ιs are index sets that refer to weak and strong enriched nodes, respectively; these are placed at the intersections of discontinuities and element edges of the FE discretization. Their corresponding enrichment functions ψi and χi , which are constructed from Lagrange shape functions in integration elements, are associated with enriched DOFs α i and β i , respectively. Finally, si is a scaling factor that modifies the weak enrichment term to improve the stability of the formulation. It is worth noting that the three-term formulation (5.26) can be used to solve problems containing both types of discontinuities. Also, in the absence of strong discontinuities, (5.26) recovers the IGFEM approximation [6]. Across strong and weak discontinuities, the primal field is C −1 - and C 0 -continuous, respectively. In the case of elasticity, because the primal field is discontinuous along sets of zero measure, the function spaces here are also different from those discussed earlier in § 2.2.2. In fact, the vector-valued function space Seh given by (5.26) is actually a subspace of " # ! "  () = v ∈ Rd " v ∈  (), v|i ∈  1 (i ) , d  d  with  () ≡ L 2 () and  1 (i ) ≡ H 1 (i ) denoting vector-valued Lebesgue and Sobolev spaces, respectively. In other words, each component vi ∈  is squareintegrable over the entire domain , but the restriction to each subdomain i is used to get around the fact that the function does not have a properly defined gradient along discontinuities. In practice we subdivide cut elements into integration elements and standard Gauss quadrature is used to compute integrals. Quadrature points are always in the interior of integration elements—we use open integration rules—and thus no gradients are evaluated along discontinuities.

Fig. 5.9 illustrates a mesh crossed by a discontinuity  C and the location of enriched nodes (red circles). For an enriched node x i , for instance, a weak enrichment ψi is constructed from Lagrange shape functions of all integration elements connected to the node. The function attains its maximum value at x i and ramps linearly to zero at all other nodes. For the strong enrichment function χj , associated to node x j , all Lagrange functions on one side of the discontinuity are negated to produce a jump As explained earlier in this chapter, we require the jump " "  in the function. χ x j = χ x j " C + − χ x j " C − = 1 so that β j physically represents the jump in the displacement at x j . Because enrichment functions vanish at the location of standard FEM nodes, DOFs uˆ i in (5.26) retain their physical meaning, e.g., the displacement at the location of the ith node.

5.4.1 Treatment of nonzero essential boundary conditions Non-homogeneous Dirichlet boundary conditions can be prescribed strongly after solving a local problem. Fig. 5.10 shows a triangular element cut by an interface that intersects the Dirichlet boundary  D at node x 4 . Note that the displacement is known

Discontinuity-enriched finite element formulations

117

Figure 5.9 Weak enrichment ψi and strong enrichment χj associated with enriched nodes x i and x j , respectively.

Figure 5.10 Boundary region  D with prescribed Dirichlet BCs aligned with an element edge.

explicitly at both x 1 and x 2 since the enrichment function associated to enriched node ¯ 1 ) and uˆ 2 = u(x ¯ 2 ). Therefore, enx 4 is exactly zero at these locations, i.e., uˆ 1 = u(x riched DOFs can be calculated simply by ¯ 4 ) − ϕ1 (x 4 )uˆ 1 − ϕ2 (x 4 )uˆ 2 . α 4 = u(x

(5.27)

These DOFs can subsequently be enforced as in standard FEM together with uˆ 1 and uˆ 2 . Here we show the case where only weak DOFs need enforcement. For the more general case of both weak and strong enriched DOFs, see Example 8.2.

5.4.2 Hierarchical space Approximations of the form (5.26) work when elements are split by a single discontinuity. However, it is possible to generalize the formulation to resolve the kinematics of several discontinuities intersecting a single finite element. To that end, enrichment functions are constructed hierarchically, a procedure that was first proposed for IGFEM and coined the Hierarchical Interface-enriched Finite Element Method (HIFEM) [8,9]. For DE-FEM, the hierarchical finite element approximation is ex-

118

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pressed as v h (x) =

 i∈ιh

ϕi (x)uˆ i +

 k∈ιH i∈ιw

ski ψki (x)α ki +



χki (x)β ki ,

(5.28)

k∈ιH i∈ιs

where ιH = {1, 2, . . . , D} is the index set of hierarchical levels resulting from D discontinuities that interact with the element. With the right choice of data structure the computer implementation is straightforward. For the bookkeeping we use an ordered tree (actually an ordered forest), where each uncut mesh finite element is a root of the tree. When a discontinuity cuts an element, the latter is subdivided into integration elements and added as leaves to the corresponding root element in the data structure. If any of these leaves is cut by a subsequent discontinuity, they then become the parents of new integration elements that are subsequently added. This process is repeated till all integration elements are created considering all discontinuities. Fig. 5.11 schematically illustrates this process for a 3-node triangular element. The first level of enrichment functions is constructed by  3 using integration elements ej(1) j =1 . If the first interface (marked in red) is a strong discontinuity, both weak and strong enrichments, ψ14 , ψ15 and χ14 , χ15 , respectively, are needed. In the implementation, double nodes are created at the location of coor-

Figure 5.11 Process of creating the element hierarchy: (a) a 3-node mesh element ei is cut by two discontinuities; (b) the first discontinuity splits the mesh element into integration subdo (1) 3 mains ej j =1 . These are used to construct the first level of enrichment functions associated with enriched nodes marked with red circles; (c) the second discontinuity creates a second  (2) 6 level of integration elements ej j =1 , which are in turn used to create the second level of enrichments. Enriched nodes at this second level are marked with blue squares; (d) schematic of ordered tree data structure used to store parent/child relationships.

Discontinuity-enriched finite element formulations

119

dinates x 4 and x 5 , and these notes are associated, respectively, with weak and strong DOFs α 14 , α 15 and β 14 , β 15 . If the first interface is a weak discontinuity, only single enriched nodes are created at x 4 and x 5 , which are associated with weak DOFs α 14 , α 15 and their corresponding enrichment functions ψ14 , ψ15 . The procedure is then repeated for the second level of enrichments, which are built with the aid of La (2) 6 grange shape functions in integration elements ej j =1 . As before, the number of enriched nodes that will be created at each coordinate depends on whether the second discontinuity is weak or strong. Notice the smaller support for the second level of enrichments. By choosing both the trial solution and the test function from eh we arrive at the discrete form of (5.23). The statement 

        ˜ wh Be v h , wh = Le w h − Be u,

e∈h

e∈h

∀ w h ∈ eh ,

(5.29)

e∈h

leads to the discrete system of equations KU = F , where global arrays are the result of assembling the contribution of all elements, i.e., K = Ae k e and F = Ae r e . The calculation of local arrays for uncut elements follows standard procedures (see Chapter 2). Cut elements are subdivided into integration elements; the local stiffness matrix and force vector for the eth integration element are given by



∂t C M d∂e , ∂δ e  C ∩∂e f e =  b de +  t¯ d∂e + M  t C d∂e , ke =

e

B  CB de +

 N ∩∂e

M

(5.30)

 C ∩∂e

respectively, where C is the constitutive matrix,  = ϕ

 ψ χ is a vector that

 stacks standard and enrichment functions, and B ≡   =  ϕ  ψ  χ is the strain–displacement matrix that is obtained by applying the differential operator given by Eq. (2.76). For the cohesive contributions, δ ≡ u, and M is a matrix that is formed by considering 1D linear shape functions (hat functions) along the cohesive segment parameterized by master coordinate ξ = [−1, 1]. Concretely,

 M = (1 − ξ )/2 (1 + ξ )/2 ⊗ I , where ⊗ denotes Kronecker product and I the ddimensional identity matrix. Noteworthy, in the computer implementation, area and boundary integrals in (5.30) are computed in different subroutines; therefore, integrals along the integration elements’ boundaries—often shared between two elements—are computed only once. Notice that the linear system of equations KU = F can be written explicitly making reference to the subdivision in standard and enriched (both weak and strong)

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Fundamentals of Enriched Finite Element Methods

Figure 5.12 Schematic of an integration element (shaded), whose local arrays are computed using an isoparametric mapping. For a given integration point ξ i , the corresponding global coordinate x is obtained and an inverse mapping is used to obtain the local coordinates ξ p in the parent element. The latter is needed because the parent shape functions are evaluated in parent’s local coordinate system.

components: ⎡ K uu

⎢ ⎢  ⎢K uα ⎣ K uβ

K uα K αα K αβ

⎤⎡ ⎤ ⎡ ⎤ F uˆ ⎥⎢ ⎥ ⎢ u⎥ ⎥⎢ ⎥ ⎢ ⎥ K αβ ⎥ ⎢α ⎥ = ⎢F α ⎥ . ⎦⎣ ⎦ ⎣ ⎦ β K ββ Fβ K uβ

(5.31)

There is a caveat, since the local arrays given by (5.30) are evaluated numerically on a master element that maps to an integration element. In addition, some components of the integrands also involve another isoparametric mapping, i.e., shape functions of the parent uncut triangular element. It is therefore important to note that there are usually two mappings involved when evaluating these arrays numerically. This is conceptually shown in Fig. 5.12, where for a given integration point ξ i on $ the master integration element , the corresponding global coordinate x = i ϕi x i is needed for obtaining the master coordinate ξ p on the parent element (by means of an inverse mapping); this is required for evaluating the shape functions of the parent element that make up ϕ. Because the strain–displacement matrix requires the derivatives with respect to global coordinates, both isoparametric mappings are involved in its computation. The derivatives of the ith parent shape function with respect to global coordinates are ∇ x ϕi = J −1 p ∇ ξ p ϕi , where J p is the Jacobian matrix associated with the parent element’s isoparametric mapping. Similarly, for the j th enrichment function defined over the integration element, we have ∇ x ψj = J −1 e ∇ ξ i ψj , with J e denoting the Jacobian matrix associated with the integration element (we denote by je its determinant). Once these are known the strain–displacement matrix is written as before,

Discontinuity-enriched finite element formulations

i.e., B =  ϕ

ψ

  χ .1 The local arrays are then computed as

B  CBje dξ +

ke =



∂t C Mje d∂ξ , ∂δ C ∩ ¯ ϕ t je d∂ξ + M  t C je d∂ξ .





fe =

121

ϕ bje dξ +

M

N ∩

(5.32)

C ∩

More details about the computer implementation, including pseudocode for element subroutines, are given in Chapter 12.

5.5 Convergence In this section we compare the convergence behavior of DE-FEM for fracture under modes I, II, and III, to that obtained by standard FEM on fitted meshes. The analytical displacement fields, provided by linear elastic fracture mechanics, are given by Mode I: ⎡

⎤

⎡ ux

⎢ ⎥ KI ⎢ ⎥ u = ⎢uy ⎥ = √ ⎣ ⎦ 2πr uz

⎤

r [(2κ − 1) cos θ2 − cos 3θ 2 ]⎥ ⎢ 4μ ⎢ r ⎥ ⎢ 4μ [(2κ + 1) sin θ2 − sin 3θ ⎥, 2 ]⎦ ⎣ 2ν1 z θ − E cos 2

(5.33)

Mode II: ⎤

⎡ KII u= √ 2πr

r [(2κ + 3) sin θ2 + sin 3θ 2 ] ⎥ ⎢ 4μ ⎥ ⎢r ⎥, ⎢ 4μ [−(2κ − 3) cos θ2 − cos 3θ 2 ]⎦ ⎣ 2ν1 z θ E sin 2

(5.34)

Mode III: ⎤

⎡ u=

KIII μ

%

0

2r π

⎥ ⎢ ⎥ ⎢ ⎢ 0 ⎥, ⎦ ⎣ sin θ2

(5.35)



1 For a 2-D problem, the strain–displacement matrix B =  ϕ1 . . .

⎡

∂ϕ1

0

∂ϕ1 ∂y

∂ϕ1 ∂y ∂ϕ1 ∂x

⎢ ∂x ⎢ B =⎢ 0 ⎣

...

∂ψ1 ∂x

0

0

∂ψ1 ∂y ∂ψ1 ∂x

∂ψ1 ∂y

...

χ ...

is

⎤

∂χ1 ∂x

0

0

∂χ1 ∂y ∂χ1 ∂x

∂χ1 ∂y

  ψ1 . . . ⎥ ⎥ . . .⎥ , ⎦

−1 where, for instance, ∂ϕ1 /∂x = ex · J −1 p ∇ ξ p ϕ1 and ∂ψ1 /∂x = ex · J e ∇ ξ e ψ—notice the choice of Jacobian matrix depending on whether the function is defined on parent or integration element.

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Fundamentals of Enriched Finite Element Methods

where r, θ are the polar coordinates with respect to a local system at the crack front aligned with the crack direction, μ is the shear modulus, κ and ν1 are equal to 3 − 4ν and 0 for plane strain, or (3 − ν)/(1 + ν) and ν for plane stress, respectively. Their corresponding stress fields around the crack front are given by Mode I: ⎡ KI θ⎢ ⎢ σ=√ cos ⎢ 2⎣ 2πr

1 − sin θ2 sin 3θ 2

⎤

sin θ2 cos 3θ 2

0

⎥ ⎥ 0 ⎥, ⎦ 2ν2

1 + sin θ2 sin 3θ 2 sym.

(5.36)

Mode II: ⎡ − sin θ2 (2 + cos θ2 cos 3θ 2 ) ⎢ KII ⎢ σ=√ ⎢ 2πr ⎣ sym.

cos θ2 (1 − sin θ2 sin 3θ 2 ) θ 2

θ 2

sin cos cos

3θ 2

⎤ 0 0

⎥ ⎥ ⎥, ⎦

−2ν2 sin θ2 (5.37)

Mode III: ⎡ 0

KIII σ=√ 2πr

0 − sin θ2

⎢ ⎢ ⎢ 0 ⎣ sym.

⎤

⎥ ⎥ cos θ2 ⎥ , ⎦ 0

(5.38)

where ν2 = ν for plane strain or ν2 = 0 for plane stress. DE-FEM is tested on a cubic domain discretized using structured FE meshes, containing 6, 162, 2058, 20 250, and 178 746 linear tetrahedral elements. The displacement fields given by (5.33)–(5.35) are prescribed on the cube’s boundary. Elastic modulus E = 10 and Poisson ratio ν = 0.3 are used. The convergence results are illustrated in Fig. 5.13, where it can be seen that DE-FEM converges at the same rate as standard FEM on fitted meshes. Nevertheless, both FEM and DE-FEM recover a convergence rate that is roughly half of the optimal convergence rate βh = 1/3 (notice the rate is with regards to the total number of DOFs nD ). DE-FEM has been further explored to extract stress intensity factors for 2-D [7] and 3-D [10] problems. Also, the condition number has been thoroughly studied [5,10].

5.6 Weak and strong discontinuities As discussed earlier, DE-FEM provides a versatile procedure to analyze both weak and strong discontinuities with a unified three-term formulation. For strong discontinuities, both weak and strong enrichment terms are required to reproduce kinematically independent fields at either side of the discontinuity. In the absence of the strong term, the

Discontinuity-enriched finite element formulations

123

Figure 5.13 Error in energy norm as a function of the total number of DOFs nD for modes I, II, and III. The figures show that both standard FEM and DE-FEM have the same convergence rates, although these are not optimal. Deformed configurations obtained after post-processing are also shown.

Figure 5.14 Analysis of cracked femur with a fitted mesh to the external boundary but unfitted from the material interface and the crack.

formulation recovered is that of IGFEM, which was proposed for discontinuous gradient fields. By means of a hierarchical implementation it is possible to interact with multiple weak and strong discontinuities. Consider, for instance, a femur bone organ (adapted from Zhang et al. [10]), where we simplistically describe the interface between cancellous (inner) and cortical (outer) bone with a perfectly bonded interface. In addition, a crack is added to partially split the bone. Fig. 5.14 shows the finite element mesh used, composed of 7355 tetrahedral elements, that was fitted to the femur boundary. The figure also shows the final deformation, where it can be seen that the DE-FEM formulation is able to seamlessly resolve the kinematics of both weak and strong discontinuities.

5.7 Recovery of field gradients The approximations discussed in this chapter provide great flexibility for the analysis of problems containing discontinuities using structured meshes. They do so by reconstructing the same approximation space as that obtained by standard FEM, whereby cut elements are subdivided into standard elements fitted to the interface. However, it

124

Fundamentals of Enriched Finite Element Methods

is well known that gradients are particularly sensitive to how fitted meshes are created, and finite elements with bad aspect ratios can be the source of large errors in gradient fields (strain and/or stress). In particular, because the finite element approximation space recovered by discontinuity-enriched formulations strongly depends on how elements are cut—which determines the shape of integration elements and subsequent enrichment functions—large errors in field gradients have been reported. A study found that stresses could be overestimated more than 150% [11]. Many techniques have been proposed over the years to obtain better field gradients. A non-exhaustive list on important contributions for standard FEM includes the works of Brauchli and Oden [12], Hinton and Campbell [13], Zienkiewicz and Zhu [14,15], Wiberg et al. [16,17], Tabbara et al. [18], Boroomand et al. [19,20], Ubertini [21] and Benedetti et al. [22]. Other procedures have also been proposed to improve the recovery of gradient fields in e-FEMs, including the works of Xiao et al. [23], Bordas et al. [24–26], Rodenas et al. [27], Prange et al. [28], and Lins et al. [29,30]. Again, this list is by no means exhaustive. Through a simple example, in this section we illustrate the issue of stress recovery by IGFEM and use a stress improvement procedure (SIP) to obtain an enhanced field. The recovery procedure relies on a mixed formulation based on the Hu–Washizu principle. In a nutshell, the enhanced stress is obtained by fitting a complete quadratic polynomial to each of the stress components in the target element. To obtain the enhanced stress, we consider a patch of elements E surrounding the target element where we seek an enhanced stress so that (i) equilibrium is satisfied in a weak manner over the patch; and (ii) the error between the enhanced stress and the directly-calculated stress is minimized. To wit, ! #  h (5.39) δ σ˜  e σ e − σ e de = 0 {e∈E} e

and  {e∈E} e

δζ  e {div (σ e ) + b} de = 0,

(5.40)

respectively. In (5.39), σ e is the eth element’s enhanced stress, σ he is the directlycalculated stress, and ζ e , σ˜ e are properly chosen vector- and tensor-valued polynomial spaces, respectively. Subsequently, the enhanced stress is obtained by solving a small system of linear equations per element (comprised of Eqs. (5.39) and (5.40)) as a post-processing step. We compared this procedure with other less sophisticated stress smoothening techniques, namely averaged and weight-averaged directly-calculated stresses over a patch of elements. These are expressed, respectively, as $ σ¯ h =

h e∈E σ e

|E|

$ h e∈E σ e Ae , and σ¯ h = $ e∈E Ae

(5.41)

Discontinuity-enriched finite element formulations

125

Figure 5.15 Schematic of Eshelby’s inclusion problem with outside radius ro = 2 and inside radius ri . For the finite element analysis, a square domain of size 2 × 2 (dashed square) is considered, with the analytical displacement u¯ prescribed on its boundary. For the material properties, E2 /E1 = 10.

Figure 5.16 Maximum von Mises stress as a function of internal radius ri : (left) directlycalculated and enhanced stresses, σeh and σe , respectively, evaluated by standard FEM with fitted meshes and IGFEM on an unfitted structured mesh; (right) comparison between IGFEM stresses (directly calculated and enhanced) and the averaged and weight-averaged smoothing formulations given by (5.41).

where | · | denotes set cardinality and thus |E| is the number of elements in the patch, and Ae is the area of the eth element. Consider in Fig. 5.15 Eshelby’s inclusion problem, where a circular inclusion, with radius ri and material constants E1 , ν1 , is contained by a larger circular matrix with radius ro and constants E2 , ν2 . For a prescribed displacement field ur = ro and uθ = 0, with polar coordinates r, θ , this problem has an exact solution [31]. The problem is then solved numerically by prescribing essential BCs on the dashed square domain shown in Fig. 5.15, which is discretized by a structured mesh of 60 × 60 × 2 constant strain triangular elements. To ensure the creation of all shapes of integration elements, we vary the internal radius in the range ri = [0.35, 0.42] and in steps of ri = 0.035. Fig. 5.16 summarizes the comparison between all stress-recovery procedures. In the left figure, the

126

Fundamentals of Enriched Finite Element Methods

Figure 5.17 von Mises stress fields for an internal radius ri = 0.3605: (a,b) standard FEM on a fitted mesh; (c,d,e,f) IGFEM on an unfitted mesh. The fields correspond to directlycalculated (a,c), recovered (b,d), averaged (e), and area-weighted (f) stresses.

solid line represents the exact von Mises stress. As apparent from the figure, directly calculated stresses obtained from IGFEM oscillate, as shown by the curve IGFEM (σeh ). Directly-calculated stresses from the standard FEM solution on fitted meshes are shown as the FEM (σeh ) curve. This curve does not oscillate but it is not very accurate when compared to the exact stress. The stress improvement procedure described above greatly improves the recovered stress field for both IGFEM and FEM on matching meshes. These curves are shown as IGFEM (σe ) and FEM (σe ), respectively. On the right figure, the directly-calculated and enhanced stresses retrieved from IGFEM are also compared to the smoothing techniques of Eq. (5.41). The most accurate stress values are provided by SIP. It is worth noting that the results of IGFEM are actually equivalent to those obtained by the Conformal Decomposition Finite Element Method (CDFEM) [32,33]. CDFEM is simply standard FEM, whereby cut elements are replaced by elements that conform to the interface (in our case we used IGFEM’s integration elements as constant strain triangles). Finally, the von Mises stress distributions for ri = 0.3605 are shown in Fig. 5.17. It was shown that for this problem, a higher convergence rate was obtained for the enhanced stress when looking at the H0 norm of the error in stress as a function of mesh size h (1.2 vs. 1 for directly-calculated stresses) [31].

Discontinuity-enriched finite element formulations

127

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[17] N.-E. Wiberg, F. Abdulwahab, S. Ziukas, Enhanced superconvergent patch recovery incorporating equilibrium and boundary conditions, International Journal for Numerical Methods in Engineering 37 (20) (1994) 3417–3440. [18] M. Tabbara, T. Blacker, T. Belytschko, Finite element derivative recovery by moving least square interpolants, Computer Methods in Applied Mechanics and Engineering 117 (1) (1994) 211–223. [19] B. Boroomand, O.C. Zienkiewicz, Recovery by equilibrium in patches (REP), International Journal for Numerical Methods in Engineering 40 (1) (1997) 137–164. [20] B. Boroomand, O.C. Zienkiewicz, An improved REP recovery and the effectivity robustness test, International Journal for Numerical Methods in Engineering 40 (17) (1997) 3247–3277. [21] F. Ubertini, Patch recovery based on complementary energy, International Journal for Numerical Methods in Engineering 59 (11) (2004) 1501–1538. [22] A. Benedetti, S. de Miranda, F. Ubertini, A posteriori error estimation based on the superconvergent recovery by compatibility in patches, International Journal for Numerical Methods in Engineering 67 (1) (2006) 108–131. [23] Q.-Z. Xiao, B.L. Karihaloo, Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery, International Journal for Numerical Methods in Engineering 66 (9) (2006) 1378–1410. [24] S. Bordas, M. Duflot, Derivative recovery and a posteriori error estimate for extended finite elements, Computer Methods in Applied Mechanics and Engineering 196 (35) (2007) 3381–3399. [25] M. Duflot, S. Bordas, A posteriori error estimation for extended finite elements by an extended global recovery, International Journal for Numerical Methods in Engineering 76 (8) (2008) 1123–1138. [26] S. Bordas, M. Duflot, P. Le, A simple error estimator for extended finite elements, Communications in Numerical Methods in Engineering 24 (11) (2008) 961–971. [27] J.J. Ródenas, O.A. González-Estrada, J.E. Tarancón, F.J. Fuenmayor, A recovery-type error estimator for the extended finite element method based on singular+smooth stress field splitting, International Journal for Numerical Methods in Engineering 76 (4) (2008) 545–571. [28] C. Prange, S. Loehnert, P. Wriggers, Error estimation for crack simulations using the XFEM, International Journal for Numerical Methods in Engineering 91 (13) (2012) 1459–1474. [29] R. Lins, M. Ferreira, S. Proença, C. Duarte, An a-posteriori error estimator for linear elastic fracture mechanics using the stable generalized/extended finite element method, Computational Mechanics 56 (6) (2015) 947–965. [30] R. Lins, S.P. Proença, C.A. Duarte, Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method, International Journal for Numerical Methods in Engineering 119 (12) (2019) 1279–1306. [31] J. Zhang, A.M. Aragón, An improved stress recovery technique for the unfitted finite element analysis of discontinuous gradient fields, International Journal for Numerical Methods in Engineering 123 (3) (2022) 639–663. [32] D.R. Noble, E.P. Newren, J.B. Lechman, A conformal decomposition finite element method for modeling stationary fluid interface problems, International Journal for Numerical Methods in Fluids 63 (6) (2010) 725–742. [33] R.M. Kramer, D.R. Noble, A conformal decomposition finite element method for arbitrary discontinuities on moving interfaces, International Journal for Numerical Methods in Engineering 100 (2) (2014) 87–110.

Part Two Applications

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GFEM approximations for fractures

6

The simulation of fracture mechanics problems has been performed for many years using standard FEMs. Crack surfaces are modeled in the FEM via meshes with double nodes to define the discontinuity of the elasticity solution across the surfaces. Degenerate quarter-point elements are often adopted along the crack front to approximate the singularity of Linear Elastic Fracture Mechanics (LEFM) problems. This is a mature and well-understood technology that is available in virtually all commercial FEM software. However, this approach can require significant computational resources and user time in model preparation. There has been much work in the development of adaptive meshing techniques [1–4] that are aimed at the generation of strongly graded meshes in crack front regions, often employing crack front template meshes to help resolve the singularities. The automatic construction of these template meshes, which utilize brick elements, is often difficult in the case of evolving 3-D crack surfaces [4], and user intervention may be required to generate a suitable mesh [5]. Care must be taken to ensure a well-structured mesh in the crack front region throughout the course of a simulation so as to avoid, for instance, elements with poor aspect ratios, which may adversely impact the resulting solution quality [2]. Furthermore, even when a well-designed FEM mesh with quarter-point elements is adopted, the convergence of the FEM is still controlled by the singularity of the elasticity solution, and thus the convergence rate is much lower than for problems with a smooth solution, as shown later in Section 6.3.2. This chapter presents GFEM approximations for 2-D and 3-D fracture mechanics problems. The focus is on LEFM for simplicity, but the case of cohesive fractures is also briefly discussed. In Section 6.2, we discuss how to select enrichment functions for the GFEM using a priori knowledge about the solution of fracture problems, such as the presence of discontinuities and singularities. These enrichments form a basis of patch approximations spaces χα (ωα ), α ∈ Ih , defined in Chapter 13. We select enrichments that can mimic the properties of the function to be approximated locally in patches ωα , α ∈ Ih . This more abstract view of the GFEM based on patch spaces is, however, not necessary to understand the material in this chapter. As such, this material is presented in remarks that can be skipped by the reader and revisited after reading the approximation theory for partition of unity methods presented in Chapter 13. High-order GFEM approximations are presented in subsections that can also be skipped if the reader is interested in linear approximations only. This is also the order of approximation adopted in the numerical examples presented in Section 6.3. The GFEM removes the limitations of the FEM for this class of problems while preserving its flexibility, such as the ability of analyzing domains with complex geometries, being able to solve a broad range of nonlinear and time-dependent problems, etc. First, the GFEM can adopt meshes that do not fit the crack surface or the crack front. This is particularly appealing for the simulation of fracture propagation problems and for the robust implementation of mesh adaptivity. The latter can be performed by simply bisecting elements intersected by the crack surface and/or front [6,7]. The mesh Fundamentals of Enriched Finite Element Methods. https://doi.org/10.1016/B978-0-32-385515-0.00014-3 Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved.

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need not be adjusted such that the elements fit the crack surface like in the case of the FEM, thus avoiding the creation of elements with poor aspect ratios. This flexibility and robustness of the GFEM greatly reduces the user time required for modeling complex fracture problems. Second, we show in Section 6.3.3 that the singular enrichments adopted by the GFEM are able to deliver optimal convergence rates—the same rates of convergence as in problems with a smooth solution. As a result, for a given target discretization error, the GFEM requires much fewer degrees of freedom than an FEM of the same polynomial order. Strategies to control the conditioning of GFEM and IGFEM matrices are discussed in Chapter 11. They lead to GFEM and IGFEM stiffness matrices with a condition number of the same order as in the FEM.

6.1 Governing equations: 3-D elasticity

We consider the problem of a body  ⊂ Rd , d = {2, 3}, with a crack surface d as illustrated in Fig. 6.1 for the 3-D case.

Figure 6.1 Schematic of a 3-D domain  with a crack surface d . The body has two crack − faces, d+ and d− , with unity normal vectors n+ d and nd , respectively. When the particular crack face is not relevant, nd represents one of these vectors.

The governing equations and boundary conditions for the class of problems considered in this chapter are the same as those in Section 2.2, namely, equilibrium equation (2.57) subjected to boundary conditions (2.58)–(2.60). In addition, boundary conditions can also be prescribed on the crack faces d+ and d− . Without loss of generality, we consider Neumann boundary conditions given by σ · nd = t¯ d

on d .

(6.1)

Tractions t¯ d on d can represent, for example, the fluid pressure in hydraulic fracture problems [8]. For simplicity, we also assume that the body force vector b = 0 and that only Neumann and/or Dirichlet boundary conditions are prescribed ( R = ∅).

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133

We also assume a linear elastic constitutive law in the body  and small deformation kinematics as described in Section 2.2.

6.1.1 Weak form The weak formulation of the problem is given by: Find uh ∈ Vh () such that ∀ v h ∈ V0h (),      + v h · t¯ d d , ∇v h : σ (uh ) d = v h · t¯ d + (6.2) 

N

d+

+ ¯+ where d+ is the crack face with normal vector n+ d , t d is the prescribed traction on d ,   − + with t¯ d = −t¯ d , and v h = v h,+ − v h,− is the virtual displacement jump across crack surface d . The integral over d is dropped in the case of problems with stress-free crack surfaces. In the case of the GFEM, the approximation space V0h () and Vh () are given by (cf. Sections 2.2.4 and 4.6 for details)   (6.3) V0h () = v ∈ SGFEM () : v| D = 0 ,

Vh () = V0h () + u¯ ,

(6.4)

where u¯ is the prescribed displacements on  D .

6.2 GFEM approximation for fractures The solution u of a fracture mechanics problem can be decomposed into u = uˆ + u˜˜ + u˘ ,

(6.5)

where uˆ is a continuous and smooth function and u˜˜ is a discontinuous function. The ˘ depends on the fracture model adopted. It is singular in behavior of the last term, u, the case of a linear elastic fracture mechanics model, while it has strong, but bounded, gradients in the case of, e.g., cohesive and smeared crack models. Therefore, each ˘ fracture model may require a different set of enrichment functions to approximate u. The first two terms, however, are common to all sharp fracture models. Next, we discuss how to select enrichment functions for the GFEM using this a priori knowledge about the solution of fracture problems. GFEM spaces for fracture problems reflect the decomposition (6.5) of the elasticity solution and is partitioned into three sets br SGFEM = ShFEM + SH GFEM + SGFEM .

Thus, in the case of fracture problems, br SeGFEM = SH GFEM + SGFEM .

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br ˜˜ and u, ˆ u, ˘ Spaces ShFEM , SH GFEM , and SGFEM provide approximations to functions u, respectively. The details of each of these approximations are presented next.

6.2.1 Approximation of uˆ Since uˆ is a continuous and smooth function, it can be approximated well by standard FEM shape functions. This approximation belongs to space ShFEM . For simplicity, consider first-order (linear) approximations. A function belonging to the FEM space ShFEM is then given by uˆ h (x) =



u α ϕα (x) ,

(6.6)

α∈Ih

where u α ∈ Rd are standard FEM degrees of freedom, d = {2, 3}, is the dimension of the problem, and ϕα (x) are 2-D or 3-D linear FEM shape functions.

6.2.1.1 High-order approximations High-order approximations of uˆ can be defined in several ways. The first option is to adopt in (6.6) high-order Lagrange shape functions for the partition of unity functions ϕα , α ∈ Ih [9]. The second option is to hierarchically enrich (6.6) with either p-hierarchical FEM [10] or GFEM shape functions [11–13]. In the first case, p-FEM approximations (cf. Chapter 3 and [14,15]) are adopted. In the second case, the GFEM shape functions are defined as in Section 4.5, but using d-dimensional polynomial enrichments. An example of polynomial enrichments of degree p = 2 in 2-D is 

Eˆ αj

 (x − xα ) (y − yα ) (x − xα )2 (y − yα )2 (x − xα )(y − yα ) , = 1, , , , , j =0 hα hα hα h2α h2α (6.7)

mˆ α

where m ˆ α is the number of polynomial enrichments, hα a scaling factor, and x α = (xα , yα ) the coordinates of the enriched node. The corresponding GFEM shape functions of degree p = 3 are given by 

φαj

 (x − xα ) (y − yα ) (x − xα )2 (y − yα )2 (x) = ϕ (x) 1, , , , , α j =0 hα hα h2α h2α (x − xα )(y − yα ) . hα

mˆ α

(6.8)

Fig. 4.2 shows an example of a polynomial GFEM shape function. Hereafter, we denote this type of GFEM shape function as a p-GFEM shape function to emphasize that, like the p-FEM shape functions, they are also hierarchical.

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135

The high-order GFEM approximation of uˆ is given by uˆ p (x) =

 α∈Ih



u α0 ϕα (x) + 

 α∈Ih



std. FEM approx.

ϕα (x)

m ˆα 

u αj Eˆ αj (x) .

j =1



(6.9)

p-GFEM enrichment

If m ˆ α = 0 and u α0 = u α , Eq. (6.9) degenerates to the linear approximation given in (6.6). Remark 6. Enrichment functions (6.7) form a basis for the patch approximation space χα (ωα ) (cf. Chapter 13). This basis can approximate well smooth functions over the patch ωα . Remark 7. Eq. (6.9) is the same as (13.1), but for a vector-valued function and mα = m ˆ α , Ihe = Ih (all nodes in the mesh are enriched).

6.2.2 Approximation of u˜˜ Most GFEM approximations of fracture problems assume that the discontinuous function u˜˜ can be decomposed in the following product: ˜˜ ˜ , u(x) = Hd (x)u(x)

(6.10)

where u˜ is a continuous and smooth function, and  1 if x is above or on the crack surface, Hd (x) = 0 otherwise,

(6.11)

with d being the crack surface. Function Hd (x) is the Heaviside enrichment adopted in Example 4.3, but generalized to higher dimensional problems. Since u˜ is a smooth ˆ it can also be approximated well by standard FEM shape functions. function like u, A first-order GFEM approximation of u˜˜ is given by   u˜˜ h (x) = Hd (x) uH uH (6.12) α ϕα (x) = α ϕα (x)Hd (x) ,



H H α∈Ih

α∈Ih

discont. GFEM sh. fn.

d where u H α ∈ R are GFEM degrees of freedom, d is the dimension of the problem, ϕα (x) is a linear FEM shape function, and IhH ⊂ Ih is the set of nodes of elements intersected by the crack surface, but not the crack front. If the crack is at the boundary of an element, only the element nodes at this element boundary are in the set IhH . Figs. 6.13 and 6.17a show this set of nodes for 2-D and 3-D edge crack problems. Fig. 6.2 shows a discontinuous GFEM shape function as defined in (6.12). The set of discontinuous GFEM functions defined via the Heaviside enrichment ˜˜ forms a basis of the GFEM space SH GFEM , which provides approximations (6.12) of u.

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Figure 6.2 Construction of a GFEM shape function using a Heaviside enrichment. The top, middle, and bottom functions are the finite element shape function, the Heaviside enrichment function, and the resulting generalized FE shape function, respectively. The FEM shape function is defined on a patch of four quadrilateral elements. The grid shown in the figure is for visualization purposes only.

Remark 8. Heaviside function (6.11) can also be defined as [16]  1 if x is above or on the crack surface, Hd (x) = −1 otherwise.

(6.13)

This definition does not change the GFEM spaces defined earlier—it only changes their bases. Therefore, either definition of Hd leads to the same GFEM solution, up to round-off errors. Remark 9. The Discontinuity-Enriched FEM (DE-FEM) presented in Chapter 5 also assumes that the discontinuous function u˜˜ can be decomposed as in (6.10), but it adopts another type of discontinuous function in the decomposition. In contrast with the Heaviside function, the discontinuous enrichments adopted in the DE-FEM have compact support and are zero at the nodes of the FEM mesh.

6.2.3 Cohesive fracture problems The solution of a cohesive fracture problem has no singularity at the fracture process zone. The elasticity solution component u˘ defined in (6.5) is not singular in this case, and GFEM approximations for this fracture model are just a combination of the

GFEM approximations for fractures

137

approximations of uˆ and u˜˜ [17,18]. This combination is given by uh (x) =

 α∈Ih



u α ϕα (x) + 

std. FEM approx.



uH α ϕα (x)Hd (x) ,

α∈IhH





(6.14)

discont. linear GFEM approx.

in the case of linear approximations. High-order approximations are also defined as a ˜˜ combination of high-order approximations of uˆ and u. No a priori information about the exact solution in the fracture process zone is used in the GFEM approximation (6.14)—function u˘ is ignored in this approximation. As a result, a sufficiently fine mesh must be used in this region of the analysis domain, as shown in [19]. While the required mesh refinement is on par with the standard FEM, the GFEM still brings the benefit of adopting meshes that do not need to fit the fracture surface. Therefore, the fracture path does not need to be known a priori, like in standard FEM simulations. It is noted that the GFEM approximation (6.14) is an extension to higher dimensions of the 1-D approximation (4.22) derived in Example 4.3, which deals with a linear cohesive law in 1-D. The span of the GFEM approximation (6.14) is the GFEM space SGFEM = ShFEM + SH GFEM . This space is related to the DE-FEM space (5.26) when there are no weak discontinuities. The proof of this relationship in 1-D is presented in Section 5.3.

6.2.3.1 High-order approximations High-order approximations of uˆ and u˜˜ can be defined using the same approach presented in Section 4.5.1, but extended to vector-valued problems in a d-dimensional space. The starting point is Eq. (6.14), which can be written as uh (x) =



u α ϕα (x) + Hd (x)

α∈Ih

= uˆ h (x) + Hd (x)u˜ h (x) ,



uH α ϕα (x)

(6.15)

α∈IhH

(6.16)

where uˆ h (x) and u˜ h (x) are continuous functions. These functions are linear FEM approximations in 2-D or 3-D. High-order approximations based on a high-order partition of unity or p-hierarchical enrichments can be used instead. An approach based on discontinuous p-hierarchical GFEM shape functions is adopted in this section. Other approaches can be found in [10,20,21]. A high-order GFEM approximation of a function with a strong discontinuity, but no singularity, is defined as up (x) = uˆ p (x) + Hd (x)u˜ p (x)

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Fundamentals of Enriched Finite Element Methods

=



ϕα (x)

α∈Ih

=

mα  α∈Ih j =0

mα 



u αj Eˆ αj (x) + Hd (x)

j =0

α∈IhH

u αj ϕα (x)Eˆ αj (x) +



cont. sh. fn.

ϕα (x)

mα 

ˆ uH αj Eαj (x)

j =0

mα  

H uH αj ϕα (x)Eαj (x) , 

H j =0

α∈Ih

(6.17)

discont. sh. fn.

H where Eαj (x) = Hd (x)Eˆ αj (x). The same number of enrichments (mα ) is adopted to represent uˆ p and u˜ p . This is required for optimal convergence [22]. Eq. (6.17) is the same as (4.39), but for vector-valued problems in a d-dimensional space. If mα = 0, α ∈ Ih , (6.17) reduces to the linear GFEM approximation (6.14) with u αo ≡ u α and H H ˆ uH α0 ≡ u α since Eα0 (x) = 1 and Eα0 (x) = Hd (x). The last term of (6.17) involves a linear combination of discontinuous p-hierarchical GFEM shape functions.

Remark 10. The enrichment functions used in (6.17) form a basis for the patch approximation spaces χα (ωα ), α ∈ IhH , i.e.,   χα (ωα ) = span Eˆ αj , Hd Eˆ αj , 0 ≤ j ≤ mα , α ∈ IhH . The basis of these spaces can approximate well uˆ and u˜˜ over patches ωα , α ∈ IhH . ˜˜ Their selection reflects the a priori knowledge we have about functions uˆ and u. Remark 11. If a node α is enriched with a Heaviside function, the patch of elements associated with the node, ωα , is cut by the crack surface into two disjoint subdomains, ωα + and ωα − , with ωα = ωα + ∪ ωα − . The discontinuity created by a Heaviside enrichment always ends at an element boundary even if the actual crack ends inside an element. This is illustrated in Figs. 6.3 and 6.4. If GFEM approximation (6.14) or (6.17) is used with an LEFM problem, the crack tip will be at a wrong location in 2-D and the approximation of curved crack fronts in 3-D will be fairly poor unless an extremely refined mesh is used. This is illustrated in Example 6.1. On the other hand, GFEM approximation (6.17) does not lead to this issue when modeling cohesive discontinuities since, in this case, the location of the cohesive process zone is part of the solution of the nonlinear problem and is not restricted to element boundaries if high-order approximations are adopted.

Figure 6.3 A Heaviside enrichment fully cuts the support of node α. The solid line cutting elements represents the crack. Figure courtesy of Professor Angelo Simone from the University of Padova.

Another limitation of the Heaviside enrichment is that it cannot approximate well the singularity at the crack front of an LEFM problem. Therefore, if an LEFM fracture

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139

Figure 6.4 Two-dimensional mesh with nodes enriched with Heaviside function. The discontinuity ends at element boundaries. Figure courtesy of Professor Angelo Simone from the University of Padova.

problem is solved using only Heaviside enrichments, extremely refined meshes are required in order to capture the singularity at the crack front/tip. In fact, the meshes have to be finer than √ FEM meshes with quarter-point elements since those elements can approximate a r singularity. The enrichments described in the next section remove these limitations of the Heaviside functions. Example 6.1: Heaviside enrichment for a half-penny crack. This example illustrates the poor approximation of curved crack fronts provided by Heaviside enrichments. The problem with a half-penny surface crack, as shown in Fig. 6.5, is used to illustrate this limitation of Heaviside enrichments. The GFEM approximation was computed using (6.17).

Figure 6.5 Half-penny surface crack. A constant traction is prescribed on the top and bottom faces of the domain.

Fig. 6.6a shows a zoom-in of the half-penny surface crack and nodes from set IhH . This surface defines the geometry of the crack surface we want to analyze. We call this surface a geometrical crack surface (cf. Appendix A of [21]). In enriched FEMs like the GFEM, the geometry of a crack is defined by the enrichment functions, not the finite element mesh. Since Heaviside enrichment leads to discontinuities that end at element boundaries, it leads to the approximation of the geometry of the crack shown in Fig. 6.6b. The approximation is clearly poor even though the 3-D mesh is refined in the neighborhood of the crack front. This crack surface, defined by the enrichment functions, is called the computational crack surface (cf. Appendix A of [21]).

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Fundamentals of Enriched Finite Element Methods

Figure 6.6 Comparison between the user-defined geometrical crack surface (left) and the computational or effective crack surface defined by Heaviside enrichments. The latter always ends at element boundaries leading to a poor approximation of the crack front geometry.



6.2.4 Approximation of u˘ This section presents GFEM approximations of function u˘ defined in (6.5). As explained earlier, the behavior of u˘ depends on the fracture model adopted. We focus on the case of linear elastic fracture mechanics. The GFEM shape functions are defined using singular enrichment functions that approximate well the solution field in a neighborhood of the crack front. These shape functions form a basis of the GFEM space Sbr GFEM . We start with a short review of classical LEFM solutions near a crack front.

6.2.4.1 Elasticity solution in the neighborhood of a crack front The singular enrichments used in the GFEM for LEFM problems are closely related to the expansion of the elasticity solution in the neighborhood of a crack front [14,23]. This expansion, in turn, is based on a further decomposition of the solution in terms of three modes of deformation [23], Mode I, II, and III, which represent opening, shearing, and tearing of the crack, as illustrated in Fig. 6.7. In a neighborhood sufficiently close to the crack front (called the asymptotic region), the displacement components in the local crack front Cartesian coordinate

Figure 6.7 Deformation modes of a fracture: opening (left), shearing (middle), and tearing (right).

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141

system (x, ¯ y, ¯ z¯ ) shown in Fig. 6.8 is given by [23,24] ⎧ ⎫ ⎧ ⎫ (κ − 12 ) cos θ2 − 12 cos 3θ ¯ θ, z¯ ) ⎬ ⎨ u(r, 2 ⎬ KI (¯z) √ ⎨ 1 θ 1 3θ v(r, ¯ θ, z¯ ) ¯ θ, z¯ ) = r u(r, = √ ⎩ (κ + 2 ) sin 2 − 2 sin 2 ⎭ ⎩ ⎭ 2G 2π w(r, ¯ θ, z¯ ) 0 ⎧ ⎫ (κ + 32 ) sin θ2 + 12 sin 3θ ⎬ 2 KII (¯z) √ ⎨ 3 θ 1 3θ + r √ −(κ − ) cos − cos 2 2 2 2 ⎭ ⎩ 2G 2π 0 ⎫  ⎧ 0 ⎬ ⎨ KIII (¯z) 2r 0 + + O(r) ⎭ G π ⎩ sin θ2  KII (¯z) √ KIII (¯z) 2r KI (¯z) √ f (θ ) rf I (θ ) + rf II (θ ) + = √ √ G π III 2G 2π 2G 2π + O(r), (6.18) where (r, θ, z¯ ) are the cylindrical coordinates indicated in Fig. 6.8 and KI (¯z), KII (¯z), and KIII (¯z) are the stress intensity factor functions along the crack front. These functions are constant for problems under plane strain conditions. Material parameter κ = 3 − 4ν is the Kolosov constant for plane strain conditions with ν being the Poisson’s ratio and G the shear modulus. It is noted that the asymptotic expansion of ¯ θ, z¯ ) is an infinite series—only the first term from each mode of the expansion is u(r, shown in (6.18).

Figure 6.8 Crack front Cartesian and cylindrical coordinate systems (adapted from [25]).

Eq. (6.18) assumes that the fracture is planar with a straight front, is in an infinite linear elastic body, and has a stress-free crack surface. The material is assumed to be homogeneous and isotropic. As long as these assumptions hold, (6.18) is valid for any geometry of a sufficiently large cracked body or boundary conditions prescribed to this body, which is quite remarkable. The only unknown parameters in (6.18) are the stress intensity factors. ˘ Functions from (6.18) are used next to define enrichments able to approximate u. Two sets of singular enrichments are considered: The Oden and Duarte [11,26] and the Belytschko and Black [27] enrichments. Virtually all GFEMs for LEFM problems use one of these two sets. It is noted that if the problem solved with these enrichments

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Fundamentals of Enriched Finite Element Methods

violates any of the assumptions listed above—such as fracture geometry—mesh refinement, in addition to enrichment, is in general required for acceptable accuracy. This is the case in most practical 3-D problems. Nonetheless, even in this case, it is possible to achieve higher convergence rates than in the FEM and to adopt meshes that do not fit the crack surface—a great advantage in the case of propagating fractures.

6.2.4.2 Oden and Duarte branch enrichment functions The Oden and Duarte [11,26,28] (OD) enrichments are based on the first term from each mode of the asymptotic expansion of the elasticity solution under the same assumptions as in (6.18). Let (x, ¯ y, ¯ z¯ ) and (r, θ, z¯ ) denote the local crack front Cartesian and cylindrical coordinate systems, respectively, shown in Fig. 6.9.

Figure 6.9 Crack front Cartesian and cylindrical coordinate systems defined at a point on the crack front [21].

An approximation of u˘ using GFEM shape functions defined with OD enrichments is given by u˘ h (x) =



  OD(I) OD(III) br OD(II) br ϕα (x)R x x¯ (x) ubr E (x) + u E (x) + u E (x) , αI front αII front αIII front

α∈Ihbr

(6.19) br where ubr αi , i = I, II, III, ∈ R, Ih ⊂ Ih is the set of nodes enriched with singular functions, and the vector-valued OD enrichments are given by  OD(I) Efront (x) = r(x)f I (θ (x)) ,  OD(II) Efront (x) = r(x)f II (θ (x)) , (6.20)  OD(III) Efront (x) = r(x)f III (θ (x)) ,

with f i , i = I, II, III, defined in (6.18). Matrix R x x¯ (x) is the rotation from the local crack Cartesian coordinate system to the global Cartesian coordinate system at point x with entries given by ⎡ ⎤ R11 R12 R13 R x x¯ (x) = ⎣R21 R22 R23 ⎦ . (6.21) R31 R32 R33

GFEM approximations for fractures

143

The columns of R x x¯ (x) represent the components of the crack coordinate system base vectors with respect to the global base vectors. The implementation of vector-valued OD enrichment functions leads to just three degrees of freedom per enriched node α ∈ Ihbr : ubr αi , i = I, II, III. However, this implementation is a bit involved since most FEM data structures cannot handle vectorvalued shape functions. An alternative scalar implementation based on OD enrichments proposed in [21] is adopted here. These OD enrichments are related to earlier scalar-value formulations presented in [6,25,29] and are capable of exactly representing Mode I, Mode II, and, if required, Mode III vectors of the OD vector-valued OD(i) enrichment, Efront , i = I, II, III, for any given rotation R x x¯ (x). This set has six singular enrichments (two per global direction) for problems that are known a priori to not contain a significant Mode III component or whenever a crack front local coordinate system has its z¯ axis parallel to any of the global coordinate Cartesian system axes. It uses nine singular enrichments (three per global direction) for problems that are known to have a significant Mode III component and a crack front local coordinate system with its z¯ axis not parallel to any of the global coordinate Cartesian system axes. These sets of scalar OD enrichments are named OD6 (denoted by E OD6-scalar ) front ), respectively, depending on the number of enrichand OD9 (denoted by E OD9-scalar front ments they use. Details are presented next. OD6 basis. The OD6 basis, as its name implies, uses six scalar singular enrichments and is given by  κ − 12 cos θ2 − 12 cos 3θ 2 ⎢  √ ⎢ ¯ θ 1 1 3θ E OD6-scalar r R(x) (x) = front ⎣ κ + 2 sin 2 − 2 sin 2 ⎡

sin θ2

 

κ+

κ−



3 2 sin 3 2 cos

θ 2 θ 2

+ 12 sin 3θ 2

⎤

⎥ ⎥, + 12 cos 3θ 2 ⎦

cos θ2 (6.22)

¯ where R(x) is defined as

¯ R(x) =

⎧⎡ R11 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎣R21 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ R31

R12 R22 R32

⎤ R13 ⎥ R23 ⎦ R33

⎡ ⎪ ⎪ ⎪ R11 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎣R21 ⎪ ⎪ ⎩ R31

R12 R22 R32

⎤ 0 ⎥ 0⎦ 0

if z¯ is parallel to x, y, or z global axis, (6.23) otherwise.

It is noted that entries and in the matrix shown on the right-hand side of (6.22) are identical to the first two components of vector f I (θ ), entries and are identical (in absolute value) to the first two components of vector f II (θ ), and entry is identical to component of vector f III (θ ). The plot of the OD6 enrichments is shown in Fig. 6.10.

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Fundamentals of Enriched Finite Element Methods

Figure 6.10 Plot of OD enrichment functions.

The crack front local coordinate system z¯ axis is parallel to either the x, y, or z global axis if and only if the third column of the rotation matrix R x x¯ (x) contains exactly two entries equal to 0 and one entry with absolute value of 1. This can be used to check the cases indicated in (6.23). The OD6 basis can represent exactly the Mode I and II vector-valued OD enrichments defined in (6.20). However, it is only able to represent the Mode III vector if the crack local z¯ axis is parallel to a global coordinate system axis. OD9 basis. The OD9 basis uses six or nine scalar singular enrichments and is given by

GFEM approximations for fractures

145

⎧ ⎪ E OD6-scalar (x) if z¯ is parallel to x, y, or z global axis , ⎪ front ⎪ ⎪ ⎪ ⎪ ⎪ ⎤ ⎪ √ ⎨⎡ OD6-scalar E front, (x) E OD6-scalar (x) r sin θ2 OD9-scalar front, E front (x) = ⎢ ⎥ ⎪ ⎢ OD6-scalar ⎪ OD6-scalar (x) √r sin θ ⎥ otherwise , ⎪ E (x) E ⎢ ⎪ front, 2⎥ ⎪ ⎦ ⎣ front, ⎪ ⎪ ⎪ √ ⎩ E OD6-scalar (x) E OD6-scalar (x) θ r sin front, front, 2 (6.24) where subscripts refer to entry in matrix E OD6-scalar defined on the left-hand front side of (6.22). The OD9 basis can represent exactly the Mode I, II, and III vector-valued OD enrichments defined in (6.20). It uses six enrichments when the crack local z¯ axis is parallel to a global coordinate system axis, and nine enrichments otherwise. It is noted that OD enrichments adopt different enrichments for each displacement direction. This is in contrast with the enrichments introduced in previous sections and with the standard FEM, which usually adopts the same shape functions for all displacement components when solving elasticity problems. However, this does not pose implementation difficulties. Remark 12. In a crack propagation problem, the selection between OD6 and OD9 bases can be done automatically using the relative values of stress intensity factors (SIFs): Adopt OD9 if Model III SIF, KIII , is not negligible relative to KI ; otherwise, adopt OD6. In two dimensions, the OD basis is given by E OD front (x) =

√

 1 cos θ2 − 12 cos 3θ κ − 2 2 2D ⎣   rR x x¯ (x) 1 θ 1 3θ κ + 2 sin 2 − 2 sin 2 ⎡

 

κ+

κ−



3 2 sin 3 2 cos

θ 2 θ 2

+ 12 sin 3θ 2 + 12 cos 3θ 2

⎤ ⎦,

(6.25) where R 2D x x¯ (x) =

R11 R21

! R12 . R22

(6.26)

Remark 13. The OD enrichments belong to the patch approximation spaces χα (ωα ), α ∈ Ihbr . It is noted that the basis of some of these spaces may also have a Heaviside function if a geometrical enrichment strategy is adopted, as discussed later. GFEM shape functions defined using OD enrichments can approximate u˘ well, the solution of LEFM problems in a neighborhood of a crack front. A GFEM approximation of u˘ based on linear FEM shape functions, ϕα (x), is given by br

h

u˘ =

mα  α∈Ihbr

i=1

  br u br

ϕ (x)E (x) , α αi αi

(6.27)

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Fundamentals of Enriched Finite Element Methods

OD6-scalar (x), E OD9-scalar (x), or E OD (x). where vector E br αi (x) is the ith column of E front front front br Parameter mα is equal to 2 in 2-D or either 2 or 3, depending on whether OD6 or d OD9 enrichments are adopted, in 3-D. Vectors u br αi ∈ R , d = {2, 3}, (for a fixed i) are GFEM degrees of freedom. Operator performs an entrywise product of two vectors (Hadamard product).

Remark 14. The set of singular GFEM functions defined in (6.27) form a basis of the GFEM space Sbr GFEM defined earlier. Remark 15. It is possible to define high-order OD enrichments [20] from the product of OD singular functions and polynomials. Another option is to adopt in the definition of OD enrichments terms of order O(r 3/2 ) and higher from each mode of the asymptotic expansion of the elasticity solution [26,28]. While these high-order singular enrichments can be effective in improving the accuracy of the GFEM approximation, they lead to several additional degrees of freedom and make the control of the conditioning of the stiffness matrix more difficult [20]. A first-order GFEM approximation, uh ∈ SGFEM , of the solution of LEFM problems is simply a combination of the approximations uˆ h , u˜˜ h , and u˘ h , i.e., uh (x) =

 α∈Ih



u α ϕα (x) + 

std. FEM approx. br

+

mα 



uH α ϕα (x)Hd (x)

α∈IhH





discont. linear GFEM approx.

  br u br αi ϕα (x)E αi (x) ,

α∈Ihbr i=1





(6.28)

singular GFEM approx. br d where u α , u H α , and u αi ∈ R , d = {2, 3}, (for a fixed i) are degrees of freedom. High-order GFEM approximations can be defined using the approaches discussed in Section 6.2.3.1 and in references [10,20,21]. For later reference, we write Eq. (6.28) in matrix notation (zero entries not shown) ⎡ ⎤ ⎡ ⎤ ϕα Hd  ϕα  ⎣ ⎦ uα + ⎣ ⎦ uH ϕα ϕα Hd uh (x) = α ϕα ϕα Hd α∈Ih α∈IhH ⎡ ⎤ mbr ϕα E br α  αi ⎣ ⎦ u br + ϕα E br αi αi br ϕα E αi α∈Ihbr i=1

=

 α∈Ih

Nα u α +

 α∈IhH

br

NH α

uH α

+

mα  α∈Ihbr i=1

br N OD αi u αi = NU ,

(6.29)

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147

br where E br αi refers to entry of vector E αi defined in (6.27) and matrix N is simH br br ply the concatenation of N α , α ∈ Ih , N α , α ∈ IhH , and N OD αi , α ∈ Ih , i = 1, . . . , mα . Vector U collects and stacks all degrees of freedom.

6.2.4.3 Belytschko and Black branch enrichment functions The Belytschko and Black [27] (BB) enrichments, like the OD enrichments, are based on the first term from each mode of the asymptotic expansion of the elasticity solution (6.18). They are given by  BB 4 {Eαi }i=1 = E BB front (x) =

√ √ θ √ θ √ θ θ r sin , r cos , r sin sin θ, r cos sin θ , 2 2 2 2 (6.30)

where (r, θ ) are the polar coordinates in the crack front coordinate system shown in Fig. 6.9. ˘ using GFEM An approximation of the singular component of an LEFM solution, u, shape functions defined with BB enrichments is given by u˘ h (x) =

4  α∈Ihbr

BB u br αi ϕα (x)Eαi (x) .

(6.31)

i=1

d Vector u br αi ∈ R , d = {2, 3}, (for a fixed i) gives the GFEM degrees of freedom. It is noted that the same functions are used to enrich all components of the displacement vector, in contrast with OD enrichments. The BB basis adds four enrichments in each Cartesian direction of an enriched node. This leads to 12 dofs in 3-D and eight in 2D, per node, which is more than in the case of OD6/OD9 enrichments, which leads to six/nine dofs in 3-D and four dofs in 2-D. This higher number of degrees of freedom of the BB basis leads to matrices that are much more ill-conditioned than in the case of the OD basis, as shown in Chapter 11 and [21,30]. However, BB enrichments are popular in the literature due to their simplified individual basis terms and ease of implementation. Enrichments (6.30) were first proposed by Fleming et al. [31] in the context of the Element Free Galerkin Method [32]. Linear combinations of BB enrichments can represent the OD enrichments. Furthermore, vector u˘ h can be used in any Cartesian coordinate system, i.e., there is no need to formally transform (rotate) the vector from the crack front to the global Cartesian system since the transformation is linear. Nonetheless, a transformation from crack front polar coordinates (r, θ ) to global coordinates (x, y, z) must be performed as in the OD enrichments.

Remark 16. The BB enrichments (6.30) belong to the patch approximation spaces χα (ωα ), α ∈ Ihbr . It is noted that, like in the case of OD enrichments, the basis of some of these spaces may also have a Heaviside function if a geometrical enrichment strategy is adopted.

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Fundamentals of Enriched Finite Element Methods

Remark 17. The set of singular GFEM functions defined in (6.31) form a basis of the GFEM space Sbr GFEM . A first-order GFEM approximation, uh ∈ SGFEM , of the solution of LEFM problems using BB approximation (6.31) is given by   uh (x) = u α ϕα (x) + uH α ϕα (x)Hd (x) α∈Ih





std. FEM approx.

+

4  α∈Ihbr



α∈IhH





discont. linear GFEM approx.

BB u br αi ϕα (x)Eαi (x) ,

i=1



(6.32)

singular GFEM approx. br d where u α , u H α , and u αi ∈ R , d = {2, 3}, (for a fixed i) are degrees of freedom. As in the case of OD enrichments, high-order GFEM approximations based on BB enrichments can also be defined. Eq. (6.32) in matrix notation is given by

u (x) = h

 α∈Ih

Nα u α +

 α∈IhH

NH α

uH α

+

4 

br N BB αi u αi ,

(6.33)

α∈Ihbr i=1

where N α and N H α are defined in (6.29) and ⎤ ⎡ BB ϕα Eαj ⎥ ⎢ BB ϕα Eαj N BB ⎦. αi = ⎣ BB ϕα Eαj The singular OD and BB enrichments remove a major limitation of the Heaviside enrichment presented in the previous section. They allow the fracture front to be located anywhere in the mesh, instead of being restricted to element boundaries, as shown in Exercise 6.1 for the case of Heaviside enrichment. The OD and BB enrichments are singular, reflecting the behavior of the exact solution along a crack front. The integration of GFEM shape functions built with these enrichments must take into account this singularity. There is vast literature on this subject. The special quadrature rules for 2-D and 3-D elements presented in [33] are adopted in the computations presented in this book, unless stated otherwise.

6.2.5 Topological and geometrical singular enrichment We consider two strategies to define the set of nodes enriched with singular functions, Ihbr , namely topological and geometrical enrichment. In the first case, the set Ihbr contains only nodes from elements intersected by the crack front in 3-D or containing

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149

the crack tip in 2-D. If the crack front/tip is at the boundary of an element, all of its nodes belong to set Ihbr . The dimension of set Ihbr is nearly mesh-independent in this case. Examples of topological enrichment are shown in Figs. 6.13a and 6.17a for a 2-D and a 3-D fracture problem, respectively. In the second case, the set Ihbr contains all nodes within a fixed (mesh-independent) subdomain around the front or tip. The dimension of the set Ihbr grows quickly with mesh refinement. In this enrichment strategy, IhH ∩ Ihbr = ∅. This leads to two additional enrichment strategies: If a node α ∈ IhH ∩ Ihbr , it is enriched with (a) both Heaviside and singular enrichment or (b) only singular enrichments. One example of case (a) is shown in Fig. 6.13b, and examples for case (b) are shown in Fig. 6.18. 2-D and 3-D examples demonstrating the effect of these enrichment options on the convergence rate of the GFEM are presented in Sections 6.3 and 6.4, respectively. Their effect on the conditioning of the stiffness matrix is discussed in Section 11.1.6. Example 6.2: GFEM enrichments for a half-penny crack. This example demonstrates how branch function enrichments can address the limitations of Heaviside enrichments identified in Example 6.1. The half-penny surface crack shown in Fig. 6.5 is solved using polynomial, Heaviside, and branch OD enrichments. Fig. 6.11a shows a zoom-in of the half-penny surface crack and nodes from sets IhH and Ihbr . This surface defines the geometry of the crack surface as discussed in Exercise 6.1. Nodes with boxes are from enrichment set IhH , while those with a spherical glyph are from set Ihbr . The crack surface is shown in green. Patches that intersect the crack front are enriched with branch functions, while those fully cut by the crack are enriched with the Heaviside function. In the GFEM, the geometry of a crack is defined by the enrichment functions, not the finite element mesh. This effective crack surface, defined by the enrichment functions, is called the computational crack surface [8]. Fig. 6.11b shows this surface. It can clearly approximate well the circular shape of the crack front.

Figure 6.11 Comparison between the user-defined geometrical crack surface (left) and the computational or effective crack surface defined by GFEM enrichments (right).



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Fundamentals of Enriched Finite Element Methods

6.2.6 Discrete equilibrium equations The weak form presented in Section 6.1.1 is discretized in this section using the first-order GFEM approximation defined in (6.28). For simplicity, polynomial GFEM enrichments are not considered. We also restrict the derivation to the case of OD branch functions. The case of BB enrichments is left as an exercise for the reader. The reader is encouraged to review the solution of Example 4.3 before reading this section. With the above simplifications, the GFEM approximation space for LEFM problems is then given by (cf. Eq. (6.29))  SGFEM =

mα "    H br u " uh = Nα u α + NH u + N OD α α αi u αi br

h"

α∈Ih

α∈IhH

α∈Ihbr i=1

#

br d u α, u H α , u αi ∈ R , d = 2 or 3 .

We further simplify the derivation by assuming that the prescribed Dirichlet boundary condition is homogeneous. This leads to the following approximation spaces (cf. Eqs. (6.4) and (6.3)):   Vh () = V0h () = v ∈ SGFEM () : v| D = 0 . The GFEM approximation uh (x) given by (6.28) satisfies the homogeneous boundary conditions on  D if the degrees of freedom at nodes on  D are eliminated, i.e., set to zero. This is not possible if the crack surface d intersects  D . In this case, it is necessary to adopt other approaches to enforce the Dirichlet boundary condition, such as a penalty [34] or Nitsche’s [35] methods. Hereafter, we assume that d ∩  D = ∅, which is a common situation. Using the above assumptions, the weak formulation of the problem is, from (6.2), given by: Find uh ∈ V0h () such that ∀ v h ∈ V0h () 

    + (v h ) : C : (uh ) d = v h · t¯ d + v h · t¯ d d . + N 

 

  d

B (uh ,v h )

(6.34)

L (v h )

Using Voigt notation, the approximation of the strain tensor  can be written as ⎡ ⎢ ⎢ ⎢ (uh ) = ⎢ ⎢ ⎢ ⎣

x y z yz xz xy

⎤ ⎥ ⎥ ⎥ ⎥ = Duh ⎥ ⎥ ⎦

GFEM approximations for fractures

=



151

DN α u α +

=

br

DN H α

uH α

α∈IhH

α∈Ih







Bα u α +

+

br DN OD αi u αi

α∈Ihbr i=1 br

BH α

uH α

+

α∈IhH

α∈Ih

mα 

mα 

br B OD αi u αi

α∈Ihbr i=1

= DNU = BU , where D is the differential operator ∇ s in matrix format ⎡

∂ ∂x

⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ D=⎢ ⎢ 0 ⎢ ⎢ ∂ ⎣ ∂z ∂ ∂y

0

0

⎤

⎥ 0 ⎥ ⎥ ∂ ⎥ ∂z ⎥ , ∂ ⎥ ⎥ ∂y ⎥ ∂ ⎥ ∂x ⎦ 0

∂ ∂y

0 ∂ ∂z

0 ∂ ∂x

and ⎡ ⎢ ⎢ ⎢ B α = DN α = ⎢ ⎢ ⎢ ⎣

(ϕα ),x 0 0 0 (ϕα ),z (ϕα ),y

⎡

H BH α = DN α

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

0 (ϕα ),y 0 (ϕα ),z 0 (ϕα ),x

(ϕα Hd ),x 0 0 0 (ϕα Hd ),z (ϕα Hd ),y

⎡

0 0 (ϕα ),z (ϕα ),y (ϕα ),x 0

0 (ϕα Hd ),y 0 (ϕα Hd ),z 0 (ϕα Hd ),x

(ϕα E br αi ),x

⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ OD B OD = DN = ⎢ αi αi ⎢ 0 ⎢ ⎢ ⎢ (ϕα E br ),z αi ⎣ br (ϕα E αi ),y

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

0 0 (ϕα Hd ),z (ϕα Hd ),y (ϕα Hd ),x 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

0

0

(ϕα E br αi ),y

0

0

(ϕα E br αi ),z

(ϕα E br αi ),z

(ϕα E br αi ),y

0

(ϕα E br αi ),x

(ϕα E br αi ),x

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎦

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Fundamentals of Enriched Finite Element Methods

H OD br Matrix B is simply the concatenation of B α , α ∈ Ih , B H α , α ∈ Ih , and B αi , α ∈ Ih , br with i = 1, mα , and U is the global vector of degrees of freedom. Similarly for virtual displacements,

vh = N V ,   $ % v h = N  V = N(d+ ) − N (d− ) V , (v h ) = DN V = BV . Plugging the above approximations into the weak form (6.34) leads to  h h T B(u , v ) = V B T CB d U = V T KU , 



K

where C is the linear elasticity (Hooke) tensor C in matrix form, and   + L (v h ) = V T N T t¯ d +V T NT t¯ d d = V T F t + V T F d . N d+





Ft

Fd

The above, together with (6.34), leads to the following system of equilibrium equations: KU = F t + F d .

(6.35)

In a computer implementation, the global matrix K and global vectors F t and F d are assembled from element stiffness matrix k e and load vectors f t,e and f d,e , respectively, as in the standard FEM (see also Section 2.1.4). The integration of element matrices and vectors requires in general nonstandard procedures and numerical quadratures due to the singularity and discontinuity of GFEM shape functions. This issue is discussed in, e.g., [33] and Appendix A of [21]. Provided that the assumptions listed at the beginning of this section hold, Dirichlet boundary conditions can be imposed by simply eliminating rows and columns from the system of equations (6.35), like in the standard FEM. ANALOGIES BETWEEN GFEM FOR 1-D AND 3-D FRACTURE PROBLEMS EXERCISE 6.1.— Compare the equations presented in Section 6.2.6 with those used in the solution of Example 4.3. STRAIN–DISPLACEMENT MATRIX FOR BB ENRICHMENTS EXERCISE 6.2.— Write the 3-D strain-displacement matrix, B BB αi , using the singular GFEM approximation defined in (6.31) and based on Belytschko–Black branch function enrichments (6.30). This matrix is analogous to OD BB matrix B OD αi . What is the key difference between B αi and B αi ? EXERCISE 6.3.— STRAIN–DISPLACEMENT MATRIX FOR VECTOR-VALUED OD ENRICHMENTS Write the 3-D strain-displacement matrix, B OD αi , if the OD enrichments are defined as vector-valued functions like in Eqs. (6.19) and (6.20).

GFEM approximations for fractures

153

EXERCISE 6.4.— GFEM FOR COHESIVE FRACTURES: WEAK FORM Modify the weak form adopted in Section 6.2.6 to handle the case of fractures with a linear cohesive law given by t¯ + d = k s u , where k s is the stiffness of the linear springs on the crack surface and u = u(d+ ) − u(d− ) is the displacement jump across the crack surface d . GFEM APPROXIMATION FOR COHESIVE FRACTURES EXERCISE 6.5.— Discuss possible modifications to GFEM approximations (6.28) and (6.32), which are required for the solution of cohesive fractures defined in Exercise 6.4. Write down the GFEM approximation for this class of problem. Hint: The elasticity solution does not have a singularity at the crack front of cohesive fractures, in contrast with the LEFM case discussed in previous sections. The GFEM enrichments should reflect that. EXERCISE 6.6.— GFEM FOR COHESIVE FRACTURES: EQUILIBRIUM EQUATIONS Derive the GFEM equilibrium equations for the weak form derived in Exercise 6.4 and the GFEM approximation derived in Exercise 6.5. EXERCISE 6.7.— GFEM FOR PRESSURIZED FRACTURES: WEAK FORM Modify the weak form adopted in Section 6.2.6 to handle the case of pressurized fractures. Assume constant tractions on the fracture surface given by [8] + − t¯ + d = −pn = pn ,

where n+ and n− are the unity normal vectors to fracture faces d+ and d− , respectively, and p is the pressure of the fluid in the fracture cavity. EXERCISE 6.8.— GFEM FOR PRESSURIZED FRACTURES: EQUILIBRIUM EQUATIONS Derive the GFEM equilibrium equations for the weak form derived in Exercise 6.7 and the GFEM approximation (6.28) (see also Eq. (6.29)). Note: Fractures subjected to constant pressure have the same type of singularity as dry fractures, justifying the adoption of the GFEM approximation (6.28).

6.3 Convergence of linear GFEM approximations: 2-D edge crack The convergence of linear GFEM approximations given by (6.28) is studied in this section. The performance of the method is compared with linear and quadratic FEMs defined on meshes with three- and six-node triangular elements, respectively. The 2-D edge crack shown in Fig. 6.12a is solved with a sequence of uniform meshes of triangular elements. The prescribed boundary conditions are also illustrated in the figure. ¯ and the crack line d are given by The problem domain  ¯ = {x|0 ≤ x ≤ 16, 0 ≤ y ≤ 16}  and d = {x|0 ≤ x ≤ 8, y = 8} , respectively. The crack tip C is taken as the endpoint of d in the interior of the domain: C = (8, 8).

154

Fundamentals of Enriched Finite Element Methods

Figure 6.12 Two-dimensional edge crack problem definition, applied tractions computed from Mode I analytical solution. Deformed shape and contour plot of von Mises stress are shown [20].

The cracked domain is subjected to traction boundary conditions computed from the first term of the Mode I asymptotic expansion of the elasticity solution in the neighborhood of a crack (see also Eq. (6.18)), ⎫ ⎧  1 θ 1 3θ ⎪ ⎪ κ − cos − cos ⎪ ⎬ 2 2 2 2 ⎪ √ ⎨ , (6.36) uI (r, θ ) = r   ⎪ ⎪ ⎪ ⎭ ⎩ κ + 1 sin θ − 1 cos 3θ ⎪ 2

2

2

2

where the arguments are given in the crack tip polar coordinates (r, θ ), −π < θ < π, κ = 3 − 4ν is the Kolosov’s constant for a plane strain problem, and ν is the Poisson’s ratio. The adopted material properties are Young’s modulus equal to unity and a Poisson’s ratio of 0.30. The displacement field given by (6.36) is the exact solution of this problem, since it satisfies the equilibrium equations (2.57) and boundary condition (2.59). The stiffness matrix for a pure Neumann problem like this one is singular and has three zero eigenvalues (in 2-D) corresponding to the three rigid body motions. To uniquely solve the problem, an iterative algorithm is applied to the perturbed and scaled stiffness matrix, as described in Section 11.1.5. A sequence of structured and uniform finite element discretizations is used for both the GFEM and the FEM. The element sizes in the GFEM meshes are given by h = 16/(2j + 1) with j = 3, 4, 5, 6, 7, 8, leading to mesh grids of (2j + 1) × (2j + 1) squared cells. Triangular elements are used with two triangles per grid cell. The crack line lies in the interior of generalized finite elements, as shown in Fig. 6.13 for the case of the 17 × 17 mesh. The nodes with Heaviside and OD singular enrichments are shown in the figure. No polynomial enrichment is adopted. Thus, the polynomial order of the approximation is equal to the linear partition of unity provided by the triangular elements. The FEM uses uniform meshes with elements fitting the crack line. Double nodes are used at the crack to approximate the displacement jump. The FEM meshes

GFEM approximations for fractures

155

have elements of sizes h = 16/2j , with j = 3, 4, 5, 6, 7, 8, leading to mesh grids of 2j × 2j cells.

Figure 6.13 Mesh with 17 × 17 cells and nodal enrichments. Nodes with a red sphere are enriched with OD singular functions, while those with a blue square are enriched with the Heaviside function. These nodes belong to sets Ihbr and IhH , respectively. Some nodes in (b) have both enrichments and are shown with a yellow sphere. No polynomial enrichment is adopted.

The convergence of linear GFEM approximations using topological and geometrical singular enrichment strategies (cf. Section 6.2.5) are compared with that of linear and quadratic FEMs in the next sections.

6.3.1 Topological enrichment In this enrichment strategy, only the nodes of the element containing the crack tip or the elements with the tip at their boundary are enriched with singular functions. Therefore, the size of the branch function enrichment region goes to zero as the mesh is refined close to the crack tip. This leads to sub-optimal convergence rates [9,36], as demonstrated in this example. Fig. 6.13a shows one example of this enrichment strategy. Fig. 6.14a shows the plot of the relative error in energy norm, er , against the number of degrees of freedom. The convergence of the FEM with linear triangular elements is also included for comparison with the GFEM. The relative error in energy norm is given by (cf. Chapter 2)

e = r

& & &u − u h &

E()

u E()

' =

% $ B (u, u) − B uh , uh . √ B (u, u)

(6.37)

156

Fundamentals of Enriched Finite Element Methods

The strain energy of the exact solution is computed using (6.36)—the exact solution for this problem—and the bilinear form given in (6.2).1

Figure 6.14 Relative error in energy norm of linear GFEM with topological enrichment. The convergence of the FEM with linear and quadratic triangular elements is also shown.

The slope of the curves shown in Fig. 6.14a reaches 0.25. This is the convergence rate of the methods, and it implies that the error of the FEM and GFEM with topological enrichment is of order O(h1/2 ) = O(N −1/4 ), where N is the number of dofs. The convergence of the methods is therefore controlled by the singularity at the crack tip. The optimal rate of convergence in energy norm for a linear approximation is 1.0, with respect to element size, and 0.5, with respect to number of dofs in 2-D. However, since the solution of the problem belongs to a Hilbert space of order k = 3/2, the convergence rate of the GFEM and the FEM is β = min(p, k − 1) = 0.5. While the convergence rates of the GFEM and the FEM are nearly identical, the error of the GFEM is smaller than the FEM. This is because the GFEM uses enrichment functions that capture the singularity at the crack tip. This reduces the error but does not increase the rate of convergence when topological enrichment is used.

6.3.2 Comparison with best-practice FEM While Fig. 6.14a shows that linear GFEM approximations are more efficient than linear FEMs, in practice, fracture mechanics problems are typically solved using quadratic finite elements with quarter-point elements around the crack tip. These elements were originally introduced by Henshel and Shaw [37] and by Barsoum [38]. 1 The crack faces are stress-free for this problem, i.e., t¯ + = t¯ − = 0. Therefore, d d



B (u, u) =

N

u · t¯ d := W ,

where W represents (twice) the virtual work of the external tractions t¯ . The computation of W involves only a surface integral, in contrast with the computation of the bilinear form B. Furthermore, the integrand of W is evaluated far from the crack tip, where the solution is singular and more difficult to numerically integrate. For these reasons, it is recommended, whenever possible, to compute the energy of the solution (exact or approximate) using the work of the prescribed Neumann boundary conditions, instead of the bilinear form B.

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√ They can approximate, like the branch functions used in the GFEM, the r singularity of the LEFM solutions at the crack tip. Quarter-point elements do not require modifications of an FEM code. The midside node of element edges with an end-node at the crack tip is simply moved closer to the crack tip in these elements. This leads to a nonlinear mapping between master and global coordinates that reproduces the square root singularity of the elasticity solution at the crack tip [38]. The definition of 2-D quarter-point elements may also involve edge collapsing of quadrilateral elements. The quarter-point elements adopted here are based on six-node triangular elements. We believe that it is necessary to compare the GFEM, or any other method, with the best-practice FEMs for fracture analysis before concluding which method is more computationally efficient. Fig. 6.14b shows the convergence of quadratic triangular FEMs with and without quarter-point elements on the same sequence of meshes defined earlier. We can observe that the convergence rate of the FEM on quadratic meshes is the same as on linear ones. Nonetheless, the curve for meshes with quarter-point elements is significantly shifted down relative to the curve without these elements. The curves for linear GFEM and quadratic FEM without quarter-point elements are nearly identical. However, the error of the FEM on meshes with quarter-point elements is significantly smaller than the error of linear GFEM approximations with topological enrichment. In the next section, we investigate under which circumstances linear GFEM approximations with geometrical enrichment are more efficient than best-practice FEMs for LEFM problems.

6.3.3 Geometrical enrichment The results presented in Fig. 6.14 show that the GFEM presented in this chapter enables the use of meshes that do not fit the crack surface while delivering more accurate solutions than the linear FEMs. However, if topological enrichment is adopted, the convergence rate of the method is the same as in the FEM and the method can be less accurate than quadratic FEMs with quarter-point elements. The convergence of the GFEM for 2-D fracture mechanics problems is analyzed in [39]. They show that if proper enrichment functions are used in a fixed region around the crack tip, the GFEM delivers optimal convergence—the same convergence rate as in problems with smooth solutions. In this so-called geometrical enrichment strategy, the size of the enrichment zone is independent of the mesh size, in contrast with topological enrichment [9,36]. An example of a mesh with geometrical enrichment is shown in Fig. 6.13b. The set of nodes enriched with singular functions in this case is given by Ihbr = {α ∈ Ih : x α ∈ B(x c , R)} ,

(6.38)

where B(x c , R) is the circle centered at point x c with radius R. Center x c is taken at the crack tip located at (8, 8) and a radius R = 2 is adopted. It is noted that in the enrichment strategy adopted here, some nodes may be enriched with both singular and Heaviside functions.

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The convergence of the GFEM with geometrical enrichment is shown in Fig. 6.15. The asymptotic convergence rate of this GFEM is 0.51, which is slightly above the optimal rate for a first-order approximation. The geometrical enrichment eliminates the effect of the crack tip singularity on the convergence of the GFEM even with a sequence of uniform meshes, which is quite remarkable. However, the error of the linear GFEM with geometrical enrichment on the first three meshes is still higher than for the quadratic FEM with quarter-point elements. Given that the slope of the curve for the GFEM with geometrical enrichment is higher than the curve for quarter-point elements, the GFEM eventually becomes, for the same problem size, more accurate than the FEM. However, at least for this problem, the level of mesh refinement required for this to happen may be not practical. This is in general the case for 3-D problems, as shown in Section 6.4. One obvious strategy to further improve the computational performance of the GFEM is to adopt quadratic approximations combined with geometrical enrichment around the crack tip. This is shown in [20] and in [21] for 2-D and 3-D fracture problems, respectively, together with strategies to control the conditioning of the GFEM approximations. Conditioning control is needed even for linear GFEM approximations. This is presented in detail in Chapter 11.

Figure 6.15 Relative error in energy norm of linear GFEM with topological and geometrical enrichment. The convergence of the FEM with linear and quadratic triangular elements is also shown.

6.4 Convergence of linear GFEM approximations: 3-D edge crack This section presents convergence studies of 3-D linear GFEM approximations given by (6.28) and (6.32), which adopt OD and BB singular enrichments, respectively. The

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convergence of the GFEM using these singular enrichments is compared with that of linear and quadratic FEMs. The 3-D edge crack shown in Fig. 6.16 is solved with a sequence of structured and uniform meshes of tetrahedron elements. Four meshes are used for the GFEM: Meshes with (17 × 17 × 4), (33 × 33 × 8), (49 × 49 × 12), and (65 × 65 × 16) sets of tetrahedron elements in the x-, y-, and z-directions, respectively. These meshes were generated by first creating a uniform mesh of hexahedron elements and then replacing each hexahedron by a set of six tetrahedrons. Mesh (17 × 17 × 4) is shown in Fig. 6.17a. These meshes are such that the crack surface cuts through the elements. The FEM uses linear and quadratic tetrahedron elements with four and ten nodes, respectively, and meshes with elements fitting the crack surface. Double nodes are used at the crack surface to represent the displacement jump. Four meshes are adopted: (16 × 16 × 4), (32 × 32 × 8), (48 × 48 × 12), and (64 × 64 × 16) sets of tetrahedrons in the x-, y-, and z-directions, respectively. ¯ and the crack surface d are given, respectively, by The problem domain   = {x | 0 ≤ x ≤ 16, 0 ≤ y ≤ 16, −4 ≤ z ≤ 0} and d = {x | 0 ≤ x ≤ 8, y = 8, −4 ≤ z ≤ 0}.

Figure 6.16 Edge crack loaded with Mode I tractions. The crack surface is shown in green.

The cracked domain is subjected to traction boundary conditions computed from the first term of the Mode I asymptotic expansion of the elasticity solution in the neighborhood of a crack, ⎫ ⎧  ⎪ ⎪ κ − 12 cos θ2 − 12 cos 3θ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨   √ 1 θ 1 3θ (6.39) uI (r, θ ) = r κ + 2 sin 2 − 2 sin 2 ⎪ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0

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where r and θ are polar coordinates at the crack front, −π < θ < π, κ is the material constant (3 − 4ν), and ν is the Poisson’s ratio. The Young’s modulus is taken as E = 1 and Poisson’s ratio ν = 0. Coordinate direction z is the direction of the crack front. The prescribed boundary conditions are illustrated in Fig. 6.16. It is noted that if a nonzero Poisson’s ratio is adopted for this problem, the traction vector t¯ at the intersection of the crack front with the domain boundary is not bounded. The stress component σzz at these points is given by $ % 2 E ν cos θ2 . (6.40) σzz = √ r (ν + 1) This leads to difficulties in applying Neumann boundary condition at these points. The displacement given by (6.39) is the exact solution of this problem, as it satisfies the equilibrium equations (2.57) and boundary condition (2.59). The solution of this quasi-3-D problem is constant along the z-direction—the direction of the crack front. This problem is the 3-D counterpart of the 2-D edge crack studied in Section 6.3. Most of the results presented here are from Section 4.1 of [25]. The convergence of GFEM approximations (6.28) and (6.32), which adopt OD and BB singular enrichments, is shown in Figs. 6.19 and 6.20, respectively. The relative error in the energy norm is computed using Eq. (6.37). Since this is a Mode I problem, the OD6 basis defined in Section 6.2.4.2 is adopted. The strain energy of the exact solution is computed using (6.39)—the exact solution for this problem—and the bilinear form given in (6.2). The convergence of the GFEM with topological and geometrical singular enrichments is shown in the figures. The convergence of linear and quadratic FEM approximations, with and without quarter-point elements, is also shown in Figs. 6.19 and 6.20. Quarter-point tetrahedron elements are used around the crack front. They are created by moving closer to the crack front, the midside node of element edges that have a single node on the crack front. Topological enrichment. An example of a GFEM mesh with topological enrichment is shown in Fig. 6.17a. The nodes with singular enrichments (OD or BB) and with the Heaviside function are shown in the figure, while the deformed configuration is shown in Fig. 6.17b. Like in the 2-D edge crack problem solved earlier, the convergence rate of the GFEM with topological enrichment and the FEM is controlled by the singularity at the crack front. The slopes of the curves for these approximations are indicated in Figs. 6.19 and 6.20, and they approach the asymptotic rate of 0.5/3 = 0.167. This is the case for the GFEM with OD or BB enrichments, as shown in the figures. This implies that for 3-D fracture problems, the error of the FEM and GFEM with topological enrichment is of the order O(h1/2 ) = O(N −1/6 ), where N is the number of dofs. This is a very low convergence rate and explains in part why solving fracture problems in 3-D is challenging. Both OD and BB singular enrichment bases span a space that contains the exact solution (6.39) of the problem at hand. However, since the dimension of the BB basis is higher than the OD basis, the approximation provided by the BB enrichments has a slightly smaller error than the one provided by the OD basis. This can be observed by

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Figure 6.17 Nodes with red boxes are from enrichment set IhH , while those with a purple spherical glyph are from set Ihbr . The crack surface is shown in green. Patches that intersect the crack front are enriched with branch functions, while those fully cut by the crack are enriched with the Heaviside function. The deformed configuration shows that the enrichments represent exactly the crack surface geometry.

comparing the curves shown in Figs. 6.19 and 6.20 for linear GFEM approximations with topological OD and BB singular enrichments. Geometrical enrichment. Examples of meshes with geometrical enrichments are shown in Fig. 6.18. The set of nodes enriched with singular functions in this case is given by Ihbr = {α ∈ Ih : x α ∈ B(C, D)} ,

(6.41)

where B(C, D) is the cuboidal region with edge length D = 4 around the crack front C and given by B(C, D) = {x | 6 ≤ x ≤ 10, 6 ≤ y ≤ 10, −4 ≤ z ≤ 0} , with C = {x | x = 8, y = 8, −4 ≤ z ≤ 0}. Therefore, singular enrichments are used in a cuboidal region of (4 × 4 × 4) units around the crack front. It is noted that for this problem, a node is enriched with either a Heaviside or branch functions, but not both. Therefore, Ihbr ∩ IhH = ∅. The convergence of the GFEM with OD and BB geometrical enrichment is shown in Figs. 6.19 and 6.20, respectively. The slope of the curves for these approximations approach 1/3 = 0.33, which is the optimal rate for linear approximations. Like in the 2-D case, the geometrical enrichment eliminates the effect of the singularity at the crack front even on a sequence of uniform meshes. The slight suboptimal rate with geometrical enrichment is likely caused by adopting Ihbr ∩ IhH = ∅. The convergence

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Figure 6.18 Geometrical enrichment around the crack front. The size of the enrichment zone is independent of the element size. Nodes with boxes are enriched with the Heaviside function, while those with a spherical glyph are enriched with singular branch functions (OD or BB). The crack surface is shown in green.

Figure 6.19 Relative error in energy norm of linear GFEM and FEM approximations against the problem size. Topological and geometrical enrichments with OD singular functions are adopted.

analysis presented in Section 11.1.6 supports this conjecture. Furthermore, it can be shown that even with geometrical enrichment, a sufficiently fine mesh must be used in order to achieve optimal convergence [40]. The required mesh size depends on the

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Figure 6.20 Relative error in energy norm of linear GFEM and FEM approximations against the problem size. Topological and geometrical enrichments with BB singular functions are adopted.

size of the geometrical enrichment zone and the polynomial order of the GFEM approximation (cf. Section 4.3 of [40]). Comparison with best-practice FEM. The convergence of quadratic FEM approximations with and without quarter-point elements is also shown in Figs. 6.19 and 6.20. Like in the 2-D case, the curve for meshes with quarter-point elements is significantly shifted down relative to the FEM curve of the same order, but without these elements. The error of the FEM on meshes with quarter-point elements is significantly smaller than the error of linear GFEM, even when a geometrical singular enrichment is adopted. At a very low error level, the GFEM curve will cross the one for the FEM with quarter-point elements. However, the crossing point may happen at an error level that is not needed in many applications. Furthermore, the corresponding problem size will be impractical. Like in the 2-D case, adopting a quadratic GFEM with geometrical enrichment is a more efficient approach, as shown in [20,21].

References [1] A. Ural, G. Heber, P. Wawrzynek, A. Ingraffea, D. Lewicki, J. Neto, Three-dimensional, parallel, finite element simulation of fatigue crack growth in a spiral bevel pinion gear, Engineering Fracture Mechanics 72 (2005) 1148–1170. [2] B. Davis, P. Wawrzynek, A. Ingraffea, 3-D simulation of arbitrary crack growth using an energy-based formulation—Part I: Planar growth, Engineering Fracture Mechanics 115 (2014) 204–220, https://doi.org/10.1016/j.engfracmech.2013.11.005.

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[3] P. Bouchard, F. Bay, Y. Chastel, Numerical modelling of crack propagation: Automatic remeshing and comparison of different criteria, Computer Methods in Applied Mechanics and Engineering 192 (35–36) (2003) 3887–3908, https://doi.org/10.1016/S0045-7825(03) 00391-8. [4] A. Paluszny, R. Zimmerman, Numerical simulation of multiple 3D fracture propagation using arbitrary meshes, Computer Methods in Applied Mechanics and Engineering 200.9–12 (2011) 953–966, https://doi.org/10.1016/j.cma.2010.11.013. [5] FRANC3D, Version 7.0, Fracture Analysis Consultants, Inc., Ithaca, NY (USA), 2016. [6] J. Pereira, C. Duarte, D. Guoy, X. Jiao, Hp-generalized FEM and crack surface representation for non-planar 3-D cracks, International Journal for Numerical Methods in Engineering 77 (5) (2009) 601–633, https://doi.org/10.1002/nme.2419. [7] J. Pereira, C. Duarte, X. Jiao, Three-dimensional crack growth with hp-generalized finite element and face offsetting methods, Computational Mechanics 46 (3) (2010) 431–453, https://doi.org/10.1007/s00466-010-0491-3. [8] P. Gupta, C. Duarte, Simulation of non-planar three-dimensional hydraulic fracture propagation, International Journal for Numerical and Analytical Methods in Geomechanics 38 (2014) 1397–1430, https://doi.org/10.1002/nag.2305. [9] P. Laborde, J. Pommier, Y. Renard, M. Salaün, High-order extended finite element method for cracked domains, International Journal for Numerical Methods in Engineering 64 (3) (2005) 354–381, https://doi.org/10.1002/nme.1370. [10] A. Byfut, A. Schroder, hp-adaptive extended finite element method, International Journal for Numerical Methods in Engineering 89 (2012) 1392–1418. [11] C. Duarte, I. Babuška, J. Oden, Generalized finite element methods for three dimensional structural mechanics problems, Computers & Structures 77 (2000) 215–232, https:// doi.org/10.1016/S0045-7949(99)00211-4. [12] J. Oden, C. Duarte, O. Zienkiewicz, A new cloud-based hp finite element method, Computer Methods in Applied Mechanics and Engineering 153 (1998) 117–126, https:// doi.org/10.1016/S0045-7825(97)00039-X. [13] R. Taylor, O. Zienkiewicz, E. Onate, A hierarchical finite element method based on the partition of unity, Computer Methods in Applied Mechanics and Engineering 152 (1998) 73–84. [14] B.A. Szabó, I. Babuška, Finite Element Analysis, John Wiley and Sons, New York, 1991. [15] B. Szabó, A. Düster, E. Rank, The p-version of the finite element method, in: Encyclopedia of Computational Mechanics, John Wiley & Sons, Ltd, 2004. [16] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) (1999) 131–150. [17] G.N. Wells, L.J. Sluys, A new method for modelling cohesive cracks using finite elements, International Journal for Numerical Methods in Engineering 50 (12) (2001) 2667–2682. [18] N. Moës, T. Belytschko, Extended finite element method for cohesive crack growth, Engineering Fracture Mechanics 69 (2002) 813–833. [19] Z. Shabir, E. Van der Giessen, C. Duarte, A. Simone, The role of cohesive properties on intergranular crack propagation in brittle polycrystals, Modelling and Simulation in Materials Science and Engineering 19 (3) (2011) 035006, https://doi.org/10.1088/0965-0393/ 19/3/035006. [20] A. Sanchez-Rivadeneira, C. Duarte, A stable generalized/extended FEM with discontinuous interpolants for fracture mechanics, Computer Methods in Applied Mechanics and Engineering 345 (2019) 876–918, https://doi.org/10.1016/j.cma.2018.11.018.

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[21] A. Sanchez-Rivadeneira, N. Shauer, B. Mazurowski, C. Duarte, A stable generalized/extended p-hierarchical FEM for three-dimensional Linear Elastic Fracture Mechanics, Computer Methods in Applied Mechanics and Engineering 364 (2020) 112970, https:// doi.org/10.1016/j.cma.2020.112970. [22] C. Duarte, L. Reno, A. Simone, A high-order generalized FEM for through-the-thickness branched cracks, International Journal for Numerical Methods in Engineering 72 (3) (2007) 325–351, https://doi.org/10.1002/nme.2012. [23] M. Kanninen, C. Popelar, Advanced Fracture Mechanics, Oxford University Press, New York, 1985. [24] M. Williams, On the stress distribution at the base of a stationary crack, Journal of Applied Mechanics 24 (1) (1956) 109–114. [25] V. Gupta, C. Duarte, I. Babuška, U. Banerjee, Stable GFEM (SGFEM): Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics, Computer Methods in Applied Mechanics and Engineering 289 (June 2015) 355–386, https:// doi.org/10.1016/j.cma.2015.01.014. [26] J. Oden, C. Duarte, Chapter: Clouds, cracks and FEMs, in: B. Reddy (Ed.), Recent Developments in Computational and Applied Mechanics, International Center for Numerical Methods in Engineering, CIMNE, Barcelona, Spain, 1997, pp. 302–321, http:// gfem.cee.illinois.edu/jmartincolor/. [27] T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (1999) 601–620. [28] J. Oden, C. Duarte, Chapter: Solution of singular problems using hp clouds, in: J. Whiteman (Ed.), The Mathematics of Finite Elements and Applications—Highlights 1996, John Wiley & Sons, New York, NY, 1997, pp. 35–54, http://gfem.cee.illinois.edu/papers/ mafelap_color.pdf. [29] J. Pereira, C. Duarte, X. Jiao, D. Guoy, Generalized finite element method enrichment functions for curved singularities in 3D fracture mechanics problems, Computational Mechanics 44 (1) (2009) 73–92, https://doi.org/10.1007/s00466-008-0356-1. [30] A. Sanchez-Rivadeneira, C. Duarte, A simple, first-order, well-conditioned, and optimally convergent generalized/extended FEM for two- and three-dimensional linear elastic fracture mechanics, Computer Methods in Applied Mechanics and Engineering 372 (2020) 113388, https://doi.org/10.1016/j.cma.2020.113388. [31] M. Fleming, Y.A. Chu, B. Moran, T. Belytschko, Enriched element-free Galerkin methods for crack tip fields, International Journal for Numerical Methods in Engineering 40 (1997) 1483–1504. [32] T. Belytschko, Y. Lu, L. Gu, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 (1994) 229–256. [33] K. Park, J. Pereira, C. Duarte, G. Paulino, Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems, International Journal for Numerical Methods in Engineering 78 (10) (2009) 1220–1257, https:// doi.org/10.1002/nme.2530. [34] J. Chessa, P. Smolinski, T. Belytschko, The extended finite element method (XFEM) for solidification problems, International Journal for Numerical Methods in Engineering 53 (2002) 1959–1977. [35] M. Griebel, M. Schweitzer, A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic PDEs, SIAM Journal on Scientific Computing 22 (3) (2000) 853–890. [36] E. Béchet, H. Minnebo, N. Moës, B. Burgardt, Improved implementation and robustness study of the X-FEM for stress analysis around cracks, International Journal for Numerical Methods in Engineering 64 (2005) 1033–1056.

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[37] R. Henshell, K. Shaw, Crack tip finite elements are unnecessary, International Journal for Numerical Methods in Engineering 9 (3) (1975) 495–507, https://doi.org/10.1002/nme. 1620090302. [38] R. Barsoum, On the use of isoparametric finite elements in linear fracture mechanics, International Journal for Numerical Methods in Engineering 10 (1) (1976) 25–37, https:// doi.org/10.1002/nme.1620100103. [39] S. Nicaise, Y. Renard, E. Chahine, Optimal convergence analysis for the extended finite element method, International Journal for Numerical Methods in Engineering 86 (2011) 528–548, https://doi.org/10.1002/nme.3092. [40] V. Gupta, C. Duarte, On the enrichment zone size for optimal convergence rate of the generalized/extended finite element method, Computers & Mathematics with Applications 72 (2016) 481–493, https://doi.org/10.1016/j.camwa.2016.04.043.

Generalized enrichment functions for weak discontinuities

7

Materials are rarely made of a single constituent; on the contrary, most natural and man-made materials are composed of different phases, each having its own chemical composition. Materials like steel, that seem to be homogeneous at the macroscopic scale, may be quite heterogeneous when looking at their microstructure. The presence of a particular phase (impurities) may have a detrimental impact on steel’s strength and ductility properties. In other cases the blending between phases may be desired; it is well known that the phases’ interplay in natural materials like bone or nacre plays a fundamental role in improving fracture resilience. In other words, high fracture toughness can be attained by combining materials whose isolated behavior may even be brittle. Modeling the behavior of multiphase materials is therefore carried out by material scientists, physicists, and engineers on a regular basis. These problems are characterized by a field that exhibits jumps in its gradient along interfaces ( 0 -continuity assuming perfect bonding between phases). They thus contain so-called weak discontinuities, since it is the field gradient that is discontinuous rather than the field itself. Problems with strong discontinuities, where the field itself is discontinuous, were the subject of Chapter 6. Jumps in the gradient of the field could also materialize even on homogeneous materials at locations where loads area applied over very narrow regions. Traditionally, problems containing weak discontinuities have been modeled by standard FEM using meshes where material interfaces align with the edges of finite elements, i.e., fitted, matching, or discontinuity-conforming discretizations. The  0 -continuous nature of the field is readily accommodated by the inherent  0 finite element approximation, as it was shown in Exercise 2.6. Nevertheless, enriched formulations provide a means to use unfitted discretizations. By using enrichment functions that incorporate the discontinuity in their gradient within elements traversed by interfaces, enriched formulations provide a flexible alternative to standard FEM for modeling these problems. While interface-enriched formulations were described earlier in Chapter 5, this chapter reviews the different enrichment functions that can be used for modeling problems containing weak discontinuities in the context of GFEM.

7.1 Formulation As done in previous chapters, we first start by introducing the model problem for treating weak discontinuities. Fig. 7.1a shows a solid composed of two phases 1 and 2 . Each phase has boundary ∂i ≡ i = i \ i , i = {1, 2}, so their common interface is defined as 12 = 1 ∩ 2 . We will denote the portion of the boundary with prescribed Dirichlet boundary conditions in each phase as iD ≡  D ∩ i . Similarly, the Neumann boundary in each phase is iN ≡  N ∩ i . Fundamentals of Enriched Finite Element Methods. https://doi.org/10.1016/B978-0-32-385515-0.00015-5 Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved.

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Figure 7.1 (a) Body composed of two phases 1 and 2 separated by interface 12 . The solid is subjected to a body force per unit volume b, to prescribed displacements u¯ on  D , and to prescribed tractions t¯ on  N ; (b) Unfitted finite element discretization h highlighting elements whose nodes are enriched to recover the kinematics of the weak discontinuity.

7.1.1 Linear elastostatics We first consider the formulation for linear elasticity. Our boundary value problem, stated in strong form is: Given bi : i → Rd , σ i : i → Rd × Rd , u¯ i : iD → Rd , and t¯ i : iN → Rd , find ui ∈  2 (i ) such that in i , i = {1, 2},

(7.1)

σ i · ni = t¯ i

on iN ,

(7.2)

ui = u¯ i

iD ,

(7.3)

∇ · σ i + bi = 0

on

with interface conditions: σ i · n12  = 0 on 12 ,

(7.4)

u = 0 on 12 ,

(7.5)

where u = 0 implies ui  = 0 for every component ui of the vector (and similarly for other vectors). In the preceding formulas it is implied that ui ≡ u|i and the same applies for other subscripted quantities. This is important because we seek a function in the entire domain, so although ui ∈  2 (i ), because of the material interface, u ∈  0 (). In other words, we seek a solution u that exhibits a jump in the gradient in the direction perpendicular to the interface 12 , i.e., ∇ui · n12  = 0. The variational form is obtained similarly to what was done in § 2.2.2: Given ui = v i + u˜ i , with u˜ i | D = u¯ i , find v i ∈ 0 (i ) (space defined in (2.63)) such that i

 i={1,2}

Bi (v i , w i ) =

 i={1,2}

Li (wi ) −

 i={1,2}

Bi (u˜ i , w i )

∀wi ∈ 0 (i ), (7.6)

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where the bilinear and linear forms are, respectively, given by  Bi (v i , wi ) = ∇wi :σ i d ,    i Li (wi ) = wi · bi d + wi · t¯ i d . iN

i

(7.7) (7.8)

Notice that, while the vector-valued function space in each phase 0 (i ) can be defined as per (2.63), because of the weak discontinuity the space over the entire domain is   d  d  0 () = w ∈ L 2 () , w|i ∈ H 1 (i ) , w| D = 0, i = {1, 2} . (7.9) i

The weak formulation given by Eqs. (7.6)–(7.8) is, in fact, the result of assuming a continuous weight function across the interface. This, however, does not necessarily have to be the case, and a variational formulation can be obtained by considering the continuity of tractions given by (7.4) in a weak sense [1]. Consider the weighted residual obtained by multiplying (7.1) and (7.4) with their corresponding vector-valued test functions:    ˆ · σ · n12  d = 0 w wi · (∇ · σ i + bi ) d − 12 (7.10) i={1,2} i ˆ ∈ ˆ (12 ), ∀wi ∈ 0 (i ), w 

where ˆ (12 ) = vˆ vˆi ∈ H 1/2 (12 ), i = {1 . . . d} and H 1/2 (12 ) is the set of sufficiently regular functions along the interface. These are square-integrable functions, i.e., the trace on functions in H 1 (i ). Using the L 2 (12 ), which are the result  of applying − ∇w · w identity wi · (∇ · σ i ) = ∇ · σ  i i : σ i results in i   i={1,2} i

    ∇ · σ − ∇w d + · w :σ + w · b i i i i i i

12

ˆ · (σ 2 − σ 1 ) · n12 d = 0 w

ˆ ∈ ˆ (12 ), ∀wi ∈ 0 (i ), w (7.11) where the jump in tractions along the interface has been explicitly stated. The divergence theorem can now be applied to the first term to yield 

    d = ∇ · σ · w w i · (σ i · ni ) d i i i={1,2} i

=

  N i={1,2} i

i={1,2} i

w i · t¯ i d +



12

(7.12) w2 · (σ 2 · n12 ) − w1 · (σ 1 · n12 ) d ,

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where we used ni · (σ  i · w i ) = w i · (σ i · ni ) and the fact that w i |iD = 0 to remove reactive tractions on the Dirichlet boundary, and n12 ≡ n2 = −n1 . The weak form is finally stated as: Given the bilinear and linear forms in Eqs. (7.7) ¯ find v i ∈ 0 (i ) such that, for all and (7.8), respectively, ui = v i + u˜ i , with u˜ i | D = u, i

ˆ ∈ ˆ (12 ), w i ∈ 0 (i ), w 

(B)

  Bi (v i , w i ) +

i={1,2}

 − 

12





w1 · (σ 1 · n12 ) − w2 · (σ 2 · n12 ) d 

12



(7.13)

ˆ · (σ 1 − σ 2 ) · n12 d = w Li (w i ) − Bi (u˜ i , wi ). i={1,2}   i={1,2} (A)

Term (A) in (7.13), as we pointed out above, is the weak statement of traction equilibrium at the interface, whereas term (B) is referred to as the interface jump. These two terms cancel one another—and the variational statement simplifies to Eqs. (7.6)–(7.8)—when ˆ = w 1 = w 2 (when the test functions along the interface are continuous). Noteworthy, w standard FEM uses fitted meshes to the interface, and therefore the jump in the field gradient is readily accommodated by the  0 -continuous nature of the approximation. With e-FEA we decouple interfaces from the finite element discretization but we keep the  0 continuity of the weight functions in order to use the simpler variational form.

7.1.2 Heat conduction For the heat equation the formulation is obtained in a similar manner, although this time we also consider a mixed boundary condition to account for convection. Consequently, the boundary of our domain is now split in disjoint regions  =  D ∪  N ∪  R . Given the thermal conductivity tensor κ i : i → Rd × Rd , heat transfer coefficient Hi : iR → R, the heat source fi : i → R, prescribed temperature u¯ : iD → R and prescribed heat flux q¯i : iN → R, and ambient temperature u∞ ∈ R, find the temperature field u ∈ C 2 such that κ i ∇ui ) + fi = 0 ∇ · (κ

in i , i = {1, 2} ,

(7.14)

ui = u¯ i

on iD ,

(7.15)

κ i ∇ui · ni = q¯i

iN , iR ,

(7.16)

κ ∇ui · ni = H (u∞ − ui )

on on

(7.17)

with conditions at the interface κ ∇ui · n12  = 0 on 12 , κ

(7.18)

ui  = 0 on 12 .

(7.19)

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171

As done previously, the weak formulation satisfies Eq. (7.14) on average by multiplying it with a function wi ∈ V0 (i ) and integrating over the domain. Then we look for a function u ∈ U such that   Bi (ui , wi ) = Li (wi ) ∀wi ∈ V0 (i ), (7.20) i={1,2}

i={1,2}

where



 κ ∇ui ) d + ∇wi · (κ

Bi (ui , wi ) = i



 Li (wi ) =

wi fi d + i

iN

iR

Hi ui wi d ,

wi q¯i d +

(7.21)

 iR

Hi wi u∞ d .

(7.22)

7.1.3 Discrete equations We focus on the elastostatics problem and for simplicity, assume Dirich homogeneous let boundary conditions. The Galerkin form is: Find uh ∈ 0h h such that   B uh , w h = L w h

 ∀w h ∈ 0h h ,

(7.23)

where both trial and weight functions are taken from the generalized finite element space (4.40), i.e., uh , w h ∈ SGFEM , defined on a discretization of our body h as shown in Fig. 7.1b. Note that finite elements need not conform to the material interface. The behavior missing from using such unfitted discretization is regained through enrichment functions that display a gradient jump across the interface. In fact, the approximation can be represented as a combination of smooth and nonsmooth components     uh (x) = u˜ h + uˆ h such that ∇ u˜ h · n = 0 and ∇ uˆ h · n = 0, where n is the normal to the interface. The smooth part can be approximated as   u˜ h (x) = u˜ ij E˜ ij (x), ϕ(x) i∈ιh

(7.24)

j ∈ιe

with polynomial enrichment set, as presented in (4.25) but extended to d-dimensional Euclidean space. In 2-D, for instance,       x − x i y − y i x − xi 2 y − yi 2 ˜ Eij (x) = 1, , , , ,... , (7.25) hi hi hi hi with x i = (xi , yi ) the coordinates associated with the ith node and hi the size of its support (e.g., the diameter of a circle circumscribing all of its elements).

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Similarly, the nonsmooth component of the approximation can be written as   uˆ h (x) = uˆ ij Eˆ ij (x), ϕ(x) (7.26) i∈ιh

j ∈ιe

where the enrichment functions Eˆ ij exhibit the required a priori knowledge about the gradient jump. Any  0 -continuous function can be used, although, as explained below, their performance in recovering an approximation with the desired accuracy varies greatly.

7.1.4 Enrichment functions for weak discontinuities Consider the signed distance function to the interface—also known as the level set function—given by  θ (x) = x − x min  sign (x − x min ) · nmin ,    a

where

 x min = arg min x − x i  , x i ∈

sign (a) =

1 for a ≥ 0, −1 for a < 0.

A discretized counter part of the level set function is usually interpolated by partition of unity shape functions as  θ h (x) = θ (x i )ϕi (x). i∈ιh

This function will be used henceforth in the definition of the weak discontinuity enrichments.

7.1.4.1 The distance function An obvious enrichment option is to use the absolute value of the signed distance function, a procedure explored by Sukumar et al. [2], i.e., S(x) = |θ (x)|. The function is shown schematically in 1-D in Fig. 7.2. Because the function is nonzero in blending elements—i.e., those that do not have all nodes enriched and thus “blend” the cut enriched elements (also known as representative elements) to the rest of the mesh—it introduces a spurious behavior that degrades the accuracy and convergence rates of the approximation (more on this below).

Figure 7.2 Distance function.

Generalized enrichment functions for weak discontinuities

173

7.1.4.2 The distance function with smoothing The aforementioned loss of accuracy can be partially alleviated by modifying the distance function such that its value is constant in blending elements (see Fig. 7.3). The enrichment is therefore ⎧ ⎪ ⎨|θ (xi )| if x ≤ xi , S2 = |θ (x)| if xi ≤ x ≤ xj , ⎪ ⎩  if x ≥ xj . θ xj

Figure 7.3 Distance function with smoothing.

7.1.4.3 The ridge function The previous two functions have their minimum value at the location of the interface. However, it is possible to create a function using the level set where the maximum value is located at the interface. The enrichment function, proposed by Möes et al. [4] and illustrated in Fig. 7.4 for 1-D, is given by   |θ (x i )|ϕi (x) − θ (x i )ϕi (x) . M(x) = i∈ιh i∈ιh       (A)

(B)

This function has three features that make it a much better choice than the previously discussed enrichments. Firstly, as the interface approaches the nodes of the element that contains the interface, the enrichment’s maximum value approaches to zero. Secondly, the function is exactly zero at mesh nodes of the element that contains the interface, and thus their associated DOFs associated retain their physical meaning, and Dirichlet boundary conditions can be prescribed as in standard FEM. Lastly, because the function is exactly zero also in blending elements, this function does not have the foregoing spurious effect.

Figure 7.4 Ridge function.

This function is remarkably similar to the enrichment function for weak discontinuities discussed in Chapter 5. While the weak enrichment in DE-FEM is constructed using Lagrange shape functions in integration elements—instead of using the level

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Fundamentals of Enriched Finite Element Methods

set function—the resulting function retains the same properties mentioned above. Noteworthy, the weak DE-FEM enrichment also needs scaling to reduct its effect as discontinuities approach standard mesh nodes. This scaling, which improves the condition number of system matrices, will be studied in detail in Chapter 11.

7.1.4.4 Corrected enrichments Some enrichment functions for weak discontinuities fail to recover optimal rates of convergence for smooth problems. This undesired behavior is due to the presence of unwanted terms in blending elements—partially enriched elements that blend the elements containing the discontinuity with the rest of the mesh. A solution to this problem for linear approximations was proposed by Fries [3], whereby the weak enrichment function is multiplied by a ramp function r(x). This function is constant in the representative element and ramps linearly to zero in blending elements. For instance, the procedure to correct the distance function is schematically illustrated in Fig. 7.5, where it can be seen that the resulting enrichment function is quadratic in blending elements—and thus the generalized enrichment function would be cubic. Although such correction works, it is worth noting that a second layer of nodes need to be enriched (those of representative and blending elements) so the computer implementation is more involved and the overall computational cost raises.

Figure 7.5 Corrected distance function.

7.1.5 Enrichment performance In this section we show the performance of enrichment functions presented earlier in the context of heat conduction. The example we follow is borrowed from Aragón et al. [5]. Consider a square plate h = [0, L]2 , as schematically illustrated in Fig. 7.6a. We consider the following manufactured function:  x(L−4x)(5L−4x)(L−2x) , x ≤ L/2, 6L3 (7.27) u(x, y) = (3L−4x)(L−2x)(L−x)(L+4x) , x ≥ L/2, 6L3  ∂u  = −1. The heat source which is shown in Fig. 7.6b. It can be verified that ∂x L/2 required to satisfy the heat equation, given a unit conductivity value, is  2 17L2 − 96Lx + 96x 2 f= . (7.28) 3L3 Several enrichment strategies are explored, including the ramp function Rr of Exercise 4.2 (and a similar function Rl that ramps towards the left), and functions S and

Generalized enrichment functions for weak discontinuities

175

Figure 7.6 (a) Line heat source example: A 2-D square domain h = [0, L]2 has an applied heat source f per (7.28) and a line heat source fs = 1 at x = L/2. The plate has a prescribed temperature u¯ along left and right edges, and insulated bottom and top edges; (b) Exact temperature field given by (7.27).

M introduced earlier in § 7.1.4. In addition, we study not only linear but also quadratic approximations by enriching with linear monomials. To simplify notation, we denote the shifted and scaled linear monomials in (4.25) as x˜ =

y − yi x − xi and y˜ = . hi hi

Therefore, a quadratic enrichment strategy that uses such monomials is, for example, E ij = {1, x, ˜ y} ˜ × {1, M} = {1, x, ˜ y, ˜ M, xM, ˜ yM}. ˜ The error with respect to the exact solution is quantified through the H 1 norm as   2 2 2         ∇u − ∇uh  d , u − uh  1 = u − uh  2 + H ()

L ()



which is similar to the error in the energy norm since the error in the field gradient is considered—and therefore we expect similar convergence rates. The finite element results are obtained by prescribing the temperature u(0, ¯ y) = u(L, ¯ y) = 0, the heat flux q(x, ¯ 0) = u(x, ¯ L) = 0, applying a heat source f as per Eq. (7.28) to the entire domain h , and a heat source of unit magnitude along the line x = L/2. The problem is then solved with fitted and unfitted meshes with increasingly smaller triangular elements. The results are summarized in Fig. 7.7. Our reference results are those corresponding to standard FEM, for which optimal rates are obtained when using matching meshes (FEM-M curves). The curve FEM-NM shows that convergence using unfitted finite element meshes is not optimal (the convergence rate is halved). For linear enrichment strategies, labeled in the figure E1 –E7 , results show that only E1 , E3 , and E5 recover optimal convergence rates. All other linear approximations fail to recover optimal convergence because of the decay of the approximation in blending elements. This is alleviated by correcting the enrichment functions as explained in § 7.1.4 to the expense of adding more DOFs. The only linear enrichment strategy that works out of

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Fundamentals of Enriched Finite Element Methods

Figure 7.7 Error in H 1 norm as a function of the total number of DOFs nD for the plate with line heat source of Fig. 7.6a.

the box is E1 , for which we notice the function is identically zero in blending elements. The behavior for quadratic enrichment strategies is quite different, as all enrichment strategies recover optimal convergence rates without the need for correction.

7.2 Discussion and further reading The modeling of voids can be traced back to the GFEM work of Strouboulis et al. [6]. Although that work briefly discussed the modeling of material interfaces as well, these were first resolved by Sukumar et al. [2] via the distance function enrichment (see § 7.1.4.1). This function, however, was shown later to yield suboptimal convergence rates due to loss of accuracy in blending elements. Depending on the type of enrichment used, a blending element may no longer be able to reproduce a linear field and

Generalized enrichment functions for weak discontinuities

177

thus accuracy may be severely degraded, which is manifested in suboptimal convergence rates. A straightforward solution to this problem is simply using enrichment functions that, while reproducing properly the kinematics in representative elements, are identically zero in blending elements. The ridge function proposed by Möes et al. [4] has this property (see § 7.1.4.3); as shown in the previous section, this function yields optimal convergence rates for linear and quadratic approximations. Another solution to the issue of loss of accuracy in blending elements was introduced with the corrected XFEM by Fries [3], at the expense of requiring an extra layer of enrichments (see § 7.1.4.4). Nevertheless, the results of the previous section, demonstrate that optimal convergence rates are still attained for quadratic approximations in GFEM without the need for any correction, as long as the discontinuous part of the solution has the same interpolation order as that of the continuous part. This fact was also demonstrated for single [7] and branched [8] cracks. Optimal rates were found even when using enrichment functions that delivered poor convergence in the linear setting. The preceding discussion is related to straight interfaces. Regarding curved interfaces, Cheng and Fries found suboptimal rates using the corrected XFEM for enriched quadratic and cubic approximations [9]. They discuss the issue of describing curved geometries appropriately and the ill-conditioning of the stiffness matrix as the polynomial approximation is increased. Dréau et al. [10] proposed a different approach for resolving curved weak interfaces, whereby the geometry was represented on a finer mesh than that used to compute the field. They proposed a new enrichment strategy to link both meshes, and investigated up to cubic interpolations, without attaining optimal convergence rates either. Legrain et al. [11] constructed high-order interpolations on a coarse mesh while decoupling the representation of the discontinuities with a finer mesh—both related to the use of the same quadtree data structure. They reported optimal convergence rates for up to quintic polynomial order, for which they used the Heaviside enrichment coupled to Nitsche’s method to enforce continuity of the displacement field. However, ill-conditioning of system matrices was still an issue when increasing the polynomial order. While in GFEM/XFEM all nodes of representative elements cut by interfaces are enriched, an alternative approach localizes the enrichments to interfaces—i.e., associate enrichment functions to nodes placed directly along interfaces; this is the main idea behind the Interface-enriched GFEM put forth by Soghrati et al. [12], discussed in detail in Chapter 5. Within interface-enriched formulations, Soghrati et al. [13] proposed a quadratic formulation for reducing the discretization error due to curved material interfaces. For the construction of the enrichment functions, they investigated both transition elements (with three nodes corresponding to the curved edge) and 6node triangular elements as integration elements. Because the quadratic interpolation was localized only to cut elements, the effect of the approach was more prominent for coarse meshes and convergence rates did not improve over linear IGFEM. Soghrati and Barrera [14] later applied the quadratic enrichment strategy in IGFEM with meshes composed of 6-node quadratic Lagrange elements and showed optimal convergence rates.

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References [1] G. Haikal, A stabilized finite element formulation of non-smooth contact, PhD thesis, University of Illinois at Urbana–Champaign, 2009. [2] N. Sukumar, D. Chopp, N. Moës, T. Belytschko, Modeling holes and inclusions by level sets in the extended finite-element method, Computer Methods in Applied Mechanics and Engineering 190 (46) (2001) 6183–6200, https://doi.org/10.1016/S0045-7825(01)002158. [3] F. Thomas-Peter, A corrected XFEM approximation without problems in blending elements, International Journal for Numerical Methods in Engineering 75 (5) (2007) 503–532, https://doi.org/10.1002/nme.2259. [4] N. Moës, M. Cloirec, P. Cartraud, J.-F. Remacle, A computational approach to handle complex microstructure geometries, Computer Methods in Applied Mechanics and Engineering 192 (28) (2003) 3163–3177, https://doi.org/10.1016/S0045-7825(03)00346-3. [5] A.M. Aragón, C.A. Duarte, P.H. Geubelle, Generalized finite element enrichment functions for discontinuous gradient fields, International Journal for Numerical Methods in Engineering 82 (2) (2010) 242–268, https://doi.org/10.1002/nme.2772. [6] T. Strouboulis, K. Copps, I. Babuška, The generalized finite element method: An example of its implementation and illustration of its performance, International Journal for Numerical Methods in Engineering 47 (8) (2000) 1401–1417, https://doi.org/10.1002/(SICI) 1097-0207(20000320)47:83.0.CO;2-8. [7] P. Laborde, J. Pommier, Y. Renard, M. Salaün, High-order extended finite element method for cracked domains, International Journal for Numerical Methods in Engineering 64 (3) (2005) 354–381, https://doi.org/10.1002/nme.1370. [8] C. Duarte, L. Reno, A. Simone, A high-order generalized FEM for through-the-thickness branched cracks, International Journal for Numerical Methods in Engineering 72 (3) (2007) 325–351, https://doi.org/10.1002/nme.2012. [9] K.W. Cheng, T.-P. Fries, Higher-order XFEM for curved strong and weak discontinuities, International Journal for Numerical Methods in Engineering 82 (5) (2010) 564–590, https://doi.org/10.1002/nme.2768. [10] K. Dréau, N. Chevaugeon, N. Moës, Studied X-FEM enrichment to handle material interfaces with higher order finite element, Computer Methods in Applied Mechanics and Engineering 199 (29) (2010) 1922–1936, https://doi.org/10.1016/j.cma.2010.01.021. [11] G. Legrain, N. Chevaugeon, K. Dréau, High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation, Computer Methods in Applied Mechanics and Engineering 241–244 (2012) 172–189, https://doi.org/10.1016/j. cma.2012.06.001. [12] S. Soghrati, A. Aragón, C. Duarte, P. Geubelle, An interface-enriched generalized finite element method for problems with discontinuous gradient fields, International Journal for Numerical Methods in Engineering 89 (8) (2012) 991–1008. [13] S. Soghrati, C.A. Duarte, P.H. Geubelle, An adaptive interface-enriched generalized FEM for the treatment of problems with curved interfaces, International Journal for Numerical Methods in Engineering 102 (6) (2015) 1352–1370, https://doi.org/10.1002/nme.4860. [14] S. Soghrati, J.L. Barrera, On the application of higher-order elements in the hierarchical interface-enriched finite element method, International Journal for Numerical Methods in Engineering 105 (6) (2016) 403–415, https://doi.org/10.1002/nme.4973.

Immerse boundary (fictitious domain) problems

8

Fully decoupling the finite element discretization from the problem’s geometry entails dealing with the boundary—in addition to material interfaces and/or cracks. Being able to chose any finite element discretization, regardless of the problem geometry, gives the analyst unparalleled flexibility with regards to mesh choices. In fact, a robust immersed finite element method could be used to fully automate analysis, bypassing completely the time spent in creating computational models (which can be higher than 80% of the total analysis time [1]). This flexibility can be used to obtain statistically significant results by analyzing many material microstructure realizations using the same (usually structured) finite element mesh, or to obtain accurate analysis when dealing with moving boundaries as in topology optimization. As a result, many methods have been proposed over the years to solve immersed boundary (or fictitious domain) problems, including the unfitted FEM [2,3], CutFEM [4–6], the finite cell method [7,8], and certainly e-FEMs [9,10]. e-FEMs gained their popularity within the computational mechanics scientific community mainly because of their ability to decouple the finite element discretization from discontinuities, primarily in fracture mechanics. To a lesser degree, e-FEMs have successfully been applied to weak discontinuities. However, e-FEMs face a number of challenges when treating the external boundary where there is a need to enforce non-homogeneous Dirichlet boundary conditions—which in this context have been called embedded Dirichlet conditions [11]. Due to the fact that enrichment functions do not usually vanish at Dirichlet regions, prescribing essential BCs is usually done in a weak sense by means of penalty [12,13], Lagrange multiplier [4,11,13–17], or Nitsche [3,5,13,18] methods. When using dual methods like the Lagrange multiplier method, much care has to be taken for the discretization of the Lagrange multiplier space. A bad choice may lead to an overly-constrained boundary and therefore to locking (mathematically the inf–sup or Ladyzhenskaya–Babuška–Brezzi (LBB) condition is not satisfied). This is problematic because an overconstrained boundary translates into oscillations in recovered traction fields [15,16,19]. This issue can be addressed by means of stabilized or stable methods [9]. While stabilization terms are added to the formulation in the former [4,5,20], in the latter the inf–sup condition is satisfied by either carefully choosing the Lagrange multiplier space [16,17,21] or by enriching the primal field [11]. In Chapter 5 we introduced discontinuity-enriched finite element approximations for problems with weak and strong discontinuities. In this chapter we demonstrate these methods can also be used to solve immersed boundary problems with both weak [10] and strong [10] enforcement of embedded Dirichlet conditions. Most importantly, we emphasize the fact that recovered reactions (traction field at Dirichlet boundaries) are devoid of numerical oscillations observed when using XFEM [24]. Fundamentals of Enriched Finite Element Methods. https://doi.org/10.1016/B978-0-32-385515-0.00016-7 Copyright © 2024 Alejandro M. Aragón and C. Armando Duarte. Published by Elsevier Inc. All rights reserved.

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8.1 Formulation Referring back to the schematic in Fig. 5.8, here we simply start with the variational statement of equilibrium for our model problem assuming homogeneous Dirichlet boundary conditions: Given the body force bi : i → Rd , prescribed displacement u¯ i : iD → 0, and prescribed traction t¯ i : iD → Rd , where i = {m, o} refers to either matrix or inclusion phases, find u ∈  such that B(u, w) = L (w) where

∀w ∈  ,

 

L (w) =

i={m,o}

i

i={m,o}

i

 

B(u, w) =

(8.1) 

wi · bi d +

N

wi · t¯ i d ,

(8.2)

 i (w i ) : σ i (ui ) d ,

(8.3)

where once again we assume a linear constitutive law between stress and strain, and we use small strain theory. The finite-dimensional counterpart of (8.1) is then solved in an immerse setting with a computational domain  that fully encloses the problem’s geometry, i.e.,  ⊃ . First, this so-called domain is subdivided into non-overlapping finite  hold-all E ei , as shown in Fig. 8.1a. Second, computational geomelements such that h = ni=1 etry operations are carried out as outlined in Chapter 12 to create enriched nodes and integration elements. Enriched nodes (shown with red circular and blue square symbols in the figure for weak and strong discontinuities, respectively) are placed along all discontinuities (boundary, interfaces, and cracks), at the intersection location between discontinuities and the edges of the background discretization h . Integration elements in cut elements (see dashed lines) are then created and added to an ordered tree data structure to keep track of the mesh hierarchy. A representation of such hierarchy was shown in Fig. 5.11. Third, elements laying completely outside the domain of interest are removed from the analysis, and the displacement at nodes of elements cut by the boundary that lie outside is prescribed to zero (see symbols). These last two steps are illustrated in Fig. 8.1b. Following a Bubnov–Galerkin procedure, the vector-valued trial and test functions are taken from the discontinuity-enriched finite element space  h ⊂  (see Chapter 5), such that uh ∈  h is given by uh =

 i∈ιh



strong

weak

       ϕi (x)uˆ i + ski ψki (x)α ki + χki (x)β ki , 

std. FEM



k∈ιH i∈ιw





k∈ιH i∈ιs

(8.4)



enrichment

where ιH ≡ Z+ = {1, 2, . . . , D} is the index set of hierarchical levels resulting from D discontinuities, and thus the hierarchical nature of the approximation is made explicit for weak and strong enrichments.

Immerse boundary (fictitious domain) problems

181

Figure 8.1 Schematic representation of an immersed boundary: (a) Elements cut by the boundary and other discontinuities are found hierarchically; (b) new enriched nodes (red circular and blue square symbols) and integration elements (dashed lines) are added to the discretization. Standard nodes ( symbols) that lie outside the domain are fixed, i.e., zero displacement prescribed ( symbols).



Then B uh , w h = L w h , ∀w h ∈  h yields the discrete system KU = F , where E as usual the global arrays are the result of an assembly procedure, i.e., K = Ane=1 ke , nE F = Ae=1 f e . At element level, the stiffness matrix and force vectors are computed per Eq. (5.30). Considering traction-free cracks, for the eth integration element the local stiffness matrix and force vector are computed as  ke = e fe =

e

 ϕ  ϕ

ψ ψ

χ



χ



b de +

 C ϕ   N ∩∂e

ψ  ϕ

ψ

 χ de , χ



(8.5) t¯ d∂e .

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Notice in (8.5) the hierarchical nature of the approximation, which will also be embedded in the resulting global arrays. In other words, the final global stiffness matrix can be arranged so that the upper-left block of elements corresponds exclusively to the standard part of the approximation, followed by coupled block-diagonal enriched components.

8.1.1 Treatment of boundary conditions The last ingredient for solving immersed problems with the enriched procedure pertains to boundary conditions. Neumann boundary conditions do not pose much of a challenge, since the boundary is known explicitly and therefore we just need to integrate f e in (8.5). Note that the resulting force vector will have contribution to standard DOFs, which are associated with nodes that are not located along the boundary with the prescribed traction (this is not the case with enriched DOFs). With regards to non-homogeneous Dirichlet boundary conditions the treatment is usually not so straightforward. As discussed in the introduction of this chapter, prescribing embedded Dirichlet conditions is usually done in a weak manner. Fortunately, because of the hierarchical nature of the enrichment functions used in DE-FEM, prescribing Dirichlet conditions can be done strongly. In other words, because enrichment functions vanish at the location of standard nodes, it is only required that we solve local problems to determine the values of enriched DOFs. Prescribing strong enriched DOFs is straightforward since these physically represent the jump in the displacement. The enriched DOFs α kj that correspond to an enriched node x j are determined by setting uh x j = u¯ x j and solving for α kj . To wit, α kj =



1 

u¯ x j − ϕi x j uˆ i − sni ψni x j α ni skj ψkj i∈ιh

−



 χni x j β ni ,

n∈h i∈ιw n