Substructuring Method for Civil Structural Health Monitoring 9819913683, 9789819913688

Citation preview

Engineering Applications of Computational Methods 15

Shun Weng Hongping Zhu Yong Xia

Substructuring Method for Civil Structural Health Monitoring

Engineering Applications of Computational Methods Volume 15

Series Editors Liang Gao, State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China Akhil Garg, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China

The book series Engineering Applications of Computational Methods addresses the numerous applications of mathematical theory and latest computational or numerical methods in various fields of engineering. It emphasizes the practical application of these methods, with possible aspects in programming. New and developing computational methods using big data, machine learning and AI are discussed in this book series, and could be applied to engineering fields, such as manufacturing, industrial engineering, control engineering, civil engineering, energy engineering and material engineering. The book series Engineering Applications of Computational Methods aims to introduce important computational methods adopted in different engineering projects to researchers and engineers. The individual book volumes in the series are thematic. The goal of each volume is to give readers a comprehensive overview of how the computational methods in a certain engineering area can be used. As a collection, the series provides valuable resources to a wide audience in academia, the engineering research community, industry and anyone else who are looking to expand their knowledge of computational methods. This book series is indexed in both the Scopus and Compendex databases.

Shun Weng · Hongping Zhu · Yong Xia

Substructuring Method for Civil Structural Health Monitoring

Shun Weng School of Civil and Hydraulic Engineering Huazhong University of Science and Technology Wuhan, Hubei, China

Hongping Zhu School of Civil and Hydraulic Engineering Huazhong University of Science and Technology Wuhan, Hubei, China

Yong Xia Department of Civil and Environmental Engineering The Hong Kong Polytechnic University Hong Kong, China

ISSN 2662-3366 ISSN 2662-3374 (electronic) Engineering Applications of Computational Methods ISBN 978-981-99-1368-8 ISBN 978-981-99-1369-5 (eBook) https://doi.org/10.1007/978-981-99-1369-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Structural health monitoring is increasingly used to describe the gradual or sudden changes of a civil structure by implementing a sensor system on full-scale civil infrastructures. Damage identification based upon changes in vibration characteristics on a global basis is one of the most difficult but significant disciplines. There are usually two approaches to damage identification discipline. Data-driven approaches establish a mathematical model to represent the state of a system, while model-driven methods establish a high-fidelity physical model of the structure, usually by finite element analysis. In the model-driven methods, parameters of the model are adjusted to reduce a penalty function based on the residuals between a measurement set and the corresponding model predictions. Model-driven methods are physically meaningful and can achieve the four-level damage identification including detection, localization, quantification, and prediction. The updated physical model can also be used for a large number of applications such as design optimization, structural control, and prediction of load behavior. Nevertheless, a civil structure is often large-scale, and its finite element model consists of a large number of degrees-of-freedom and uncertain parameters. The response analysis, model updating, and damage identification of the large-scale structures are time-consuming or even prohibited to be performed. Substructuring methods possess many merits in dynamic analysis, model updating, and damage identification of large-scale structures. With the substructuring methods, a global structure is divided into a number of independent substructures. Only one or more substructures are repeatedly analyzed and the re-analysis of the global structure is thereby avoided. The substructuring methods are advantageous to analyze a large-scale structure in piece-wise manner. The first author researched on the substructuring methods from her Ph.D. study supervised by the second and third authors and was deeply attracted by the fantastic performance of the substructuring methods. Afterward, the authors and their research students make great effort to the development of the substructuring methods, including the forward and inverse substructuring methods, the frequency-domain and time-domain substructuring methods, the linear and nonlinear substructuring methods, and deterministic and indeterministic substructuring methods. In addition, the substructuring methods have been extended to a large number of disciplines. With these new v

vi

Preface

improvement and findings on the substructuring methods, it is time to systematically integrate the previous work together to show a complete view on characteristics of the substructuring methods in civil structures. This book mainly addresses the forward substructuring method. Part I presents the linear substructuring methods, including our improvement on the substructuring methods for eigensolutions, eigensensitivity, dynamic response, response sensitivity, and model updating. Part II talks about the model condensation methods, which are regarded as the substructuring method reduced in the physical space. The final part of the book is the substructuring methods for nonlinear analysis and nonlinear model updating. The purpose of writing this book is to illustrate the fundamentals of the substructuring methods and to make more researchers or engineers benefit from the substructuring methods. In writing the book, the authors are always reminded that the book mainly serves as a textbook or a reference book for the graduate students or practicing engineers to understand the substructuring methods. The readers are supposed to have some background in structural analysis, structural dynamics, random vibration, and mathematics. We would be very happy and grateful to receive constructive comments and suggestions from the readers, and discussions and collaborations on substructuring methods are warmly welcomed. Wuhan, China Wuhan, China Hong Kong, China

Shun Weng Hongping Zhu Yong Xia

Acknowledgements

The writing of this book has been a challenging and laborious task that could not have been completed without the help of many individuals. We are grateful to many people who helped in the preparation of this book. First, it is greatly appreciated to Dr. Jiajing Li, Dr. Wei Tian, and Dr. Zhidan Chen for their participation in research works presented in this book and for their essential work on editing and language correction of the book. The present and former research students at Huazhong University of Science and Technology (HUST) contributed to some research works in this book as well. Special gratitude to Dr. Ling Mao, Dr. Ling Ye, Dr. Hong Yu, Dr. Ke Gao, Mr. Yongyi Yan, Mr. Feng Liang, Ms. Yue Zuo, Mr. Huixian Zhao, etc. Our work presented in this book has been largely supported by the National Key R&D Program of China (2021YFF0501001), National Natural Science Foundation of China (51922046, 51778258, 51108205, 51838006, 51629801), Natural Science Foundation of Hubei Province (2020CFA047), and the foundation from HUST (2023JCYJ014). The China Railway Eryuan Engineering Group Co. Ltd. and China Railway Siyuan Survey and Design Group Co. Ltd. provided a large number of opportunities for the practical application of the research. All the supports are gratefully acknowledged. We are grateful to the School of Civil and Hydraulic Engineering of HUST for the substantial support and prolific resource. We are very grateful to all colleagues and staffs at HUST, who have helped us both in our research and daily work. It is difficult to list all of them on this single page. A vote of thanks must go to Mr. Mengchu Huang, senior editor at Springer Nature Press, for his patience and encouragement from the beginning and during the preparation of this book, and to all editors at Springer Nature Press for their patience and scrutiny in the editing of this book. Finally, we are grateful to our families for their love, encouragement, and endurance.

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Objective of Substructuring Method in Structural Health Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Category of Substructuring Model Updating Methods . . . . . 1.3 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I 2

3

1 1 3 5 7

Linear Substructuring Methods

Substructuring Method for Eigensolutions . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Methods for Eigensolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Subspace Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Lanczos Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Substructuring Method for Eigensolutions . . . . . . . . . . . . . . . . . . . 2.3.1 Component Mode Synthesis . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Kron’s Substructuring Method . . . . . . . . . . . . . . . . . . . . . . 2.3.3 First-Order Residual Flexibility Based Substructuring Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Second-Order Residual Flexibility Based Substructuring Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Residual Flexibility for Free Substructure . . . . . . . . . . . . 2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A Three-Span Frame Structure . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The Balla Balla River Bridge . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 13 14 15 16 19

Substructuring Method for Eigensensitivity . . . . . . . . . . . . . . . . . . . . . . 3.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic Methods for Eigensensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Eigenvalue Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 48 48

22 25 27 29 29 35 44 45

ix

x

Contents

3.2.2 Eigenvector Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . Substructuring Method for Eigensensitivity . . . . . . . . . . . . . . . . . . 3.3.1 Eigenvalue Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Eigenvector Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Derivative of Residual Flexibility . . . . . . . . . . . . . . . . . . . 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Three-Span Frame Structure . . . . . . . . . . . . . . . . . . . . 3.4.2 The Balla Balla River Bridge . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 51 51 53 55 56 56 59 65 65

4

Substructuring Method for High-Order Eigensensitivity . . . . . . . . . . 4.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Basic Method for High-Order Eigensensitivity . . . . . . . . . . . . . . . . 4.2.1 Second-Order Eigensolution Derivatives . . . . . . . . . . . . . 4.2.2 General High-Order Eigensolution Derivatives . . . . . . . . 4.3 Substructuring Method for High-Order Eigensensitivity . . . . . . . . 4.3.1 Second-Order Eigensolution Derivatives . . . . . . . . . . . . . 4.3.2 High-Order Eigensolution Derivatives . . . . . . . . . . . . . . . 4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 67 67 69 70 70 76 78 82 83

5

Iterative Bisection Scanning Substructuring (IBSS) Method for Eigensolution and Eigensensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 IBSS Method for Eigensolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 IBSS Method for Eigensensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3.1 Eigenvalue Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3.2 Eigenvector Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.1 A Cantilever Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.2 The Canton Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6

Simultaneous Iterative Substructuring Method for Eigensolutions and Eigensensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 SIS Method for Eigensolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 SIS Method for Eigensensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Eigenvalue Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Eigenvector Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 A Frame Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Wuhan Yangtze River Navigation Center . . . . . . . . . . . . .

3.3

103 103 104 107 107 109 110 110 113

Contents

xi

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7

8

9

Substructuring Method Considering Elastic Effects of Slave Modes in the Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Basic Method for Time History Dynamic Response and Response Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Substructuring Method for Time History Dynamic Response and Response Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 A Three-Bay Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Wuhan Yangtze River Navigation Center . . . . . . . . . . . . . 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Substructuring Method Considering Inertial Effects of Slave Modes in the Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Substructuring Method for Time History Dynamic Response and Response Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 A Three-Bay Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Wuhan Yangtze River Navigation Center . . . . . . . . . . . . . 8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Substructuring Method for Finite Element Model Updating . . . . . . . 9.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Fundamentals of Sensitivity-Based FE Model Updating Using Modal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Fundamentals of Sensitivity-Based FE Model Updating Using Time History Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 FE Model Updating by Substructuring Method Using Modal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 FE Model Updating by Substructuring Method Using Time History Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 The Balla Balla Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Wuhan Yangtze River Navigation Center . . . . . . . . . . . . . 9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 125 131 131 133 140 141 143 143 143 147 147 151 152 153 155 155 156 157 158 158 158 158 164 170 170

xii

Part II

Contents

Dynamic Condensation Methods

10 Dynamic Condensation for Eigensolutions and Eigensensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Static Condensation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 IOR Method for Eigensolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 IOR Method for Eigensensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Eigenvalue Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Eigenvector Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 GARTEUR Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 A Cantilever Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 174 176 178 179 183 184 184 189 191 191

11 Dynamic Condensation to the Calculation of Structural Responses and Response Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 IOR Method for Structural Responses . . . . . . . . . . . . . . . . . . . . . . . 11.3 IOR Method for Response Sensitivities . . . . . . . . . . . . . . . . . . . . . . 11.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 A Three-Span Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 A Cantilever Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 193 197 198 198 205 209 210

12 Dynamic Condensation Approach to Finite Element Model Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Dynamic Condensation-Based FE Model Updating Using Modal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Dynamic Condensation-Based FE Model Updating Using Time History Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Junshan Yangtze River Bridge . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Jiangyin Yangtze River Bridge . . . . . . . . . . . . . . . . . . . . . . 12.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211 211 211 214 216 216 218 223 224

Part III Nonlinear Substructuring Methods 13 Substructuring Method for Responses and Response Sensitivities of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 13.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 13.2 Substructuring Method for Structural Responses of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Contents

13.3 Substructuring Method for Response Sensitivities of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 A Nonlinear Spring–Mass System . . . . . . . . . . . . . . . . . . . 13.4.2 A Nonlinear Frame Model . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Model Updating of Nonlinear Structures Using Substructuring Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Procedure of the Substructure-Based Nonlinear Model Updating Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Example: A Nonlinear Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Model Updating Without Measurement Noises . . . . . . . . 14.3.2 Model Updating with Measurement Noises . . . . . . . . . . . 14.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A Modal Derivative Enhanced Kron’s Substructuring Method for Response and Response Sensitivities of Geometrically Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Substructuring Method for Responses of Geometrically Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Substructuring Method for Response Sensitivities of Geometrically Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Computational Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Example: A Hinged Plate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

234 238 238 244 251 251 253 253 253 254 254 261 262 263

265 265 266 271 275 278 283 284

16 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 16.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 16.2 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Symbols

{} {}T , []T {}−1 , []+ I Diag( ) K M F R E u ωi , f i λi , /\ φi, o ¯ ¯ o φ, J S r α β χ f ˙ x¨ x, x, x¨ g N NP NS NT τ g C

Vector Transpose of the vector or matrix Inverse, pseudo-inverse matrix Unity matrix Diagonal-block assembly of the substructures, primitive matrices Stiffness matrix Mass matrix Residual flexibility matrix Rigid body modes Kron’s receptance matrix Equivalent stiffness matrix of reduced eigenequation ith modal frequency (rad/s, Hz) ith eigenvalue, matrix of eigenvalues ith eigenvector (mode shape), matrix of eigenvectors ith expanded mode shape, matrix of expanded mode shapes Objective function Sensitivity matrix Elemental physical parameter Elemental stiffness parameter Elemental mass parameter Dynamic response Force Displacement, velocity, acceleration Earthquake excitation Degrees of freedom of the global structure Size of the primitive matrix Number of the substructures Number of interface DOFs Interface force along the boundaries of the substructures Connection force from the adjacent substructures Damping matrix xv

xvi

D T µ V Z | f˜ ˜ K ˜ M ˜ C o || Ts γ KC MD

Symbols

Connection matrix Modal transformation matrix Transformation matrix for master and slave DOFs Physical transformation matrix Modal modes participation factor Connection matrix of the substructural eigenmodes Reduced force Reduced stiffness matrix Reduced mass matrix Reduced damping matrix Modal derivative tensor Reduction basis of geometrically nonlinear system Transformation mode Regularization parameter Equivalent stiffness matrix of the reduced eigenequation Equivalent mass matrix of the reduced eigenequation

Superscripts A E p U O L N

Analytical results Experimentally measured results Assembled matrices or vectors Updated results Initial results Linear substructure Nonlinear substructure

Subscripts m s m s d r g B I

Master DOFs Slave DOFs Master modes Slave modes Deformational modes Rigid body modes Variables associated with the global structure Boundary of a substructure Inner part of a substructure

Symbols

R S D

xvii

Reduced model Static condensation Dynamic condensation

Abbreviations CFT CMS CPU DOF FE FEM FRFS IBSS IRS MAC MD RBM SHM SIS SRF SRFS SV TDEES TDIES

Concrete filled tube Component mode synthesis Central processing unit Degree of freedom Finite element Finite element model First-order residual flexibility substructuring method Iterative bisection scanning substructuring method Improved reduced system Modal assurance criteria Modal derivative Rigid body mode Structural health monitoring Simultaneous iterative substructuring method Stiffness reduction factor Second-order residual flexibility substructuring method Similarity of vectors Time-domain elastic effect-based substructuring method Time domain inertial effect-based substructuring method

Chapter 1

Introduction

1.1 The Objective of Substructuring Method in Structural Health Monitoring Civil structures, including buildings, bridges, tunnels, dams, pipelines, and so on, provide fundamental essentials for a country and/or society. The safety and serviceability of these civil structures are basically crucial to a civilized society and a productive economy, and also the ultimate goals of engineering, academic, and management communities. Structural health monitoring (SHM) technologies have been developed for monitoring, evaluation, and maintenance of these civil structures during the past half-century. A high-fidelity finite element model (FEM) is frequently required by response prediction, optimization design, vibration control, and damage identification. The FEM is simplified and parameterized from a real structure, which introduced uncertainties in the material properties, geometry size, and boundary conditions. And the linear or nonlinear models of a civil structure are usually idealized hypothesis. The dynamic responses of a highly idealized numerical model are usually deviated from the measured counterparts. Brownjohn et al. (2001) reported that the maximum relative differences of the numerical and experimental modal frequencies of a curved cable-stayed bridge reached 40%, and relative differences of most modes are larger than 10%. Jaishi and Ren (2005) show that the differences of the natural frequencies measured from a steel arch bridge and those computed by an FEM counterpart reach 20%, and the modal assurance criteria (MAC) values of mode shapes between practical measurement and numerical model are as low as 62%. Zivanovic et al. (2007) found that the average difference of natural frequencies of a footbridge predicted by an FEM and their experimental counterparts is 29.8%. Therefore, the parameters of the FEMs need to be adjusted effectively to obtain a more truthful model that can be used for various disciplines. Model updating is a technique that updates the FE model of a structure so that it can predict dynamic properties of the structure more accurately. Parameters of © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_1

1

2

1 Introduction

the FEM are adjusted to reduce a penalty function of residuals between the analytical prediction and measurement data in an optimal way. Model updating methods (Friswell and Mottshead 1995) are usually categorized into one-step methods and iterative methods according to whether the system matrix is updated directly or indirectly. The former directly reconstructs the system matrices of the analytical model, which is precise and computationally cheap but may have poor physical meanings. The latter modifies the physical parameters of the FEM repeatedly to minimize the discrepancy between analytical predictions and their measured counterparts. Iterative methods preserve positive-definiteness, symmetry, and sparseness in the updated system matrices, and more importantly, the predicted parameters are physically meaningful to be understood. In the iterative model updating methods, the prespecified physical parameters of the FE model are updated repeatedly to minimize the residuals between the measurements and model predictions in an optimal manner. The dynamic properties and associated sensitivity matrices of the analytical model are calculated in each iteration repeatedly. The FEM of a large-scale structure usually consists of a large number of degrees of freedom (DOFs) and many uncertain parameters. Repeatedly calculating the solutions from such a large-scale FEM is very expensive or even prohibited in terms of computational time and memory. First, the large system matrices, including stiffness and mass matrices, occupy a large amount of storage space. Second, dynamic analysis based on the large system matrices is a time-consuming process. Calculation of sensitivity with respect to a large number of parameters wastes even more computational resources and time. Nowadays, the model updating process is usually incorporated with the statistical analysis or nonlinear analysis, and thus the process is heavier or even prohibited. Third, there are a lot of uncertain parameters to be adjusted in a large-scale FEM. It takes a long time to calculate the sensitivity matrices with respect to a large number of updating parameters of the model contains. The large number of parameters hinders the convergence of the large-scale optimization problem as well. The iterative model updating of a civil structure usually involves a heavy work. For example, Xia et al. (2008) performed the model updating of the Balla Balla Bridge in Western Australia. It was modeled with 907 elements, 949 nodes, and 5400 DOFs. Within each iteration, calculating the eigensolutions cost about 10 s, and calculating the eigensensitivities with respect to the 1130 uncertain parameters took more than two hours. Calculation of the eigensensitivity costs dominant computation time during the model updating. The optimization converged within 155 iterations, which cost about 420 h in total. The computation time for the model updating of this middle-scale bridge is tremendous. It is impossible to perform model updating on a large-scale civil structure. For example, a fine FEM was established for the Tsing Ma Suspension Bridge. The FEM consists of about 300,000 nodes, 450,000 elements, and 1.2 million DOFs. It took about five hours to extract the first 100 eigensolutions using a 64-bit Itanium server with eight CPUs of 1.5 GHz each (Duan et al. 2011). One has to simplify the model to achieve a more efficient model updating on the sacrifice of accuracy. It is challenging to update the model for such a large-scale structure using a conventional approach.

1.2 The Category of Substructuring Model Updating Methods

3

Substructuring methods are effective to deal with large-size civil structures. The substructuring methods partition the global structure into several substructures, which are analyzed independently. A reduced model is constructed from the substructural solutions to recover the global dynamic properties. The substructuring method merits some distinct advantages over the global methods that analyze the global structure as a whole: (1) It is much easier and quicker to store and analyze the small substructural system matrices independently than analyze the global structure as a whole (2) Substructuring methods allow for the specific substructures to be analyzed solely to recognize local dynamic behaviors more easily, avoiding the analysis of other substructures or the global structure (3) Substructuring methods reduce the number of uncertain parameters to be updated and thus alleviate the ill-condition problems and accelerate the convergence of the model updating (4) In practical testing, the experimental instruments can be saved if it is necessary to measure the whole structure only for one or more substructures (5) Substructuring methods allow sharing and combining substructures from different project groups, or from modeled parts and experimental parts. The civil structure is usually large-sized, whereas the damage is usually located in a local area. The substructuring method analyzes the large-scale structure in a piece-wise manner rather than as a whole. The time-consuming optimization calculation, uncertainty analysis, and nonlinear calculation are constrained to the local substructure, which can effectively improve the accuracy and efficiency of model updating and damage identification.

1.2 The Category of Substructuring Model Updating Methods According to the model updating process, the substructuring approach can be categorized into the forward and inverse methods (Weng et al. 2020). In the forward substructuring method, the global FEM is divided into several substructural models. The partitioned substructures are analyzed independently to obtain their designated solutions. Afterward, the substructural solutions are assembled to recover the solutions of the global structure by imposing constraints at the interfaces. In model updating, these global solutions are compared with the experimental counterparts to construct the objective function. The process is shown in Fig. 1.1. The sensitivity matrices with respect to an elemental parameter are computed within one substructure that contains the elemental parameter. The global sensitivity is recovered from sensitivity matrices of one specific substructure, while the sensitivity matrices of other substructures are zeros. The elemental parameters are iteratively adjusted by minimizing the objective function in accordance with the sensitivity matrices. In this forward substructure-based model updating method, if a

4

1 Introduction

Fig. 1.1 Forward substructuring method for model updating

local area is changed, only one or several substructures are analyzed independently without repeatedly analyzing the large-size matrices of the global structure. On the other hand, the substructuring approach can be used in an inverse manner performed on the experimental data (Fig. 1.2). First, substructural properties of one substructure, for example, flexibility, frequencies, and mode shapes, are extracted from the experimental data on the global structure by imposing the constraints of displacement compatibility and force compatibility. After the substructural experimental properties are obtained, the focused substructure can be updated independently using a conventional model updating method. This model updating process involves solely one substructure and thus improving the efficiency of the optimization process. The main work of this inverse substructuring method depends on an effective decoupling algorithm to extracting the effective substructural properties which validly represent the real local area. In substructuring methods, the global model is reduced into a modal space spanned by a few substructural master modes, based on which the dynamic analysis, sensitivity analysis, and model updating are implemented accurately and efficiently. In this book,

1.3 Organization of the Book

5

Fig. 1.2 Inverse substructuring method for model updating

a few master modes of substructures are used to recover the eigensolutions or dynamic response of the global structure, and modal truncation is adopted to speed up the analysis of independent substructures. This strategy can also be used in the physical space, where the global model is reduced to a few DOFs. By using a transformation matrix, the global model is reduced into a low-dimensional physical space spanned by the master DOFs. The latter strategy is usually named as the model reduction technique and is frequently employed as a companion of substructuring method. Both the master modes in the modal space and the master DOFs in the physical space are much fewer than that of the global model. Consequently, computational resources and time are saved.

1.3 Organization of the Book This book will introduce the substructuring method and its applications in FE model updating. The book comprises 16 chapters as Fig. 1.3. Chapter 1 provides the objective and background of the substructuring methods in SHM, and the other 15 chapters are grouped into three topics. Chapters 2–8 present the linear substructuring methods. Chapter 2 introduces the substructuring method for the eigensolutions, and the first-order and second-order residual flexibility compensation will be adopted to improve the accuracy. Chapter 3 presents the substructuring method for the eigensensitivity, and Chap. 4 describes the substructuring method for the high-order eigensensitivity. The improvement on the iterative substructuring method is presented in Chaps. 5 and 6 to achieve higher accuracy. In particular, Chap. 5 proposes an iterative bisection scanning substructuring method to calculate the eigensolutions and eigensensitivity modes by modes, and

6

1 Introduction 1 Introduction

Part A: Linear substructuring methods 2 Substructirng method for eigensolutions

Part B: Dynamic condensation methods

7 Substructuring method for response and sensitivity considering elastic effect

3 Substructirng method for eigensensitivity 4 Substructirng method for high-order eigensensitivity 5 iterative substructuring method with mode by mode

8 substructuring method for response and sensitivity considering inertial and elastic effect

6 Iterative substructuring method with all modes simultaneously

Part C: Nonlinear substructuring methods

10 Dynamic condensation for eigensolutions and eigensensitivity

13 Substructuring method for material nonlinear system

11 Dynamic condensation for response and sensitivity

14 Substructuring method for geometrically nonlinear system

12 Dynamic condensation for model updating

15 nonlinear substructuring method for model updating

9 Substructuring method for model updating

16 Epilogue

Fig. 1.3 Organization of the book chapters

Chap. 6 develops an iterative substructuring method to compute the eigensolutions and eigensensitivity for all modes simultaneously. Subsequently, the substructuring method is extended to calculate the dynamic response and response sensitivity in the time domain in Chaps. 7 and 8, by including the elastic effects and inertial effects of substructures, respectively. The fast calculations of eigensolutions, eigensensitivities, dynamic response, and response sensitivity are applied to the sensitivity-based model updating process performed on two practical structures in Chap. 9. Chapters 11 and 12 involve the second topic, namely the model condensation methods, which are usually regarded as the substructuring method reduced in the physical space. A new dynamic condensation method is developed in Chap. 11, which is developed for the fast calculation of eigensolutions and eigensensitivities in the frequency domain. The method is proposed to calculate the dynamic response and response sensitivity in the time domain in Chap. 12. The final topic of the book is the substructuring methods for nonlinear analysis and nonlinear model updating. In particular, Chap. 13 presents a substructuring method for calculating the response and response sensitivity of nonlinear systems, and Chap. 14 proposes the associated substructure-based nonlinear model updating. Chapter 15 develops a modal derivative-enhanced substructuring method for calculating the response and response sensitivity of geometrically nonlinear structures. Chapter 16 concludes the book and discusses possible future research.

References

7

References Brownjohn, J.M.W., Xia, P.Q., Hao, H., Xia, Y.: Civil structure condition assessment by FE model updating: methodology and case studies. Finite Elem. Anal. Des. 37(10), 761–775 (2001) Duan, Y.F., Xu, Y.L., Fei, Q.G., et al.: Advanced finite element model of Tsing Ma bridge for structural health monitoring. Int. J. Struct. Stab. Dy. 11(2), 313–344 (2011) Friswell, M.I., Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics. Kluwer Academic Publishers, Norwell, MA (1995) Jaishi, B., Ren, W.X.: Structural finite element model updating using ambient vibration test results. J. Struct. Eng. 131(4), 617–628 (2005) Weng, S., Zhu, H., Xia, Y., et al.: A review on dynamic substructuring methods for model updating and damage detection of large-scale structures. Adv. Struct. Eng. 23(3), 584–600 (2020) Xia, Y., Hao, H., Deeks, A., et al.: Condition assessment of shear connectors in slab-girder bridges via vibration measurements. J. Bridg. Eng. 13(1), 43–54 (2008) Zivanovic, S., Pavic, A., Reynolds, P.: Finite element modelling and updating of a lively footbridge: the complete process. J. Sound Vib. 301(1–2), 126–145 (2007)

Part I

Linear Substructuring Methods

Chapter 2

Substructuring Method for Eigensolutions

2.1 Preview In dynamic analysis, the frequencies and mode shapes are the eigensolutions of stiffness and mass matrices, which serve to construct the objective function in frequency domain model updating. The elemental parameters in the FEM are iteratively modified to minimize the residuals of frequencies and mode shapes between the analytical predictions and the measured counterparts in an optimal way. In the iterative model updating, the eigensolutions are computed repeatedly from the stiffness and mass matrices of the analytical FEM. To simulate the real structure accurately, the FEM of a large-scale structure includes a large number of elements, nodes, and parameters. The stiffness and mass matrices of the analytical FEM are large in size. It is a big challenge to efficiently obtaining the eigensolutions from large-size system matrices. Sparse matrix techniques, order reduction methods, and substructuring methods are effective mathematical scheme developed for the fast analysis of large-scale matrices. Sparse matrix techniques, such as the subspace iteration method and the Lanczos algorithm or (Bath 1982), are widely used in commercial software such as ABAQUS and ANSYS. It is an efficient way to alleviate the computation load of large-size system matrices. Sparse matrix technique exploits the sparsity of the assembled mass and stiffness matrices, which perform numerical operations directly on the nonzero components of the large-size system matrices. Order reduction methods reduce the size of the system matrices by removing some DOFs of the original FEM and retaining a much smaller set. It transforms the dynamic behavior of the discarded DOFs to those of the retained DOFs. The reduced eigenequation is then solved to approximate the eigensolutions of the original structure. The substructuring technology can be a promising solution to accelerate the calculation of eigensolutions of a large-scale structure. Substructuring methods divide a structure into smaller independent substructures. The stiffness and mass matrices of substructures are constructed independently, from which the eigensolutions of

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_2

11

12

2 Substructuring Method for Eigensolutions

substructures are calculated. Subsequently, the eigensolutions of the original structure are recovered from the eigensolutions of the independent substructures by constraining the interface of the adjacent substructures (Klerk et al. 2008). The substructuring methods are efficient than the global method in three main respects. First, the global structure is divided into smaller substructures, and the small substructural system matrices is faster and easier to be analyzed. Second, the specific substructures (local area) can be analyzed independently without studying the entire structure as a whole. This significantly reduces the size of the system matrices. Third, the substructuring method allows for the combination of different parts and will be more efficient to be used together with parallel computation techniques. In this chapter, the commonly used Lanczos algorithm and subspace iteration methods for eigensolutions are first introduced. Also, the traditional substructuring methods for eigensolutions, such as the component mode synthesis method and Kron’s substructuring method, are introduced. The Kron’s substructuring method requires the complete eigensolutions of all substructures to recover the eigensolutions of the global structure. This is time-consuming for a large-scale structure, which may have many large-size substructures. To improve this computational inefficiency, a modal truncation approximation is proposed and elaborated. Only the lowest eigensolutions of the substructures need to be calculated. The discarded higher eigensolutions are compensated by the first-order residual flexibility or the second-order residual flexibility. The division of substructures and the selection of master modes in each substructure are also investigated.

2.2 Basic Methods for Eigensolutions By using the basic FE method, the stiffness and mass matrices of a structure with N DOFs are constructed. The classical eigenequation for the structure has the form of K{φi } = λi M{φi }

(2.1)

where K and M are the N × N symmetric stiffness and mass matrices, respectively, λi represents the ith eigenvalue, and {φi } is the corresponding eigenvector. λi and {φi } form a pair of eigensolutions of the system. In structural dynamics, the eigenvalue is the square of the circular frequency, and the eigenvector is the mode shape of the structure. The subspace iteration method and Lanczos method are commonly used to solve the large-size eigenequation. These two methods are introduced in Sect. 2.1 and Sect. 2.2. K and M are assembled by the contribution of all n elements in the discrete FEM. In particular, K=

n E j=1

Kj =

n E j=1

α j Kej , M =

n E j=1

Mj =

n E j=1

β j Mej

(2.2)

2.2 Basic Methods for Eigensolutions

13

where Kj and Mj are the jth elemental stiffness matrix and elemental mass matrix, respectively, and α j and β j are respectively the ‘elemental stiffness parameter’ and ‘elemental mass parameter’.

2.2.1 Subspace Iteration Method The subspace iteration method extracts the eigensolutions of the large-size system matrices by projecting them onto an orthogonal subspace to reduce the computational cost. As solving the complete eigensolutions is expensive and usually not necessary for the large-scale structures, the approximated solution spanned by the lowest eigenpairs is favorable. Figure 2.1 demonstrates the general procedure of the subspace iteration method. The eigenproblem in Eq. (2.1) spans an N-dimensional space. The subspace iteration method solves this eigenproblem in an iterative way with a set of l linearly independent vectors (l is much less than N). The subspace is spanned by these l vectors. By iterating with l vectors, the subspace is gradually improved so that it will ultimately span the l-dimensional subspace of the original N-dimensional space. The subspace iteration method is time-consuming in searching the optimized l-dimensional space, especially for a large-scale structure. Fig. 2.1 Basic subspace iteration method

14

2 Substructuring Method for Eigensolutions

2.2.2 Lanczos Method The Lanczos algorithm has been developed as a powerful tool to extract eigenvalues of a real symmetric matrix. The Lanczos algorithm transforms a generalized N-dimensional eigenproblem into a standard tridiagonal matrix with a smaller dimension m, as described in Fig. 2.2. The Lanczos algorithm constructs an orthogonal basis for the Krylov subspace as ) ( ( ( ) )m−1 {q1 } Qm = span {q1 } K−1 M {q1 } · · · K−1 M = span({q1 } {q2 } · · · {qm })

(2.3)

{q1 } is an arbitrary starting vector, and {qj } is a Lanczos vector orthogonal to the previous {qj−1 } Lanczos vectors with respect to mass matrix M, and m is the dimension of the Krylov subspace. The tridiagonal matrix in Fig. 2.2 is formulated by

Fig. 2.2 Basic Lanczos algorithm

2.3 Substructuring Method for Eigensolutions

15

⎤ α1 β2 ⎥ ⎢ β2 α2 β3 ⎥ ⎢ ⎥ ⎢ . . . . ⎥ ⎢ Tm = ⎢ β3 . . ⎥ ⎥ ⎢ .. .. ⎣ . . βm ⎦ βm αm ⎡

(2.4)

This block version of Lanczos method is more powerful, which naturally produces a block tridiagonal matrix Tm . As compared with the subspace iteration method, the Lanczos method is preferable to compute a large number of eigenpairs for very large sparse matrices.

2.3 Substructuring Method for Eigensolutions Substructuring methods are promising to improve computational efficiency. Calculating eigensolutions with the substructuring method consists of three basic steps: division of a global structure into substructures, calculation of the eigensolutions for the independent substructures, and reconnection of the substructures to the global structure with compatibility equations. The global structure is divided into N S independent substructures, each with N (j) DOFs (j = 1, 2,…, N S ). After division, one node of the global structure at the interface belongs to two or more nodes of the separated substructures. Each interface DOF in the original global structure is shared by two or more substructures that are connected to it. In consequence, there are N T interface DOFs if one interface is shared by two separate substructures. The total number of DOFs of all substructures increases to N p , which is larger than N. For example, if the nth (n = 1, 2, …, N T ) interface DOF is shared by tn substructures, it has N E T

N =N+ P

n=1

(tn − 1) =

NS E

N ( j)

(2.5)

j=1

After division, each independent substructure, for example, the jth substructure, has a stiffness matrix K( j) and mass matrix M( j) . The generalized eigenequation of the jth substructure can be written as { } { } ( j) ( j) ( j) = λi M( j) φi K( j) φi

(2.6)

Both K( j ) and M( j ) are of order N (j) × N (j) . The eigenequation Eq. (2.6) can be solved by the subspace iteration method or Lanczos method introduced } 2.2, { in Sect. ( j) ( j) are the by treating the jth substructure as an independent structure. λi and φi

16

2 Substructuring Method for Eigensolutions

ith eigenvalue and eigenvector of the jth substructure, respectively. Equation (2.6) can be decoupled into N (j) pairs of eigenvalues and eigenvectors as | | | | ( j) ( j) ( j) ( j) ( j) ( j) /\( j) = Diag λ1 , λ2 , . . . , λ N ( j ) , φ( j ) = φ1 , φ2 , . . . , φ N ( j )

(2.7)

Those eigenpairs satisfy the orthogonal properties, which are normalized with respect to the mass matrix as {| |T φ( j ) M( j) φ( j) = I N ( j ) | ( j ) |T ( j) ( j) φ K φ = /\( j )

(2.8)

The substructural matrices are diagonally assembled to the primitive form like | | | | Mp = Diag M(1) , M(2) , . . . , M(Ns ) , Kp = Diag K(1) , K(2) , . . . , K(Ns ) | | | | φp = Diag φ(1) , φ(2) , . . . , φ(Ns ) , /\p = Diag /\(1) , /\(2) , . . . , /\(Ns )

(2.9)

where superscript ‘p’ denotes the variables of the primitive form. The primitive matrices include the substructural matrices stacked directly without constraining them. The size of the primitive matrices is N p × N p . Since the substructural eigenpairs satisfy the mass-normalized orthogonality in Eq. (2.8), the primitive matrices satisfy the orthogonality conditions that {| | T φp M p φp = I N p | p |T p p φ K φ = /\p

(2.10)

The primitive system can be reconnected to the global system by constraining the geometric compatibility and force equilibrium at the interface points of the adjacent substructures. The methods of constraining the independent substructures have been studied extensively. The component mode synthesis (CMS) and Kron’s substructuring method are two representative ones.

2.3.1 Component Mode Synthesis The CMS method combines the component-wise analysis and model reduction techniques. Instead of describing the substructure at all DOFs, this method represents the dynamic displacement of a substructure in terms of the reduced orthogonal basis expressed by the most dominant component modes. Thus, the reduced substructure is obtained. Afterward, the reduced substructures are assembled to obtain a reduced model of the global structure, from which the global eigensolutions are efficiently calculated.

2.3 Substructuring Method for Eigensolutions

17

In the CMS method, the component modes of the independent substructures can be classified into four groups according to the boundary conditions, namely the rigid body modes, the normal modes, the constraint modes, and the attachment modes. (1) The rigid body modes (RBMs) describe the rigid body movement of a freefree substructure, which can be calculated either by extracting the null basis of the free-free stiffness K, or by constructing the self-equilibrium equation. The former method requires the rank-deficient free-free stiffness K rearranged according to the inner and interface DOFs as | K=

| KII KIB KBI KBB

(2.11)

where the subscript ‘I’ indicates the inner DOFs and the subscript ‘B’ represents the interface DOFs. After separating the interface DOFs, KII is a square matrix with a full rank. The generalized inverse of the free-free stiffness and the rigid body modes are computed from the inverse of KII as (K)+ =

|

| | | KII−1 0 −KII−1 KIB ,R= I 0 0

(2.12)

where R is the RBMs. For a free-free structure with a large number of DOFs, the inverse of KII is not only expensive but also inaccurate. Alternatively, the selfequilibrium method can be used. For a two-dimensional structure having N nodes, the three independent RBMs are the x translation (Rx = 1, Ry = 0, Rz = 0), the y translation (Rx = 0, Ry = 1, Rz = 0) and the z rotation (Rx = − y, Ry = x, Rz = 1), i.e., ⎡

⎤ 1 0 0 1 ··· 0 0 RT = ⎣ 0 1 0 0 · · · 1 0 ⎦ −y1 x1 1 −y2 · · · x N 1

(2.13)

The columns |of R are naturally orthogonal and can be normalized by dividing )1/2 | E( 2 2 1/2 xi + yi + 1 N , N , and , respectively. 1/2

(2) The normal modes are calculated by the general eigenequation for both the fixed boundary condition and free boundary condition. The fixed interface normal modes are obtained by restraining all boundary DOFs, which is equivalent to solve an eigenequation formed by the stiffness and mass matrices of the inner DOFs, i.e., (KII − λi MII ){φI }i = {0} The complete fixed interface normal modes are written as

(2.14)

18

2 Substructuring Method for Eigensolutions

|

φn N ×Ni

φI = IB

| (2.15)

where N i is the number of the normal modes, equal to the number of the inner DOFs. Only the inner DOFs are used to construct the normal modes for the fixed interface condition. The fixed interface modes are normalized with respect to the mass matrix MII of inner DOFs as φTI MII φI = I, φTI KII φI = /\

(2.16)

The free interface normal modes are computed from the system matrices by releasing all boundary DOFs and solving the eigenequation of ( ) K − λ j M φ j = 0( j = 1, 2 . . . , , . . . , Nd = N − Nr )

(2.17)

where N d is the quantity of the deformational modes and N r is the quantity of the rigid body modes, respectively. (3) A constraint mode represents the force that generate a unit displacement at one coordinate of a specified ‘constraint’ coordinate D, while the remaining coordinates of that set D are restrained, and the remaining DOFs of the structure are free. The force equilibrium satisfies | KII KIB KBI KBB

||

φIB IBB

|

| =

|

0

(2.18)

f BB

The constraint mode matrix is given by | φc =

N ×Nb

φIB IBB

|

| =

−KII−1 KIB IBB

| (2.19)

where N b is the number of the constraint modes, which is equal to the number of the inner DOFs. The constraint modes are stiffness orthogonal to all fixed interface normal modes, that is φTn Kφc = 0

(2.20)

(4) An attachment mode is regarded as the displacement vector due to a single unit force applied at one of the given coordinates A, which is therefore computed by | KII KIB KBI KBB

||

φI IB

|

|

0 = IB

| (2.21)

2.3 Substructuring Method for Eigensolutions

19

CMS methods are mainly classified into the fixed interface method and the free interface CMS method. According to the boundary condition, mode components are selected from the above four groups to represent the displacement of an independent substructure and to recover the global eigensolutions. The fixed interface CMS method was first proposed by Hurty (1965). The global eigensolutions are projected onto the space spanned by the substructural constraint modes and fixed normal modes. Craig and Bampton (1968) simplified the fixed interface CMS method by partitioning the interface forces into statically determinate and indeterminate ones. Since this simplification is efficient and easy-understand, it is widely used for eigensolutions calculation. Since it is difficult or even unable to acquire constraint modes from experiments conveniently, the fixed interface CMS method is sometimes limited to be used. To overcome this limitation, the free interface CMS method was first proposed by MacNeal (1971) for structures with flexible boundary conditions. The substructural displacements are represented by the free interface normal modes, attachment modes and rigid body modes, from which the global eigensolutions are recovered. Conducting a modal test on a free-free structure is considerably more convenient than on one with fixed constraints. Rubin (1975) retained partial free interface normal modes, and included the inertial effect of truncated modes with a second-order Maclaurin-series expansion. The free interface CMS method is less accurate than the fixed interface CMS method, since the free interface constraint is weaker than the fixed interface. However, the free interface method is considerably more efficient in handling complicated substructural interfaces. The complete eigenmodes of the substructures with changed interface have to be recomputed in the fixed interface method. Synthesizing the features of both methods, researchers have developed the hybrid interface (mixed interface) CMS method to make full use of the advantages of the fixed and free interface CMS methods.

2.3.2 Kron’s Substructuring Method Kron first proposed a dynamic substructuring method in the book Diakoptics (Kron 1963) to compute the eigensolutions of the large matrices in a piece-wise manner. The independent substructures are analyzed independently and constrained by imposing displacement constraints at the interface coordinates of the adjacent substructures via the Lagrange multiplier technique. Kron’s substructuring method has distinct advantages in handling large-scale systems since it is highly accurate under complicated interface conditions. The complete substructural modes are calculated from the stiffness and mass matrices of substructures directly without considering the substructural boundary, and thus avoid computing the constrained modes, normal modes, attachment modes, and rigid body modes.

20

2 Substructuring Method for Eigensolutions

The displacement of the original global structure x of size N × 1 is expanded to x p of size N p × 1 after being divided into substructures, which includes the identical displacements in the interface DOFs. The geometric compatibility is sufficed by applying the displacement constraints at the interfaces { } D xp = 0

(2.22)

where D is a rectangular matrix containing the general implicit constraints to ensure the nodes at the interfaces move jointly. The independent substructures have identical displacements at the interfaces. In matrix D, each row contains two nonzero entries. For a rigid connection, the two entries are 1 and − 1. If the connected points x 1 and x 2 are not rigidly connected, for example, they have the linear relation x 1 = ax 2 , then the two nonzero entries are 1 and − a. Kron’s method considers the connection condition by matrix D directly instead of including additional items such as the constraint modes or linking force. According to the virtual work theorem, the motion equation of an undamped structure is { } { } Mp x¨ p + K p x p = f p + g p

(2.23)

|T | x¨ p = x¨ (1) , · · · , x¨ (i ) , · · · , x¨ (Ns )

(2.24)

|T | x p = x (1) , · · · , x (i ) , · · · , x (Ns )

(2.25)

|T | f p = f (1) , · · · , f (i ) , · · · , f (Ns )

(2.26)

|T | g p = g (1) , · · · , g (i ) , · · · , g (Ns )

(2.27)

where

where x¨ p and x p are the acceleration and displacement vectors of the independent substructures, g p is the connection force from the adjacent substructures, and f p is the external force. For a free vibration system, f p = 0. The virtual work conducted by the connection force along x p is { }T δW = g p δx p

(2.28)

The virtual work theorem assumes the connection to be incompleted, and the geometric compatibility in Eq. (2.22) is violated at the interface coordinates by an amount of {η} like { } D x p = {η}

(2.29)

2.3 Substructuring Method for Eigensolutions

21

In the interface coordinates, an associated force vector {τ } is used to represent the internal connection forces due to the ‘misfit’. Equation (2.29) gives } { δW = {τ }T {δη} = {τ }T D δx p

(2.30)

From Eqs. (2.28) and (2.30), one can obtain {

gp

}T

} { δx p = {τ }T D δx p

(2.31)

and thus g p = DT τ

(2.32)

Consequently, Eq. (2.23) is transformed into |

Mp 0 0 0

|{

x¨ p τ¨

}

| +

Kp −DT −D 0

|{

xp τ

} =

{ } 0 0

(2.33)

Since the time-domain can be expressed in the modal space by ( / vibration ) }T { {x p , τ }T = φ, τ exp j λt , the expanded mode shape of the global structure is represented by the primitive form of the mode shapes φp via the modal coordinates z as { } | p |{ } φ φ 0 z = (2.34) τ 0 I τ where φ is the expanded mode shape of the global structure containing the identical values in the interface DOFs. Due to the orthogonality relations in Eqs. (2.8) and (2.33) can be simplified into the eigenequation of the global structure |

/\p − λI −| 0 −| T

|{ } { } z 0 = τ 0

(2.35)

where | = (Dφp )T is the normal connection matrix that constrains the nodes to move jointly at the interface points of the adjacent substructures. Therefore, the eigenvalue λ obtained from Eq. (2.35) is equivalent to the eigenvalue λ of the original global structure. The eigenvectors of the global structure φ can be obtained after combining the identical DOFs in the expanded eigenvectors φ. | takes the size of N p × (N p − N ), and (N p − N ) is the number of the constraint relations. The first line of Eq. (2.35) gives ( )−1 z = /\p − λI |τ

(2.36)

22

2 Substructuring Method for Eigensolutions

Substituting Eq. (2.36) into the second line of Eq. (2.35) to eliminate the modal coordinates z, one has ( )−1 | T /\p − λI |τ = 0 or Eτ = 0

(2.37)

)−1 ( where E = | T D| and D = /\p − λI . Matrix E with a size of (N p − N ) × (N p − N ) is known as the Kron matrix or receptance matrix. Since the above analysis has no approximation in the derivation of E, the eigenvalues obtained from the receptance matrix are exactly those of the original global structure. In the original Kron’s method, λ is obtained by scanning the determinant of receptance matrix E. This scanning process is very time-consuming since E is dependent on the unknown item λ. It is onerous to calculate the whole eigensolutions of each substructure to assemble /\p and φp . Moreover, the final eigenequation in searching the eigensolutions has a size of N p × N p , which is very large for a large-scale structure.

2.3.3 First-Order Residual Flexibility Based Substructuring Method 2.3.3.1

Method Description

From the viewpoint of energy conservation, the complete modes of all substructures contribute to the eigenmodes of the global structure, i.e., the complete eigensolutions of all substructures are required to assemble the primitive form of /\p and φp . This is inefficient and not worthwhile as only a few eigenmodes are generally of interest for a large-scale structure. Considering the lower modes usually contain most energy in the vibration problem, the complete eigenmodes of each substructure are divided into the ‘master’ eigensolutions (lower modes) and the ‘slave’ eigensolutions (higher modes). Only the master modes are calculated to assemble the eigenequation of the global structure, while the slave modes are discarded and compensated by an energy index in the later calculations. In this connection, the eigenequation (Eq. 2.35) is rewritten according to the master modes and slave modes as ⎡ p ⎤⎧ ⎫ ⎧ ⎫ 0 −| m ⎨ zm ⎬ ⎨ 0 ⎬ /\m − λI ⎣ (2.38) 0 /\ps − λI −| s ⎦ zs = 0 ⎩ ⎭ ⎩ ⎭ T T −| m −| s 0 τ 0 where | | | | ( j) ( j) ( j) (2) (Ns ) , /\(mj ) = Diag λ1 , λ2 , . . . , λ ( j ) /\pm = Diag /\(1) m , /\m , . . . , /\m Nm

(2.39)

2.3 Substructuring Method for Eigensolutions

23

| | | | ( j) ( j) ( j) (2) (Ns ) , φ(mj ) = φ1 , φ2 , . . . , φ ( j ) φpm = Diag φ(1) m , φm , . . . , φm Nm

| | | ( j) ( j) (2) ( j) (Ns ) , /\ /\ps = Diag /\(1) , /\ , . . . , /\ = Diag λ ( j ) , λ ( j) s s s s Nm +1

| ( j) | (2) (Ns ) , φs φps = Diag φ(1) s , φs , . . . , φs

| ( j) ( j) = φ ( j) , φ ( j) Nm +1

Nm +2

Nm +2

,...,λ

,...,φ

(2.40) ( j) ( j) ( j) Nm +Ns

|

(2.41) |

( j) ( j) ( j) Nm +Ns

(2.42) Nmp =

Ns E

Nm( j) , Nsp =

j=1

Ns E

Ns( j) , Nm( j) + Ns( j) = N ( j) ( j = 1, 2, . . . , Ns ) (2.43)

j=1

|T |T | | | m = Dφpm , | s = Dφps

(2.44)

Hereinafter subscript ‘m’ indicates the items belonging to the ‘master’ modes, and the subscript ‘s’ for the ‘slave’ modes. According to the second line of Eq. (2.38), the slave coordinate zs can be expressed by ( )−1 {zs } = /\ps − λI | s {τ }

(2.45)

Substituting Eq. (2.45) into Eq. (2.38) leads to |

/\pm − λI −| ( p m )−1 T T −| m −| s /\s − λI | s

|{

zm τ

} =

{ } 0 0

(2.46)

The required eigenvalues λ usually correspond to the lowest modes of the global structure, and far less than the items in /\ps when a proper size of the master modes is ( )−1 ( )−1 chosen. In that case, | Ts /\ps − λI | s ≈ | Ts /\ps | s . Representing τ with zm from the second line of Eq. (2.46) and substituting it into the first line, the eigenequation is reduced to ( ( ) )−1 −1 | Tm u{zm } = λ{zm }, u = /\pm + | m | Ts /\ps | s

(2.47)

The eigenvalue of the global structure is λ. The eigenvector of the global structure is recovered by φ = φpm Zm . The size of the reduced eigenequation is equal to the number of the retained master modes, which is much smaller than the original one ( )−1 (Eq. 2.38). It is noted that | Ts /\ps | s can be calculated from the first-order residual flexibility.

24

2 Substructuring Method for Eigensolutions

( )−1 | Ts /\ps | s = DFp DT ( )−1 | p |T φs Fp = φps /\ps ⎡( ) ( (1) )−1 | (1) |T −1 /\s φs − φ(1) K(1) s ⎢ .. =⎢ . ⎣

⎤ ⎥ ⎥ ⎦

( (N ) )−1 ( (Ns ) )−1 | (N ) |T s) K s /\s φs s − φ(N s (2.48)

The first-order residual flexibility-based substructuring method (FRFS) is advantageous in two aspects. First, only the master modes of substructures are required without computing the complete substructural modes, and the contribution of the discarded slave modes are compensated by the first-order residual flexibility. Second, the size of the eigenequation is significantly reduced to the number of the master modes.

2.3.3.2

Error Quantification

( ( )−1 )−1 In the proposed substructuring method, /\ps − λI is replaced by /\ps since the required λ is far less than the values in /\ps . The error due to this approximation is ⎡ ( p )−1 ( p )−1 ⎢ ⎢ /\s − λI − /\s =⎢ ⎣ ⎡

1

( )

/\ps 1 −λ

−

1 (/\ps )1

⎤ ..

. 1

(/\ps ) Nsp −λ λ

p p ⎢ ((/\s )1 −λ)(/\s )1 .. ⎢ =⎢ . ⎣

(

−

1

⎥ ⎥ ⎥ ⎦

(/\ps ) Nsp ⎤

λ)

⎥ ⎥ ⎥ ⎦

(/\ps ) Nsp −λ (/\ps ) Nsp

λ )( ) = Diag (( p ) /\s i − λ /\ps i And the relative error is ⎛ ⎞ ( ) λ p p ) ( λ /\s )i −λ)((/\s )i ) (( ⎠ = Diag ( ) Diag⎝ i = 1, 2, . . . , Nsp p 1 /\s i (/\ps )i −λ

(2.49)

(2.50)

Therefore, the largest relative error depends on min λ/\p . Usually, the large-size ( s) global structure has much smaller frequencies than the small-size substructure. Given

2.3 Substructuring Method for Eigensolutions

25

a suitable selection of master modes, the required eigenvalues λ are much smaller than the minimum value of /\ps , and thus the error introduced will be insignificant. In other words, the minimum eigenvalue of slave modes in the substructures controls the accuracy of the proposed substructuring method. In practice, the substructures inherently have larger natural frequencies than the global structure, as the former is of much smaller size than the latter. This property can guarantee the precision of the proposed substructuring method. In addition, the accuracy of the proposed method can be improved in two manners. On the first hand, we can improve the accuracy ( ) by including more modes as master modes, which increases the values of min /\ps . It is suggested ( ) to select the lowest modes as master modes to increase the magnitude of min /\ps . The number of the master modes in the substructures is usually suggested to be 2–3 times ) modes ( the required of a large-scale structure to ensure λ much smaller than min /\ps . On the second hand, more items of the Taylor expansion can be retained, which results in the second-order residual flexibility substructuring method, as described in the next section.

2.3.4 Second-Order Residual Flexibility Based Substructuring Method 2.3.4.1

Method Description

( )−1 If the first two items of the Taylor expansion of | Ts /\ps − λI | s are retained, Eq. (2.46) becomes |

/\pm − λI −| Tm

) ( ( ) −| m ( ) −2 −1 − | Ts /\ps | s + λ| Ts /\ps | s

|{

zm τ

} =

{ } 0 0

(2.51)

After arranging Eq. (2.51), the eigenequation can be expressed as the standard form like | | |{ } |{ } I 0 −| m /\pm zm zm ( p )−1 ( p )−2 (2.52) =λ T T T τ τ 0 | s /\s | s −| m −| s /\s | s where {

( )−1 ( )−1 | p |T T φ D | Ts /\ps | s = Dφps /\ps ( ) ( )−2 | sp |T T −2 φs D | Ts /\ps | s = Dφps /\ps

(2.53)

( )−2 | p |T φs is the second-order residual flexibility. Similar to the first-order φps /\ps residual flexibility, the primitive form of the second-order residual flexibility can

26

2 Substructuring Method for Eigensolutions

also be obtained by the diagonal assembly of the system matrices and master modes of the substructures as |(( ( )−2 | p |T )−1 ( )−1 ( (1) )−2 | (1) |T ) , φs =Diag K(1) M(1) K(1) φm φps /\ps − φ(1) m /\m (( )−1 ( )−1 ( (Ns ) )−2 | (Ns ) |T )| s) . . . , K(Ns ) M(Ns ) K(Ns ) /\m φm − φ(N m (2.54) Including first- and second-order residual flexibility, the eigensolutions of the global structure are calculated based on the eigenequation of Eq. (2.52). Since this second-order residual flexibility-based substructuring method (SRFS) includes the contribution of the second item in the Taylor expansion, SRFS method is more accurate than the FRFS method. The difference of the cost of computational load between the SRFS and FRFS lies in two aspects: (i) The SRFS method requires some additional effort to calculate the second-order ( )−2 | p |T φs residual flexibility matrix φps /\ps (ii) The eigenequation of SRFS method (Eq. (2.52)) contains the ‘misfit’ displacements at the interface points, which size is a little larger than that of the FRFS method. If the same quantity of master modes is retained by FRFS method and SRFS method, the SRFS method is more accurate and takes more computational sources. However, if the same precision is mandatory by FRFS method and SRFS method, the SRFS method requires less master modes. The computational work required by extracting the substructural eigenmodes is reduced, and the assembled eigenequation might be much smaller than that of the FRFS method.

2.3.4.2

Error Quantification

The error in the SRFS method is introduced by the truncation of Taylor expansion as well, which is evaluated as ⎛ )−1 ( )−1 ( )−2 ( p p p ⎜ ¯ − /\s − λ¯ /\s = Diag⎝ ( /\s − λI ⎛(

⎞ p

/\s

1 ) i

− λ¯

) (( ) )( ) (( ) )⎞ p 2 p p p /\ − /\s − λ¯ /\s − λ¯ /\s − λ¯ ⎜ s i ⎟ i i i = Diag⎝ ⎠ (( ) )( )2 p p ¯ /\s − λ /\s i i ⎛

⎜ = Diag⎝ ((

⎞

p

/\s

) i

2 λ¯

¯ − λI

The relative error is

)(

⎟

) ⎠ p 2 i

/\s

−(

1 p

/\s

λ¯

⎟ ) −( ) ⎠ p 2 /\s i i

(2.55)

2.3 Substructuring Method for Eigensolutions

⎛

27

⎞

⎛( )2 ⎞ ⎜ (( ) ) ( ) ⎟ ) ( ⎟ = Diag⎝ ( λp ) ⎠ i = 1, 2, . . . , Nsp Diag⎜ 1 ⎠ ⎝ /\s i (/\ps )i −λ 2

λ (

/\ps i −λ

2 /\ps i

)

( Therefore, the relative error of the SRFS method is dependent on

(2.56)

λ min(/\ps )

)2 ,

which is the square of the errors by the FRFS method. The minimum value of /\ps controls the accuracy of the SRFS method, similar to the FRFS method. Since the eigenvalue λ required by the global structure is far less than the minimum value of slave modes in substructures in /\ps , the error of the FRFS method is much smaller than that of the SRFS method, i.e., the SRFS method is more accurate than the FRFS method.

2.3.5 Residual Flexibility for Free Substructure In substructuring method, the global structure is divided into the free substructures and/or fixed substructures. The stiffness matrix of a free substructure is rank-deficient, from which the inverse of stiffness is inaccurate to be computed directly. In this section, the residual flexibility matrix is derived for the free structures based on the orthogonal property of the stiffness and modal flexibility matrices. The first-order residual flexibility matrix will be derived first, followed by the general formulae of the high-order residual flexibility matrices. The complete eigenmodes of a free substructure are divided into N m master modes φm and N s slave modes φs . It is noted that the master modes φm consist of the N r rigid body modes R and the (N m -N r ) deformational master modes φm−r . The deformational modes include the deformational master modes φm−r and deformational slave modes φs . The relation between master modes, slave modes, rigid body modes, and deformational modes is illustrated in Fig. 2.3.

Fig. 2.3 Relationship between different kinds of modal modes

28

2 Substructuring Method for Eigensolutions

The generalized stiffness matrix can be expressed by the rigid body modes, master modes, and slave modes as ( )−1 K + RRT =

NE m −Ns

i =1 φ ∈ φm

1 φi φiT + λi

Ns E

i =1 φ ∈ φs

1 φi φiT + RRT λi

−1 T T T = φm−r /\−1 m−r φm−r + φs /\s φs + RR

(2.57)

Therefore, the first-order residual flexibility matrix for the free structure can be expressed by the master modes as ) ( T T −1 T T φs /\−1 − φm−r /\−1 s φs = K + RR m−r φm−r − RR

(2.58)

Accordingly, the second-order residual flexibility matrix for the free structure is )( ) ( −1 T −1 T T φs /\−2 s φs = φs /\s φs φs /\s φs (( ) )−1 T T = K + RRT − φm−r /\−1 m−r φm−r − RR (( ) )−1 T T K + RRT − φm−r /\−1 m−r φm−r − RR ( )−2 T T = K + RRT − φm−r /\−2 m−r φm−r − RR

(2.59)

which leads to ( ) T T −2 T T φs /\−2 − φm−r /\−2 s φs = K + RR m−r φm−r − RR

(2.60)

The general formula for the k-order residual flexibility is derived as )( ) ( −1 T −(k−1) T T φs /\−k φs s φs = φs /\s φs φs /\s (( ) )−1 T T = K + RRT − φm−r /\−1 m−r φm−r − RR (( ) )−(k−1) T T K + RRT − φm−r /\−(k−1) φ − RR m−r m−r ) ( −k T −k T − φm−r /\m−r φm−r − RRT = K + RR

(2.61)

) ( T T −k T T φs /\−k − φm−r /\−k s φs = K + RR m−r φm−r − RR

(2.62)

i.e.,

If the mass-normalized eigenvectors are considered, the first-order residual flexibility matrix for the free structure is

2.4 Examples

29

) ( T T −1 T T φs /\−1 − φm−r /\−1 s φs = K + (MR)(MR) m−r φm−r − RR

(2.63)

The second-order residual flexibility matrix can be obtained with the master modes as ) ( ) ( −1 T −1 T T φs /\−2 s φs = φs /\s φs M φs /\s φs ( )−1 ( )−1 = K + (MR)(MR)T M K + (MR)(MR)T T T − φm−r /\−2 m−r φm−r − RR

(2.64)

Generally, the k-order residual flexibility is given by )( ( )) ( −1 T −1 T k−1 T φs /\−k s φs = φs /\s φs M φs /\s φs )−1 ( ( )−1 )k−1 ( M K + (MR)(MR)T = K + (MR)(MR)T T T − φm−r /\−k m−r φm−r − RR

(2.65)

2.4 Examples 2.4.1 A Three-Span Frame Structure The three-span frame (Fig. 2.5) is numerically modeled by 160 two-dimensional beam elements, each 2.5 m long. The model has 140 nodes and 408 DOFs in total. The material constants of the beam elements are bending rigidity (EI) = 170 × 106 Nm2 , axial rigidity (EA) = 2500 × 106 N, mass per unit length (ρA) = 110 kg/m, and Poisson’s ratio = 0.3. The elements are labeled in Fig. 2.5a. The frame is disassembled into three substructures at eight interface nodes, as shown in Fig. 2.5b. In consequence, the three substructures have 51, 55, and 42 nodes, respectively. After division at eight nodes, there are 48 interface DOFs (each node has three DOFs) with 24 shared ones. The first 20 eigensolutions of the global structure are calculated using five approaches. First, the entire structure is analyzed with the conventional Lanczos eigensolver, which is regarded as the exact results for comparison. Second, the frame is analyzed by the original Kron’s substructuring method, in which the complete eigensolutions of all substructures are calculated to assemble the primitive matrices. The primitive matrices /\p and φp of size 432 × 432 are solved by the standard Lanczos eigensolver. In this substructuring method, the complete modes of all substructures are included without any approximation during the whole process, the obtained eigensolutions are accurate but the computation will be inefficient. Third, the first 50 modes of each substructure are computed as the master modes, while the residual high modes are discarded directly without any compensation. The size of

30 Fig. 2.5 FEM of the three-span frame structure (Unit: m)

2 Substructuring Method for Eigensolutions

2.4 Examples

31

the eigenequation is 150 × 150. Fourth, the frame is analyzed by the proposed FRFS method. The first 50 modes in each substructure are chosen as master modes, and the higher modes are compensated by the first-order residual flexibility. The FRFS method is performed following the procedure of: (1) The global structure is divided into three substructures. The nodes and elements of each substructure are labeled individually by regarding it as an independent structure. (2) The first 50 eigensolutions of the three substructures are computed as the master modes, from which the first-order residual flexibility of each substructure is obtained. (3) The primitive form of the master eigensolutions /\pm and φpm is assembled from the master modes of the three substructures. The master eigensolutions /\pm and φpm have the size of 150 × 150 and 150 × 432, respectively. (4) The connection matrix D is established. There are eight interface points (each has 3 DOFs) and 24 connections to assemble the global structure. The values in D are 1 and − 1 at the connections, and zeros at other coordinates. Consequently, the connection matrix has a size of 24 × 432. (5) The matrix u is of order 150 × 150 according to Eq. (2.47), and the reduced eigenequation is solved with the standard Lanczos method to calculate the first 20 eigenpairs (λ and zm ). (6) The expanded eigenvectors are constructed by the superposition of substructural p master modes like φ = φm zm , and the eigenvectors of the global structure are recovered by merging the identical values at the interface coordinates of the expanded eigenvectors φ. Finally, the frame is analyzed with the SRFS method. As before, the first 50 modes in each substructure are computed as the master modes. The SRFS method calculates both the first-order residual flexibility and second-order residual flexibility to construct the eigenequation (Eq. 2.52), and the eigenequation has a size of 174 × 174 including the ‘misfit’ displacement. The first 20 frequencies of the global structure are compared in Table 2.1. ‘Lanczos’, ‘Original’, ‘Original-Partial’, ‘FRFS’, and ‘SRFS’ denote the aforementioned five methods. The second line of the table presents the computation time (in seconds) consumed by the central processing unit on a personal computer with 2 GB memory and 1.86 GHz Intel Core 2 Duo processor. Table 2.1 adopts the similarity indices and relative errors to evaluate the accuracy of the eigenvectors from different approaches. The SV value estimates the similarity of two sets of mode shapes { }) ( SV {φi }, φ˜ i =

| { }|2 | | |{φi }T φ˜ i | ({ } { }) T ) ( {φi }T {φi } φ˜ i φ˜ i

(2.66)

1.7843

1.7843

5.5365

1

2

16.6188

18.8074

25.4704

26.1799

28.0818

29.7843

30.9747

31.4907

33.5952

33.8409

16.6188

18.8074

20.1977

22.6170

25.4704

26.1799

28.0818

29.7843

30.9747

31.4907

32.3470

32.3574

33.5952

33.8409

5

6

7

8

9

10

11

12

13

14

15

16

17

18

9.8198

4

32.3574

32.3470

22.6170

20.1977

14.6864

9.8198

14.6864

3

5.5365

Freq (Hz)

Original

0.5640

Lanczos

0.1703

Freq (Hz)

Mode

CPU time (second)

32.3354

32.0649

31.0906

30.8539

30.1980

29.8720

28.5257

27.6134

27.0610

25.4569

25.0778

21.1509

19.8156

18.8166

14.5231

9.7959

5.5495

1.7898

Freq (Hz)

0.1671

0.693

0.379

0.635

0.410

0.303

1.632

1.047

3.494

6.533

2.184

12.328

7.214

5.997

13.396

0.415

0.582

0.539

0.341

Relative error (%)

Original-partial

33.8432

33.5979

32.3604

32.3492

31.4924

30.9789

29.7877

28.0839

26.1808

25.4753

22.6235

20.1979

18.8122

16.6223

14.6865

9.8199

5.5365

1.7843

Freq (Hz)

0.1978

FRFS

0.014

0.007

0.012

0.006

0.004

0.032

0.017

0.008

0.018

0.040

0.111

0.003

0.060

0.034

0.001

0.001

0.000

0.000

Relative error (%)

Table 2.1 Frequencies and mode shapes of the frame structure by different methods

0.034

0.019

0.023

0.007

0.009

0.069

0.043

0.016

0.058

0.099

0.236

0.006

0.130

0.081

0.002

0.003

0.000

0.000

(1–SV) (%)

0.039

0.067

0.047

0.047

0.020

0.141

0.109

0.062

0.019

0.085

0.029

0.018

0.018

0.000

0.003

0.006

0.000

0.000

Difference norm (%)

Mode shape error

SRFS

33.8409

33.5952

32.3574

32.3470

31.4907

30.9748

29.7843

28.0818

26.1799

25.4705

22.6172

20.1977

18.8075

16.6188

14.6864

9.8198

5.5365

1.7843

Freq (Hz)

0.2413

0.000

0.000

0.000

0.000

0.000

0.001

0.000

0.000

0.000

0.001

0.002

0.000

0.001

0.000

0.000

0.000

0.000

0.000

Relative error (%)

0.033

0.018

0.023

0.007

0.009

0.068

0.043

0.015

0.044

0.094

0.236

0.006

0.130

0.081

0.002

0.003

0.000

0.000

(1–SV) (%)

(continued)

0.033

0.066

0.039

0.046

0.019

0.139

0.108

0.062

0.011

0.040

0.028

0.018

0.021

0.000

0.002

0.005

0.000

0.000

Difference norm (%)

Mode shape error

32 2 Substructuring Method for Eigensolutions

34.4871

34.4871

34.5654

19

20

34.5654

Freq (Hz)

Freq (Hz)

Mode

33.1796

33.0881 1.036

0.760

Relative error (%)

0.1671

0.5640

0.1703

Freq (Hz)

Original-partial

Original

Lanczos

CPU time (second)

Table 2.1 (continued)

34.5690

34.4924

Freq (Hz)

0.1978

FRFS

0.025

0.015

Relative error (%)

0.051

0.037

(1–SV) (%) 0.048

0.085

Difference norm (%)

Mode shape error

34.5657

34.4873

Freq (Hz)

0.2413

SRFS

0.001

0.001

Relative error (%)

0.048

0.034

(1–SV) (%)

0.041

0.084

Difference norm (%)

Mode shape error

2.4 Examples 33

34

2 Substructuring Method for Eigensolutions

In Eq. (2.66), {φi } is the ith eigenvector accurately obtained on the global structure by the Lanczos method and is taken for the reference. {φ˜ i } represents the ith eigenvector by the other four types of substructuring methods. An SV value of identity means that the two vectors are identical, and the eigenvector from substructuring method is accurate. A value of 0 indicates that the two vectors are perpendicular. A higher SV value indicates a more accurate eigenvector. The second index estimates the relative difference of the eigenvectors in terms of

Difference Norm =

{ }) ( norm {φi } − φ˜ i norm({φi })

(2.67)

The comparison of several substructuring methods in Table 2.1 shows that, (1) As compared with the global Lanczos method, the original Kron’s substructuring method takes much longer time. Since the original Kron’s substructuring method includes the complete modes of all substructures, the size of eigenequation is enlarged, although the original Kron’s substructuring method is advantageous to possess the sparsity of system matrices (2) Discarding the slave mode directly without any compensation introduces a significant error in the eigensolutions. Since the substructures are constrained based on the principle of virtual work, completely discarding the energy contribution of the slave modes leads to inaccurate eigensolutions (3) The proposed substructuring method includes the contribution of higher modes via the residual flexibility and improves the accuracy of the eigenvalues effectively. The relative errors of the first 20 frequencies are less than 0.1% with the FRFS method and less than 0.002% with the SRFS method. The proposed method can achieve not only high-precision eigenvalues but also good eigenvector results. When the same quantity of master modes is retained, the SRFS method achieves a higher precision but costs a little more computational time and memory than the FRFS method. The accuracy is quite sufficient for usual engineering applications (4) As compared with the traditional Kron’s substructuring method, the proposed method calculates only 50 master modes of each substructure, without computing the complete modes of all substructure. This also reduces the size of eigenequation from 432 × 432 to 150 × 150, and thus further decreases the computational loads significantly. The proposed substructuring method takes a little longer time than the global Lanczos method. This is because the frame structure is small, and the Lanczos eigensolver is fast to perform on this small structure. On the other hand, the analysis of each substructure, especially the calculation of the residual flexibility, costs a small amount of computational effort. This makes the substructurebased eigensolutions consume more computational time than the global method, although the substructure-based eigenequation of size 150 × 150 is smaller than the global eigenequation of size 432 × 432. In addition, these interim results

2.4 Examples

35

Table 2.2 Size of the eigenequation with various methods Sub 1

Lanczos method

Original Kron’s method

FRFS

SRFS

–

153 × 153

50 × 50

50 × 50

Sub 2

–

165 × 165

50 × 50

50 × 50

Sub 3

–

142 × 142

50 × 50

50 × 50

Global structure

408 × 408

432 × 432

150 × 150

174 × 174

in substructures will be re-used in calculating the eigensensitivity and model updating in subsequent chapters. If the proposed substructuring method is applied to model updating or damage identification, the calculation of eigensolutions and sensitivity matrix is required for the concerned substructures only. Besides, the eigenequation size of the proposed method is much smaller than that of the global Lanczos method and the original Kron’s substructuring method, as listed in Table 2.2. This example indicates that the improvement in the substructuring method can reduce the computational load significantly while maintaining high precision. Although the SRFS method has a better precision, the second-order residual flexibility is required to be computed and the eigenequation is more complicated. Since the FRFS method can satisfy most engineering applications and cost much less computational resources, the FRFS method might be preferable in practice due to its simple formulation. In the next example, the FRFS method will be utilized in a practical bridge.

2.4.2 The Balla Balla River Bridge The Balla Balla River Bridge (Fig. 2.6) is a three-span continuous reinforced concrete bridge located on Coastal Highway on the Balla Balla River in the Shire of Roebourne, Western Australia. The FEM is constructed according to the design drawings, which is comprised of 907 elements, 947 nodes, and 5420 DOFs in total, as shown in Fig. 2.7. The beam elements and shell elements used in the model are listed in Table 2.3. The analytical model of the Balla Balla River Bridge is analyzed to study the computational efficiency of the substructuring method in the calculation of eigensolutions. The different master modes and division formation of the substructures are used to investigate their influence on the computation accuracy and efficiency. The global model is firstly divided into five substructures at around 10, 20, 30, and 40 m along the longitudinal direction, as shown in Fig. 2.7. The information on the location, elements, and nodes of the five substructures is listed in Table 2.4. In this example, the FRFS method is utilized to calculate the first 20 eigensolutions of the global structure, where the first 40 modes in each substructure chosen as the

36

2 Substructuring Method for Eigensolutions

Fig. 2.6 General view of Balla Balla River Bridge

Fig. 2.7 FEM of the Balla Balla River Bridge with five substructures Table 2.3 FEM information of Balla Balla River Bridge

Bridge component

Element type

Quantity

Bearing

Beam

56

Slab

Shell

288

Girder

Shell

252

Stirrup

Beam

231

Diaphragm

Shell

Total

80 907

2.4 Examples

37

Table 2.4 Division formation with five substructures Index of substructures

Sub 1

Sub 2

Sub 3

Sub 4

Sub 5

Geometric range (m)a

0–10

10–20

20–30

30–40

40–54

No. of elements

187

182

132

182

224

No. of nodes

205

212

161

212

251

No. of DOFs

1095

1260

966

1260

1371

23

No. of interface nodes

23

23

23

Note a In the longitudinal direction

master modes. The frequencies are listed in Table 2.5, together with the relative errors compared with the exact results using the global Lanczos method. Table 2.5 Frequencies and computational time of the Balla Balla Bridge with different master modes retained CPU time (s)

Exact

Original

40 master modes

60 master modes

90 master modes

8.0253

238.8509

10.3725

10.9643

13.0231

Mode Frequency Frequency Frequency Relative Frequency Relative Frequency Relative index (Hz) (Hz) (Hz) error (%) (Hz) error (%) (Hz) error (%) 1

5.8232

5.8232

5.8288

0.097

5.8281

0.084

5.8269

0.063

2

5.9998

5.9998

6.0028

0.051

6.0028

0.051

6.0028

0.051

3

6.0007

6.0007

6.0038

0.052

6.0038

0.051

6.0037

0.051

4

6.2635

6.2635

6.2691

0.089

6.2677

0.066

6.2669

0.053

5

6.8621

6.8621

6.8656

0.051

6.8655

0.051

6.8655

0.051

6

6.8987

6.8987

6.9023

0.052

6.9023

0.052

6.9022

0.051

7

6.9975

6.9975

7.0034

0.084

7.0022

0.067

7.0012

0.052

8

7.7391

7.7391

7.7465

0.095

7.7449

0.075

7.7432

0.053

9

8.6063

8.6063

8.6142

0.092

8.6128

0.075

8.6109

0.053

10

8.7145

8.7145

8.7205

0.069

8.7197

0.059

8.7191

0.052

11

9.4460

9.4460

9.4535

0.079

9.4525

0.068

9.4510

0.053

12

10.9814

10.9814

10.9870

0.051

10.9870

0.051

10.9870

0.051

13

10.9816

10.9816

10.9872

0.051

10.9872

0.051

10.9872

0.051

14

12.1302

12.1302

12.1511

0.172

12.1417

0.094

12.1375

0.059

15

13.0048

13.0048

13.0227

0.137

13.0167

0.091

13.0122

0.057

16

13.2693

13.2693

13.2868

0.132

13.2810

0.088

13.2767

0.056

17

14.9312

14.9312

14.9431

0.080

14.9421

0.073

14.9399

0.058

18

15.8194

15.8194

15.8880

0.434

15.8610

0.263

15.8337

0.090

19

16.9266

16.9266

16.9515

0.147

16.9463

0.116

16.9370

0.062

20

17.5480

17.5480

17.6043

0.321

17.5646

0.095

17.5602

0.070

38

2 Substructuring Method for Eigensolutions

The computational accuracy is certainly influenced by the master modes retained in each substructure. 40 modes, 60 modes, and 90 modes in each substructure are chosen as ‘master’, to investigate the effect of the number of master modes on the computational accuracy. The obtained 20 eigensolutions and the corresponding errors are compared in Table 2.5. The relative errors of the frequencies and the SV values of the eigenvectors are compared in Figs. 2.8 and 2.9, respectively. The accuracy of the frequencies and eigenvectors is improved when more master modes are included in each substructure, especially for the higher modes. At the same time, the computational time increases as more master modes are included. This is because including more master modes not only increases the computational endeavor on extracting the substructural master modes but also enlarges the size of the eigenequation. The eigenequation sizes resulting from the 40 master modes, 60 master modes, and 90 master modes are 200 × 200, 300 × 300, and 450 × 450, respectively. The number of master modes required in each substructure depends on the accuracy requirement. Based on the error analysis described previously, it is suggested to make the minimum value of /\ps as large as possible. The master modes can be chosen -2

Relative errors (Log)

10

-4

10

-6

10

-8

10

40 master modes 60 master modes 90 master modes

-10

10

0

1

2

3

4

5

6

7

8

9

10

14

13

12

11

15

16

17

18

19

Mode

Fig. 2.8 Accuracy of frequencies with different master modes

Similarity of eigenvector

1

0.998

0.996

0.994

40 master modes 60 master modes 90 master modes

0.992

0.99 0

1

2

3

4

5

6

7

8

9

10

11

12

13

Mode

Fig. 2.9 Accuracy of eigenvectors with different master modes

14

15

16

17

18

19

20

20

2.4 Examples

39

by each substructure uniformly or by a predefined boundary. Sturm’s sequence check can be employed to determine the number of eigenvalues smaller than a specified value, i.e., all the smallest eigenvalue modes among all substructures are selected as the master modes. In this study, 940,000 are selected as the threshold value for Sturm’s sequence check. Accordingly, there are 61, 87, 76, 92, and 84 master modes, whose eigenvalues are smaller than 940,000 in the five substructures. Figure 2.10 compares the relative errors of eigensolutions from equally selecting 80 master modes in each substructure and using Sturm’s sequence check. For the lower eigenmodes, there is almost no discrepancy between the two selecting strategies. Sturm’s sequence checks is helpful to choose the master modes with lowest eigensolutions, and selecting master modes with Sturm’s sequence check has a slightly better result than selecting master modes from each substructure equally, especially when the global structure deserves higher modes. Since the substructures are usually similarly divided, the eigenvalues in the substructures are comparable. Equally selecting master modes from each substructure is a convenient practice. On the other hand, the accuracy and efficiency are also affected by the substructural division formation. It is better to reduce the interface joints for a smaller transformation matrix D. From a practical point of view, dividing a building across the columns is better than through the slabs, and dividing a bridge across the slab is better than across the piers. Dividing a structure into too many substructures or few substructures might not be preferable. One needs to trade off the number of substructures and the size of each

Fig. 2.10 Computation accuracy including different master modes

40

2 Substructuring Method for Eigensolutions

Table 2.6 Different division formation with three substructures Index of substructures

Sub 1

Sub 2

Sub 3

Geometric range (m)a

0–16.5

16.5–34.5

34.5–54

No. of elements

275

292

340

No. of nodes

394

327

371

No. of interface nodes Note

a In

23

23

the longitudinal direction

substructure. To investigate the influence of the division formation of the substructures, four division strategies are considered, and the bridge is divided into 3, 5, 8, and 11 substructures. The detailed substructure information about these division formations is given in Tables 2.6, 2.7 and 2.8. In the first scheme, the first 80 modes in each substructure are chosen as master modes to calculate the first 40 eigensolutions of the global structure. The master modes in each substructure, the size of the reduced eigenequation, and the corresponding CPU time are recorded in Table 2.9. The relative errors of the frequencies are compared in Fig. 2.11. Except the division of three substructures, the other three division formations achieve similar accuracy, and the division of more substructures leads to slightly better accuracy. If retaining the same number of master modes in each substructure, dividing the global structure into more substructures definitely improves the accuracy since more master modes are included. The division of three substructures and 11 substructures cost more computational time than that of five substructures and eight substructures, as listed in Table 2.9. If a structure is divided into insufficient substructures, the independent substructure has a large size as well. The calculation of the master eigensolutions and the residual flexibility matrix of each substructure with a large number of elements and nodes will cost supplementary CPU resources. On the other hand, when the global structure is divided into a large number of substructures, numerous substructures need to be analyzed. This not only increases the computation for the analysis of numerous substructures, but also enlarge the quantity of interface coordinate. In consequence, the global eigenequation has a larger size. Among the four division formations, dividing the global structure into five substructures can not only reach high precision but also save computational resources. Dividing the global structure into excessive substructures or too few ones is both inefficient. In the second scheme, around 400 master modes of all substructures are selected. This scheme makes the reduced eigenequation having a similar size but the divided substructures having different master modes. Table 2.9 compares the CPU time for calculating the first 40 eigensolutions of the global structure with the four division formations. The relative errors of the frequencies computed from Scheme 2 are compared in Fig. 2.12. If the total number of the master modes among all substructures is similar, the division of more substructure will have fewer master modes in each substructure. The division with more substructures results in lower precision, since fewer master

143

159

No. of elements

No. of nodes

Note

a In

the longitudinal direction

No. of interface nodes

Sub 1

0–7

Substructure index

Geometric range (m)a

23

138

110

7–14

Sub 2

23

Table 2.7 Division formation with eight substructures Sub 3

143

116

14–21

23

Sub 4

115

88

21–27

23

Sub 5

138

110

27–34

23

Sub 6

143

116

35–41

23

Sub 7

115

88

41–47

23

Sub 8

159

136

47–54

2.4 Examples 41

23

115

No. of nodes 113

Note a In the longitudinal direction

No. of interface nodes

88

99

No. of elements

5–10

0–5

Sub 2

Geometric range (m)a

Substructure Sub 1 index

23

92

66

10–15

Sub 3

23

Table 2.8 Division formation with 11 substructures

143

116

15–20

Sub 4

23

92

66

20–25

Sub 5

23

92

66

25–30

Sub 6

23

92

66

30–35

Sub 7

23

143

116

35–40

Sub 8

23

92

66

40–45

Sub 9

23

92

66

45–50

Sub 10

23

113

99

50–54

Sub 11

42 2 Substructuring Method for Eigensolutions

2.4 Examples

43

Table 2.9 Matrix size and computational time with different division formations No. substructures No. of master modes in each substructure

Scheme 1

Scheme 2

3

5

8

11

80

80

80

80

Size of the global eigenequation

240

400

640

880

CPU time (second)

20.8

12.8

16.7

26.1

No. of master modes in each substructure

133

80

50

37

Size of the global eigenequation

399

400

400

407

CPU time (second)

24.9

12.8

13.4

22.7

-1

10

Relative error(Log)

3 substructures 5 substructures 8 substructures 11 substructures -2

10

-3

10

-4

10

0

5

10

15

20

25

30

35

40

Modes

Fig. 2.11 Relative errors of frequencies with various substructure division formations (Scheme 1)

( ) modes in each substructure decrease the minimum value of slave modes in min /\ps . By contrast, the division of fewer substructures includes more master modes in each substructure, and thus achieves higher accuracy. Nevertheless, the division of fewer substructures increases the CPU time in calculating the eigensolutions and the residual flexibility matrix for the large-size substructures. When applying the substructuring method in model updating, the calculation of the sensitivity matrix in each substructure will be heavier, and the substructuring technology may lose its promising advantages. In practice, division of substructures needs to a few trials to determine the optimal number of substructures before model updating is performed. In the present study, if the bridge is divided into five substructure and 40 master modes are used in each substructure (Table 2.10), the eigenequation size is reduced from 5400 × 5400 to 200 × 200 with the proposed FRFS method. This will benefit

44

2 Substructuring Method for Eigensolutions -1

10

Relative error(Log)

3 substructures 5 substructures 8 substructures 11 substructures -2

10

-3

10

-4

10

0

5

10

20

15

25

30

35

40

Modes

Fig. 2.12 Relative errors of frequencies with various substructure division formations (Scheme 2)

Table 2.10 Size of eigenequation with various master modes Lanczos Each substructure Global structure

5400 × 5400

40 master modes

60 master modes

90 master modes

40 × 40

60 × 60

90 × 90

200 × 200

300 × 300

450 × 450

the model updating process, in which the eigenequation is repeatedly constructed and solved.

2.5 Summary In this chapter, a substructuring method is proposed to calculate the eigensolutions of a large-scale structure. The original Kron’s substructuring method is improved by a modal truncation approximation to reduce the computational load of. Only a few eigensolutions of each substructure are retained to assemble the global eigenequation, while the higher modes are compensated with the first- and second-order residual flexibilities. The SRFS method achieves better results than that of the FRFS method, whereas it increases the computational effort in computing the second-order residual flexibility and enlarge the eigenequation. The division formation needs to trade off the number of master modes in each substructure and the number of substructures. Including more master modes will improve the accuracy but decrease the efficiency.

References

45

Divisions with much excessive or an insufficient number of substructures are both undesirable.

References Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, New Jersey (1982) Craig, R.R., Bampton, M.M.C.: Coupling of substructures for dynamic analysis. AIAA J. 6(7), 1313–1319 (1968) Felippa, C.A., Park, K.C., Filho, M.R.J.: The construction of free-free flexibility matrices as generalized stiffness inverses. Comput. Struct. 68, 411–418 (1998) Fox, R.L., Kapoor, M.P.: Rate of change of eigenvalues and eigenvectors. AIAA J. 6(12), 2426–2429 (1968) Hurty, W.C.: Dynamic analysis of structural systems using component modes. AIAA J. 3(4), 678– 685 (1965) Klerk, D.D., Rixen, D.J., Voormeeren, S.N.: General framework for dynamic substructuring: history, review, and classification of techniques. AIAA J. 46(5), 1169–1181 (2008) Kron, G.: Diakoptics. Macdonald and Co., London (1963) MacNeal, R.H.: A hybrid method of component mode synthesis. Comput. Struct. 1, 581–601 (1971) Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14, 1201–1205 (1976) Rubin, S.: Improved component-mode representation for structural dynamic analysis. AIAA J. 8(8), 995–1005 (1975) Rubin, S.: Improved component mode representation for structural dynamic analysis. AIAA J. 13(8), 995–1006 (2012) Weng, S., Xia, Y., Xu, Y.L., et al.: Improved substructuring method for eigensolutions of large-scale structures. J. Sound Vib. 323, 718–736 (2009) Xia, Y., Hao, H., Deeks, A.J., Zhu, X.: Condition assessment of shear connectors in slab-girder bridges via vibration measurements. J. Bridg. Eng. 13(1), 43–54 (2008)

Chapter 3

Substructuring Method for Eigensensitivity

3.1 Preview Eigensensitivity is the derivative of eigensolutions (eigenvalues and eigenvectors) with respect to the elemental parameters. It is usually used to estimate the most sensitive parameters that contribute to the objective function, and thus accelerate the convergence of the optimization process. In model updating, the eigensensitivity needs to be calculated with respect to the elemental parameters one by one repeatedly. The cost of calculating the eigensensitivity is the dominant contributor to model updating and many optimization procedures. Since the practical engineering structures nowadays are usually large-scale in nature, efficient computation of the eigensensitivity with respect to various design parameters is a critical requirement in the model updating analysis. The substructuring method benefits to accelerate the calculation of eigensensitivity from the large-size system matrices. As illustrated in Chap. 2, the substructures are analyzed independently, which are assembled to compute the eigensolutions of the global structure by imposing constraints at the interfaces of the adjacent substructures. Since the substructures are independent, the eigensensitivity with respect to one parameter is related to one substructure only. The substructural derivative matrices are computed in one substructure, while those in other substructures are zeros. The eigensensitivity of the global structure is recovered by the substructural derivative matrices with most of zeros. Since the substructure has a much smaller size than the entire structure, the computation efficiency is improved significantly. In this chapter, the eigensensitivity equation is derived based on the reduced eigenequation of the substructuring method developed in Chap. 2. Only the derivatives of the master modes of the substructures are calculated, and the derivatives of the discarded slave modes are compensated by the derivative of residual flexibility. In consequence, the derivatives of the master modes and residual flexibility are assembled to compute the eigensensitivity of the global structure. The first-order

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_3

47

48

3 Substructuring Method for Eigensensitivity

derivatives of eigensolutions are formulated first, and the second-order derivatives and the general high-order derivatives will be introduced in the next chapter.

3.2 Basic Methods for Eigensensitivity 3.2.1 Eigenvalue Derivatives Fox and Kapoor (1968) first derived the eigenvalue derivatives with respect to elemental parameters by the modal superposition method. The elemental parameters can be the physical parameter of an element, such as the stiffness, density, and thickness. In this book, the flexural rigidity of an element is chosen as the elemental parameter, which is denoted by r. By differentiating Eq. (2.1) with respect to r, the equation of the eigensensitivity is expressed as ( ) ∂λi ∂M ∂K ∂{φi } {φi } = {0} + − M − λi + (K − λi M) ∂r ∂r ∂r ∂r

(3.1)

where i = 1, 2, …, m, and m is the number of the interest eigenpairs of the global T structure.{ Since } the eigenpairs of a structure satisfy {φi } (K − λi M) = {0}, premulT tiplying φi on both sides of Eq. (3.1) leads to the first-order eigenvalue derivative with respect to r ( ) ∂M ∂K ∂λi T } {φ {φi } −λi = i + ∂r ∂r ∂r

(3.2)

According to the basic finite element method, the stiffness matrix K is assembled from the discrete elemental stiffness matrices of all n elements, and mass matrix M by the discrete element stiffness matrices of all n elements, i.e., K=

n E j=1

Kj =

n E j=1

α j Kej , M =

n E

Mj =

j=1

n E

β j Mej

(3.3)

j=1

where Kj and Mj are respectively the jth element stiffness matrix and element mass matrix, and α j and β j are respectively the ‘elemental stiffness parameter’ and ‘elemental mass parameter’. If the designated parameter r is an elemental stiffness parameter, for example, Young’s modules, Eq. (3.2) is further simplified to ∂λi = {φi }T Kej {φi } ∂r j

(3.4)

Likewise, if r is an element mass parameter, for example, the density, the eigenvalue derivative is formed as

3.2 Basic Methods for Eigensensitivity

49

∂λi = −λi {φi }T Mej {φi } ∂r j

(3.5)

3.2.2 Eigenvector Derivatives Once the eigenvalue derivative is achieved, Eq. (3.1) can be re-arranged in terms of the eigenvector derivative as ui

∂{φi } = {Yir } ∂r

(3.6)

where {Yir } =

∂λi M{φi } − ∂r

(

) ∂K ∂M {φi } − λi ∂r ∂r

u i = K − λi M i} can be solved with the modal method (Fox and The eigenvector derivative ∂{φ ∂r Kapoor 1968) or Nelson’s method (Nelson 1976).

3.2.2.1

Modal Method for Eigenvector Derivatives

In the modal method, the\ eigenvector derivative is expressed as a linear summation | of all eigenvectors o = φ1 φ2 · · · φ N like ∂{φi } E { } = φ j c ji = o{ci } ∂r j=1 N

(3.7)

}T { where ci = c1i c2i · · · c N i is the participation factors of the orthogonal modes. Substituting Eq. (3.7) into Eq. (3.6) and premultiplying oT on both sides gives oT (K − λi M ) o {ci } = oT {Yi }

(3.8)

Due to the orthogonal condition of oT Mφ = I and oT Kφ = /\, Eq. (3.8) is simplified to (/\ − λi I)ci = oT {Yi } Decoupling Eq. (3.9) to N − 1 equations of

(3.9)

50

3 Substructuring Method for Eigensensitivity

( ) { } λ j − λi c ji = φ Tj {Yi }

(3.10)

where j = 1, 2, …, N but j /= i. Therefore, the coefficient of {ci } except cii can be solved by

c ji =

{ } φ Tj {Yi } λ j − λi

(3.11)

In the case of j = i, the coefficient cii can be solved from the mass orthogonal condition of the eigenvectors, which satisfies {

} φiT M {φi } = 1

(3.12)

Equation (3.12) is differentiated with respect to r as { } ∂{φi } { T } ∂M {φi } = 0 + φi 2 φiT M ∂r ∂r

(3.13)

Substituting Eq. (3.7) into Eq. (3.13) and noticing that {

| { } \ } φiT M o {ci } = φiT M {φ1 } · · · {φi−1 } {φi } {φi+1 } · · · {ci } = cii

(3.14)

cii is obtained as cii = −

1 { T } ∂M {φi } φ 2 i ∂r

(3.15)

Equation (3.7) indicates that all N eigenvectors are required to calculate the firstorder derivative of the ith eigenvector.

3.2.2.2

Nelson’s Method for Eigenvector Derivatives

Nelson’s method expresses the ith eigenvector derivative in terms of the sum of the particular and homogeneous vectors as ∂{φi } = {vi } + ci {φi } ∂r

(3.16)

where ci is the participation factor. Substituting Eq. (3.16) into Eq. (3.6) gives (K − λi M) {vi } = {Yi }

(3.17)

If there are no repeated frequencies, the rank of (K − λi M) is (N − 1). The above equation is rank deficient. To solve this rank-deficient equation, the kth item of {vi }

3.3 Substructuring Method for Eigensensitivity

51

is set to zero, and the kth row and column of (K − λi M) and kth item of {Yi } are removed. The full-rank equation in solving {vi } is thus established as ⎫ ⎤⎧ ⎫ ⎧ (K − λi M)11 0 (K − λi M)13 ⎨ vi1 ⎬ ⎨ Yi1 ⎬ ⎣ ⎦ vik = 0 1 0 0 ⎩ ⎭ ⎩ ⎭ vi3 Yi3 (K − λi M)31 0 (K − λi M)33 ⎡

(3.18)

where the kth item(referred to as the pivot) is usually chosen at the DOF corresponding to the maximum value in {φ i }. Subsequently, Eq. (3.17) is substituted into the orthogonal equation in Eq. (3.13) leads to { } { } ∂M {φi } = 0 2 φiT M({vi } + ci {φi }) + φiT ∂r

(3.19)

Therefore, the participation factor ci can be computed by { } 1 { } ∂M {φi } ci = − φiT M {vi } − φiT 2 ∂r

(3.20)

Once the vector {vi } and the participation factor {ci } are computed, the eigeni} can be obtained. The advantage of Nelson’s method over the vector derivative ∂{φ ∂r modal method is that the former needs one eigenvector only.

3.3 Substructuring Method for Eigensensitivity 3.3.1 Eigenvalue Derivatives The reduced eigenequation (Eq. 2.47) is rewritten for the ith mode as ( ) u − λi I {zi } = 0 ( ( ) )−1 −1 u = /\pm + | m | Ts /\ps | s | Tm

(3.21)

The eigenvalue λi and eigenvector {zi } have been solved in Chap. 2. Equation (3.21) is differentiated with respect to an elemental parameter r as follows ) ( } { ( ) ∂zi ∂ u − λi I {zi } = {0} + u − λi I ∂r ∂r

(3.22)

Premultiplying {zi }T on both sides of Eq. (3.22) gives the following equation.

52

3 Substructuring Method for Eigensensitivity

) ( } { ( ) ∂zi T ∂ u − λi I {zi } u − λi I {zi } = 0 + {zi } ∂r ∂r T

(3.23)

Since u − λi I is symmetric and u − λi I=0, the first item on the left-hand side of Eq. (3.23) is zero. In consequence, Eq. (3.23) is arranged, and the derivative of eigenvalue λi with respect to r is ∂λi ∂u {zi } = {zi }T ∂r ∂r

(3.24)

where ∂u = ∂r

| ( ( ) )−1 | p p −1 T ∂ /\m + | m | s /\s | s | Tm ∂r

∂/\pm ∂| m ( T ( p )−1 )−1 T + | s /\s | s |m + |m ∂r ∂r ( ( ) )−1 ∂| T −1 m + | m | Ts /\ps | s ∂r

=

| |( ( p )−1 )−1 T ∂ | s /\s | s ∂r

| Tm (3.25)

∂/\pm ∂r

is the diagonal gathering of the eigenvalue derivatives of the T ∂ op master modes of all substructures, ∂|m = [ m ] DT is associated with the diagonal In Eq. (3.25),

∂r

∂r

∂

|( )−1 | −1 | Ts (/\ps ) | s

assembly of the master eigenvector derivatives of all substructures. ∂r is associated with the derivative matrices of the first-order residual flexibility of all substructures, which can be obtained from the derivative of master modes as )−1 ( ( ) −1 ∂ | Ts /\ps | s ∂r where ) ( ( ) −1 ∂ | Ts /\ps | s ∂r

) ( ( ) −1 ( ( ) )−1 ∂ | Ts /\ps | s ( ( ) )−1 −1 −1 = − | Ts /\ps | s | Ts /\ps | s ∂r (3.26)

=D

( ( ) \ | ) −1 T ∂ ops /\ps ops

DT ∂r( ⎡ ( )−1 ( )−1 \ ( j) |T ) ⎤ ∂ K( j) om − o(mj ) /\(mj) ⎦ × DT = D × Diag⎣ ∂r (3.27)

3.3 Substructuring Method for Eigensensitivity

53

For the free substructures, the first-order residual flexibility with respect to an elemental parameter r is demonstrated in Sect. 3.3.3. Since the substructures are independent, the eigensensitivity with respect to r is related to one substructure that contains r (for example, the Qth substructure). In consequence, the derivative matrices of the eigenvalues, the eigenvectors, and the residual flexibility are nonzero and calculated only in the Qth substructure. These entries in the other substructures are zero matrices, i.e., ⎡ ⎡ ⎤ ⎤ ( p )−1 \ p |T 0 0 0 0 0 0 p p p os ∂/\m ⎢ ∂/\(Q) ⎢ ∂φ(Q) ⎥ ∂om ⎥ ∂os /\s m m = ⎣ 0 ∂r 0 ⎦, = ⎣ 0 ∂r 0 ⎦, ∂r ∂r ∂r 0 0 0 0 0 0 ⎡ ⎤ 0 0 0 (Q) −1 (Q) (Q) T ⎢ ⎥ = ⎣ 0 ∂os (/\s ) [os ] 0 ⎦ (3.28) ∂r

0

0

0

The substructural eigenvalue derivatives and eigenvector derivatives can be obtained using traditional Nelson’s method in Sect. 3.2.2.2, by treating the Qth substructure as an independent structure. The eigenvalue derivatives of the global structure with respect to the elemental parameter rely solely on the reduced eigenequation and one particular substructure, not on other substructures. As the size of a substructure is much smaller than the global structure, the computational efficiency in calculating the eigenvalue derivatives is improved. This feature has a significant advantage when the substructuring method is applied to the iterative model updating process because only the modified substructures are required to be re-analyzed and other substructures remain untouched.

3.3.2 Eigenvector Derivatives The ith eigenvector of the global structure can be recovered from the substructural master modes by φi = opm {zi }

(3.29)

Differentiating the above equation with respect to the structural parameter r leads to { } ∂opm ∂φi p ∂zi {zi } + om = ∂r ∂r ∂r

(3.30)

54

3 Substructuring Method for Eigensensitivity ∂op

where opm represents the master eigenvectors of all substructures, and ∂rm is the eigenvector derivatives of the substructural master modes, which is nonzero only in ∂op the Qth substructure. ∂rm can be computed using Nelson’s method by treating the Qth substructure as an{ independent structure. {zi } is the eigenvector of the reduced } i is computed, the eigenvector derivative of the ith mode eigenequation. After ∂z ∂r of the global structure can be obtained. { i} Similar to Nelson’s method, ∂z is separated into the sum of a particular part ∂r and a homogeneous part as {

∂zi ∂r

} = {vi } + ci {zi }

(3.31)

where ci is a participation { i }factor. Substituting Eq. (3.31) into Eq. (3.22) gives the derivative equation of ∂z ∂r ) ( ( ) ∂ u − λi I {zi } u − λi I ({vi } + ci {zi }) = − ∂r

(3.32)

( {zi } is) the eigenvector of the reduced eigenequation, which satisfies u − λi I {zi } = {0}. Equation (3.32) can be simplified to ) ( u − λi I {vi } = {Yi }

(3.33)

and ) ( ∂ u − λi I {zi } {Yi } = − ∂r u and {Yi } have been computed during the calculation of the eigenvalue derivatives in the previous section. If there are no repeated frequencies, the reduced system matrix u takes size N pm and rank (N pm –1). By setting the kth row and column of u and kth item of {Yi } to zero, the following full-rank equation is established to solve {vi }. ⎫ ⎤⎧ ⎫ ⎧ u 11 0 u 13 ⎨ vi1 ⎬ ⎨ Yi1 ⎬ ⎣ 0 1 0 ⎦ vik = 0 ⎩ ⎭ ⎩ ⎭ vi3 u 31 0 u 33 Yi3 ⎡

(3.34)

where the pivot, k, is chosen at the maximum entry in {zi }. In consequence, {vi } can be solved from Eq. (3.34). The eigenvector of the reduced eigenequation satisfies the orthogonal condition of: {zi }T {zi } = 1

(3.35)

3.3 Substructuring Method for Eigensensitivity

55

Differentiating Eq. (3.35) with respect to r gives ∂{zi } ∂{zi }T {zi } + {zi }T =0 ∂r ∂r

(3.36)

Substituting Eq. (3.31) into Eq. (3.36) results in ( ) {vi }T + ci {zi }T {zi } + {zi }T ({vi } + ci {zi }) = 0

(3.37)

The participation factor ci is thus obtained as ci = −

) 1( {vi }T {zi } + {zi }T {vi } 2

(3.38)

Finally, the first-order derivative of {zi } with respect to r is {

∂zi ∂r

} = {vi } −

) 1( {vi }T {zi } + {zi }T {vi } {zi } 2

(3.39)

As far as Eq. (3.30) is concerned, the eigenvector derivative of the global structure ∂op is spanned by the eigenvectors opm and eigenvector derivatives ∂rm of the substruc{ i} ∂op and z as the weights. Since ∂rm has been computed in the eigenvalue tures with ∂z ∂r derivatives, the eigenvector derivatives of the global structure are achieved quickly through Nelson’s transformation on the reduced eigenequation. This is a significant merit of the substructuring method since the calculation of the eigensensitivity dominates computational resources in the model updating process.

3.3.3 Derivative of Residual Flexibility Differentiating the first-order residual flexibility (Eq. 2.58) with respect to an elemental parameter r, the derivative matrix has the form of ) )−1 ) ( ( ( T T ∂ os /\−1 ∂ K + RRT ∂ om-r /\−1 s os m-r om-r = − ∂r ∂r ∂r ) ( T −2 ∂K = − K + RR ∂r ) ( ( T ) ( −1 ) ∂ o ∂ /\ m-r m-r + om-r oTm-r − 2om-r /\−1 m-r ∂r ∂r

(3.40)

Accordingly, the second-order residual flexibility (Eq. 2.60) is differentiated as

56

3 Substructuring Method for Eigensensitivity

)−2 ) ) ( ( ( T T ∂ os /\−2 ∂ K + RRT ∂ om-r /\−2 s os m-r om-r = − ∂r ∂r ∂r ) ( T −3 ∂K = −2 K + RR ∂r ( ) ( T ) ( −2 ) ∂ o ∂ /\ m-r m-r − 2om-r /\−2 + om-r oTm-r m-r ∂r ∂r

(3.41)

In general, the kth-order residual flexibility (Eq. 2.62) has the derivative matrix ( ) )−k ) ( ( T T ∂ os /\−k ∂ K + RRT ∂ om-r /\−k s os m-r om-r = − ∂r ∂r ∂r ) ( T −(k+1) ∂K = − k K + RR ∂r ( ) ) ( T ∂ /\−k m-r −k ∂om-r T + om-r om-r − 2om-r /\m-r ∂r ∂r

(3.42)

If the mass-normalized eigenmodes are involved, the derivative matrix of the kth-order residual flexibility has the form of: ( ) ( ) ( ( ) )k−1 T −1 T −1 ( ) ∂ K + (MR)(MR) M K + (MR)(MR) T ∂ os /\−k s os = ∂r ∂r ( ) ) ( T ) ( ∂ o ∂ /\−k m-r m-r −k T + om-r om-r − 2om-r /\m-r (3.43) ∂r ∂r

3.4 Examples 3.4.1 The Three-Span Frame Structure The three-span frame in Fig. 2.5 described in Chap. 2 serves to illustrate the procedure using the proposed substructuring method to calculate the eigensensitivity. As shown in Fig. 3.1, the flexural rigidity of Element 103 in Substructure 2, denoted as r 1 in the figure, is chosen as the design parameter. The first 30 modes of each substructure are retained as the master modes to calculate the eigensensitivity of the first 10 modes of the global structure with respect to r 1 . The influence of the master modes on computational accuracy and efficiency will be also investigated later. The substructure-based eigensensitivity can be obtained from the following steps.

3.4 Examples

57

Fig. 3.1 Three-span frame and the designated parameter r 1 (2) (3) (1) (2) (3) (1) The eigensolutions of each substructure, /\(1) m , /\m , /\m , om , om , and om (m = 1, 2, …, 30), are calculated. The eigensolutions of the global structure are computed with the reduced eigenequation Eq. (3.21) as λi and {zi }, and the eigenvector of the global structure is recovered through φi = opm {zi } (i = 1, 2, …, 10) (2) Parameter r 1 is located in Substructure 2. The eigenvalue and eigenvector derivatives of the first 30 modes of Substructure 2 with respect to param∂/\(2) ∂o(2) eter r 1 ( ∂rm1 , ∂rm1 ) are calculated, from which the derivative of the residual flexibility with respect to r 1 is calculated within Substructure 2, namely ) ( (2) −1 (2) T ∂ o(2) s (/\s ) [os ] . Since Substructure 2 is free after partition, the rigid body ∂r1

58

3 Substructuring Method for Eigensensitivity

modes have to be included to calculate the derivative of the residual flexibility matrix as described in Eq. (3.40) (3) The derivatives of the eigensolutions and residual flexibility of the other )two ( −1 T ( j) ( j) ( j) ∂ o (/\(s j ) ) [o(s j ) ] s ∂o ∂/\ = substructures are set to zeros, i.e., ∂rm1 = [0], ∂rm1 = [0], ∂r1 [0], (j = 1, 3), and the primitive form of the derivative matrices is then constructed as ⎤ ⎡ ⎡ ⎤ \ p| \ p| 0 0 0 0 0 0 ∂ /\m (2) (2) ⎥ ∂ om ⎢ ⎢ ⎥ m m = ⎣ 0 ∂/\ = ⎣ 0 ∂o 0 ⎦, 0 ⎦, ∂r ∂r 1 1 ∂r1 ∂r1 0 0 0 0 0 0 ⎤ ⎡ |( ( ) \ | )| −1 T 0 ( 0 0 ) ∂ ops /\ps ops (2) −1 (2) (2) T ⎥ ⎢ = ⎣ 0 ∂ os (/\s ) [os ] 0 ⎦. ∂r1 ∂r1 0 0 0 The first-order eigenvalue derivatives of the global structure, 10), are obtained using Eq. (3.24).

∂λi ∂r1

(i = 1, 2, …,

(4) The first-order derivative{ of }substructural mode participation factor {zi } with ∂zi , is calculated using Eq. (3.39) respect to parameter r 1 , ∂r 1 (5) The eigenvector derivatives of the global structure with respect to parameter r 1 i at the are formed using Eq. (3.30) and eliminating the identical values of ∂φ ∂r interfaces of the substructures. For comparison, the traditional Nelson method is employed to calculate the eigensensitivity of the global structure directly, which is regarded as the accurate one. Table 3.1 compares the first 10 eigenvalue derivatives computed by the substructuring method and the global method. The relative errors of the eigenvalue derivatives are less than 3%, which is satisfactory for most practical engineering applications. Following modal assurance criteria (MAC), the similarity of the eigenvector derivatives obtained with the global method and the proposed substructuring method is denoted as the similarity of vector (SV) and is given by ({ SV

|{ } { }|2 | ∂φ T ∂ φ˜ | i { }) | | i } | ∂r1 ∂r1 | ∂ φ˜ i ∂φi , = ({ }T { })({ }T { }) ∂r1 ∂r1 ∂φi ∂ φ˜ i ∂ φ˜ i ∂φi ∂r1

∂r1

∂r1

(3.44)

∂r1

{ } ∂φi represents the eigenvector derivative obtained with the global method, where ∂{r { }1 ˜ ∂ φi and ∂{r represents the eigenvector derivative with the substructuring method. As 1

3.4 Examples

59

Table 3.1 Comparison of eigenvalue derivative and eigenvector derivative with respect to r 1 Mode Eigenvalue derivatives

Correlation of eigenvector Nelson’s method Substructuring method Difference (%) derivatives (SV)

1

0.876

0.876

0.00

0.999

2

3.621

3.622

0.02

0.999

3

3.431

3.433

0.07

0.992

4

49.478

49.567

0.18

0.997

5

72.918

73.650

1.00

0.997

6

292.125

294.986

0.98

0.995

7

219.068

220.354

0.59

0.999

8

742.183

756.540

1.93

0.995

9

675.675

689.697

2.08

0.987

10

526.273

535.207

1.70

0.981

shown in Table 3.1, the SV values for most modes are above 0.99, indicating that the proposed method can achieve good accuracy in calculating eigenvector derivatives. The accuracy of the eigensensitivity computation is certainly affected by the number of master modes retained in the substructures. To investigate the influence of master modes, 10 master modes and 50 master modes are retained in the substructure to calculate the eigensensitivity with respect to r 1 . Figure 3.2 compares the relative errors of the eigenvalue derivatives using 10, 30 and 50 master modes. The SV values of eigenvector derivatives is compared in Fig. 3.3. It can be found that the use of more master modes improves the accuracy of the eigensolution derivatives, especially for the higher modes, since including more master modes increase the minimum eigenvalue of slave modes. Retaining 50 master modes in each substructure slightly improves the results as compared with 30 master modes but consumes more computation time. Too few master modes may result in undesirable results, and too many master modes have slight improvement on precision but decrease the efficiency. Trading off the accuracy and computational time, the number of master modes retained in each substructure is usually 2–3 times that of the interest modes of the global structure. The computational efficiency of the proposed method will be investigated using a relatively large structure in the next section.

3.4.2 The Balla Balla River Bridge The Balla Balla River Bridge is divided into 11 substructures as shown in Fig. 3.4 to illustrate the accuracy and efficiency of the present substructuring method. There are 50 master modes retained in each substructure to recover the eigensensitivity of the global structure. Details of these 11 substructures are provided in Table 2.8.

60

3 Substructuring Method for Eigensensitivity

Relative error (Log)

10

10

10

10

10

0

-2

-4

-6

10 master modes 30 master modes 50 master modes -8

1

2

3

4

5

6

7

8

9

10

9

10

Mode

Fig. 3.2 Accuracy of the eigenvalue derivatives using different master modes

1

0.9

0.8

sV 0.7

0.6

0.5

10 master modes 30 master modes 50 master modes 1

2

3

4

5

6

7

8

Mode

Fig. 3.3 Accuracy of the eigenvector derivatives using different master modes

The designed elemental parameters refer to Young’s moduli of the four shell elements denoted as r 1 ~ r 4 in the figure, which are intentionally selected in different substructures. First, the eigensensitivities of the first 20 modes with respect to the four elemental parameters are directly calculated using the traditional Nelson method on the global method for comparison. Afterward, the eigensensitivities of the global structure are calculated from the proposed substructuring method and are compared in Table 3.2. When the global structure is divided into 11 substructures and the first 50 master modes are retained in each substructure, the errors of the first 20 eigenvalue derivatives are less than 0.1% mostly, and all the SV values of eigenvectors are greater than 0.99. Afterward, computational time in calculating the eigensensitivities with respect to the four designed element parameters is recorded. To investigate the effects of

3.4 Examples

61

r4 r2

r5

r3

r1

Fig. 3.4 Substructure model of the Balla Balla River Bridge and the designated parameters

the division formation, the bridge is also divided into 5, 8, and 15 substructures. The computational efficiency is definitely affected by the division formation of the substructures. Details of the different division formations are provided in Tables 2.4, 2.7, and 3.3. For the four division formations, selection of different master modes leads to different levels of precision. As stated in Chap. 2, if identical number of master modes are retained in each substructure, division of more substructures achieves better results and cost longer computation time undoubtedly. In this study, the number of master modes is selected to make the relative errors of the first 20 eigenvalue derivatives between substructuring methods and global method less than 3%. Based on this criterion, 80 master modes are required by each substructure with the division of five substructures, 60 master modes in division of eight substructures, 50 master modes in division of 11 substructures, and 50 master modes in division of 15 substructures. Figure 3.5 compares the computation time of eigensensitivity required by using the conventional global method and the proposed substructuring method with the four division schemes. The results show the following conclusion. (1) Since only a specific substructure and the reduced eigenequation are analyzed to calculate the eigensensitivity of the global structure, the proposed substructuring method reduces the computation time in all cases. (2) The computational time by the substructuring method is heavily dependent on the substructure division. Since large substructures take longer to handle than smaller ones, the division of five or eight substructures requires more computational time than division of 11. However, an excessive number of substructures lead to a large connection matrix C and large primitive matrices of the substructures, which renders the transformations among these matrices computationally

0.16

239.01

805.98

567.23

48.57

238.87

805.96

566.15

48.60

16

17

0.03

0.05

152.01

141.00

321.59

112.26

241.75

5.20

14

0.03

13

2.43

121.02

15

151.98

0.05

11

140.96

10

12

112.24

321.56

8

241.73

7

9

0.16

5.19

5

6

2.43

120.99

3

4

121.24

0.67

121.13

0.67

1

2

0.06

0.19

0.00

0.06

0.00

0.00

0.02

0.02

0.01

0.02

0.01

0.26

0.00

0.03

0.00

0.00

0.09

Mode Global Present Relative method method error (10–2 ) (10–2 ) (%)

0.970 678.54

0.998 760.02

0.998 874.12

0.993 179.95

1.000 0.05

1.000 0.06

0.969 73.59

0.997 23.86

0.990 566.79

0.976 468.14

0.960 140.42

0.995 9.17

0.997 0.48

0.973 329.45

1.000 7.35

1.000 0.18

0.989 223.79

Global method (10–2 )

677.02

760.64

873.25

179.80

0.05

0.06

73.62

23.84

566.83

468.18

140.44

9.19

0.48

329.46

7.34

0.18

223.99

Present method (10–2 )

0.22

0.08

0.10

0.22

0.00

0.00

0.03

0.09

0.01

0.02

0.02

0.26

0.00

0.00

0.14

0.00

0.09

Relative error (%)

r2 Eigenvalue derivatives

Eigenvalue derivatives

SV

r1

r3 Global method (10–2 ) 188.20

241.09

0.02

0.03

66.51

43.89

121.08

155.04

39.38

5.43

0.33

247.21

5.84

0.05

0.03

0.00

0.00

0.00

0.00

0.08

0.00

0.05

0.00

0.00

0.03

0.11

0.00

0.10

Relative error (%)

0.954 572.98

0.998 894.03

573.61

898.99

0.11

0.05

0.998 1090.01 1088.32 0.02

0.992 241.79

1.000 0.02

1.000 0.03

0.995 66.51

0.999 43.89

0.999 121.18

0.995 155.04

0.992 39.40

0.999 5.43

0.999 0.33

0.990 247.13

1.000 5.85

1.000 0.05

Present method (10–2 )

Eigenvalue derivatives

0.995 188.36

SV

Table 3.2 Eigensensitivity with respect to the four designed structural parameters

Global method (10–2 ) 378.89

4.20

14.33

0.37

373.96

6.89

0.02

0.26

0.14

0.00

0.06

0.28

0.00

0.03

0.948 4.78

0.999 227.23

0.998 76.32

0.996 754.21

1.000 0.18

1.000 0.00

0.999 314.84

0.999 559.87

0.999 8.72

4.74

226.81

76.66

745.73

0.18

0.00

314.73

559.77

8.73

0.82

0.02

0.39

1.12

0.00

0.00

0.03

0.02

0.13

0.984

0.998

0.997

0.990

1.000

1.000

0.999

0.999

0.999

0.999

0.998

0.999

0.999

0.998

1.000

1.000

0.998

SV

(continued)

Relative error (%)

0.999 1193.00 1194.77 0.15

0.997 4.21

0.999 14.31

0.999 0.37

0.997 374.17

1.000 6.91

1.000 0.02

Present method (10–2 )

Eigenvalue derivatives

r4

0.997 378.76

SV

62 3 Substructuring Method for Eigensensitivity

649.76

20

651.47

611.41

590.99

611.82

589.08

18

19

0.26

0.32

0.06

Mode Global Present Relative method method error (10–2 ) (10–2 ) (%)

Eigenvalue derivatives

r1

Table 3.2 (continued) r2 Global method (10–2 )

Present method (10–2 )

Relative error (%)

Eigenvalue derivatives

0.987 792.71

0.984 11.90 790.86

11.77 0.29

1.06

0.979 1498.32 1496.84 0.11

SV

r3 Global method (10–2 ) 639.04 41.49

0.66

2.57

Relative error (%)

0.993 1265.73 1260.49 0.45

0.976 41.22

Present method (10–2 )

Eigenvalue derivatives

0.963 655.92

SV

r4 Global method (10–2 )

Present method (10–2 )

Relative error (%)

Eigenvalue derivatives

169.89

0.51

0.996 3364.71 3360.68 0.16

0.997 170.75

0.969 1277.09 1281.57 0.36

SV

0.982

0.986

0.971

SV

3.4 Examples 63

66

92

77

No. of elements

No. of nodes 90

Note a In the longitudinal direction

23

3–7.5

0–3

Geometric range (m)a

No. of interface nodes

Sub 2

Substructure Sub 1 index

23

69

44

7.5–10.5

Sub 3

23

69

44

10.5– 13.5

Sub 4

23

Table 3.3 Division formation with 15 substructures

69

44

13.5– 16.5

Sub 5

23

97

72

16.5– 19.5

Sub 6

23

69

44

19.5– 22.5

Sub 7

23

92

66

22.5–27

Sub 8

23

69

44

27–30

Sub 9

23

92

66

30– 34.5

Sub 10

23

97

72

34.5– 37.5

Sub 11

23

69

44

37.5– 40.5

Sub 12

23

92

66

40.5– 45

Sub 13

23

92

66

45– 49.5

Sub 14

23

113

92

49.5– 54

Sub 15

64 3 Substructuring Method for Eigensensitivity

References

65

Fig. 3.5 Computational time with different division formations

expensive. In consequence, division of 15 substructures is less efficient than division of 11 substructures. This phenomenon is similar to the calculation the eigensolutions in Chap. 4. It is better to trade off the number of substructures against and the size of each, before applying the proposed method to a model updating process.

3.5 Summary The eigensensitivity equation is derived based on the reduced eigenequation developed in Chap. 2, and the first-order eigenvalue derivatives and eigenvector derivatives are calculated by the improved substructuring method. The eigensensitivity of a global structure is assembled from the eigensensitivity matrices of a particular substructure and a reduced eigensensitivity equation by emulating Nelson’s method. The number of master modes and the division formation of the substructures affects the accuracy and the efficiency. Retaining more master modes in the substructures can achieve better accuracy but consume more computational expense. The proposed method can achieve a good accuracy when proper master modes are retained. The division of excessive or too few substructures are both undesirable. It is suggested to trade-off needs between the number of substructures and the size of each.

References Felippa, C.A., Park, K.C., Filho, M.R.J.: The construction of free-free flexibility matrices as generalized stiffness inverses. Comput. Struct. 68, 411–418 (1998) Fox, R.L., Kapoor, M.P.: Rate of change of eigenvalues and eigenvectors. AIAA J. 6(12), 2426–2429 (1968)

66

3 Substructuring Method for Eigensensitivity

Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14, 1201–1205 (1976) Xia, Y., Weng, S., Xu, Y.L., Zhu, H.P.: Calculation of eigenvalue and eigenvector derivatives with the improved Kron’s substructuring method. Struct. Eng. Mech. 36, 37–55 (2010)

Chapter 4

Substructuring Method for High-Order Eigensensitivity

4.1 Preview The second- and high-order derivatives of the eigenpairs are particularly important in the case when large variations in design parameters exist or the natural frequencies are closely spaced (Brandon 1984; Li et al. 2012). The eigensensitivity with repeated or close eigenvalues usually requires second-order or even highorder derivatives. In addition, the second-order eigensensitivity is also required by the high-precision damage identification or statistical damage identification. For the case of large changes in design parameters, the linear approximation in the use of first-order derivatives may be inadequate for the sensitivity-based model updating, and the high-order derivatives need to be taken into account (Friswell 1995; Wu et al. 2007; Weng et al. 2013). A large-scale practical structure is usually represented by a complex model, involving a large number of degrees of freedom (DOFs) and undetermined structural parameters. In model updating or optimization process, structural parameters in the analytical model are iteratively modified to satisfy the objective functions in an optimal way. For a large-scale structure, it is computationally expensive to repeatedly extract the second- and high-order eigensensitivity from the large-size global system matrices. This chapter extends the substructure-based eigensensitivity to calculate the second- and high-order eigensensitivity.

4.2 Basic Method for High-Order Eigensensitivity 4.2.1 Second-Order Eigensolution Derivatives The second-order eigensolution derivatives can be achieved by differentiating Eq. (3.1) twice. If the two parameters are denoted as r j and r k , respectively, the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_4

67

68

4 Substructuring Method for High-Order Eigensensitivity

eigenequation is differentiated as (K − λi M)

∂(K − λi M) ∂{φi } ∂(K − λi M) ∂{φi } ∂ 2 {φi } + + ∂r j ∂rk ∂rk ∂r j ∂r j ∂rk +

∂ 2 (K − λi M) {φi } = 0 ∂r j ∂rk

(4.1)

{ } Premultiplying φiT on both sides of Eq. (4.1) gives the second-order derivative of the ith eigenvalue { } ∂(K − λi M) ∂{φi } { T } ∂(K − λi M) ∂{φi } ∂ 2 λi = φiT + φi ∂r j ∂rk ∂rk ∂r j ∂r j ∂rk | | 2 2 { T} ∂ K ∂ M ∂λi ∂M ∂λi ∂M {φi } − λi − − + φi ∂r j ∂rk ∂r j ∂rk ∂r j ∂rk ∂rk ∂r j

(4.2)

Since the second-order derivatives of K and M with respect to r are zero, Eq. (4.2) is simplified to { } ∂(K − λi M) ∂{φi } { T } ∂(K − λi M) ∂{φi } ∂ 2 λi = φiT + φi ∂r j ∂rk ∂rk ∂r j ∂r j ∂rk | | { T } ∂λi ∂M ∂λi ∂M {φi } − φi + ∂r j ∂rk ∂rk ∂r j

(4.3)

The second-order eigenvector derivative is calculated from Eq. (4.1) as well, which is rewritten as ui

∂ 2 {φi } = Yi( j,k) ∂r j ∂rk

(4.4)

and u i = (K − λi M) ) ( ∂(K − λi M) ∂{φi } ∂(K − λi M) ∂{φi } ∂ 2 (K − λi M) {φi } Yi( j,k) = − + + ∂rk ∂r j ∂r j ∂rk ∂r j ∂rk As before, the second-order eigenvector derivative can be solved with the modal method and Nelson’s method. Only Nelson’s method is used here for its high efficiency. The second-order derivative of the ith eigenvector can be expressed as the sum of the particular and homogeneous solutions as { } ∂ 2 {φi } = vi( j,k) + ci ( j,k) {φi } ∂r j ∂rk

(4.5)

4.2 Basic Method for High-Order Eigensensitivity

69

{ } where vi ( j,k) is not unique but can be calculated from { } { } u i vi( j,k) = Yi ( j,k)

(4.6)

{ } to the expression of Eq. (3.43), the line and column of vi( j,k) , u i and { Equivalent } Yi ( j,k) corresponding to the largest magnitude in the eigenvector are set { to zero; } thus, a full-rank equation is formulated to uniquely determine the vector vi( j,k) . The participation factor ci(j,k) is calculated from the second-order derivative of the mass orthogonal condition of Eq. (3.44) { } { T } ∂M ∂{φi } { T } ∂M { T } ∂M ∂{φi } ∂ φiT ∂{φi } {φi } + 2 φi φi + 2 φi +2 M ∂r j ∂rk ∂r j ∂rk ∂rk ∂r j ∂rk ∂r j { T } ∂ 2 {φi } =0 (4.7) + 2 φi M ∂r j ∂rk Substitution of Eq. (4.5) into Eq. (4.7) produces the following equation for the solution of the participation factor ci(j,k) . ( { } ∂M ∂{φi } { T } ∂M ∂{φi } 1 { T } ∂M {φi } + φiT φi ci ( j,k) = − + φi 2 ∂r j ∂rk ∂r j ∂rk ∂rk ∂r j ) { T} ∂ φi ∂{φi } { T } M + φi Mvi( j,k) + ∂rk ∂r j

(4.8)

4.2.2 General High-Order Eigensolution Derivatives Calculation of the second-order eigensolution derivatives can be generalized to the higher-order eigensolution derivatives. The hth-order eigensolution derivative is achieved by differentiating the eigenequation (Eq. (3.1)) h times with respect to the parameters r 1 , r 2 , …, r h as (K − λi M)

{ } ∂ h λi ∂ h {φi } − M{φi } = Yi (1,2,...,h) ∂r1 ∂r2 · · · ∂rh ∂r1 ∂r2 · · · ∂rh

(4.9)

{ } where vector Yi (1,2,...,h) contains { }the terms involving the derivatives of order (h-1) and the lower. Premultiplying φiT on both sides of Eq. (4.9) gives the hth-order derivative of the ith eigenvalue in the form of } { ∂ h λi = − Yi(1,2,...,h) ∂r1 ∂r2 · · · ∂rh

(4.10)

70

4 Substructuring Method for High-Order Eigensensitivity

The hth-order eigenvector derivative can be obtained following Eqs. (4.5)–(4.8). As before, the eigenvector derivative is expressed as the sum of a particular part and a homogeneous part as } { ∂ h {φi } = vi(r1 ,r2 ,...,rh ) + ci(r1 ,r2 ,...,rh ) {φi } ∂r1 ∂r2 · · · ∂rh

(4.11)

{ } The unique vi (r1 ,r2 ,...,rh ) can be achieved using the equivalent method to Eq. (3.43) and Eq. (4.6). The participation factor ci(r1 ,r2 ,...,rh ) can be obtained from the hth derivative of the mass orthogonal condition Eq. (3.44) {

} φiT M

∂ h {φi } = di(1,2,...,h) ∂r1 ∂r2 · · · ∂rh

(4.12)

where scalar di(1,2,...,h) contains the terms involving derivatives of order (h-1) and the lower. Thereby, the participation factor ci (r1 ,r2 ,...,rh ) is obtained as } { } { ci(1,2,...,h) = di(1,2,...,h) − φiT M vi(1,2,...,h)

(4.13)

4.3 Substructuring Method for High-Order Eigensensitivity 4.3.1 Second-Order Eigensolution Derivatives Due to the symmetric property and simple form of the reduced eigenequation by substructuring method (Eq. 3.21), it is easy to derive the second-order derivatives of the eigensolutions by directly redifferentiating this reduced eigenequation. In this section, the second-order eigenvalue and eigenvector derivatives are formulated with the substructuring method.

4.3.1.1

Second-Order Eigenvalue Derivatives

Without losing generality, Eq. (3.21) is differentiated with respect to two design variables r j and r k as ) ) ) ( ( ( ) ∂ 2 {zi } ∂ 2 u − λi I ∂ u − λi I ∂{zi } ( ∂ u − λi I ∂{zi } {zi } + + + u − λi I =0 ∂r j ∂rk ∂r j ∂rk ∂rk ∂r j ∂r j ∂rk

(4.14)

4.3 Substructuring Method for High-Order Eigensensitivity

71

{ } Premultiplying ziT on both sides of Eq. (4.14) and re-arranged it, the secondorder eigenvalue derivative can be obtained as ) ) ( ( { T } ∂ u − λi I ∂{zi } { T } ∂ u − λi I ∂{zi } { T} ∂ 2u ∂ 2 λi {zi } + zi = zi + zi ∂r j ∂rk ∂r j ∂rk ∂r j ∂rk ∂rk ∂r j (4.15) In Eq. (4.15), the second-order eigenvalue derivative comprises two parts: the 2 u and the multiplication of the firstcomponent of the second-order derivative ∂r∂ j ∂r k order derivative matrix. The latter could be obtained from the previous section, while 2 u the former ∂r∂ j ∂r is calculated as follows. k ∂2u ∂r j ∂rk

is contributed by the second-order eigenvalue derivatives, and the residual flexibility derivatives of the substructures as ∂ 2u = ∂r j ∂rk

| ( ( ) )−1 | p −1 ∂ 2 /\pm + | m | Ts /\s | s | Tm

∂ 2 /\pm + = ∂r j ∂rk

{ ∂2

∂r j ∂rk } ( ) |−1 | p −1 T | m | s /\s | s | Tm (4.16)

∂r j ∂rk

( ( ) )−1 −1 Concerning on Eq. (3.34), the second-order derivative of | m | Ts /\ps | s | Tm can be decoupled as ( ( ) ( )−1 )−1 T ∂ 2 | m | Ts /\ps | s |m ⎡ ⎢ =⎢ ⎣|m

∂r j ∂rk |( | ( p )−1 )−1 T 2 | s /\s | s ∂ ∂r j ∂rk

| Tm + 2| m

| |( ( p )−1 )−1 T ∂ | s /\s | s ∂r j

⎤ ∂| Tm ⎥ ⎥ ∂rk ⎦

⎡

|−1 } ( ) p −1 T } | T /\ ∂ | | s s −1 ( ) s ∂| m ∂| Tm ⎢ | m T p −1 + 2⎣ | s /\s | s + |m ∂rk ∂r j ∂rk ∂r j } ( ) |−1 ∂ 2 | T | m p −1 T +| m | s /\s | s ∂r j ∂rk

(4.17)

72

4 Substructuring Method for High-Order Eigensensitivity

In Eq. (4.17), the pivot work is to solve the second-order derivative matrix of the inverse of the residual flexibility, which is calculated by |( ∂

( )−1 )−1 p |T |s s /\s ∂r j ∂rk

| (

(

p

= − |T s /\s

)−1

|s

)−2 ∂

( ( ) p −1 + 2 |T |s s /\s ) ( −1 T ∂ opm (/\pm ) [opm ]

where ∂r j ∂rk the substructures as

( ( ) } | ) ( p )−1 p p −1 p T K − om /\m om

∂r j ∂rk ) ( ( ) ) ( ( ) −1 −1 p T /\p )−3 ∂ | T | ∂ | | /\ s s s s s s ∂r j

∂rk

(4.18)

can be obtained from the derivatives of the master modes of

( )−1 ( p )−1 ∂/\pm ∂ /\pm | p |T ( p )−1 ∂ 2 /\pm ( p )−1 | p |T p /\m om − 2om /\m om /\m = ∂r j ∂rk ∂r j ∂rk | |T ( p )−1 ∂/\pm ( p )−1 ∂ opm p − 2om /\m /\m ∂r j ∂rk | p | |T | | p |T ( p )−1 | p |T ( p )−1 ∂ 2 opm ∂ om om ( p )−1 ∂ om p ∂ /\m p /\ +2 + om + om /\m ∂rk m ∂r j ∂rk ∂r j ∂r j ∂rk (4.19) −opm

u is contributed by two kinds of derivative Based on the detailed decoupling, ∂r∂ j ∂r k matrices. The first group is the second-order derivative matrices of the substructures, including the second-order derivative of eigenvalues, eigenvectors, and residual flexi|( )−1 | −1 ∂ 2 | Ts (/\ps ) | s 2 p 2 T ∂ | ∂ /\ . The second group is the cross multibility, that is, ∂r j ∂rmk , ∂r j ∂rmk , and ∂r j ∂rk plication of the first-order derivatives of the residual flexibility and eigenvectors, that |( )−1 | −1 ∂ | Ts (/\ps ) | s ∂| T is and ∂rkm , and the multiplication of the first-order eigenvectors ∂r j ( T ) ∂| m ∂| T and ∂rkm . ∂r j 2

∂ 2 /\p

∂ 2 |T

In the former group, ∂r j ∂rmk and ∂r j ∂rmk are, respectively, the second-order eigenvalue derivatives and eigenvector derivatives of the master modes in substructures, and |( )−1 | −1 ∂ 2 | Ts (/\ps ) | s are associated with the second-order derivatives of the substruc∂r j ∂rk tural residual flexibility. If r j and r k are located in the same substructure (e.g., the rth substructure), the three items are calculated for the rth substructure solely, while those corresponding to other substructures are zero, i.e.,

4.3 Substructuring Method for High-Order Eigensensitivity

73

⎤ 0 2 T (r ) ∂ ∂ 2 opm ⎥ ∂ |m ∂ 2 /\m , = C 0 ⎦ ∂r j ∂rk ∂r j ∂rk ∂r j ∂rk ∂r j ∂rk 0 0 0 ⎡ ⎤ 0 0 0 (r ) ⎢ ∂ 2 om ⎥ = C⎣ 0 ∂r 0 ⎦, j ∂rk ⎡

2

/\pm

( ( ) ) −1 ∂ 2 | Ts /\ps | s ∂r j ∂rk

0 ⎢ = ⎣0

0

0 0 0 ( ( ) | | ) −1 T ∂ 2 ops /\ps ops

=C ⎡

∂r j ∂rk

0

⎢ = C⎢ ⎣0 0

∂2

(

CT

⎤ 0 0 ) ⎥ (r ) −1 ) (r ) T o(r s (/\s ) [os ] ⎥ 0 ⎦C ∂r j ∂rk 0 0

(4.20)

In case that r j and r k are located in different substructures, the second( ) p −1 T 2 | ∂ /\ | 2 p 2 T ( ) s s s ∂ | ∂ /\ are zero-matrix without order derivative matrices ∂r j ∂rmk , ∂r j ∂rmk , and ∂r j ∂rk computation. The latter group includes the multiplication of first-order derivatives. The primitive form of the ) derivative matrices of the residual flexibility and eigenvectors, ( first-order −1 ∂ | Ts (/\ps ) | s ∂| T and ∂rkm , is diagonal assembly of the derivative matrices of the namely ∂r j substructures. The sub-block corresponding to the two substructures, which include r j or r k as stated in Sect. 3.3, is nonzero only, and all other sub-blocks are zero-matrix. Therefore, if r j and r k are located in different substructures, the multiplication of them gives zero-matrix as well. In conclusion, if the designed variables r j and r k are located in two different 2 u are zeros since the substructures are independent substructures, all items in ∂r∂ j ∂r k and unrelated to each other, i.e., | ( | ( p )−1 )−1 T T 2 | | ∂ /\ | | m s s s m ∂ 2u ∂ 2 /\pm = + =0 (4.21) ∂r j ∂rk ∂r j ∂rk ∂r j ∂rk In this case, the second-order derivative of the eigenvalue is simplified into the multiplication of first-order derivative matrices ( ( ) ) { T } u − λi ∂{zi } { T } u − λi ∂{zi } { T} ∂ 2u ∂ 2 λi {zi } + zi ∂ = zi + zi ∂ ∂r j ∂rk ∂r j ∂rk ∂r j ∂rk ∂rk ∂r j ) ( ∂ u − λi ∂{zi } (4.22) = 2{zi }T ∂r j ∂rk

74

4 Substructuring Method for High-Order Eigensensitivity

If the design variables r j and r k are located in the same substructure (e.g., the ( ) p −1 T 2 | ∂ /\ | 2 p 2 T ( ) s s s ∂ | ∂ /\ are assembled using the rth substructure). Items ∂r j ∂rmk , ∂r j ∂rmk , and ∂r j ∂rk ∂| T

∂ 2 /\p

derivative matrices of the rth substructure solely. ∂r j ∂rmk and ∂r j ∂rm k can be calculated with the traditional method addressed in Sect. 4.2.1 by treating the rth substructure 2 u can be formed by the second-order as an independent structure. Subsequently, ∂r∂ j ∂r k derivative matrices and the multiplication of the first-order derivative matrices within the rth substructure. The substructures are independent and unrelated to each other. If the two parameters (r j and r k ) are located in two different substructures, the second-order eigenvalue derivative is calculated by the first-order derivative matrices (substructural eigensolutions and residual flexibility) of two substructures. If r j and r k are located in the same substructure, the second-order eigenvalue derivative is calculated by the first- and second-order derivative matrices of only one substructure (the master eigensolutions and residual flexibility).

4.3.1.2

Second-Order Eigenvector Derivatives

The second-order eigenvector derivative can be computed by double-differentiating the reduced eigenequation directly. The ith eigenvector of the global structure can be recovered by the substructural master modes as oi = opm {zi }

(4.23)

The second-order eigenvector derivative of the ith mode with respect to two parameters r j and r k is acquired by differentiating this expression as ∂opm ∂{zi } ∂opm ∂{zi } ∂ 2 oi ∂ 2 {zi } ∂ 2 opm {z i } + = + + opm ∂r j ∂rk ∂r j ∂rk ∂r j ∂rk ∂rk ∂r j ∂r j ∂rk All items, including the second-order derivative ∂opm

∂2u ∂r j ∂rk

(4.24)

and the first-order deriva-

i} tives of ∂r and ∂{z , are available from the previous ∂r ∂ 2 {zi } derivatives. Only ∂r j ∂rk is required to be solved here.

analysis in eigenvalue

The double-differentiated eigenequation (Eq. (4.14)) is rewritten as ( } { ) ∂{zi } u − λi I = Yi ( j,k) ∂r j ∂rk where

(4.25)

4.3 Substructuring Method for High-Order Eigensensitivity

75

( ( ( ) ) ) { } ∂ 2 u − λi I ∂ 2 u − λi I ∂{zi } ∂ 2 u − λi I ∂{zi } {zi } + Yi( j,k) = + (4.26) ∂r j ∂rk ∂r j ∂rk ∂rk ∂r j {

} Yi ( j,k) can be obtained directly using the interim results in calculating the eigenvalue derivatives. ∂ 2 (zi ) According to Nelson’s method, ∂r is expressed as the sum of a particular part j ∂rk and a homogeneous part as } { ∂ 2 {zi } = vi( j,k) + ci ( j,k) {zi } ∂r j ∂rk

(4.27)

{ } where vi ( j,k) is not unique but may be calculated by substituting Eq. (4.27) into Eq. (4.25) as ( } { } ){ u − λi I vi( j,k) = Yi( j,k)

(4.28)

The eigenvector of the reduced eigenequation satisfies the orthogonality property of {zi }T {zi } = 1

(4.29)

Double differentiating Eq. (4.29) with respect to r j and r k and substituting Eq. (4.27) into it gives ( 2

) ) ({ } ∂{zi }T ∂{zi } + {zi }T vi ( j,k) + ci( j,k) {zi } = 0 ∂rk ∂r j

The participation factor ci(j,k) is obtained as ci( j,k) = −

{ } ∂{zi }T ∂{zi } − {zi }T vi( j,k) ∂rk ∂r j

(4.30)

Given {vi(j,k) } and ci(j,k) , the second-order eigenvector derivative can be recovered from Eqs. (4.24) and (4.27). It is seen that the second-order eigenvector derivative is obtained by solving the reduced eigenequation solely. As the calculation of eigenvector derivatives usually dominates computational resources in the common global method, this substructuring method improves the computational efficiency significantly.

76

4 Substructuring Method for High-Order Eigensensitivity

4.3.2 High-Order Eigensolution Derivatives Due to the symmetric property and simple form of the reduced eigenequation (Eq. 3.21), it is easy to derive high-order derivatives of the eigensolutions by directly redifferentiating the reduced eigenequation.

4.3.2.1

High-Order Eigenvalue Derivatives

Equation (3.21) is differentiated with respect to k design parameters (r 1 , …, r k ) as ) ) ({ } ∂{zi }T ∂{zi } T + {zi } vi ( j,k) + ci( j,k) {zi } = 0 2 ∂rk ∂r j (

(4.31)

{ } Remultiplying Eq. (5.38) by ziT , the kth-order eigenvalue derivative is obtained as ) ( { T } ∂ k−1 u − λi I ∂{zi } { T} ∂ k λi ∂ku {zi } + zi = zi ∂r1 ∂r2 · · · ∂rk ∂r1 ∂r2 · · · ∂rk ∂r1 ∂r2 · · · ∂rk−1 ∂rk ) ( k−1 { T } ∂ u − λi I ∂ {zi } (4.32) + · · · + zi ∂r1 ∂r2 ∂r3 · · · ∂rk In Eq. (5.39), the kth-order eigenvalue derivative comprises two parts: the kthk order derivative ∂r1 ∂r∂ 2u···∂rk and the multiplication of the (k–1)th and lower order derivatives which can be re-used directly from the calculation of lower order eigensensitivity. k Similar to the second-order derivative, the kth-order derivative ∂r1 ∂r∂ 2u···∂rk is nonzero only if the k parameters (r 1 , r 2 , …, r k ) are located in the same substructure (e.g., the Qth substructure). In this case, the kth-order eigensensitivity of the global structure is calculated from the kth-order derivatives of substructural master modes of only the Qth substructure. Otherwise, if the k parameters (r 1 , r 2 , …, r k ) are located in different substructures, the kth-order eigensensitivity is determined by the (k–1)th and lower order derivatives of substructural master modes of difference substructures. It is seen that if m design parameters of (r 1 , r 2 , …, r k ) are located in the Mth substructure, the mth-order substructural derivatives are calculated within the Mth substructure to form the kth-order eigensensitivity of global structure.

4.3.2.2

High-Order Eigenvector Derivatives

Differentiating Eq. (4.23) with respect to k parameters (r 1 , r 2 , …, r k ), the kth-order eigenvector derivative of the ith mode is acquired as

4.3 Substructuring Method for High-Order Eigensensitivity

∂ k−1 opm ∂{zi } ∂ k oi ∂ k opm {zi } + = ∂r1 ∂r2 · · · ∂rk ∂r1 ∂r2 · · · ∂rk ∂r1 ∂r2 · · · ∂rk−1 ∂rk ∂ k {zi } ∂opm ∂ k−1 {zi } + ··· + + opm ∂r1 ∂r2 ∂r3 · · · ∂rk ∂r1 ∂r2 · · · ∂rk

77

(4.33)

∂ k {zi } ∂r1 ∂r2 ···∂rk

is required to calculate the kth-order eigenvector derivative of the global structure. It is written as the particular part and homogeneous part as } { ∂ k {zi } = vi (∂r1 ,∂r2 ,··· ,∂rk ) + ci(∂r1 ,∂r2 ,··· ,∂rk ) {zi } ∂r1 ∂r2 · · · ∂rk

(4.34)

{ } Substituting Eq. (4.34) into Eq. (4.31) and remultiplying Eq. (4.31) by ziT , the eigenequation is written as ( } { } ){ u − λi I vi(∂r1 ,∂r2 ,··· ,∂rk ) = Yi(∂r1 ,∂r2 ,··· ,∂rk )

(4.35)

where {

Yi(∂r1 ,∂r2 ,··· ,∂rk )

}

( ( ) ) ∂ k−1 u − λi I ∂{zi } ∂ k u − λi I {zi } + = ∂r1 ∂r2 · · · ∂rk ∂r1 ∂r2 · · · ∂rk−1 ∂rk ) ( ∂ u − λi I ∂ k−1 {zi } + ··· + ∂r1 ∂r2 ∂r3 · · · ∂rk

(4.36)

{ } Afterward, vi (∂r1 ,∂r2 ,··· ,∂rk ) can be calculated from Eq. (4.35). ci(∂r1 ,∂r2 ,··· ,∂rk ) is calculated by differentiating the orthogonal equation {zi }T {zi } = 1 with respect to k parameters (r 1 , r 2 , …, r k ), which gives { T} zi

∂ k {zi } = di(∂r1 ,∂r2 ,··· ,∂rk ) ∂r1 ∂r2 · · · ∂rk

(4.37)

And di (∂r1 ,∂r2 ,··· ,∂rk ) contains terms involving the (k–1)th and lower order derivatives as well. Thus, substituting Eq. (4.34) into Eq. (4.37) gives } { }{ ci (∂r1 ,∂r2 ,··· ,∂rk ) = di (∂r1 ,∂r2 ,··· ,∂rk ) − ziT vi(∂r1 ,∂r2 ,··· ,∂rk )

(4.38)

In consequence, { the kth-order } eigenvector derivative of the global structure is formulated from vi (∂r1 ,∂r2 ,··· ,∂rk ) and ci (∂r1 ,∂r2 ,··· ,∂rk ) . In summary, the proposed substructuring method calculates the first-, second, and high-order eigenvector derivatives by analyzing the independent substructures containing the design parameters and a reduced eigenequation. Since the calculation of eigenvector derivatives usually dominates the computational resource in the common global model updating, this substructuring method improves the computational efficiency considerably.

78

4 Substructuring Method for High-Order Eigensensitivity

4.4 Examples The Balla Balla River Bridge (Fig. 2.6) is also employed to demonstrate the accuracy and efficiency of the proposed substructuring method in calculating the second-order eigensensitivity. The second-order eigensensitivities with respect to parameter pairs (r 5 , r 5 ), (r 5 , r 1 ), and (r 5 , r 2 ) shown in Fig. 3.4 are calculated. First, the second-order eigenvalue and eigenvector derivatives of the first 20 modes are calculated using the global method, which are regarded as the accurate results for comparison. The procedure of second-order eigensensitivity is described in Sect. 4.2. Afterward, the global structure is divided into 11 substructures, each retaining the first 50 modes as the master. The second-order eigensensitivity is also calculated by the proposed substructuring method and is compared with the exact results in Table 4.1. In Table 4.1, the second-order eigensensitivity with respect to parameter pair (r 5 , r 5 ) represents the case that the two elemental parameters are identical. That with respect to (r 5 , r 1 ) gives the result of two different parameters in the same substructure, whereas that with respect to (r 5 , r 2 ) is the case of two parameters in different substructures. Table 4.1 demonstrates that the relative differences of the second-order eigenvalue derivatives between the proposed substructuring method and the global method are smaller than 3% for most modes. Following the previous first-order eigenvector derivative, the accuracy of the second-order eigenvector derivative is denoted by SV, which is defined by ({ SV

| { 2 }) ∂ φ˜ i ∂ 2 φi , = ({ ∂r j ∂rk ∂r j ∂rk

{

∂ 2 φi ∂r j ∂rk

∂ 2 φi ∂r j ∂rk

| |{ | ∂ 2 φ }T { ∂ 2 φ˜ }|2 i i | | | ∂r j ∂rk ∂r j ∂rk | }T { 2 })({ 2 }T { ∂ φi ∂r j ∂rk

∂ φ˜ i ∂r j ∂rk

∂ 2 φ˜ i ∂r j ∂rk

}) (4.39)

}

represents the second-order eigenvector derivative from the global } φ˜ i from the substructuring method. Table 4.1 reports the SV values method and ∂r∂ j ∂r k of most modes are above 0.95, indicating a good accuracy of the second-order eigenvector derivatives. In addition, the first elemental parameters r j are designated to Young’s moduli of the 24 slab element in the first substructure, and the second elemental parameters r k are Young’s moduli of all 288 slab elements across the whole structure. Among the 288 parameters of r k , 24 parameters are located in the first substructure, and the remainders are in the other substructures. The second-order eigensensitivities with respect to these two groups of parameters are calculated in Table 4.2. where

{

2

− 0.0002

− 0.0001

− 1.1497

− 0.0002

− 0.0001

− 1.1484

12

14

− 0.1864

− 0.1860

11

13

− 0.2168

− 0.1999

− 0.2160

− 0.1997

9

− 0.0643

− 0.0643

8

10

− 0.0259

− 0.4022

− 0.0258

− 0.4020

6

− 0.0008

− 0.0008

5

7

− 0.0031

− 0.1536

− 0.0030

− 0.1537

3

4

− 0.7012

− 0.0016

− 0.7010

− 0.0016

1

0.11

0.00

0.00

0.22

0.10

0.36

0.00

0.05

0.23

0.00

0.02

0.66

0.00

0.03

Relative error (%)

0.991

0.993

0.981

0.988

0.974

0.996

0.987

0.981

0.992

0.992

0.985

0.990

0.999

0.998

SV

− 0.2129

0.0000

0.0000

− 0.0063

0.0610

0.0161

0.0014

0.0938

− 0.0025

− 0.0001

0.0339

0.0008

0.0001

− 0.0730

Global method

− 0.2139

0.0000

0.0000

− 0.0062

0.0609

0.0159

0.0014

0.0937

− 0.0025

− 0.0001

0.0340

0.0008

0.0001

− 0.0729

Substructuring method

Eigenvalue derivative

Substructuring method

Eigenvalue derivative

Global method

∂ 2 {φ} ∂2λ ∂r5 ∂r1 , ∂r5 ∂r1

∂ 2 {φ} ∂2λ ∂r5 ∂r5 , ∂r5 ∂r5

2

Mode

Table 4.1 Second-order eigenvalue derivatives and eigenvector derivatives

0.48

0.00

0.00

0.03

0.16

1.42

0.00

0.12

0.00

0.00

0.29

0.00

0.00

0.12

Relative error (%)

0.971

0.970

0.974

0.985

0.986

0.985

0.986

0.986

0.991

0.994

0.995

0.983

0.998

0.996

SV

− 0.1031

0.0000

0.0000

0.0051

0.0502

− 0.0062

− 0.0010

0.0231

− 0.0012

0.0000

0.0036

0.0003

− 0.0003

− 0.0295

Global method

− 0.1037

0.0000

0.0000

0.0053

0.0502

− 0.0065

− 0.0010

0.0230

− 0.0012

0.0000

0.0035

0.0003

− 0.0003

− 0.0296

Substructuring method

Eigenvalue derivative

∂ 2 {φ} ∂2λ ∂r5 ∂r2 , ∂r5 ∂r2

0.59

0.00

0.00

3.22

0.00

4.90

0.00

0.55

0.00

0.00

1.07

0.00

0.00

0.21

0.985

0.975

0.969

0.978

0.992

0.982

0.985

0.982

0.994

0.988

0.990

0.995

0.993

0.996

SV

(continued)

Relative error (%)

4.4 Examples 79

Relative error (%)

SV

− 0.5337

− 4.9587

− 0.9247

− 1.3274

− 0.5327

− 4.9578

− 0.9347

− 1.3160

17

18

19

20

− 3.9901

− 2.4320

− 3.9496

− 2.4723

15

1.03

0.86

1.07

0.02

0.20

1.63

0.985

0.988

0.985

0.977

0.982

0.986

− 0.5541

0.1271

− 0.6186

0.0462

− 0.2371

− 0.7531

Global method

− 0.5642

0.1231

− 0.6166

0.0455

− 0.2343

− 0.7539

Substructuring method

Eigenvalue derivative

Substructuring method

Eigenvalue derivative

Global method

∂ 2 {φ} ∂2λ ∂r5 ∂r1 , ∂r5 ∂r1

∂ 2 {φ} ∂2λ ∂r5 ∂r5 , ∂r5 ∂r5

16

Mode

Table 4.1 (continued)

1.82

3.16

0.32

1.34

1.17

0.11

Relative error (%)

0.975

0.983

0.981

0.977

0.973

0.980

SV

− 0.3337

− 0.0093

− 0.1997

− 0.0066

0.0102

− 0.3398

Global method

− 0.3346

− 0.0094

− 0.1996

− 0.0065

0.0105

− 0.3401

Substructuring method

Eigenvalue derivative

∂ 2 {φ} ∂2λ ∂r5 ∂r2 , ∂r5 ∂r2

0.27

1.15

0.07

1.67

2.74

0.08

Relative error (%)

0.974

0.979

0.980

0.990

0.987

0.985

SV

80 4 Substructuring Method for High-Order Eigensensitivity

288

13,427

No. of elements

Time (second)

Global method

3594

434.4

24

Sub 1 382.3

32

Sub 2 286.3

24

Sub 3

Location of parameter r k

Substructuring method

Table 4.2 Computational time of the global and substructuring methods

398.7

32

Sub 4 286.7

24

Sub 5 283.6

24

Sub 6 283.0

24

Sub 7

390.3

32

Sub 8

282.9

24

Sub 9

377.6

32

Sub 10

188.8

16

Sub 11

4.4 Examples 81

82

4 Substructuring Method for High-Order Eigensensitivity

The computational time is shown in Table 4.2. The global method totally takes up 13,427 s to calculate the second-order eigensensitivities with respect to the 24 parameters r j and the 288 parameters r k . The substructuring method takes 3594 s. It is noted that, in both the global method and the substructuring method, the computational time cost by the second-order eigensensitivity involves the time in calculating the first-order eigensensitivity. In particular, if parameters r j and parameters r k are both located in the first substructure, second-order eigensensitivities with respect to the 24 parameters of r j and 24 parameters of r k take 434.4 s. If parameters r j and r k are located in different substructures, for example, r j is located in first substructure and r k in the third, five, six, seven, or nine substructure calculating the second-order eigensensitivity takes about 280 s. The third, five, six, seven, and nine substructures have 24 parameters r k as well, and they have a similar size with the first substructure. As expected, r j and r k located in the same substructure take longer time than 2 u , which they are in different substructures, since the former requires the item ∂r∂ j ∂r k is zero in the latter. Similar to the calculation of the first-order eigensensitivity, the number of the master modes and the division formation of the substructures affect the computational accuracy and efficiency, which deserves several trials beforehand.

4.5 Summary The second-order eigensensitivity equation is derived by further differentiating the reduced eigenequation, from which the second-order eigenvalue and eigenvector derivatives with respect to two parameters are computed. If the two parameters are located in the same substructure, the second-order derivative matrices are required in a specific substructure that containing the two parameters, to calculate the secondorder eigensensitivity of the global structure. If the two parameters are located in different substructures, the first-order derivative matrices of the two substructures are calculated to assemble the second-order sensitivity matrix of the global structure. This substructuring method can be generalized to calculate the high-order eigensensitivity by further differentiating the reduced eigenequation. The high-order eigensensitivity of the global structure is determined by the derivative matrix of particular substructures containing the designated parameters. The high accuracy and efficiency of the substructuring method for the second-order eigensensitivity are demonstrated with a bridge structure. The substructuring method will be applied to the model updating process in the following chapter.

References

83

References Brandon, J.A.: Derivation and significance of second-order modal design sensitivities. AIAA J. 22, 723–724 (1984) Friswell, M.I.: Calculation of second- and higher order eigenvector derivatives. J. Guid. Control. Dyn. 18, 919–921 (1995) Li, L., Hu, Y., Wang, X.: A parallel way for computing eigenvector sensitivity of asymmetric damped systems with distinct and repeated eigenvalues. Mech. Syst. Signal Process. 30, 61–77 (2012) Weng, S., Zhu, H.P., Xia, Y., et al.: Substructuring approach to the calculation of higher-order eigensensitivity. Comput. Struct. 117, 23–33 (2013) Wu, B.S., Xu, Z.H., Li, Z.G.: Improved Nelson’s method for computing eigenvector derivatives with distinct and repeated eigenvalues. AIAA J. 45, 950–952 (2007)

Chapter 5

Iterative Bisection Scanning Substructuring (IBSS) Method for Eigensolution and Eigensensitivity

5.1 Preview In Chaps. 2–4, the master modes in the substructures are assembled to represent the eigensolutions and eigensensitivities of the global structure. The reduced eigenequation is obtained by approximating a non-linear item with the truncation of the Taylor expansion. Consequently, it results in some slight errors in the eigensolutions and associated eigensensitivities. This chapter introduces the IBSS method (Weng et al., 2011) to achieve higher accuracy of eigensolutions and eigensensitivities. The method estimates the contribution of the higher modes in an iterative form, from which the eigensolutions can be obtained. The eigensensitivities can also be derived in an iterated form based on this iterative eigenequation. Upon convergence, the iterative scheme accurately reproduces the eigensolutions and eigensensitivities of the original structure. The accuracy and efficiency of the method are finally verified by a cantilever plate and the Canton Tower.

5.2 IBSS Method for Eigensolution According to Chap. 2, the eigenequation of Kron’s substructuring method can be dissembled according to the master and slave modes as ⎤⎧ ⎫ ⎧ ⎫ 0 −| m ⎨ zm ⎬ ⎨ 0 ⎬ /\pm − λI ⎣ 0 /\ps − λI −| s ⎦ zs = 0 ⎩ ⎭ ⎩ ⎭ T −| m −| Ts 0 τ 0 ⎡

(5.1)

From the second line of Eq. (5.1), one has

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_5

85

86

5 Iterative Bisection Scanning Substructuring (IBSS) Method …

)−1 ( zs = /\ps − λI | s τ = tτ

(5.2)

This relationship introduces (Lin and Xia, 2003; Xia and Lin, 2004) (

/\ps − λI

)−1

|s = t

(5.3)

Equation (5.3) can be rewritten as ( )−1 ( )−1 t = /\ps | s + λ /\ps t

(5.4)

Considering Eq. (5.2), the complete set of the eigenvectors in Eq. (5.1) is then expressed as ⎤ ⎧ ⎫ ⎡ { } I0 { } ⎨ zm ⎬ z ⎢ ⎥ zm = T1 m =⎣0t⎦ z ⎩ s⎭ τ τ τ 0I

(5.5)

Substituting Eq. (5.5) into Eq. (5.1) and premultiplying T1 on both sides of Eq. (5.1) reduces the eigenequation to |

/\pm − λIm −| m −| Tm −| Ts t

|{

zm τ

}

{ } 0 = 0

(5.6)

The second line of Eq. (5.6) gives ( )−1 τ = − | Ts t | Tm zm

(5.7)

Accordingly, the eigenvector of the reduced eigenequation (Eq. 5.6) is expressed as {

zm τ

|

} =

| I ( )−1 zm = T2 zm − | Ts t | Tm

(5.8)

Substituting Eq. (5.8) into Eq. (5.6) and premultiplying T2 on both sides of Eq. (5.6) gives |(

) ( )−1 | /\pm − λIm + | m | Ts t | Tm {zm } = 0

(5.9)

As | Ts t = Dops t in Eq. (5.9) and D is a constant matrix, 0ps t is required to solve the reduced eigenequation. Premultiplying 0ps on both sides of Eq. (5.4) gives ( )−1 | p |T T ( )−1 0s D + λ0ps /\ps t ξ = 0ps t = 0ps /\ps

(5.10)

5.2 IBSS Method for Eigensolution

87

| |T Due to the orthogonality 0ps Mp 0ps = In s , it has ( )−1 | p |T ( )−1 | p |T T 0s 0s D + λ0ps /\ps ξ = 0ps t =0ps /\ps Mp 0ps t = Fp DT + λFp Mp ξ

(5.11)

( )−1 | p |T 0s is the residual flexibility of the substructures, as given where Fp = 0ps /\ps in Eq. (2.48). Finally, the reduced eigenequation Eq. (5.9) can be rewritten in a simple form as [Kd ]{zm } = λ{zm }

(5.12)

| | where Kd = /\pm + | m (Dξ)−1 | Tm . This reduced eigenequation (Eq. 5.12) can reproduce the eigensolutions of the global structure exactly. As ξ includes unknown λ, an iterative process is required to solve Eq. (5.12). From Eq. (5.11), the iteration starts with ξ[1] as ξ[1] = Fp DT

(5.13)

where the number in the square bracket indicates the iteration step. With the initial value ξ[1] , the eigensolutions can be calculated simultaneously for all modes by | ( )−1 |{ [1] } [1] { [1] } /\pm + | m DFp DT | Tm zm = λ zm

(5.14)

From Eq. (5.11), the iteration formulae can be established (k ≥ 2) as follows. (1) [k−1] p

ξ[k] = Fp DT + λ

F Mp ξ[k−1]

(5.15)

(2) ( )−1 Kd[k] = /\pm + | m Dξ[k] | Tm

(5.16)

[k]

(3) Calculate the eigenvalue λ in the kth iteration by the QR algorithm or Cholesky factorization (Bathe and Wilson 1989; Anderson et al. 1999) of Kd[k] . When the eigenvalue in the kth iteration reaches the required accuracy, the [k] [k] [k] [k] [k] = λ zm is solved to estimate both λ and zm . The eigenequation Kd[k] zm p [k] eigenvector of the global structure is then recovered by o = 0m zm . It should be noted that the initial eigenequation at Eq. (5.14) is equivalent to the FRFS method and the second iteration is equivalent to the SRFS method as introduced in Chap. 2.

88

5 Iterative Bisection Scanning Substructuring (IBSS) Method … [k−1]

In Eq. (5.15), ξ[k] = Fp DT + λ Fp Mp ξ[k−1] is calculated at the substructure ( ( j) )[k] [k−1] ( j ) ( j ) ( ( j) )[k−1] level. In other words, ξ F M ξ can be calculated = F( j) DT +λ for the jth substructure ( j = 1, 2,…, N S ) independently and then assembled in the diagonal form. Equation (5.11) reveals that ξ depends on λ, which varies for different modes. The iteration must thus be performed mode by mode. In practice, the eigensolutions of the lower modes generally converge faster than those of the higher modes. During the iteration process, only item ξ needs to be recalculated, while other items, such as /\pm and | m , remain unchanged. Furthermore, the size of the reduced system matrix Kd is equal to the number of the master modes of the substructures. Thus, the iteration process does not require much additional computational power.

5.3 IBSS Method for Eigensensitivity This subsection derives the first-order derivatives of the eigenvalues and eigenvectors with respect to an elemental parameter r. The reduced eigenequation (Eq. 5.9) can be rewritten for the ith mode as |( p ) | /\m − λi Im + | m (Dξ)−1 | Tm {zi } = {0}

(5.17)

Equation (5.17) is differentiated with respect to r as |(

)

| ∂{z } ∂ p i /\m − λi Im + | m (Dξ)−1 | T + m ∂r

|(

) | p /\m − λi Im + | m (Dξ)−1 | T m ∂r

{zi } = {0}

(5.18)

5.3.1 Eigenvalue Derivatives Premultiplying {zi }T on both sides of Eq. (5.18) gives |( ) | T ∂{zi } {zi }T /\pm − λi Im + |m (Dξ)−1 |m + {zi }T ∂r ) | |( p T ∂ /\m − λi Im + |m (Dξ)−1 |m {zi } = {0} ∂r

(5.19)

Given Eq. (5.17), Eq. (5.19) can be reduced to | )| ( p ∂ | m (Dξ)−1 | Tm ∂λi T ∂/\m {zi } = {zi } + ∂r ∂r ∂r

(5.20)

5.3 IBSS Method for Eigensensitivity

89

where ) ( T ∂ |m (Dξ)−1 |m ∂|m T − |m (Dξ)−1 = (Dξ)−1 |m ∂r ∂r ∂(Dξ) ∂| T T + |m (Dξ)−1 m (Dξ)−1 |m ∂r ∂r ∂0p ∂| Tm =D m ∂r ∂r

(5.21)

(5.22) ∂/\p

| m , {zi }, and (Dξ)−1 were obtained during the calculation of the eigensolutions. ∂rm ∂0p and ∂rm are associated with the eigensolution derivatives of the master modes of the substructures except the specific substructure containing r, as given in Eq. (3.38). ∂/\(Q) ∂0(Q) Assume r is located in the Qth substructure, ∂rm and ∂rm can be calculated using traditional approaches such as the modal method (Fox and Kapoor, 1968) or Nelson’s method (Nelson 1976; Wu et al., 2007) by treating the Qth substructure as an indeis thus required to calculate the first-order derivative of the pendent structure. ∂(Dξ) ∂r eigenvalues. According to Eq. (5.11), ) ( p ∂Fp T ∂λ p p ∂ξ ∂F p p p ∂ξ = D +λ M ξ+F M + F M ξ ∂r ∂r ∂r ∂r ∂r

(5.23)

∂Fp ∂r

where

can be computed from the Qth substructure according to Eq. (3.38). ( )[1] ∂ξ ∂ξ An iterative process is required to calculate accurate ∂r . In the first step, ∂r =

∂Fp T D . ∂r

(

∂ξ ∂r

Afterward,

)[k]

∂ξ ∂r

in the kth (k ≥ 2) iteration is expressed as

( ( )[k−1] ) [k−1] ∂Fp p ∂Fp T ∂λ p p ∂ξ D +λ M ξ+F M Fp Mp ξ (5.24) = + ∂r ∂r ∂r ∂r

and the eigenvalue derivative in the kth step is obtained as (

∂λi ∂r

)[k]

|

p

∂/\m ∂|m T + − |m (Dξ)−1 (Dξ)−1 |m ∂r ∂r ( ( ) ) | T ∂ξ [k] −1 T −1 ∂|m {zi } D (Dξ) |m + |m (Dξ) ∂r ∂r

={zi }

T

(5.25)

Similar to eigensolutions, the eigensensitivities can be calculated for all modes simultaneously in the initial iteration. Thereafter, the iterations are performed mode ∂ξ needs to be recalculated, as the other items remain by mode. In each iteration, only ∂r unchanged.

90

5 Iterative Bisection Scanning Substructuring (IBSS) Method …

5.3.2 Eigenvector Derivatives The ith eigenvector of the global structure can be recovered by calculating oi = 0pm {zi }

(5.26)

Differentiating Eq. (5.26) with respect to parameter r gives } { ∂oi ∂0mp p ∂zi {zi } + 0m = ∂r ∂r ∂r

(5.27) ∂0p

In Eq. (5.27), {zi } and 0pm have been obtained by method in Chap. 2. ∂rm are the eigenvector derivatives of the substructural master modes. { iThe } derivative primcan be calculated itive matrices are nonzero only for the Qth substructure. ∂z ∂r directly from Eq. (5.18) by applying Nelson’s method (Nelson 1976) to the reduced eigenequation. It is noted that since only one substructure (the Qth substructure) needs to be analyzed to recover the eigensensitivities with respect to parameter r, the IBSS method can be efficient. The calculation of the eigenvector derivatives of the global structure does not need to be iterative. The calculation of the eigenvector derivatives usually costs a great deal more computational time and resources than the calculation of the eigensolutions and eigenvalue derivatives. The several iterations required by the eigenvalue derivative are negligible. In this chapter, the eigenvalue and eigenvector derivatives are derived with respect to the stiffness parameters. The derivative of the stiffness matrix is used in deriving the eigensensitivity formulas, and the derivative of the mass matrix is zero. The formulas can also be generalized to calculate the eigensensitivity with respect to the mass parameters, and both Fox and Kappor’s method and Nelson’s method can be employed similarly while the derivative to the stiffness parameters is zero (Xia 2002).

5.4 Examples 5.4.1 A Cantilever Plate The cantilever plate in Fig. 5.1 is utilized here to investigate the accuracy of the proposed IBSS method for the eigensolutions and eigensensitivities. The dimensions of the plate are 4000 × 2000 × 10 mm3 . It is modeled by 40 × 20 = 800 elements, each with the size of 100 × 100 × 10 mm3 . The plate structure includes 861 nodes and 4920 DOFs in total. The material properties of the plate are Young’s modulus (E) = 206 GPa, mass density (ρ) = 7800 kg/m3 , and Poisson’s ratio = 0.3. The plate is partitioned into eight substructures averagely (N S = 8) as demonstrated in Fig. 5.1.

5.4 Examples

91

Fig. 5.1 The cantilever plate

The interface nodes are shared by two substructures at the sides of substructures, and some interface nodes are shared by four substructures at the corners of substructures. The traditional Lanczos method (Bathe and Wilson 1989) introduced in Sect. 2.2.2 is first performed on the global structure to calculate the first 10 eigensolutions, which are treated as the exact values for reference. Afterward, the proposed IBSS method is utilized to calculate the eigensolutions of the global structure. The first 20 modes in each substructure are chosen as the master modes, where the number of master modes in each substructure is normally about twice or thrice the number of modes required by the global structure. The iteration terminates when the relative difference of the frequencies between two consecutive iterations is less than 10−6 , with convergence criterion set to Tol = 10−6 . Table 5.1 demonstrates the convergence of the first 10 frequencies. The initial step in Table 5.1 denotes the results of the substructuring method without iteration. The frequencies in the initial step are insufficiently accurate for the higher modes. The predefined criterion of 10−6 can be achieved for all modes after a few iterations. Table 5.1 also compares the SV values (Eq. (2.66)) of the eigenvectors obtained using the proposed substructuring method with those derived using the global method. The SV values of the eigenvectors are about 0.99 in the initial step and are improved to greater than 0.999 after a few iterations. The eigenvalue derivatives and eigenvector derivatives for the first 10 modes of the global structure with respect to parameter r 1 are then calculated, and parameter r 1 is the flexural rigidity of the element denoted in Fig. 5.1. The results from Nelson’s method (Fox and Kapoor, 1968; Nelson 1976) performed at the global structure level is treated as exact for comparison. The convergence process of the eigenvalue derivatives is detailed in Table 5.2. The eigenvalue derivatives achieve high precision in

3.306581

6.284149

0.9999

1.0000

SV (Initial step)

SV (Final step)

1.0000

0.9999

1.0000

0.9995

0.9999

0.9982

9.49 × 10−8

11.753305

1.0000

0.9972

1.52 × 10−7

13.459178

13.459186 13.459181

11.753309 11.753306

6.284150

13.459277

13.459552

5

11.753401

11.753511

4

6.284152

6.284166

3

Relative 1.61 × 10−8 1.93 × 10−8 1.12 × 10−7 error

Global method

5

4

2.158828

3.306582

3.306581

2.158829

2.158828

1

2

3

2

1

Iteration Natural frequencies (Hz)

Table 5.1 Convergence of the natural frequencies of the cantilever plate

0.9998

0.9967

2.52 × 10−7

15.201194

15.201197

15.201205

15.201391

15.201747

6

0.9999

0.9955

6.21 × 10−8

0.9999

0.9931

1.55 × 10−7

20.955026

20.955029

19.330453 19.330451

20.955039

20.955092

20.956028

20.957231

8

19.330458

19.330548

19.330951

19.331835

7

0.9997

0.9937

4.49 × 10−7

26.479861

26.479873

26.479891

26.479953

26.480871

26.483308

9

1.0000

0.9975

2.78 × 10−7

32.394239

32.394248

32.394258

32.394389

32.396158

10

92 5 Iterative Bisection Scanning Substructuring (IBSS) Method …

5.4 Examples

93

just a few runs. The accuracy of the eigenvalue derivatives depends on the eigenso∂ξ , and ξ, and the slight error on eigensolutions will influence the precision lutions, ∂r of eigensensitivity. In consequence, the eigenvalue derivatives are not as accurate as the eigenvalues. Without loss of generality, the eigensensitivities with respect to parameter r 2 , which is the flexural rigidity of the element located in the 5th substructure (Fig. 5.1), are calculated. Table 5.3 compares the eigensensitivities obtained from three approaches, namely the global method, the substructuring method without iteration (static substructuring method), and the substructuring method with iteration (IBSS method). The SV values given in Eq. (3.51) are used to estimate the similarity of the eigenvector derivatives obtained using the substructuring method and traditional global method. Table 5.3 again shows that the IBSS method improves the accuracy of eigensensitivities significantly. In the next example, the computational efficiency of the IBSS method is investigated through a relatively large structure.

5.4.2 The Canton Tower The Canton Tower (previously known as Guangzhou New TV Tower) is a super-tall structure 610 m high that consists of a main tower (454 m) and an antennary mast (156 m), as shown in Fig. 5.2a. The structure comprises a reinforced concrete inner tube and a steel outer tube with concrete-filled-tube columns (Ni et al., 2009). The FEM of the structure includes 8738 three-dimensional elements, 3671 nodes (each of which has six DOFs), and 21, 690 DOFs in total (Fig. 5.2b). The global structure is divided into 10 substructures along the vertical direction (Fig. 5.2c). The nodes and elements included in each substructure are listed in Table 5.4. To evaluate the accuracy and computational efficiency of the IBSS method, the traditional methods performed on the global structure without substructuring are also employed for comparison and the results are treated as exact. The first 20 master modes are retained in each substructure to calculate the eigensolutions and eigensensitivities of the first 10 modes of the global structure by the proposed IBSS method. Again, the convergence criterion is set to Tol = 10−6 . Figure 5.3 illustrates the relative errors of the first and tenth frequencies obtained with the proposed method in each iteration as compared to the global method, and Fig. 5.4 shows the relative errors of the first and tenth eigenvalue derivatives with respect to the flexural rigidity of an arbitrarily selected element. The natural frequency and eigenvalue derivative of the first mode have a high accuracy at the initial step, whereas those of mode 10 have a relatively larger error, as expected. After a few iterations, the accuracy of the tenth mode is improved significantly. The computation time in calculating the first 10 eigensolutions and eigensensitivities of the global structure is presented in Table 5.5. The global method takes 11.6 s to calculate the first 10 eigensolutions of the global structure, whereas using the IBSS method the initial step takes about 34.7 s and each iteration adds about 3.0 s. In total, 32 iterations are required to satisfy the predefined accuracy, which takes about 131.0 s.

3.53 × 10−7

19.695404

19.695411

9.259601

9.259600

19.695423

19.695705

19.697504

4

9.259603

9.259619

9.260046

3

Relative 4.13 × 10−8 3.25 × 10−7 1.21 × 10−7 error

Global method

6

5

2.744548

3

0.437232

2.744549

0.437232

2

4

2.744570

2.744553

0.437234

0.437233

1

2

1

Iteration Eigenvalue derivative

2.52 ×10−7

14.630253

14.630257

14.630267

14.630374

14.631448

14.637250

5

7.41 × 10−7

51.591211

51.591250

51.591277

51.592160

51.611272

6

5.52 × 10−7

102.011633

102.011689

102.012184

102.012452

102.017643

102.062836

7

Table 5.2 Convergence of the eigenvalue derivatives with respect to parameter r 1 of the cantilever plate

1.91 × 10−6

15.151120

15.151149

15.151259

15.151466

15.152592

15.157321

15.172530

8

2.48 × 10−6

146.635493

146.635857

146.636403

146.636961

146.646961

146.762695

9

6.57 ×10−7

90.621631

90.621691

90.621751

90.621974

90.622886

90.654473

10

94 5 Iterative Bisection Scanning Substructuring (IBSS) Method …

5.4 Examples

95

Table 5.3 Eigensensitivity of the cantilever plate with respect to parameter r 2 Mode Global method

Static substructuring method

IBSS method

Eigenvalue Eigenvalue Relative Eigenvector Eigenvalue Relative Eigenvector derivative derivative error derivative derivative error derivative (SV) (SV) 1

0.112006

0.112004

1.17 × 10−5

0.998

0.112006

3.67 × 10−7

1.000

2

0.272867

0.272862

1.87 × 10−5

0.999

0.272867

1.20 × 10−7

1.000

3

2.045356

2.045367

5.38 × 10−6

0.998

2.045357

2.09 × 10−7

0.999

4

6.308775

6.310119

2.13 × 10−4

0.989

6.308780

8.16 × 10−7

0.998

5

7.966156

7.965806

4.39 × 10−5

0.996

7.966150

6.95 × 10−7

1.000

6

38.637509

38.635993

3.92 × 10−5

0.993

38.637402

2.78 × 10−6

1.000

7

38.933940

38.935761

4.68 × 10−5

0.990

38.933941

2.36 × 10−8

0.999

8

15.135030

15.130333

3.10 × 10−4

0.975

15.135007

1.50 × 10−6

0.997

9

46.854527

46.859268

1.01 × 10−4

0.989

46.854727

4.27 × 10−6

1.000

10

6.655491

6.593817

9.27 × 10−3

0.985

6.655468

3.53 × 10−6

0.998

Although the IBSS method takes longer than the global method for eigensolutions, it contributes to calculating eigensensitivities, which is more time-consuming. The global method takes about 197.6 s to calculate the first 10 eigensensitivities with respect to one parameter. The proposed IBSS method takes about 13.2 s to perform the initialization step, and each iteration adds just 2.6 s to the computation time. Moreover, the calculation of the eigenvector derivatives, which usually consumes most of the computation time in the global eigensensitivity method, takes only 0.8 s. In this case study, 39 iterations are required to achieve the predefined accuracy for the eigensensitivities of the ten modes, which takes about 115.4 s in total. This improvement in computational efficiency can be significant when applied to a practical model updating process, as actual structures always include a large number of uncertain parameters. For example, as regards this Canton Tower structure, the column of the steel outer tube is composed of 1104 three-dimensional beam elements. If the stiffness of the 1104 elements is chosen as updated candidates, the global method requires 51.9 h to calculate the first 10 eigensensitivities, whereas

96

5 Iterative Bisection Scanning Substructuring (IBSS) Method …

(a) Landscape view

(b) Global model

(c) Substructures

Fig. 5.2 Canton Tower and the FE model

the proposed substructuring method requires 32.4 h only. Given the time needed to calculate the eigensensitivities, the time spent in deriving the eigensolutions is negligible. The substructuring method also reduces the computational memory required. For example, the global method needs to handle the global stiffness and mass matrices, which are 22, 026 × 22, 026 in size. Even if the matrices are sparse, up to 2151 MB of memory is needed to acquire the eigensolutions and eigensensitivities. However, using the substructuring method, 10 substructures are analyzed independently for eigensolutions, each with a size of about 2200 × 2200 and a half-bandwidth of about

No. of tearing nodes

56

456

336

657

No. of nodes

No. of elements

945

Sub 2

Sub 1

56

873

432

Sub 3

56

873

432

Sub 4

Table 5.4 Division formation for the FE model of the Guangzhou Tower

56

786

336

Sub 5

56

786

336

Sub 6

56

873

432

Sub 7

56

846

440

Sub 8

56

990

488

Sub 9

56

1109

487

Sub 10

5.4 Examples 97

98

5 Iterative Bisection Scanning Substructuring (IBSS) Method …

Fig. 5.3 Convergence of the frequencies

10

Relative error (Log)

10

10

10

10

10

10

-2

1st mode 10th mode

-3

-4

-5

-6

-7

-8

1

2

3

4

5

6

Iterative step

Fig. 5.4 Convergence of the eigenvalue derivatives

Relative error (Log)

10

10

10

10

10

1st mode 10th mode

0

-2

-4

-6

-8

1

2

3

4

5

6

7

Iterative step

600. The assembled eigenequation of the global structure is only 200 × 200 in size. To calculate the eigensensitivities with respect to an elemental parameter, only one substructure with a size of 2200 × 2200 and a reduced eigenequation with a size of 200 × 200 are analyzed. The substructuring method thus requires only 338 MB of computer memory to estimate the eigensolutions and eigensensitivities, as listed in Table 5.5. These findings indicate that the IBSS method will be very useful for large-size structures. Afterward, the IBSS method is compared with the conventional substructuring method (FRFS method in Chap. 2.3.3). The FRFS method approximately recovers the eigensolutions and eigensensitivities of the global structure by some master modes and the first-order residual flexibility of the substructures. It is equivalent to the initial step of the proposed IBSS method without iteration. Although including more master

195.8 16.8

209.6 (207.2 + 2.4)

0.8

Eigenvector derivative

1.8

2.6 × 39

Iteration

The numbers in bold are used to clearly show the results of the total computational time and memory consumed by different methods

6.7

423.3

416.6

131.0

13.2

0.3

3.0 × 32

Conventional FRFS method (500 master modes)

34.7 11.6

Proposed IBSS method

Global method

Initialization

Iteration

Eigenvalue derivative

Initialization

Eigensensitivity Total

Eigenvalue

Eigenvector

Eigensolution

CPU time (s)

226.4

197.6

115.4

Total

304

2151

338

Memory (MB)

Table 5.5 Comparison of computation time and storage memory required by the proposed IBSS method, global method, and conventional FRFS method

5.4 Examples 99

100

5 Iterative Bisection Scanning Substructuring (IBSS) Method …

modes can improve the accuracy of the FRFS method, the cost of computation time for this precision improvement is luxurious. To demonstrate this, the master modes retained in each substructure of the Canton Tower increase from 50 to 500 gradually to calculate the first 10 eigensolutions and eigensensitivities of the global structure using the conventional approach. The computational accuracy and computation time for the first 10 eigensolutions and eigensensitivities with respect to the number of master modes are illustrated in Figs. 5.5 and 5.6. For clarity, only the first mode and the tenth mode are plotted. At least 500 master modes need to be retained in each substructure by the FRFS method to achieve the accuracy of the relative error of 10−6 . That is, 500 master modes need to be extracted from each substructure, and the resulting reduced eigenequation has the size of 5000 × 5000. Including more master modes improves the accuracy but increases the computation time heavily. -3

10

450 1st mode 10th mode

Computation time

-4

Relative error (Log)

10

350 300

-5

10

250 200

-6

10

150

Computation time(Second)

400

100 -7

10

0

50

100

150

200

250

300

350

400

450

500

50 550

Master modes

Fig. 5.5 Accuracy and computation time of eigensolutions with respect to the number of master modes using the FRFS method

5.4 Examples

101

-1

10

250

1st mode 10th mode

Computation time

-2

200

-3

10

150

-4

10

100

-5

10

50

-6

10

0

Computation time(Second)

Relative error (Log)

10

50

100

150

200

250 300 Master modes

350

400

450

500

0 550

Fig. 5.6 Accuracy and computation time of eigensensitivities with respect to the number of master modes using the FRFS method

The computation time and computer memory consumed by the proposed IBSS method and conventional FRFS method are compared in Table 5.5, both satisfying the tolerance of 10−6 . As compared with the FRFS method, the IBSS method is a little more complicated and has to perform some iterations to achieve the accurate results. Nevertheless, the IBSS method takes much shorter computation time considering two reasons: (1) only 20 master modes are required from the independent substructures, avoiding extraction of 500 master modes by the FRFS method, and (2) the 20 master modes lead to the reduced eigenequation of the IBSS method of size 200 × 200, which is much smaller than the size of the FRFS method of 5000 × 5000. In this example, the FRFS method costs about 416.6 s to extract the 500 master eigenmodes and calculate the large-size residual flexibility matrix from the independent substructures and costs 6.7 s to solve the eigenequation of 5000 × 5000. The FRFS method consumes 207.2 s to compute the substructural derivative matrices, and then, 2.4 s are required to assemble the eigenvalue derivatives and 16.8 s to solve the eigenvector derivatives. Although the IBSS performs several iterations for eigensolutions and their derivatives, it requires fewer masters modes in substructures which leads to a smaller size of eigenequation. Based on the above observation, the proposed IBSS method is more efficient than the conventional FRFS method, when high-accuracy eigensolutions and eigensensitivities are required.

102

5 Iterative Bisection Scanning Substructuring (IBSS) Method …

5.5 Summary This chapter introduces the IBSS method to compute the eigensolutions and eigensensitivities. The eigensolutions and eigensensitivities of the global structure are computed by a small amount of the master modes of the substructures, and the contribution of the higher modes is compensated by as a residual flexibility matrix in an iterative form. The iterative process is mainly performed at the substructure level, which adds only a small amount of extra computation time. The proposed method can predict the eigensolutions and eigensensitivities accurately by a small amount of master modes and a small-size eigenequation in just a few iterations. The method is thus computationally efficient, especially for large structural systems. Different from the previously introduced FRFS methods in Chap. 2, the IBSS method can achieve high accuracy using a few master modes on a reduced eigenequation by an iterative scheme. The computational accuracy of IBSS method is achieved by iterations on small-size eigenequation without enlarging the size of the retained modes of the substructures. The computational effort in analyzing the independent substructures is reduced, and the assembled eigenequation is kept in small size. The IBSS method is more efficient than the FRFS method when highly accurate results are required.

References Anderson, E., Bai, Z. et al.: LAPACK User’s Guide, Third Edition, SIAM, Philadelphia (1999). http://www.netlib.org/lapack/lug/lapack_lug.html Bathe, K.J., Wilson, E.L.: Numerical Methods in Finite Element Analysis. Prentice-Hall Inc., Englewood Cliffs, New Jersey (1989) Fox, R.L., Kapoor, M.P.: Rate of change of eigenvalues and eigenvectors. AIAA J. 6, 2426–2429 (1968) Lin, R.M., Xia, Y.: A new eigensolution of structures via dynamic condensation. J. Sound Vib. 266(1), 93–106 (2003) Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14(9), 1201–1205 (1976) Ni, Y.Q., Xia, Y., Liao, W.Y., et al.: Technology innovation in developing the structural health monitoring system for Guangzhou New TV Tower. Struct. Control. Health Monit. 16(1), 73–98 (2009) Weng, S., Xia, Y., Xu, Y.L., et al.: An iterative substructuring approach to the calculation of eigensolution and eigensensitivity. J. Sound Vib. 330, 3368–3380 (2011) Wu, B.S., Xu, Z.H., Li, Z.G.: Improved nelson’s method for computing eigenvector derivatives with distinct and repeated eigenvalues. AIAA J. 45, 950–952 (2007) Xia, Y., Lin, R.M.: A new iterative order reduction (IOR) method for eigensolutions of large structures. Int. J. Numer. Meth. Eng. 59, 153–172 (2004) Xia, Y.: Condition Assessment of Structures Using Dynamic Data. Ph.D Thesis, Nanyang Technological University, Singapore (2002)

Chapter 6

Simultaneous Iterative Substructuring Method for Eigensolutions and Eigensensitivity

6.1 Preview In Chap. 5, the IBSS method is introduced to improve the accuracy of the FRFS method in calculating eigensolutions and eigensensitivities of large-scale structures. The method compensates the contribution of higher modes in an iterative form to construct a reduced eigenequation, based on which the eigensolutions and eigensensitivities can be calculated more accurately with fewer master modes. However, the reduced system matrices in the IBSS method are frequency-dependent; thus, the eigensolutions and eigensensitivities have to be calculated in individual iterations mode by mode. This is computationally demanding when numerous modes are required. In this context, this chapter introduces a simultaneous iterative substructuring (SIS) method (Tian et al. 2019) to compute the eigensolutions and eigensensitivities of many modes simultaneously in the same iterations. The method derives a modal transformation matrix relating the master modes to the slave modes. The system matrices, which are frequency-dependent in the original eigenequation, are transformed into frequency-independent ones. The modal transformation matrix is obtained with an iterative process, and it has a much smaller order than the iterative variables employed in the IBSS method (Weng et al. 2011). Therefore, the method improves computational efficiency significantly. Consequently, the eigensolutions and eigensensitivities of required modes are calculated simultaneously in the same iterations. Finally, the precision and efficiency of different methods in calculating eigensolutions and eigensensitivities are compared using a numerical frame model and the Wuhan Yangtze River Navigation Center.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_6

103

104

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

6.2 SIS Method for Eigensolution The SIS method is developed on the basis of Eqs. (5.1)–(5.7) in Chap. 5. By substituting Eq. (5.7) into Eq. (5.2), zs is expressed by zm as ( )−1 zs = −t | Ts t | Tm zm

(6.1)

Substituting Eq. (5.7) into the first line of Eq. (5.6), one can obtain ( ) ( )−1 /\pm + | m | Ts t | Tm − λIm zm = 0

(6.2)

Equation (6.2) is the reduced eigenequation of Kron’s substructuring method p with the order of Nm , which is equal to the number of master modes of all substruc( T )−1 T tures. | m | s t | m is a frequency-dependent matrix, and consequently, Eq. (6.1) cannot be solved directly. The IBSS method implemented an iterative process based on Eq. (5.4) to solve Eq. (6.2), which is precise but inefficient as the eigensolutions and eigensensitivities have to be calculated mode by mode. In this chapter, a simplified process will be derived to transform the frequency-dependent matrix ( )−1 | m | Ts t | Tm into a frequency-independent one. This ensures that eigensolutions and eigensensitivities can be calculated efficiently for all modes simultaneously. According to Eq. (5.4), the matrix inversion lemma is used to expand the )−1 ( frequency-dependent item | Ts t as \ |−1 )−1 ( T )−1 ( T \ p |−1 |s t = | s /\s | s + λ| Ts /\ps t )−1 )−1 \ | ( ( \ | ( \ | )−1 −1 −1 −1 | Ts /\ps t | Ts t = | Ts /\ps | s − λ | Ts /\ps | s

(6.3)

\ |T )T ( Considering the orthogonal condition of 0ps Mp 0ps = In s and | s = D0ps , Eq. (6.3) is rewritten as (

| Ts t

)−1

( \ |−1 \ p |T T )−1 0s D = D0ps /\ps ( \ |−1 \ p |T T )−1 \ |−1 \ p |T p p ( T )−1 0s D 0s M 0s t | s t D0ps /\ps − λ D0ps /\ps ( p T )−1 p p p ( T )−1 ( p T )−1 = DF D − λ DF D DF M 0s t | s t (6.4)

)−1 ( As a consequence, the frequency-dependent item | m | Ts t | Tm zm in Eq. (6.2) is rewritten as ( )−1 ( ( )−1 )−1 | m | Ts t | Tm zm = | m DFp DT | Tm zm − λ| m DFp DT DFp Mp ηzm

(6.5)

6.2 SIS Method for Eigensolution

105

where ( )−1 η = 0ps t | Ts t | Tm

(6.6)

It is evident from Eq. (6.1) that ηzm = −0ps zs . Here, a modal transformation matrix T is introduced to represent the contribution of the slave modes in terms of master modes. It satisfies (Cui et al. 2016) Tzm = −0ps zs = ηzm

(6.7)

p

Accordingly, the entire Nm master modes satisfy | \ | p p p | 1 2 TZm = Tz1m Tz2m · · · TzmNm = η1 zm η2 zm · · · η Nm zmNm

(6.8)

where the superscript in η and zm|represents the substructural mode that the variables | p 1 2 Nm are associated with; and Zm = zm , zm , . . . , zm encloses the mode participation p

factors zm of the entire Nm substructural master modes. Zm is a full-rank matrix p p with the size of Nm × Nm . Accordingly, T is associated with all substructural master modes as \ p p | 1 2 T = η1 zm η2 zm · · · η Nm zmNm Z−1 m

(6.9)

In Eqs. (6.7)–(6.9), ηzm is replaced by Tzm . The transformation matrix T is associated with all the substructural master modes rather than a specific mode, which is helpful for all modes to be calculated simultaneously in later studies. Substituting Eqs. (5.4) and (6.6) into Eq. (6.7) leads to ( )−1 Tzm = ηzm = 0ps t | Ts t | Tm zm (\ | \ |−1 )( )−1 −1 = 0ps /\ps | s + λ /\ps t | Ts t | Tm zm \ |−1 \ p |T T ( T )−1 T = 0ps /\ps 0s D | s t | m zm \ | \ p |T p p ( T )−1 T −1 + λ0ps /\ps 0s M 0s t | s t | m zm ( T )−1 T p T = F D | s t | m zm + λFp Mp Tzm

(6.10)

( )−1 According to Eq. (6.4), the first item Fp DT | Ts t | Tm zm in Eq. (6.10) is expressed as ( ( )−1 )−1 Fp DT | Ts t | Tm zm = Fp DT DFp DT | Tm zm ( )−1 ( )−1 − λFp DT DFp DT DFp Mp 0ps t | Ts t | Tm zm

106

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

( )−1 = Fp DT DFp DT | Tm zm ( )−1 − λFp DT DFp DT DFp Mp Tzm

(6.11)

Substituting Eq. (6.11) into Eq. (6.10) leads to ) ( ( )−1 ( )−1 Tzm = Fp DT DFp DT | Tm zm + λ Fp − Fp DT DFp DT DFp Mp Tzm ( ) = TC + λSMp T zm (6.12) ( )−1 ( )−1 where TC = Fp DT DFp DT | Tm and S = Fp − Fp DT DFp DT DFp . From Eqs. (6.5) and (6.7), the original eigenequation Eq. (6.2) is simplified into | | || ( )−1 ( )−1 p /\pm + | m DFp DT | Tm − λ Im + | m DFp DT DFp Mp T zm = 0

(6.13)

\ |−1 (\ p |T p p )\ p |−1 \ p |T \ |−1 \ p |T 0s K 0s /\s 0s = 0ps /\ps 0s = Fp , As Fp Kp Fp = 0ps /\ps it has ( )−1 ( )−1 ( )−1 TCT Kp TC = | m DFp DT DFp Kp Fp DT DFp DT | Tm = | m DFp DT | Tm (6.14) Accordingly, Eq. (6.13) is further simplified into ( ) KC − λM D zm = 0

(6.15)

( )−1 KC = /\pm + | m DFp DT | Tm = /\pm + TCT Kp TC

(6.16)

( )−1 p p M D = Im + | m DFp DT DFp Mp T = Im + TCT Mp T

(6.17)

where

In this chapter, the original eigenequation (Eq. 6.2) is simplified into a new one p (Eq. 6.15). Equation (6.15) has the same order as Eq. (6.2) of Nm , but its system matrix MD is frequency-independent. KC is a symmetric matrix calculated from the constant value TC . Only MD is unknown and required to solve the new eigenequation (Eq. (6.15)). This ensures the eigensolutions of all required modes to be solved simultaneously. Equation (6.17) shows that MD is relevant to the modal transformation matrix T. An iterative process will be proposed as follows to calculate T.

6.3 SIS Method for Eigensensitivity

107

Equation (6.15) implies λzm = M−1 D KC zm . Accordingly, Eq. (6.12) is therefore simplified into ) ( Tzm = TC + SMp TM−1 D KC zm

(6.18)

Equation (6.18) is satisfied for master modes, and T is therefore expressed as T = TC + SMp TM−1 D KC

(6.19)

Consequently, the modal transformation matrix T is calculated iteratively from p p Eqs. (6.17) and (6.19). T has the size of Nm × Nm , which is much smaller than the p p iterative variable t of N × Nm utilized in the IBSS method. The order of the iterative variables in the IBSS method is reduced greatly, which is significant to improve the computational efficiency. Once the modal transformation matrix T is available, λ and zm are calculated from Eq. (6.15) simultaneously for all master modes. Based on Eq. (6.7), the eigenvector of the assembled global structure is expressed in the modal coordinate as ( ) 0 = 0pm zm + 0ps zs = 0pm − T zm

(6.20)

Finally, the eigenvector of the global structure φ is extracted from φ by merging the identical interface DOFs.

6.3 SIS Method for Eigensensitivity In this subsection, the eigensensitivities will be calculated by the SIS method. The first-order eigenvalue and eigenvector derivatives with respect to an elemental parameter r located in the Qth substructure will be derived.

6.3.1 Eigenvalue Derivative Equation (6.15) is differentiated with respect to r as ( ) ∂zm + KC − λM D ∂r

|

| ∂KC ∂λ ∂M D −λ − M D zm = 0 ∂r ∂r ∂r

(6.21)

T Premultiplying zm on both sides of Eq. (6.21) and considering Eq. (6.15) and the T M D zm = 1, one can obtain orthogonal condition zm

108

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

) ( ∂λ ∂M D T ∂KC = zm −λ zm ∂r ∂r ∂r Differentiating Eqs. (6.16) and (6.17) with respect to r, calculated by

(6.22) ∂KC ∂r

and

∂M D ∂r

and are

∂TCT p ∂Kp ∂TC ∂KC ∂/\pm = + K TC + TCT TC + TCT Kp ∂r ∂r ∂r ∂r ∂r

(6.23)

∂TCT p ∂M D ∂Mp ∂T = M T + TCT T + TCT Mp ∂r ∂r ∂r ∂r

(6.24)

where | | ( )−1 ∂ Fp DT DFp DT D0pm ∂Fp T ( p T )−1 ∂TC = = D DF D D0pm ∂r ∂r ∂r ) ( p ( p T )−1 ∂Fp T ( p T )−1 ∂0m p T p − D DF D D0m + F D DF D D ∂r ∂r

(6.25)

Since variables zm , λ, Kp , Mp , Fp , TC , T, D, /\pm , and 0pm are available in the calculation of eigensolutions, they are reused here directly. Each substructure is p p p ∂/\p ∂0p , ∂M , ∂F , ∂rm , and ∂rm treated to be independent, and the derivative matrices ∂K ∂r ∂r ∂r p p ∂/\ ∂0p are therefore zeros except for the Qth substructure containing r. ∂F , m , and ∂rm ∂r p ∂r p and ∂M take are calculated within the Qth substructure according to Eq. (3.38). ∂K ∂r ∂r the form of ⎡ ∂K =⎣ ∂r p

⎡ ∂M =⎣ ∂r p

∂K(Q) ∂r

⎤

0

⎦

∂K(Q) ∂r

(6.26)

0 ⎤

0

⎦

∂M(Q) ∂r

(6.27)

0

(Q)

and ∂M∂r and are the elemental stiffness and mass matrices with respect to the designed parameter r, respectively. D is required to calculate ∂M in Eq. (6.24) and then solve Eq. (6.22). Now, only ∂T ∂r ∂r T is available with an iterative algorithm based on Eqs. (6.17) and (6.19) in the is available with an iterative scheme according to previous subsection. Similarly, ∂T ∂r the derivatives of Eqs. (6.17) and (6.19) with respect to r as ∂TCT p ∂M D ∂Mp ∂T = M T + TCT T + TCT Mp ∂r ∂r ∂r ∂r

(6.28)

6.3 SIS Method for Eigensensitivity

109

(

) ∂S p ∂Mp p ∂T M T+S T + SM M−1 D KC ∂r ∂r ∂r ( ) ∂KC ∂M D −1 + SMp TM−1 − M D KC D ∂r ∂r

∂TC ∂T = + ∂r ∂r

(6.29)

where | | ( )−1 ∂ Fp − Fp DT DFp DT DFp

∂Fp ∂Fp T ( p T )−1 p DF − D DF D ∂r ∂r ∂r p ( )−1 ∂F T ( p T )−1 p + Fp DT DFp DT D DF D DF D ∂r p ( )−1 ∂F − Fp DT DFp DT D (6.30) ∂r

∂S = ∂r

=

Superior to the IBSS method in Chap. 5 that calculates the eigenvalue derivative mode by mode, the eigenvalue derivative of all the required modes can be calculated is available. In addition, simultaneously from Eq. (6.22) in the SIS method once ∂T ∂r p p the iterative variable ∂T employed in the SIS method has a size of Nm × Nm , much ∂r p ∂t p smaller than ∂r of N × Nm used in the IBSS method. Therefore, the SIS method is much more computationally efficient for eigenvalue derivatives, which will be verified in the later subsection.

6.3.2 Eigenvector Derivative By differentiating Eq. (6.20) with respect to the elemental parameter r, one can obtain the eigenvector derivatives of the assembled global structure as ∂0 = ∂r

(

) ( ) ∂zm ∂0pm ∂T − zm + 0pm − T ∂r ∂r ∂r

(6.31)

∂0p

have been obtained in calculating the eigenvalue derivaIn Eq. (6.31), ∂rm and ∂T ∂r tive. 0pm , T, and zm are available when computing the eigensolutions in the previous subsection. Therefore, the remaining task is to calculate ∂z∂rm . ∂zm is decomposed into a particular item and a general item as (Nelson 1976; ∂r Weng et al. 2011) ∂zm = v + czm ∂r

(6.32)

where v is a particular item and c is a participation factor of a specific mode. Substituting Eq. (6.32) into Eq. (6.21) and considering Eq. (6.15) gives

110

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

| | ) ( ∂KC ∂λ ∂M D −λ − M D zm KC − λM D v = − ∂r ∂r ∂r

(6.33)

C D Since KC , MD , ∂K , ∂M , λ, ∂λ , and zm are available beforehand in the calculation ∂r ∂r ∂r of eigensolutions and eigenvalue derivatives, v can be solved from Eq. (6.33) directly. The new eigenequation Eq. (6.15) satisfies the orthogonal condition of

T zm M D zm = 1

(6.34)

Differentiating Eq. (6.34) with respect to r gives T ∂zm ∂zm T ∂M D T M D zm + zm zm + zm =0 MD ∂r ∂r ∂r

(6.35)

By substituting Eq. (6.32) into Eq. (6.35), the participation factor c is thus calculated by ( ) 1 T T ∂M D T zm + zm M D v c = − v M D zm + zm 2 ∂r

(6.36)

Once the vector v in Eq. (6.33) and the participation factor c in Eq. (6.36) are calculated, ∂z∂rm is computed according to Eq. (6.32). In consequence, the eigenvector is calculated from Eq. (6.31). The derivative of the assembled global structure ∂0 ∂r can be computed from ∂0 by merging eigenvector derivative of the global structure ∂0 ∂r ∂r the identical interface DOFs.

6.4 Examples 6.4.1 A Frame Model The SIS method is applied to a frame mode in Fig. 6.1. The frame is composed of bars, and the lengths of the horizontal, vertical, and diagonal bars are 4.0 m, 3.0 m, and 5.0 m, respectively. The frame is modeled by beam element, with each element of 1.0 m length. The properties of the elements are Young’s modulus E = 210 GPa, mass density ρ = 7850 kg/m3 , the cross-sectional area A = 2.4 × 10−4 m2 , moment of inertia of the section I = 9.0 × 10−9 m4 , and Poisson’s ratio v = 0.3. The frame comprises 69 nodes, 76 two-dimensional beam elements, and 195 DOFs in total. The eigensensitivities with respect to the bending rigidities of two randomly selected elements, r 1 and r 2 (Elements 58 and 8 in Fig. 6.1a), will be investigated. The detailed procedures of the SIS method to calculate the eigensolutions and eigensensitivities are described as follows:

6.4 Examples

111 4000×5=20000

21

22

23

24 25 60 44

3000

59 58 57 1

2

r

3

41 61 4 5

27

28 29 64 48

63

43 42

1

26

46

r 7

2

45 65 9

8

31

32 33 68 52

67

47

62

6

30

50 49 69 12 13

11

35

36 37 72 56

71

51

66

10

34

70

14

53 73 16 17

39

40 76

75

55 54

15

38

74

18

19

20

(a) FEM

(b) Partitioned substructures

Fig. 6.1 The frame model

(1) The global structure is partitioned into three substructures (Fig. 6.1b). After partition, the first, second, and third substructures have 60, 93, and 60 DOFs, respectively. Each substructure is treated as an independent structure, and then, their first 12 eigensolutions are calculated by K( j) 0(mj) = /\(mj ) M( j ) 0(mj) |−1 \ (j = 1, 2, 3). The residual flexibility is calculated as F( j) = K( j ) − ( ( j) )−1 ( ( j) )T ( j) 0m /\m 0m . (2) Some constant values are assembled from those of the substructures, i.e., Kp , p ∂0pm p ∂Fp ∂/\m p ∂Kp ∂Mp p p M , F , /\m , 0m , ∂r , ∂r = 0, ∂r , ∂r , and ∂r . (3) Some intermediate variables are calculated to avoid repeated computation in ( )−1 p ∂0p C DT , D ∂rm , ∂S , ∂T and later steps, such as Fp DT , DFp DT , D0pm , S, KC , ∂F ∂r ∂r ∂r ∂KC . ∂r (4) The iteration starts with the FRFS method (Weng et al. 2009). The variables are initiated as T[0] = 0,

(

∂T ∂r

)[0]

=0

(5) The modal transformation matrix T and its derivative matrix iteratively. In the kth (k = 1, 2, 3, …) iteration,

(6.37) ∂T ∂r

are calculated

T p [k−1] M[k] D = Im + TC M T

(6.38)

( )−1 T[k] = TC + SMp T[k−1] M[k] KC D

(6.39)

112

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

( )[k−1] ∂TCT p [k] ∂T (6.40) M T + TCT Mp ∂r ∂r | ( )[k−1] |( )−1 ∂S p [k] ∂TC p ∂T M[k] = KC + M T + SM D ∂r ∂r ∂r | | ( )[k] ( ( )−1 ∂K )−1 ∂M D C [k] [k] − MD + SMp T[k] M D KC (6.41) ∂r ∂r (

(

∂T ∂r

)[k]

∂M D ∂r

)[k]

=

(6) The iterations stop once the relative differences of the first 10 lowest eigen−1 values λ[k] = eig((M[k] D ) KC ) from two consecutive iterations are less than the required tolerance (1 × 10−8 ) | [k] | | λ − λ[k−1] | −8 | | | λ[k−1] | < 1 × 10

(6.42)

In this example, five iterations are required to achieve this tolerance. (7) The first 10 lowest eigenvalues λ and their corresponding mode participation factors zm are calculated simultaneously by Eq. (6.15). The first 10 lowest eigenvectors are then calculated by Eq. (6.20). D , the first 10 eigenvalue derivatives are calculated (8) On the basis of the updated ∂M ∂r ∂zm according to Eq. (6.22). ∂r is computed following the procedures described in Sect. 6.3.2. The eigenvector derivatives are finally calculated by Eq. (6.31). The traditional method performed on the global model, which is treated as exact results for comparison, is also used to calculate the eigensolutions and eigensensitivities of the frame. The eigensolutions of the global structure are calculated using the Lanczos method (Lanczos 2008), and the eigensensitivities are computed with Nelson’s method (Nelson 1976). Figure 6.2 shows the iterative processes of the 1st, 6th, and 10th natural frequencies. The eigensolutions and eigensensitivities are also calculated by the FRFS method (Weng et al. 2009). In the FRFS method, the modal are initiated with zeros. The transformation matrix T and its derivative matrix ∂T ∂r FRFS method is equivalent to the initial step of the SIS method without iteration performed. Figure 6.2 shows that the relative errors of the 1st, 6th, and 10th natural frequencies from the FRFS method are about 1.0 × 10−3 , 2.5 × 10−3 , and 3.8 × 10−3 , respectively. After five iterations, the relative error curves decline sharply. The SIS method converges to accurate eigensolutions with only five iterations and 12 master modes in each substructure. The three natural frequencies achieve high precision with relative errors of about 9 × 10−9 , 8 × 10−9 , and 5 × 10−8 . Table 6.1 compares the first 10 eigensolutions calculated by the FRFS method and the SIS method. The relative errors of the former method are about 10−4 to 10−3 . Using the SIS method with five iterations, the relative errors drop to the order of 10−10 to 10−8 . Moreover, the SV values indicate the similarity between the calculated eigenvectors and the exact ones (Eq. (2.66)). All modes reach 1.0000 for the SIS

6.4 Examples

113

Fig. 6.2 Convergence of three natural frequencies

method. It implies that the SIS method achieves very accurate eigensolutions with only 12 master modes retained in each substructure and five iterations performed. The eigenvalue and eigenvector derivatives of the first 10 modes with respect to r 1 are then calculated. The eigensensitivities calculated by the FRFS method and the SIS method are compared in Table 6.2. The SV value in Eq. (3.51) is used to estimate the similarity of eigenvector derivative obtained by the SIS and traditional global methods. The relative errors of the eigenvalue derivatives calculated by the FRFS method are larger than 0.5% for most modes, and some are even significantly larger than 1%, which are then improved to about 1 × 10−4 or less for all modes when five iterations are performed in the SIS method. The SV values of some modes (7th, 8th, and 9th) are less than 0.95 in the initial step and are then improved to 0.99 or above after five iterations. Without losing generality, the eigensensitivities with respect to the bending rigidity of another randomly selected element r 2 in the free substructure (the second substructure in Fig. 6.1) are also calculated. The first 10 eigenvalue and eigenvector derivatives are listed in Table 6.3. It shows again that the SIS method improves the accuracy of the eigensensitivities significantly with 12 master modes included in each substructure and five iterations performed on the reduced eigenequation.

6.4.2 Wuhan Yangtze River Navigation Center The SIS method is then applied to the main building of the Wuhan Yangtze River Navigation Center located in Wuhan, P.R. China, as shown in Fig. 6.3a, to investigate its computational efficiency. It comprises a frame-core tube structure at the bottom and a steel frame on the top. The outer frame is a square with 50.1 m in length

114

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

Table 6.1 Eigensolutions using FRFS and SIS method (unit of frequencies: Hz) Mode

Exact freq.

FRFS method

SIS method

Freq.

Relative error

SV

Freq.

Relative error

SV

1

3.4423796

3.4463848

1.163 × 10−3

0.9659

3.4423795

8.593 × 10−9

1.0000

2

3.5605498

3.5768921

4.590 × 10−3

0.9136

3.5605495

6.720 × 10−8

1.0000

3

3.6893098

3.7060122

4.527 × 10−3

0.9248

3.6893091

1.849 × 10−8

1.0000

4

3.8726147

3.8743956

4.599 × 10−3

0.9841

3.8726147

2.289 × 10−10

1.0000

5

3.9097548

3.9107862

2.638 × 10−4

0.9898

3.9097548

1.291 × 10−9

1.0000

6

4.6071718

4.6188231

2.529 × 10−3

0.9467

4.6071719

7.208 × 10−9

1.0000

7

5.2884021

5.3137166

4.787 × 10−3

0.9119

5.2884019

3.491 × 10−8

1.0000

8

5.7707323

5.8066368

6.222 × 10−3

0.8807

5.7707323

6.980 × 10−9

1.0000

9

5.8594642

5.8853546

4.419 × 10−3

0.9010

5.8594636

8.863 × 10−8

1.0000

10

6.1796498

6.2029548

3.771 × 10−3

0.9297

6.1796500

5.244 × 10−8

1.0000

Table 6.2 Eigensensitivities with respect to r 1 using the FRFS and SIS methods Mode Exact ∂λ ∂r1

FRFS method

SIS method

∂λ ∂r1

∂λ ∂r1

Relative error SV (%)

Relative error

SV

1

0.0602020 0.0643355 6.866

0.9775 0.0602029 1.581 × 10−5

0.9925

2

0.2003865 0.2100268 4.811

0.9877 0.2003997 6.580 × 10−5

0.9910

3

0.5232413 0.5256877 0.468

0.9919 0.5232529 2.220 × 10−5

0.9978

4

35.442676 35.546226 0.292

0.9907 35.442746 1.950 × 10−6

1.0000

5

40.186920 40.244195 0.143

0.9822 40.186978 1.450 × 10−6

1.0000

10−5

0.9983

6

14.852659 14.777479 0.506

0.9618 14.852883 1.510 ×

7

24.385219 24.514315 0.529

0.9498 24.386325 4.530 × 10−5

0.9959

8

1.6281492 1.4882773 8.591

0.8918 1.6283244 1.076 × 10−5

0.9955

9

25.239712 25.820806 2.302

0.9241 25.241381 6.610 ×

10−5

0.9890

10

2.1960462 1.9320962 12.02

0.9517 2.1960889 1.105 × 10−4

0.9906

6.4 Examples

115

Table 6.3 Eigensensitivities with respect to r 2 using the FRFS and SIS methods Mode Exact ∂λ ∂r2

FRFS method

SIS method

∂λ ∂r2

∂λ ∂r2

Relative error SV (%)

Relative error

SV

1

2.5148072 2.6879542 6.885

0.8998 2.5148437 1.454 × 10−5

0.9928

2

5.8151784 6.0247243 3.603

0.9247 5.8161752 1.714 × 10−4

0.9913

3

9.8781838 10.262921 3.895

0.8934 9.8790403 8.671 × 10−5

0.9801

10−4

0.9898

4

0.4574927 0.4295982 6.097

0.9148 0.4574451 1.042 ×

5

1.2592437 1.2142801 3.571

0.8610 1.2592545 8.604 × 10−6

0.9971

6

5.0624470 4.9624174 1.976

0.5677 5.0628625 8.207 × 10−5

0.9919

10−4

0.9934

7

19.486079 16.250567 16.60

0.7391 19.490979 2.515 ×

8

30.333931 30.188090 0.481

0.8521 30.345402 3.782 × 10−4

0.9925

9

7.9697139 6.3694166 20.08

0.9028 7.9717123 2.508 × 10−4

0.9952

50.190686 48.445240 3.478

10−4

0.9937

10

0.9606 50.190834 4.014 ×

and width. The structure is 334.6 m tall from the bottom to the top of steel frame. The structure is modeled by 3950 nodes, 9112 elements, and 23, 364 DOFs in total (Fig. 6.3b). The bandwidth of the FEM is 461. The eigensensitivity with respect to the bending rigidity of a randomly selected shear wall element is calculated. The global structure is divided into 9 substructures along the vertical direction (Fig. 6.3c). The number of nodes, elements, and interface tearing nodes is listed in Table 6.4. In the SIS method, there are 30 master modes of each substructure retained to compute the first 10 eigensolutions and eigensensitivities of the global structure. The convergence criterion of iteration is set to 1 × 10−6 , based on which three iterations are required to achieve this criterion. The eigensolutions and eigensensitivities are also calculated by the IBSS method introduced in Chap. 5. The eigensolutions and eigensensitivities calculated by the global method are treated as the exact results for comparison. The IBSS method presents an iterative process based on t (Eq. (5.4)) to solve the original frequency-dependent eigenequation (Eq. (6.1)). It calculates the required eigensolutions and eigensensitivities iteratively mode by mode. In this example, two or three iterations are required for each mode to reach the same convergence criterion. Consequently, 25 and 23 iterations are employed in total for calculating the eigensolutions and eigensensitivities, respectively. Tables 6.5 and 6.6 compare the first 10 eigensolutions and eigensensitivities calculated with the aforementioned methods. The SIS method and the IBSS method can achieve very accurate eigensolutions and eigensensitivities. However, only three iterations are employed in the SIS method, whereas 25 and 23 iterations are required in the CIS method to gain a similar high precision of eigensolutions and eigensensitivities, respectively. The computational time consumed in a desktop computer with a 4.00 GHz CPU and 16 GB RAM by the aforementioned methods is compared in Table 6.7. The MATLAB platform is used to implement these methods, allowing the system matrices

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

...

116

(a) Perspective view

(b) Global model

(c) Substructures

Fig. 6.3 Wuhan Yangtze River Navigation Center and its FE model

to be stored and operated in sparse matrices. The global method takes 20.78 s and 9.03 s to compute the required eigensolutions and eigensensitivities, respectively, while the SIS method takes 4.49 s and 5.49 s only. Therefore, the SIS method is much more efficient than the global method in calculating eigensolutions and eigensensitivities. This is because the system matrices considering sparse nature have a size of 23, 364 × 461 in the global method, much larger than 270 × 270 in the SIS method. The IBSS method takes 145.43 s and 202.96 s for eigensolutions and eigensensitivities. Specifically, the IBSS method takes 5.795 s and 8.689 s in each iteration for the eigensolutions and eigensensitivities, respectively, while the SIS method consumes only 0.98 s and 0.66 s. The computation time for IBSS method is much longer than employed in the SIS method have a the SIS method. The iterative variables T and ∂T ∂r ∂t (25, 812 × 270) in the IBSS method. size of 270 × 270, much smaller than t and ∂r The order of the iterative variables in the IBSS method is reduced greatly.

No. of tearing nodes

56

392

336

659

No. of nodes

No. of elements

828

2

1

Substructure

56

828

392

3

56

1104

500

4

56

1104

500

5

56

Table 6.4 Substructural information of the FEM of the Wuhan Yangtze River Navigation Center

1104

500

6

56

1104

500

7

56

1153

500

8

44

1428

738

9

6.4 Examples 117

118

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

Table 6.5 Eigensolutions using the SIS and IBSS methods (unit of frequencies: Hz) Mode Exact freq. SIS method (3 iterations) Freq.

Relative error 10−8

IBSS method (25 iterations) SV

Freq.

Relative error

1.0000 0.1237088 4.495 ×

10−8

SV

1

0.1237088

0.1237088 4.496 ×

2

0.1301105

0.1301105 6.020 × 10−8 1.0000 0.1301105 6.081 × 10−8 1.0000

3

0.4190332

0.4190332 3.506 × 10−8 1.0000 0.4190332 3.557 × 10−8 0.9999

4

0.4656060

0.4656061 9.946 × 10−8 1.0000 0.4656061 1.038 × 10−7 0.9996

5

0.4728615

0.4728616 2.557 × 10−7 1.0000 0.4728616 2.615 × 10−7 0.9995

6

0.8711250

0.8711251 1.168 × 10−7 1.0000 0.8711251 1.346 × 10−7 0.9986

7

1.0498168

1.0498169 1.055 × 10−7 1.0000 1.0498169 1.495 × 10−7 0.9978

8

1.0569798

1.0569809 1.008 × 10−7 1.0000 1.0569809 1.024 × 10−7 0.9956

9

1.3508096

1.3508106 7.209 × 10−7 1.0000 1.3508106 7.244 × 10−7 0.9992

10

1.4008466

1.4008466 1.965 × 10−8 1.0000 1.4008471 3.684 × 10−7 1.0000

1.0000

Table 6.6 Eigensensitivities using the proposed SIS method and IBSS method Mode

Exact

∂λ ∂r

SIS method (3 iterations) ∂λ ∂r

IBSS method (23 iterations)

Relative error

SV

∂λ ∂r

Relative error

SV

1

0.000948482

0.000948481

3.858 × 10−7

1.0000

0.000948481

4.427 × 10−7

1.0000

2

0.002302778

0.002302777

5.900 × 10−7

1.0000

0.002302778

2.515 × 10−7

0.9999

3

0.005783008

0.005783005

4.920 × 10−7

1.0000

0.005783006

3.213 × 10−7

0.9997

4

0.015016383

0.015016347

2.375 × 10−6

0.9989

0.015016329

3.577 × 10−6

0.9995

5

0.034870584

0.034870497

2.501 × 10−6

0.9998

0.034870561

6.485 × 10−7

0.9995

6

0.046238807

0.046235502

7.148 × 10−5

0.9996

0.046238071

1.592 × 10−5

0.9983

7

0.141894690

0.141890836

2.716 × 10−5

0.9983

0.141889806

3.442 × 10−5

0.9957

8

0.088886487

0.088885614

9.827 × 10−6

0.9999

0.088880827

6.368 × 10−5

0.9977

9

0.012941839

0.012941237

4.654 × 10−5

0.9988

0.012941642

1.525 × 10−5

0.9957

10

0.000088383

0.000088381

1.996 × 10−5

1.0000

0.000088379

4.209 × 10−5

0.9978

MC

Time (s)

Methods

5.75 × 109

4.41 × 1011

IBSS method

FRFS method

Global method

3.35 × 1010

17.81

FRFS method

SIS method

0.33

1.48

IBSS method

Global method

SIS method

Total

Relative ratio (%)

1.16 × 1011 × 25 = 2.90 × 1012 7.05 × 109

3.97 × 109

2.41 × 6.97 × 108 10 10 × 3 = 7.23 × 1010

0.54

5.795 × 25 0.22 = 144.88

0.98 × 3 = 0.07 2.94

4.48 × 1011

15.40

100.00

16.56

4.82 × 1011 2.91 × 1012

3.64

12.62

100.00

14.29

3.09

1.06 × 1011

18.35

145.43

20.78

4.49

2.45 × 1011

6.46 × 1010

1.45 × 108

7.55 × 1010

9.63

2.87

0.05

3.15

Initiation

Eigenvector derivatives

1.91 × 1011 × 23 = 4.39 × 1012

1.39 × 1010

3.38 × 109

2.28 × 1011

2.04 × 6.38 × 109 10 10 × 3 = 6.12 × 1010

1.44

8.689 × 23 0.24 = 199.85

8.98

0.66 × 3 = 0.36 1.98

Iteration

Eigenvalue derivatives

Eigenvectors

Eigenvalues

Initiation

Eigensensitivities

Eigensolutions

Iteration

Table 6.7 Computational time and MC of eigensolutions and eigensensitivities of different methods

2.59 × 1011

4.45 × 1012

2.28 × 1011

1.43 × 1011

11.07

202.96

9.03

5.49

Total

5.82

100.00

5.12

3.21

5.45

100.00

4.45

2.70

Relative ratio (%)

6.4 Examples 119

120

6 Simultaneous Iterative Substructuring Method for Eigensolutions …

Moreover, the computational efficiency of the FRFS method introduced in Chap. 3 is also compared in Table 6.7. The method is equivalent to the SIS method without iteration. There are 300 master modes retained in each substructure, to reach the similar precision as the SIS method. Inclusion of more master modes enlarges the size of eigenequation. The size of the system matrices in the FRFS method is 2700 × 2700. Consequently, 18.35 s and 11.07 s are consumed for eigensolutions and eigensensitivities, respectively, which are much longer than the SIS method. Therefore, superior to the FRFS method retaining a large number of master modes to ensure accurate eigensolutions and eigensensitivities, the SIS method is much more efficient with only a few iterations and much fewer master modes. The size of the system matrices (KC , MD , etc.) in the FRFS method is 2700 × 2700, much larger than the SIS method of size 270 × 270. Besides the CPU time, Table 6.7 also compares the multiplication count (MC) consumed by different methods. MC is another basic index that can be used to evaluate the efficiency of different methods. It avoids the random errors in running time caused by different operating environments. The SIS method consumes much fewer MCs than other methods in calculating eigensolutions and eigensensitivities. It proves again that the SIS method is much more efficient than other commonly used methods. The result of the CPU time is consistent with MC. It is reliable to evaluate the efficiency of different methods with the CPU time in this example.

6.5 Summary This chapter introduces a simultaneous iterative substructuring method to compute the eigensolutions and eigensensitivities of large-scale structures. The original frequency-dependent system matrices of substructures are transformed to a frequency-independent one using the modal transformation matrix. The eigensolutions and eigensensitivities of all modes are to be calculated simultaneously. In addition, the modal transformation matrix is estimated with an iterative process performed on a small equation. SIS method has a much smaller size than the iterative variables used in the IBSS method and thus improves the computational efficiency significantly. As compared with the global method, FRFS method, and IBSS method, the proposed SIS method is capable of obtaining accurate eigensolutions and eigensensitivities efficiently with a small number of master modes in each substructure and a few iterations performed.

References

121

References Cui, J., Guan, X., Zheng, G.T.: A simultaneous iterative procedure for the Kron’s component modal synthesis approach. Int. J. Numer. Meth. Eng. 105, 990–1013 (2016) Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand. 45, 255–282 (2008) Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14, 1201–1205 (1976) Tian, W., Weng, S., Xia, Y., et al.: An iterative reduced-order substructuring approach to the calculation of eigensolutions and eigensensitivities. Mech. Syst. Signal Process. 130, 361–377 (2019) Weng, S., Xia, Y., Xu, Y.L., et al.: Improved substructuring method for eigensolutions of large-scale structures. J. Sound Vib. 323, 718–736 (2009) Weng, S., Xia, Y., Xu, Y.L., et al.: An iterative substructuring approach to the calculation of eigensolution and eigensensitivity. J. Sound Vib. 330, 3368–3380 (2011)

Chapter 7

Substructuring Method Considering Elastic Effects of Slave Modes in the Time Domain

7.1 Preview Structural responses and response sensitivities are frequently used in SHM. Using the time history of structural response and response sensitivity has several attractive features (Lu and Law 2004; Lu and Law 2007; Lu et al. 2017). First, time history data can provide an unlimited source of structural information as long as the test continues. When the quantity of the measurement data exceeds the unknown design parameters, the under-determined nonlinear optimization problem will become overdetermined (Zhan et al. 2011; Zhu et al. 2013). Moreover, time history data contain the time information, and thus, the response sensitivity method can be used for real-time SHM. Last but not least, the response sensitivity method can be directly extended to update the FEM of nonlinear structures (Lu et al. 2019). The commonly defined modal features (frequencies, mode shapes, etc.) are difficult to be used to update nonlinear structures as they only provide a good characterization of linear structures. This chapter extends the substructuring methods to the time domain for fast and precise computation of structural responses and response sensitivities. The independent substructural responses are decoupled onto the space spanned by a few master (lower) eigenvectors. The vibration equation of the global structure is expressed by a small number of master eigenvectors, while the elastic effects of slave (higher) counterparts are compensated by the residual flexibility. This ensures high precision of structural responses and response sensitivities using a few master eigenvectors only. Consequently, the size of the vibration equation of the global structure is greatly reduced, based on which the structural responses are calculated efficiently. Moreover, the proposed method is efficient in calculating response sensitivities, which are computed from the related substructure containing the designed parameters solely. Finally, the efficiency and accuracy of the proposed substructuring method are verified by a frame structure and a high-rise building. This substructuring method is referred to as the time-domain elastic effects-based substructuring (TDEES) method. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_7

123

124

7 Substructuring Method Considering Elastic Effects of Slave Modes …

7.2 Basic Method for Time History Dynamic Response and Response Sensitivity The vibration equation of a global structure with N degrees of freedom is written as M x¨ + C x˙ + Kx = f

(7.1)

where M and K are the mass matrix and stiffness matrix of the structure, respectively; C is the Rayleigh damping matrix with the form of C = a1 M + a2 K (a1 and a2 are damping coefficients); x¨ , x˙ , and x are the acceleration response, velocity response, and displacement response of the structure, respectively; and f is the external excitation imposed on the structure. This equation can be solved by the Newmark method. Differentiating the motion equation with respect to the design parameter r as M

∂ x˙ ∂x ∂M ∂C ∂K ∂ x¨ +C +K =− x¨ − x˙ − x ∂r ∂r ∂r ∂r ∂r ∂r

(7.2)

The response sensitivity in Eq. (7.2) can also be solved by the Newmark method. Newmark’s method is appropriate to solve the second-order differential equation with the constant coefficient matrix as (Newmark 1965) A x¨ (t) + B x˙ (t) + Cx(t) = L(t)

(7.3)

The basic steps of the Newmark method are as follows: (1) Give the initial conditions at time t = 0, namely x(0), x˙ (0), and x¨ (0); (2) Choose time increment /\t and parameters γ and β to ensure that the method is unconditionally stable, usually γ = 0.5 and β = 0.25; (3) Calculate the following constants, 1 β 1 1 β , α2 = , α3 = − 1, α4 = − 1, , α1 = 2 γ /\t γ /\t γ /\t 2γ γ ( ) /\t β α5 = − 2 , α6 = /\t (1 − β), α7 = β/\t 2 γ

α0 =

(4) Form the equivalent stiffness matrix K = C + α0 A + α1 B and calculate its −1 inverse matrix K ; (5) Calculate the following equivalent load vector at the initial time step, F(0) = F(0) + A(α0 x(0) + α2 x˙ (0) + α3 x¨ (0)) + B(α1 x(0) +α4 x˙ (0) + α5 x¨ (0))

7.3 Substructuring Method for Time History Dynamic Response …

125

Fig. 7.1 The flowchart of Newmark’s method

−1

(6) Calculate the displacement at the first time step, x(1) = K F(0) (7) Calculate the acceleration and velocity at the first time step, x¨ (1) = α0 (x(1) − x(0)) − α2 x˙ (0) − α3 x¨ (0) x˙ (1) = x˙ (0) + α6 x¨ (0) + α7 x¨ (1) (8) Repeat steps (5), (6), and (7) for n iterations (n is the time step in total). The dynamic response at the entire time interval T is then obtained. The flowchart of Newmark’s method is illustrated in Fig. 7.1.

7.3 Substructuring Method for Time History Dynamic Response and Response Sensitivity The substructuring method requires dividing an N DOFs global structure into N s substructures. The motion equation of the ith (i = 1, 2, …, N s ) independent substructure with ni DOFs is expressed as M(i) x¨ (i ) + C(i) x˙ (i) + K(i) x (i) = f

(i )

+g

(i )

(7.4)

126

7 Substructuring Method Considering Elastic Effects of Slave Modes …

where M(i) , C(i ) , and K(i) are the mass, damping, and stiffness matrices of the ith substructure, respectively; C(i ) takes the form of C(i ) = a1 M(i ) +a2 K(i) ; x¨ (i) , x˙ (i ) , and x (i ) are the acceleration, velocity, and displacement responses of the ith substructure, respectively; f (i ) is the external excitation imposed on the ith substructure; and g (i ) is the interface force from the adjacent substructures. The substructural variables are assembled to form the motion equations of the global structure as Mp x¨ p + Cp x˙ p + Kp x p = f p + g p

(7.5)

The system matrices are diagonally assembled from the independent substructures, which are the same to Chap. 2 and rewritten here as | | Mp = Diag M(1) , · · · , M(i ) , · · · , M(Ns )

(7.6)

| | Cp = Diag C(1) , · · · , C(i) , · · · , C(Ns )

(7.7)

| | Kp = Diag K(1) , · · · , K(i) , · · · , K(Ns )

(7.8)

|T | x¨ p = x¨ (1) , · · · , x¨ (i ) , · · · , x¨ (Ns )

(7.9)

|T | x˙ p = x˙ (1) , · · · , x˙ (i ) , · · · , x˙ (Ns )

(7.10)

|T | x p = x (1) , · · · , x (i ) , · · · , x (Ns )

(7.11)

| |T g p = g (1) , · · · , g (i ) , · · · , g (Ns )

(7.12)

|T | f p = f (1) , . . . , f (i ) , . . . , f (Ns )

(7.13)

To ensure the nodes at the interfaces from adjacent substructures moving jointly, the interface DOFs from connected substructures should satisfy the displacement compatibility as Dx p = 0

(7.14)

D is a Boolean matrix that operates on the interface DOFs of the ith substructure. Each row of matrix D contains two nonzero elements, 1 and − 1, for a rigid connection. According to Eq. (2.32), the connection force has the form of g p = DT τ

(7.15)

7.3 Substructuring Method for Time History Dynamic Response …

127

Considering Eqs. (7.5), (7.14), and (7.15), the global motion equation can be written from substructural variables as | p |{ p } | p |{ p } | p |{ p } { p } M 0 C 0 K −DT x f x¨ x˙ + + = (7.16) τ¨ τ˙ −D 0 τ 0 0 0 0 0 The structural response is expressed in the modal coordinate by the superposition of substructural eigenvectors as | | | | | | x p = op z, x˙ p = op z˙ , x¨ p = op z¨

(7.17)

where z is the mode participation factor. The eigenvectors and eigenvalues of the independent substructures satisfy the orthogonal conditions in terms of mass and stiffness matrices by | (i ) |T (i ) (i) |T |T | | o M o = I(i) , o(i ) K(i) o(i ) = /\(i ) , o(i) C(i ) o(i) = a1 I(i) + a2 /\(i) (7.18) Consequently, the primitive modes and system matrices, assembled from substructural modes and system matrices, also satisfy the orthogonal conditions of | |T | |T | p |T p p o M o = Ip , op Kp op = /\p , op Cp op = a1 Ip + a2 /\p

(7.19)

Substituting Eq. (7.17) into Eq. (7.16) leads to |

Mp 0 0 0

|{

} | p |{ p } | p |{ p } { | p |T p } C 0 o z˙ K −DT oz op z¨ f o + + = τ¨ τ˙ −D 0 0 0 τ 0 (7.20)

|T | Premultiplying op I on both sides of Eq. (7.20) gives || | |{ } | | | |{ } T T z¨ z˙ op Cp op 0 op M p o p 0 + τ¨ τ˙ 0 0 0 0 || | |T |{ } { | p |T p } | T z op Kp op − Dop f o + = p τ −Do 0 0

(7.21)

Considering the orthogonal conditions as Eq. (7.19), Eq. (7.21) is thus simplified to

128

7 Substructuring Method Considering Elastic Effects of Slave Modes …

|

|{ } | p |{ } | |T |{ } | Ip 0 z z¨ a1 I + a2 /\p 0 z˙ /\p − Dop + + 0 0 τ¨ 0 0 τ˙ τ −Dop 0 {| | } T op f p = (7.22) 0

The size of Eq. (7.22) is equal to the number of substructural modes and the interface DOFs. If all modes are included, the number of the substructural modes equals the number of the substructure DOFs, i.e., the DOFs E Ns of the global structure. n i (n b is the number of This means that the eigenequation has the size of n b + i=1 interface DOFs), which is the sum of all substructural internal and interface DOFs. For a large-scale structure with numerous DOFs, the size of the motion equation can be very large. Moreover, this motion equation is rank deficient and cannot be solved directly. A modal truncation technique is employed to solve these problems. The technique retains the first few lower modes of substructures and discards the higher modes, which avoids calculating the complete substructural eigenvectors from the largesized eigenequation. It conforms to the fact that the lower modes contribute most to structural kinetic energy. Similar to the frequency-domain methods in Chaps. 2–4, the retained lower modes are denoted as master modes, and the higher modes to be discarded are denoted as slave modes. The derivation of the proposed IDEES method starts with an undamped structure. The motion equation is |

Ip 0 0 0

|{ } | |T |{ } { | p |T p } | z z¨ /\p − Dop f o + = τ¨ τ −Dop 0 0

(7.23)

Expanding the motion equation in terms of the master and slave modes leads to )T ⎤⎧ ⎫ ⎧ | p |T p ⎫ ( ⎤⎧ ⎫ ⎡ p ⎪ o 0 − Dopm f ⎪ Im 0 0 ⎨ z¨ m ⎬ /\pm z ( )T ⎥⎨ m ⎬ ⎨ | mp |T p ⎬ ⎣ 0 Isp 0 ⎦ z¨ s + ⎢ ⎣ 0 /\ps − Dops ⎦ zs = os f ⎪ ⎩ ⎭ ⎩ ⎭ ⎪ ⎭ ⎩ τ¨ τ 0 0 0 −Dopm −Dops 0 0 (7.24) ⎡

where zm and zs are the mode participation factors of the master and slave modes, respectively. The second row of Eq. (7.24), which is the motion equation related to the slave modes, is taken out ( | |T )T z¨ s + /\ps zs − Dops τ = ops f p

(7.25)

The third row of Eq. (7.24) provides the relation between the master and slave modes as

7.3 Substructuring Method for Time History Dynamic Response …

Dopm zm + Dops zs = 0

129

(7.26)

In Eq. (7.25), the inertial term z¨ s is much smaller and negligible than other restoring term, which is set to zero. Since the slave eigenvectors make a negligible contribution | |T to the system vibration, the term ops f p is also assumed to be zero in the following derivation. As a result, combining Eqs. (7.25) and (7.26) leads to )−1 ( )−1 | p |T ( os τ = − DFp DT Dopm zm , F = ops /\ps | p |−1 | p |T p −1 p om = (K ) − om /\m

(7.27)

Submitting Eq. (7.27) into the first line of Eq. (7.24) gives ~ zm + Kz ~ m=~ M¨ f

(7.28)

where )T ( )−1 | |T ( p ~ ~ = Im M , K = /\pm + ϒm , ~ f (t) = opm f p , ϒm = Dopm DFp DT Dopm (7.29) Equation (7.28) is the reduced equation of the undamped structure. The reduced motion equation of a damped structure can be similarly derived as ~ zm + ~ ~ m=~ MR CP zm + Kz f

(7.30)

~ ~ + a2 K ~ C = a1 M

(7.31)

where

Different from the original motion equation (Eq. 7.23) with a large size, the reduced equation has a much smaller order. The size of the present motion equation (Eq. 7.30) is equal to the number of retained master eigensolutions of all substructures. Based on the reduced motion equation, it will be very efficient to calculate zm , z˙ m , and z¨ m with Newmark’s method. On the other hand, as the contribution of discarded slave modes is compensated by the residue item ϒm in Eq. (7.29), the proposed method will also be accurate in calculating zm , z˙ m , and z¨ m . Accordingly, the structural responses are calculated by the linear combination of the master modes as | | | | | | x(t)p = opm zm , x˙ (t)p = opm z˙ m , x¨ (t)p = opm z¨ m

(7.32)

Response sensitivities are the derivatives of the structural responses with respect to designed parameters. Differentiating Eq. (7.30) with respect to a designed parameter r on both sides gives

130

7 Substructuring Method Considering Elastic Effects of Slave Modes …

~ ~ ∂~ C ∂ z˙ m ~ ∂ z¨ m + ∂ M z¨ m + ~ ~ ∂zm + ∂ K zm = + z˙ m + K M C ∂r ∂r ∂r ∂r ∂r ∂r

( | |T ) ∂ opm f p (7.33) ∂r

r is usually chosen as stiffness parameters. It is noted that the size of the sensitivity equation (Eq. 7.33) is equal to the number of master modes as well. Rearranging Eq. (7.33) leads to ∂ z˙ m ~ ∂zm ~ ∂ z¨ m + ~ C +K = M ∂r ∂r ∂r

( | |T ) ~ ~ ∂ opm ∂M ∂~ C ∂K z¨ m − z˙ m − zm (7.34) fp − ∂r ∂r ∂r ∂r

where | |T ~ ~ ~ ∂~ ~ ∂ opm ∂K ∂/\pm ∂ϒm ∂ ~ C f ∂M ∂K ∂M = 0, = + , = a1 + a2 , = fp ∂r ∂r ∂r ∂r ∂r ∂r ∂r ∂r ∂r (7.35) ( p T )−1 ( )T ( p T )−1 ) ( ∂ opm ∂ϒm T p p T ∂ DF D = D DF D Dopm Dom + Dom ∂r ∂r ∂r ( )T ( )−1 ∂op (7.36) + Dopm DFp DT D m ∂r The right-hand side of Eq. (7.34) is treated as an equivalent force. The coordinate derivatives ( ∂∂rz¨ m , ∂∂rz˙ m , and ∂z∂rm ) related to the response sensitivities are calculated from Eq. (7.34) by the Newmark method. The derivatives of the reduced system matrices ~ ~ ~ ( ∂∂rM , ∂∂rK , and ∂∂rC ) are related to the master mode derivatives and the derivative of first-order residual flexibility ( ∂F ). ∂r Since the substructures are independent, the derivative matrices of the eigenvalues, the eigenvectors, and the residual flexibility are calculated in one specific substructure only (e.g., the Qth substructure) that contains the elemental parameter r. These quantities in the other substructures are zero, i.e., ⎡

⎡ ⎤ ⎤ 0 0 0 0 p (Q) ⎢ ⎥ ⎥ ∂om ∂/\(Q) m = ⎣ 0 ∂o∂rm 0 ⎦, 0 ⎦, ∂r ∂r 0 0 0 0 0 ⎡ ⎤ ( p )−1 | p |T 0 0 0 p os ∂os /\s (Q) −1 (Q) (Q) T ⎢ ⎥ = ⎣ 0 ∂os (/\s ) [os ] 0 ⎦ ∂r ∂r 0 0 0 0 ⎢ = ⎣0 ∂r 0

∂/\pm

0

(7.37)

The substructural eigenvalue derivatives and eigenvector derivatives can be obtained using traditional Nelson’s method by treating the Qth substructure as an independent structure.

7.4 Examples

131

In consequence, the structural responses of the global structure are recovered from the derivative of the substructural master modes by ∂opm ∂zm ∂ xp = zm + opm ∂r ∂r ∂r

(7.38)

∂opm ∂ x˙ p ∂ z˙ m = z˙ m + opm ∂r ∂r ∂r

(7.39)

∂ x¨ p ∂ z¨ m ∂opm z¨ m + opm = ∂r ∂r ∂r

(7.40)

The coordinate derivatives ( ∂z∂rm , ∂∂rz˙ m , and ∂∂rz¨ m ) are calculated from a small vibration equation. Only one substructure that contains the designed parameter is analyzed to calculate the response sensitivities of the global structure, avoiding analyzing the global structure as a whole. This will greatly improve computational efficiency.

7.4 Examples 7.4.1 A Three-Bay Frame The three-span frame (Fig. 7.2) is modeled by 160 two-dimensional beam elements, each 2.5 m long, resulting in 140 nodes and 408 DOFs. The material constants of the beam elements are chosen as: bending rigidity (EI) = 170 × 106 N m2 , axial rigidity (EA) = 2500 × 106 N, mass per unit length (ρA) = 110 kg/m, and Poisson’s ratio = 0.3. The elements are labeled as in the figure. The frame is disassembled into three substructures, as shown in Fig. 7.2b, resulting 51, 55, and 42 nodes in the three substructures. The eight interface nodes introduce 48 interface DOFs (each node has three DOFs) with 24 identical ones. The damping constants for Rayleigh damping are a1 = 0.1350 s−1 and a2 = 0.0137 s. An earthquake excitation is imposed on the frame in the horizontal direction. The loading is applied to all nodes of the frame with the magnitude as f (i ) = m (i) x¨ g

(7.41)

where f (i) and m (i) are the external force and mass of ith node in the horizontal direction, respectively, and x¨ g is the earthquake acceleration, as shown in Fig. 7.3. The sampling rate of the earthquake excitation is 100/3 Hz, and its duration is 30 s. Three methods, the global method, the substructuring method without consideration of elastic effects of slave modes, and the proposed TDEES method, are employed to calculate the structural responses and response sensitivities. In the global method, the structural responses and response sensitivities are calculated directly

132

7 Substructuring Method Considering Elastic Effects of Slave Modes …

Substructure 3

Substructure 2

Substructure 1

Fig. 7.2 FEM of the three-span frame structure (unit: m)

Fig. 7.3 Earthquake excitation

7.4 Examples

133

based on Eqs. (7.1) and (7.2), respectively, which are regarded as exact ones. The proposed TDEES method retains 50 master modes in each substructure. The traditional substructuring method also retains 50 master modes in each substructure, whereas the slave modes are discarded without compensation. The calculated structural responses of a randomly selected node (Node 108 in the X direction) are compared in Fig. 7.4. The response sensitivity with respect to the bending rigidity of element 110 is compared in Fig. 7.5. The results of the proposed method are nearly overlapped with those of the global method. In contrast, the results of the traditional substructuring method without consideration of the elastic effects of slave modes deviate from the exact ones. The relative errors of the calculated responses and response sensitivity are quantified as SV(x G , x Sub ) = (

∂ x G ∂ x Sub , SV ∂r ∂r

) =

||x G − x Sub ||2 ||x G ||2

||∂ x G /∂r − ∂ x Sub /∂r ||2 ||∂ x G /∂r ||2

(7.42) (7.43)

where subscripts ‘G’ and ‘Sub’ represent the results calculated by the global and substructuring methods, respectively. The relative error of the horizontal displacement at each node is illustrated in Fig. 7.6. The averaged relative error of the horizontal displacement sensitivity with respect to all stiffness parameters is shown in Fig. 7.7. The maximum and average values of Figs. 7.6 and 7.7 are listed in Table 7.1. Figures 7.6 and 7.7 and Table 7.1 show that all relative errors of the traditional substructuring method are larger than the proposed method. Therefore, it can be concluded that the proposed TDEES method, which compensates for the elastic effects of slave modes, is accurate in calculating the structural responses and response sensitivity.

7.4.2 Wuhan Yangtze River Navigation Center The main building of the Wuhan Yangtze River Navigation Centre, a 334.6 m super high-rise building in P.R. China, is used to verify the computational efficiency of the proposed substructuring method. This building is a frame-core tube structure, as demonstrated in Fig. 7.8. The FEM of this building is built in accordance with its design drawings. The model has 9112 elements, 3950 nodes, and 23,364 DOFs. The damping constants for Rayleigh damping are a1 = 0.0063 s−1 and a2 = 0.3940 s. An earthquake excitation is imposed in the horizontal direction (Y direction) of all nodes. The earthquake excitation time history is the same as in Fig. 7.3. It lasts 30 s at a sampling rate of 100/3 Hz. The total computational time steps are 1001. The FEM

134

7 Substructuring Method Considering Elastic Effects of Slave Modes …

(a) Displacement

(b) Velocity

(c) Acceleration

Fig. 7.4 Horizontal response of Node 108 by global and substructuring methods

7.4 Examples

135

(a) Displacement sensitivity

(b) Velocity sensitivity

(c) Acceleration sensitivity

Fig. 7.5 Response sensitivity by global and substructuring methods

136

7 Substructuring Method Considering Elastic Effects of Slave Modes …

Fig. 7.6 Relative error of displacement by conventional and proposed substructuring methods

Fig. 7.7 Relative error of displacement sensitivity by conventional and proposed substructuring methods

Table 7.1 Relative error of response and response sensitivity by two substructuring methods

Maximum Average

Relative error of response

Relative error of response sensitivity

Proposed (50 modes)

Conventional (50 modes)

Proposed (50 modes)

Conventional (50 modes)

4.20 × 10−3

0.19

1.44 × 10−2

0.26

10−2

0.17

3.80 ×

10−3

8.50 ×

10−2

1.26 ×

is divided into nine substructures along the vertical direction, as shown in Fig. 7.8, with the detailed substructural information listed in Table 7.2. At first, the structural response and response sensitivity are calculated by two substructuring methods (the traditional substructuring method without consideration of elastic effects of slave modes and the proposed TDEES method) to compare their accuracy. Element 1508, a shell wall element on the 12th floor of Substructure 3, is

137

…

7.4 Examples

(a) Perspective view

(b) Global model

(c) Substructures

Fig. 7.8 Wuhan Yangtze River Navigation Center and its FEM

randomly chosen to calculate the response sensitivity. The results calculated by the global method are treated as exact and used as the reference. Figures 7.9 and 7.10, respectively, plot the displacement response of Node 1008 (located in the 18th floor) in the Y direction and its sensitivity with respect to the element stiffness parameter of Element 1508. The displacement/displacement sensitivity obtained by the proposed TDEES method with only 130 master modes agrees well with the reference. However, the displacement/displacement sensitivity computed by the traditional substructuring method has to retain 300 master modes for similar accuracy. The proposed TDEES method can achieve high accuracy with a small number of master modes to calculate the structural response and response sensitivity by taking the inertial effects of slave modes into account.

2352

336

1680

No. of tearing DoFs

No. of DoFs

No. of nodes

828

392

659

336

No. of elements

2

1

Substructure

336

828 392 2352

3

336

3000

500

1104

4

336

3000

500

1104

5

Table 7.2 Division information of substructures of Wuhan Yangtze River Navigation Center

336

3000

500

1104

6

336

3000

500

1104

7

336

3000

500

1153

8

264

4428

738

1428

9

138 7 Substructuring Method Considering Elastic Effects of Slave Modes …

7.4 Examples

139

Fig. 7.9 Horizontal displacement by global and substructuring methods

Fig. 7.10 Displacement sensitivity by global and substructuring methods

Table 7.3 compares the computational time of the aforementioned methods to calculate the structural response and response sensitivity. The time consumed by the two substructuring methods is less than the global method. The proposed method takes only 7.17 s to calculate the response, just 3.24% of that consumed by the traditional global method (221.26 s). Regarding the response sensitivity, the proposed method takes 10.51 s only, about 4.57% of the global method (229.97 s). The size of the system matrices in the global method is 23,364 × 23,364, which is reduced to 1170 × 1170 in the proposed method. Since the computational effort of the structural analysis is approximately proportional to the size of system matrices, it is reasonable that the proposed method is much more efficient than the global method. In the traditional substructuring method with 300 master modes retained in each substructure, the running time for calculating the response and response sensitivity

140

7 Substructuring Method Considering Elastic Effects of Slave Modes …

Table 7.3 The computational time in calculating structural response and response sensitivity (unit: s) Running time

Response

Relative time ratio (%)

Response sensitivity

Relative time ratio (%)

Global method

221.26

100.00

229.97

100.00

Traditional substructuring (300 modes)

50.14

22.66

67.33

29.28

Proposed substructuring (130 modes)

7.17

3.24

10.51

4.57

is 50.14 s and 67.33 s, respectively, accounting for 22.66% and 29.28% of those consumed by the global method. The traditional substructuring method takes seven times the computational time of the proposed method. This is because the system matrices employed in the traditional substructuring method have a size of 2700 × 2700, much larger than that of the proposed method of 1170 × 1170. Therefore, the proposed method can achieve precise results very efficiently with only a small number of master modes when the elastic effect of slave modes is considered.

7.5 Summary In substructuring method of this chapter, the independent substructural responses are projected onto the range space of a few master eigenvectors, and the contribution of slave eigenvectors is compensated by a residue space. The motion equation and sensitivity equation are transformed into much smaller ones. Afterward, the response sensitivities with respect to a designed parameter are calculated from the master eigenvector derivatives of the related substructure solely. This improves computational efficiency. The TDEES method simultaneously ensures computational precision and enhances the computational efficiency in calculating the structural responses and response sensitivities. The proposed TDEES method is compared with the traditional global method in terms of accuracy and efficiency. Application to a frame example proves that the results from the proposed method are in good agreement with those of the global method when the residue is retained. When the residue is neglected in the traditional substructuring method, the accuracy will decline significantly. Application to a large-scale high-rise building verifies that, when the residue is neglected in the traditional substructuring method, a large number of master substructural eigenvectors are required to achieve high accuracy. When the residue is reserved by the proposed TDEES method, only a small number of master substructural eigenvectors are required to reach high accuracy. Inclusion of fewer master modes reduces

References

141

the size of the vibration equation and thus significantly improves the computational efficiency of the substructuring method.

References Lu, Z.R., Law, S.S.: State-space approach to calculate sensitivity of dynamic response. In: Proceedings of IMECE04, ASME International Mechanical Engineering Congress and Exposition, Anaheim, California, USA (2004) Lu, Z.R., Law, S.S.: Features of dynamic response sensitivity and its application in damage detection. J. Sound Vib. 303(1–2), 305–329 (2007) Lu, Z.R., Yao, R.Z., Wang, L., et al.: Identification of nonlinear hysteretic parameters by enhanced response sensitivity approach. Int. J. Non Linear Mech. 96, 1–11 (2017) Lu, Z.R., Liu, G., Liu, J., et al.: Parameter identification of nonlinear fractional-order systems by enhanced response sensitivity approach. Nonlinear Dyn. 95, 1495–1512 (2019) Newmark, N.M.: A method of computation for structural dynamics. Proc. ASCE 85(1), 67–94 (1965) Zhan, J.W., Xia, H., Chen, S.Y., et al.: Structural damage identification for railway bridges based on train-induced bridge responses and sensitivity analysis. J. Sound Vib. 330, 757–770 (2011) Zhu, H.P., Mao, L., Weng, S.: Calculation of dynamic response sensitivity to substructural damage identification under moving load. Adv. Struct. Eng. 16(9), 1621–1631 (2013)

Chapter 8

Substructuring Method Considering Inertial Effects of Slave Modes in the Time Domain

8.1 Preview In Chap. 7, the substructuring method was extended to compute the structural response and response sensitivity (Zhu et al. 2019). In TDEES method, a few lower modes are retained to form a reduced model and the contribution of discarded higher modes is compensated by static residual flexibility. The inertial effects related to the discarded modes are neglected, which leads to a slight loss of accuracy in calculating the structural response and response sensitivity. Some errors in structural response and response sensitivity may lead to the incorrect searching direction and thus result in some convergence problems in model updating. To solve this problem, an enhanced substructuring method is proposed in this chapter (Zhu et al. 2021). First, a transformation mode is derived to relate the discarded modes to the retained modes. Next, an equivalent modal mass matrix is derived to relate to the mass matrix and transformation mode, which is included in the reduced mass matrix to compensate for the inertial effect of the discarded higher modes. The accuracy of the proposed substructuring method is improved by this compensation for the inertial effect of the discarded higher modes. This substructuring method is entitled the time-domain inertial effects-based substructuring (TDIES) method.

8.2 Substructuring Method for Time History Dynamic Response and Response Sensitivity In the TDEES method presented in Chap. 7, the inertial effects of slave modes are deleted for simplification. This inevitably introduces some errors in the calculation of the dynamic response and response sensitivity. To improve the accuracy of the TDEES method, the motion equation derived by the TDIES method considers the

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_8

143

144

8 Substructuring Method Considering Inertial Effects of Slave Modes …

 p   inertial effect (¨zs ), thedamping effect a1 Is + a2 ps z˙ s , and the external force  T related to slave modes p f p . The relationship between master modal coordinates zm and slave modal coordinates zs can be derived from the expanded motion equation (Eq. (7.24)) as  −1 ps zs = −s zm , s = Fp DT DFp DT Dpm

(8.1)

s is named the transformation mode,  Ns  Ns which relates the slave mode to the master m i × i=1 m i (ni is the number of DOFs in the ith mode. s has a size of i=1 substructure, and mi is the number of master modes retained in the ith substructure). of slave Fp is the static residual flexibility matrix to compensate  Ns for the contribution Ns m i × i=1 mi . modes. It is a symmetric matrix with the size of i=1 The second row of Eq. (7.24), which is a motion equation related to slave modes, has the form of   T  T  z¨ s + a1 Isp + a2 ps z˙ s + ps zs = ps DT τ + ps f p

(8.2)

 −1 Premultiply Dps ps on both sides of Eq. (8.2) leads to    −1  −1 Dps ps z¨ s + a1 Dps ps + a2 Dps z˙ s + Dps zs = DFp DT τ + DFp f p (8.3) τ is obtained from Eq. (8.3) as    −1  −1  −1 Dps ps z¨ s + a1 Dps ps τ = DFp DT + a2 Dps z˙ s

+Dps zs − DFp f p

(8.4)

The substructural eigenmodes are orthogonal to the mass matrices (Eq. (7.19)), which can be rewritten in terms of the slave and master modes as 

pm

T ps

 Mp

pm

ps

 =



p

Im p

Is

(8.5)

Based on Eq. (8.5), the relationship is satisfied as  p T p p s M s = Isp

(8.6)

Given that the orthogonal condition of Eq. (8.6), Eq. (8.4) is then transformed to −1 p p p

 DF M s z¨ s + (a1 DFp Mp + a2 D)ps z˙ s + Dps zs − DFp f p τ = DFp DT (8.7)

8.2 Substructuring Method for Time History Dynamic Response …

145

To eliminate the slave mode-related terms in Eq. (8.7), ps zs is approximately expressed by the master mode-related term −s zm (Eq. (8.1)). Substituting Eq. (8.1) into Eq. (8.7), τ can be rewritten as −1 p p

 DF M s z¨ m + (a1 DFp Mp + a2 D)s z˙ m + Ds zm + DFp f p τ = − DFp DT (8.8) There is a relationship as  −1 Ds = DFp DT DFp DT Dpm =Dpm

(8.9)

s is usually not equal to pm because the matrix D is of row full rank. Based on Eq. (8.9), Eq. (8.8) is transformed to −1 p p  DF M s z¨ m + a1 DFp Mp s z˙ m + a2 Dpm z˙ m τ = − DFp DT

+Dpm zm + DFp f p

(8.10)

With the substitution of Eq. (8.10) into the first row of Eq. (8.2), the motion equation is reduced to ˜ zm + C˙ ˜ zm + Kz ˜ m= M¨ f

(8.11)

where  p    p ˜ = pm + ϒm ˜ = Im ˜ = a 1 Im M + Ts Mp s , C + Ts Mp s + a2 pm + ϒm , K (8.12)    T  (8.13) f = pm − Ts f p ˜ C, ˜ and K ˜ are reduced mass, damping matrix and stiffness matrix, respecwhere M,  tively, and f is the reduced external force. Consequently, zm , z˙ m , and z¨ m are computed from the reduced motion equation  (Eq. (8.11))  by theNewmark-β algorithm. TypiNs Ns cally, the size of Eq. (8.11), i=1 m i × i=1 m i , is considerably smaller than      Ns  Ns that of Eq. (7.16), n b + i=1 n i × n b + i=1 ni . The major difference between the proposed IDIES method and TDEES method equation considers the inertial effect (¨zs ), the in Chap. 7 is that the pproposedp motion  ˙ a I , and the external force related to the slave modes damping effect m + a2 m z 1 s    p T p s f . Thus, the vectors of master modal coordinates, zm , z˙ m , and z¨ m , are more accurate than those obtained by the TDEES method. The reduced motion equation of the TDEES and proposed TDIES methods is compared in Table 8.1. In the proposed TDIES method, the inertial effect related to the slave modes is considered by adding an equivalent modal mass matrix (Ts Mp s ) to the reduced stiffness matrix. The

146

8 Substructuring Method Considering Inertial Effects of Slave Modes …

Table 8.1 Comparison between TDEES and proposed TDIES methods TDEES method

p

Im + Ts Mp s

Im

Reduced stiffness

p m + ϒm  p p a1 Im + a2 m

Reduced damping Reduced external force Structural response

Proposed TDIES method

p

Reduced mass



p T m

f

p

p m zm

p

+ ϒm



m + ϒm  p   p  a1 Im + Ts Mp s + a2 m + ϒm    p T m − Ts f p  p  m − s zm

damping effect   of the slave modes is compensated by adding a dissipative term, a1 Ts Mp s + a2 ϒm , to the reduced damping matrix. The proposed TDIES method considers these effects of discarded slave modes, thereby only a small number of master modes is required to ensure accuracy. Afterward, the structural response is calculated as follows       x p = pm zm + ps zs = pm − s zm

(8.14)

Accordingly, the velocity and acceleration responses are expressed as     x˙ p = pm − s z˙ m , x¨ p = pm − s z¨ m

(8.15)

The response sensitivity is the first-order derivative of the structural response with respect to the design parameter, which helps to speed up the optimization process in the model updating. The sensitivity equation is derived by differentiating the reduced motion equation (Eq. (8.11)) with respect to a design parameter r as ˜ ˜ ˜ ˜ ∂ z¨ m + ∂ M z¨ m + C ˜ ∂ z˙ m + ∂ C z˙ m + K ˜ ∂zm + ∂ K zm = M ∂r ∂r ∂r ∂r ∂r ∂r

  T  ∂ pm ∂Ts − fp ∂r ∂r (8.16)

Rearranging Eq. (8.16) leads to ˜ ∂ z˙ m + K ˜ ∂zm = ˜ ∂ z¨ m + C M ∂r ∂r ∂r

  T  ˜ ˜ ˜ ∂ pm ∂Ts ∂M ∂C ∂K − z¨ m − z˙ m − zm fp − ∂r ∂r ∂r ∂r ∂r (8.17)

Here, the master modal coordinates zm , z˙ m , and z¨ m were obtained in the previous part. ˜ ˜ ˜ s In the right-hand side of the equation, the derivative matrices ( ∂∂rM , ∂∂rC , ∂∂rK , and ∂ ) ∂r p p ∂m ∂m are related to the sensitivity of master modes ( ∂r and ∂r ) and the sensitivity of p static residual flexibility, ∂F . The substructures are independent to each other. If the ∂r design parameter r j is located in the ith substructure, only the sensitivity matrices of

8.3 Examples

147

the ith substructure need to be calculated, while those of the remaining substructures p ∂p ∂p are zeros. Therefore, ∂rm , ∂rm , and ∂F present a zero-block form. ∂r Afterward, the response sensitivity of the assembled global structure is regained by    ∂zm  ∂ pm − s ∂ xp = zm + pm − s ∂r ∂r ∂r    ∂ z˙ m  ∂ pm − s ∂ x˙ p = z˙ m + pm − s ∂r ∂r ∂r    ∂ z¨ m  ∂ pm − s ∂ x¨ p = z¨ m + pm − s ∂r ∂r ∂r

(8.18)

(8.19)

(8.20)

In the proposed TDIES method, ∂z∂rm , ∂∂rz˙ m , and ∂∂rz¨ m are computed from the reduced sensitivity equation, which includes the inertial and damping effects related to the slave modes. Therefore, the response sensitivity calculated by the proposed TDIES method has high accuracy. The sensitivity equation derived by the proposed substructuring method has a small size, the sensitivity matrices of one specific substructure need to be calculated, and the proposed TDIES method is efficient in calculating response sensitivity. In consequence, the proposed TDIES method can substantially alleviate the computational burden during the sensitivity-based model updating.

8.3 Examples 8.3.1 A Three-Bay Frame A three-bay frame shown in Fig. 7.2 is used to test the accuracy of the proposed TDIES method in calculating the structural response and response sensitivity. The global method and TDEES method are used for comparison. The global method analyzes the FEM of the entire structure, and the results are treated as the accurate ones. The TDEES method retains 10 or 50 master modes for each substructure. The elastic effects of the slave modes are compensated by static residual flexibility without considering the inertial effect of slave modes. In the proposed TDIES method, 10 master modes are retained for each substructure, and both the elastic effect and the inertial effect related to the slave modes are included. Figure 8.1 shows the horizontal displacement of Node 101. Figure 8.2 shows the response sensitivity with respect to the element stiffness factor of Element 1. Element 1 is the column located in the 1st floor near the first left fixed bearing. Figure 8.3 demonstrates the relative error of the horizontal response of each node. Figure 8.4 demonstrates the average relative error of the displacement sensitivity of

148

8 Substructuring Method Considering Inertial Effects of Slave Modes …

Fig. 8.1 Horizontal response of Node 101 by global and substructuring methods

8.3 Examples

Fig. 8.2 Response sensitivity by global and substructuring methods

149

150

8 Substructuring Method Considering Inertial Effects of Slave Modes …

each node with respect to all stiffness parameters. Table 8.2 lists the maximum and average values of Figs. 8.3 and 8.4. For the TDEES method retaining 10 master modes, both the displacement response and response sensitivity show a noticeable shift from the reference. When

Fig. 8.3 Relative error of displacement response by TDEES and TDIES methods

Fig. 8.4 Relative error of displacement sensitivity by TDEES and TDIES methods

Table 8.2 Relative error of response and response sensitivity by two substructuring methods Relative error of response

Relative error of response sensitivity

TDIES (10 TDEES (10 TDEES (50 TDIES (10 TDEES (10 TDEES (50 modes) (%) modes) (%) modes) (%) modes) (%) modes) (%) modes) (%) Maximum 0.11 Average

0.00003

8.68

0.42

0.04

13.19

1.44

8.06

0.38

0.004

12.04

1.26

8.3 Examples

151

the number of master modes increases to 50, the displacement response and response sensitivity overlap with the reference. The average relative error of the displacement response decreases from 8.06% to 0.38%, and the average relative error of the response sensitivity decreases from 12.04% to 1.26%. The TDEES method has to retain additional master modes to improve the accuracy, which would enlarge the motion/sensitivity equation from 30 × 30 to 150 × 150. The displacement response computed by the proposed TDIES method with 10 master modes agrees well with the reference. The average relative error of the displacement response is around 0.00003%, and the average relative error of the displacement sensitivity is 0.004%. Therefore, the proposed TDIES method can accurately calculate the structural response and response sensitivity even with only 10 master modes.

8.3.2 Wuhan Yangtze River Navigation Center The main building of the Wuhan Yangtze River Navigation Centre in Fig. 7.8 is used to verify the accuracy and efficiency of the proposed TDIES method. The structural response and response sensitivity are calculated by the proposed TDIES method and TDEES method to compare their accuracy. Element 1508 is randomly chosen for calculating the response sensitivity. Element 1508 is located in Substructure 3 and is a shell wall element on the 12th floor. Figures 8.5 and 8.6 plot the displacement response in the Y direction of a randomly selected node and its sensitivity with respect to the element stiffness parameter of Element 1508. The results calculated by the Newmark method on global structure are treated as exact and regarded as the reference. The displacement/displacement sensitivity obtained by the proposed TDIES method with only 30 master modes agrees well with the reference. However, the displacement/displacement sensitivity computed by the TDEES method with the same quantity of master modes deviates from the reference. Even when the number of master modes retained for each substructure increases to 130 master modes, the displacement and its sensitivity calculated by the TDEES method still show some obvious errors. Table 8.3 compares the computational time of the TDEES method TDIES method in calculating the structural response and response sensitivity. Both substructuring methods are more efficient than the global method. Specifically, the time consumed by the substructuring methods is less than 5% of the global method. The proposed TDIES method requires extra time to calculate the equivalent modal mass matrix to compensate for the inertial effect of slave modes. When 30 master modes are retained in each substructure, the TDEES method takes a little less time than the proposed TDIES method. This extra time consumption is negligible as compared to the large amount of time steps in time domain. The TDEES improves the efficiency by including more master modes. Including more master modes also enlarges the size of motion equation. When 130 master modes are retained in each substructure, the time consumed by the TDEES method is almost two times the TDIES method

152

8 Substructuring Method Considering Inertial Effects of Slave Modes …

Fig. 8.5 Displacement response by global and substructuring methods

Fig. 8.6 Displacement sensitivity by global and substructuring methods

containing 30 master modes in each substructure. For both substructuring methods, master modes and their derivatives are required to form the reduced motion/sensitivity equation, which takes most of the computational time. Including more master modes would greatly reduce computational efficiency.

8.4 Summary This chapter proposes a substructuring method to calculate the structural response and response sensitivity by considering both the elastic and inertial effects of slave modes. The transformation mode is derived to relate higher modes to lower modes.

References

153

Table 8.3 The computational time in calculating structural response and response sensitivity (unit: second) Running time

Response Relative time ratio (%) Sensitivity Relative time ratio (%)

Global method

221.26

100.00

229.97

TDEES (30 modes)

3.50

1.58

2.21

100.00 0.96

TDEES (130 modes)

7.17

3.24

10.51

4.57

TDIES (30 modes)

3.60

1.63

2.85

1.24

The motion equation is reduced using a few lower substructural modes and the transformation mode. The inertial effect of discarded higher modes is compensated by deriving an equivalent modal mass matrix with respect to the transformation mode in the reduced matrix. Consequently, the accuracy of the proposed method is considerably improved without retaining more substructural modes. Afterward, the response sensitivity is calculated from the reduced sensitivity equation. The proposed method is first applied to a three-bay plane frame. The results show that the proposed substructuring method can accurately calculate the structural response and response sensitivity with a small number of modes retained. The proposed method is then applied to a large-scale high-rise building with hundreds of design parameters and more than 20,000 DOFs. The computational time consumed by the proposed method is much less than that by the global method.

References Zhu, H.P., Li, J.J., Weng, S., et al.: Calculation of structural response and response sensitivity with improved substructuring method. J. Aerosp. Eng. 32(3), 1–17 (2019) Zhu, H.P., Li, J.J., Tian, W., et al.: An enhanced substructure-based response sensitivity method for finite element model updating of large-scale structures. Mech. Syst. Signal Process. 154, 107359 (2021)

Chapter 9

Substructuring Method for Finite Element Model Updating

9.1 Preview Due to uncertainties in geometry, material properties, and boundary conditions, the dynamic properties or responses (e.g., frequencies, mode shapes, acceleration, velocity, displacement) of a structure predicted from a highly idealized numerical model usually differ from the practical measurements. An effective and efficient model updating is necessary to obtain a more accurate FEM for various purposes, such as damage identification, structural modification, and vibration control. In the model updating process, the physical parameters of an analytical FEM are iteratively adjusted to minimize the discrepancy between the model predicted dynamic properties or responses and the measured counterparts in an optimal manner. The eigensolutions or structural responses are used to construct the objective function. The eigensensitivities or response sensitivities provide a rapid search direction for the optimization. The substructuring methods possess some benefits for the model updating process. The substructuring methods expand the eigenequation/vibration equation of the global structure in the modal coordinates and reduce the size of the eigenequation/vibration equation with modal truncation techniques. In particular, using the substructuring methods, the optimization process handles one or several substructures only instead of the global structure and thus improves the computational efficiency significantly. It is efficient to calculate the dynamic properties and sensitivity matrices of the substructures, which can be analyzed easier and quicker than the global structure. During the model updating process, some specific substructures can be re-analyzed and assembled with other unchanged substructures to recover the solutions of the global structure, thereby avoiding repeated computation of the global structural properties. This will benefit the model updating process, where the dynamic properties and sensitivity matrices are required repeatedly. In addition, the substructures contain much fewer uncertain parameters than the global structure. This assists in accelerating the convergence of large-scale optimization problems. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_9

155

156

9 Substructuring Method for Finite Element Model Updating

In this chapter, the substructuring method introduced in the previous chapters for dynamic and sensitivity analyses will be developed to implement the sensitivitybased FE model updating of large-scale structures.

9.2 Fundamentals of Sensitivity-Based FE Model Updating Using Modal Properties The objective function using the modal properties (frequencies and mode shapes) is usually denoted as (Weng et al. 2009) J (r ) =

 i

 Wλi2

λi ({r })A − λiE λiE

2 +

 i

2 Wφi

 2  φ ji ({r })A − φ Eji   max φiE j

(9.1)

In Eq. (9.1), the eigenvalue and the associated mode shape are normalized with respect to the mass matrix; λiA and φ Aji denote the ith eigenvalue and mode shape from the analytical FEM, which are expressed as the function of the uncertain physical parameters {r }; λiE represents the ith eigenvalue from experiment (the square of the circular frequency); φ Eji is the associated mode shape at the jth point; and Wλi and Wφi are the weight coefficients considering the uncertainties of the measured frequencies and mode shapes. The objective function is minimized by continuously adjusting parameters {r} of the analytical model through optimization techniques. To find the optimal searching direction, the derivative matrices of the eigenvalues and mode shapes with respect to parameter r can be formulated as [Sλ (r )] =

∂φ(r ) ∂λ(r ) , Sφ (r ) = ∂r ∂r

(9.2)

The elemental stiffness reduction factor (SRF), pi , is employed to indicate the change ratio of the updated parameter to the initial value before updating pi =

rU − rO ri = i O i ri ri

(9.3)

where r i (i = 1, 2, …, m) is vector composed of the uncertain parameters. Superscript ‘O’ symbolizes the original values of parameters before updating, and superscript ‘U’ represents the updated values after updating. A positive SRF indicates that the updated parameter is stronger than the original one. A negative SRF indicates that the updated parameter is weaker, which specifies the location and severity of damage.

9.3 Fundamentals of Sensitivity-Based FE Model Updating Using Time …

157

The lower and upper bounds are enforced on the SRF values during the updating procedure. For different purposes, the bounds usually differ. In updating an initial model in the undamaged state, the bounds can be set to −0.5 < pi ≤ 0.5

(9.4)

In the damage detection implementation, the damaged elements are usually weaker than its undamaged state. Consequently, the bounds are set to −1 < pi ≤ 0

(9.5)

A negative SRF indicates the location of the damage, and the magnitude quantifies the damage severity. A zero of SRF value means undamaged, and −1 indicates complete loss of stiffness.

9.3 Fundamentals of Sensitivity-Based FE Model Updating Using Time History Data In the response sensitivity-based model updating, the objective function is formed as (Zhu et al. 2021) J (r ) =



   T     χ A (r ) − χ E W χ A (r ) − χ E + γ ({r } − {r0 })1

(9.6)

    where χ A is the dynamic responses obtained from the FEM, χ E represents the experimental ones, W denotes the weighting matrix of different responses, and γ stands for the regularization parameter. The objective function is minimized by adjusting the elemental parameters r starting from its initial values r 0 in an optimal manner. To indicate the searching direction in the optimization process, the sensitivity matrix is defined as S(r ) =

∂χ A (r ) ∂r

(9.7)

Calculating structural responses and response sensitivity of a large-scale structure with numerous DOFs and design parameters is time-consuming. Thus, an improved substructuring method introduced in Chap. 8 is used to rapidly calculate the structural response and response sensitivity.

158

9 Substructuring Method for Finite Element Model Updating

9.4 FE Model Updating by Substructuring Method Using Modal Data The substructure-based model updating using modal data is illustrated in Fig. 9.1. In each iteration, the eigensolutions are calculated from the independent substructures and then assembled by the reduced eigenequation to calculate the global eigensolutions. The residuals between the global eigensolutions (natural frequencies and mode shapes) and the experimental counterparts are used to construct the objective function. The eigensensitivity is calculated to indicate the searching direction in each step. In the present method, the eigensensitivity is computed from substructure-based sensitivity matrices with respect to an elemental parameter which are calculated from one substructure that contains the elemental parameter. The objective function is minimized by iteratively adjusting the elemental parameters r. In the substructure-based model updating method, only one substructure is handled independently without repeatedly analyzing the large-size matrices of the global structure when a local area is changed. The substructure-based model updating can be easily performed on the parallel computation.

9.5 FE Model Updating by Substructuring Method Using Time History Data Based on the structural response and response sensitivity calculated by the improved substructuring method in Chap. 8, the time-domain substructure-based model updating method is performed as given in Fig. 9.2. The structural time history response is used to form the objective function, and response sensitivity is used to speed up the optimization algorithm.

9.6 Examples 9.6.1 The Balla Balla Bridge The Balla Balla River Bridge, shown in Sect. 2.4.2 and Fig. 9.3, is employed to illustrate the substructure-based model updating (Xia et al. 2008). The FEM of this bridge (Fig. 9.4) has 907 elements, 947 nodes, and 5420 DOFs in total. The elements used in the model are listed in Table 9.1. In the field vibration testing, an instrumented hammer (DYTRAN 5803A, 12 LB) was used to excite the bridge structure, and accelerometers (Kistler 8330A2.5 and 8330A3) were used to collect the bridge responses. The accelerometers were placed in seven rows corresponding to the seven girders. There were 19 measurement points

9.6 Examples

159

Fig. 9.1 FE model updating by substructuring method

in each row and 133 in total, as shown in Fig. 9.5. Ten natural frequencies and mode shapes were extracted from the measured data by the rational fraction polynomial method (McConnell 2010). The mode shapes are illustrated in Fig. 9.6. The FEM will be updated using both the traditional global method and the proposed substructuring method presented in Sect. 9.4 to investigate the effectiveness and efficiency of the substructuring method in practical model updating. 1289 physical parameters are selected as updating candidates, which include Young’s modulus (E) of diaphragms, girders, slabs, and the axial rigidity (EA) and bending rigidity (EI xx , EI yy ) of the shear connectors. The objective function in this example combines the residuals in the frequencies and mode shapes between the experimental data and the analytical model. The weight coefficients are set to be 0.1 for the mode shapes and 1.0 for the frequencies. By using the traditional model updating, the eigensolutions are calculated by the Lanczos method, and eigensensitivities are calculated by Nelson’s method respectively, based on the system matrices of the global structure. The first 30 global eigenmodes are extracted from the FEM to match the 10 experimental modes. The model updating process is converged after 69 iterations. The convergence process

160

9 Substructuring Method for Finite Element Model Updating

Fig. 9.2 FE model updating by substructuring method using time history data

Fig. 9.3 General view of Balla Balla River Bridge

9.6 Examples

161

Fig. 9.4 FEM of the Balla Balla River Bridge

Table 9.1 Elements of the FEM of Balla Balla River Bridge

Bridge component

Element type

Bearing

Beam

Quantity 56

Slab

Shell

288

Girder

Shell

252

Stirrup

Beam

231

Diaphragm

Shell

Total

80 907

Fig. 9.5 Locations of sensors

in terms of the norm of the objective function is demonstrated in Fig. 9.7 (Xia et al. 2008). One iteration takes about 1.26 h, and the whole process takes about 86.16 h on an ordinary personal computer. Afterward, the FRFS substructuring method is employed to calculate the eigensolutions and eigensensitivities for model updating. The optimization algorithm, updating parameters, and convergence criterion are the same as those used in the global model updating. The global structure is divided into 11 substructures

162

9 Substructuring Method for Finite Element Model Updating

Mode 3 Freq = 10.0572 Hz Damping = 1.98 %

Mode 4 Freq = 10.7510 Hz Damping = 1.08 %

Mode 5 Freq = 11.0323 Hz Damping = 1.08 %

Mode 6 Freq = 12.6418 Hz Damping = 0.88 %

Mode 7 Freq = 14.7137 Hz Damping = 1.35 %

Mode 8 Freq = 15.7606 Hz Damping = 0.86 %

Mode 9 Freq = 16.3876 Hz Damping = 0.78 %

Mode 10 Freq = 20.1832 Hz Damping = 0.98 %

Fig. 9.6 Measured frequencies and mode shapes of the Balla Balla River Bridge

9.6 Examples 10

0.1

Objective function(Log)

10 10 10 10 10 10 10

163

0

40 master modes

-0.1

60 master modes

80 master modes

Global method Substructuring method

90 master modes

-0.2

-0.3

-0.4

-0.5

-0.6

0

10

20

30

40

50

60

70

80

90

Time(Hour)

Fig. 9.7 Convergence of the model updating process with the substructuring method and the global method

along the longitudinal direction as shown in Fig. 3.4 and Table 2.8. As proved in Chap. 3, dividing the bridge structure into 11 substructures can calculate the eigensensitivity more efficiently than the other division formations, and calculation of eigensensitivities dominates the computational time in the model updating process. As stated in Chaps. 2 and 3, the quantity of master modes retained in the substructures influences the accuracy of the eigensolutions and eigensensitivities. Inclusion of more master modes improves the accuracy but decreases the efficiency. To balance the accuracy and efficiency, the quantity of master modes in the substructures adaptively increases during the model updating process. In the beginning, the first 40 modes of each substructure are retained by the substructuring method as master modes to calculate the first 30 eigensolutions and eigensensitivities of the global structure. After several iterations, the solution approaches the optimum, and the gradient is close to zero. A small error in the calculated eigensolutions and eigensensitivities will cause a significant error in the solution. More accurate eigensolutions and eigensensitivities are required in the later stage of the model updating. Consequently, the number of master modes in the substructures increases gradually as the convergence slows down. At the final steps, 90 modes are retained in each substructure to improve the accuracy of the eigensolutions and eigensensitivities. By using this adaptive scheme, the substructuring-based model updating process converges to the predefined tolerance within 76 iterations as shown in Fig. 9.7. The computational time spent on different stages totals about 48.07 h and is about 56% of that using the global model updating, as listed in Table 9.2. The frequencies and mode shapes of the updated structure are compared with those values before updating and listed in Table 9.3. After updating, the average frequency difference between the updated model and the experimental measurement is less than 1%. The MAC values are improved from 0.85 to 0.93. The substructuring method achieves similar results as the global method. Therefore, for the practical structure with a large number of updating parameters, the proposed substructure-based model updating is computationally accurate and efficient.

164

9 Substructuring Method for Finite Element Model Updating

Table 9.2 Computation time and number of iterations with the different master modes Global method

Substructuring method 40 master modes

60 master modes

80 master modes

90 master modes

0.43

0.57

0.69

0.84

No. of iterations 69

16

18

31

11

Total for the 86.16 updating process (h)

48.07

Time for each iteration (h)

1.26

9.6.2 Wuhan Yangtze River Navigation Center The main building of the Wuhan Yangtze River Navigation Center is used to demonstrate the computational efficiency of the time-domain substructure-based model updating method. This building is a frame-core tube structure, as shown in Fig. 9.8. The FEM of this building has 9112 elements, 3950 nodes, and 23,364 DOFs. The damping constants for Rayleigh damping are a1 = 0.0063 and a2 = 0.3940. The model is excited by a horizontal earthquake excitation, as shown in Fig. 9.9. The earthquake excitation lasts 30 s with a sampling rate of 0.03 s. The total computational time steps are 1001. The FEM is divided into nine substructures along the vertical direction in Fig. 9.8c, with the detailed substructural information listed in Table 9.4. The experimental data are simulated by intentionally reducing the bending rigidity of Elements 21 and 77 (shear wall, main load-bearing component) of Substructure 3 by 20% and 30%, respectively. Based on the damaged model, the dynamic responses of the structure subject to the earthquake are calculated by the global method and are used as the experimental data. The proposed substructuring method and global method are separately used to update the FEM of the large-scale building. The global method analyzes the structure as a whole, and the results are regarded as the reference. The bending rigidity of all columns and shear wall elements of Substructure 3 is updated. Consequently, 336 design parameters have to be updated. The objective function is the residual of the analytical displacement responses in Y direction of all nodes in Substructure 3 calculated from the FEM and the experimental displacement responses. The iteration process stops when the norm of the objective function is less than 2 × 10−15 . The stiffness reduction factor is defined as SRF =

rU − rO rO

(9.8)

As aforementioned, the number of master modes in substructures affects the accuracy of the calculated structural response and response sensitivity. To ensure the efficiency and convergence of the model updating, the number of master modes adaptively varies during the model updating process. Initially, 30 master modes are

24.91

7.95

10.06

10.75

11.03

12.64

14.71

15.76

16.39

20.18

2

3

4

5

6

7

8

9

10

Averaged

18.74

6.76

1

18.52

17.55

13.27

9.45

12.13

8.71

7.74

6.26

Freq. (Hz)

0.86 0.85

12.77

0.82

0.88

0.92

0.85

23.42

14.35

17.49

19.29

4.98

0.76

0.71

−13.37 0.80

0.96

−0.27 12.84

0.93

−7.34

−14.36

MAC

Diff (%)

Before updating

Measured freq. (Hz)

Mode

20.23

16.38

15.77

14.77

12.58

10.86

11.01

10.02

7.93

6.53

Freq. (Hz)

Global method

After updating

Table 9.3 Frequencies and mode shapes of the bridge before and after updating

0.93

0.93

0.92

0.95

−0.07 0.24

0.93

0.06

0.90

0.97

−0.45 0.38

0.82

0.89

0.94

0.99

0.95

MAC

−1.56

2.42

−0.42

−0.27

−3.47

Diff (%)

20.28

16.39

15.77

14.78

12.59

10.85

11.03

10.02

7.92

6.55

Freq. (Hz)

0.95

0.50

0.00

0.06

0.45

−0.38

−1.60

2.60

−0.39

−0.33

−3.17

Diff (%)

Substructuring method

0.93

0.93

0.95

0.94

0.90

0.96

0.81

0.89

0.94

0.99

0.95

MAC

9.6 Examples 165

9 Substructuring Method for Finite Element Model Updating

…

166

(a) Perspective view

(b) Global model

Fig. 9.8 Wuhan Yangtze River Navigation Center and its FEM

Fig. 9.9 Earthquake excitation

(c) Substructures

2352

336

1680

No. of tearing DOFs

No. of DOFs

No. of nodes

828

392

659

336

No. of elements

2

1

Substructure

336

828 392 2352

3

336

3000

500

1104

4

336

3000

500

1104

5

Table 9.4 Substructure division information of Wuhan Yangtze River Navigation Center

336

3000

500

1104

6

336

3000

500

1104

7

336

3000

500

1153

8

264

4428

738

1428

9

9.6 Examples 167

168

9 Substructuring Method for Finite Element Model Updating

included in each substructure to compute the structural response and response sensitivity. After six iterations, the number of master modes increases. In the final three iterations, 130 master modes are retained in each substructure. Figure 9.10 shows the identified result by the proposed substructure-based model updating method. SRF values of Elements 21 and 77 are identified with − 0.2 and − 0.3, respectively. The SRF values of the other elements are nearly zeros. The identified location and magnitude of stiffness reduction agree well with the presumed damage case. Considering the inertial effect related to the slave modes, the structural response and response sensitivity from the substructuring method are accurate for updating a large-scale structure with hundreds of design parameters. In Table 9.5, the computational efficiency is investigated by comparing the computational time and the number of iterations between the proposed substructure-based and global-based model updating methods. A desktop computer with a 4.40 GHz Intel CPU and 128 GB memory is used. The convergence of model updating with the global and substructuring methods is plotted in Fig. 9.11. In the substructure-based model updating method, 30 master modes are retained in each substructure in the first 6 iterations, and the size of the motion/sensitivity equation is 270 × 270. Only 0.18 h are consumed for each iteration. As the model updating approaches the optimum, the number of retained master modes increases gradually. Finally, 130 master modes are retained in each substructure. The size of the system matrix is 1170 × 1170. Each iteration consumes about 0.90 h. By using this adaptive scheme, the time-domain substructure-based model updating converges to the predefined convergence criterion within 23 iterations and 9.78 h. In contrast, the size of the system matrix of global method is as large as 23,364 × 23,364. The global-based model updating method spends 14.17 h for each iteration and converges by total of 14 iterations and 198.38 h. The computational time of the substructurebased model updating method is approximately 4.93% of those consumed by the global-based model updating method.

Actual stiffness reduction location

Fig. 9.10 Identified SRF values by the proposed substructure-based model updating method

9.6 Examples

169

Table 9.5 Comparison of computational time and iterations Type of method Global

Substructure method 30 modes

60 modes

100 modes

130 modes

Size of system matrix

23,364 × 23,364

270 × 270

540 × 540

900 × 900

1170 × 1170

Time for each iteration (h)

14.17

0.18

0.35

0.65

0.90

No. of iteration 14

6

6

6

3

Total no. of iteration

14

23

Total time (h)

198.38

9.78

Relative ratio (%)

100.00

4.93

Fig. 9.11 Convergence of global-based and proposed substructure-based model updating methods

The time-domain substructure-based model updating is much more efficient than the global-based approach for several reasons. First, the size of the response equation is greatly reduced from 23,364 × 23,364 to 270 × 270 by the substructuring method. The structural response and response sensitivity are rapidly calculated from the reduced equations. A total of 336 elemental parameters have to be updated in this example. In model updating, the response sensitivity is to be calculated element by element. It means that the sensitivity equation is repeatedly solved 336 times for 1001 time steps in each iteration. Calculating the response sensitivity consumes the dominant time for model updating. The fast calculation of response sensitivity by the substructuring method can noticeably reduce the computational time of model updating. Second, the structural response and response sensitivity for model updating have high accuracy due to the compensation of inertial effect related to slave modes.

170

9 Substructuring Method for Finite Element Model Updating

Inclusion of fewer modes not only reduces the computation time for analysis of independent substructures but also decreases the size of motion equation. Third, substructures are independent of each other. The response sensitivity of the global structure is computed from sensitivity matrices of the one specific substructure (Substructure 3), while those matrices of the other eight substructures are zeros. Therefore, the time-domain substructure-based model updating is accurate and efficient.

9.7 Summary This chapter presents a substructuring-based model updating procedure. The modal data or time history response data are calculated by substructuring method to construct the objective function. The substructuring method are used to compute the sensitivity for providing the search direction. The application of the proposed method to a practical bridge demonstrates that the method is more efficient than the global-based method in updating a large-scale structure with a large number of design parameters. The effectiveness of the substructuring-based model updating method using time history response data was verified by using a super-tall building with hundreds of design parameters and more than 20,000 DOFs. The computational time of the proposed substructure-based model updating method is less than 5% of that consumed by the global-based model updating method. Although the substructuring method introduces some slight errors in calculating the eigensolutions/time history response and eigensensitivities/time history response sensitivity, an adaptive scheme is employed by including the different number of master modes during the optimization process.

References McConnell, K.G.: Modal testing. Philos. Trans. Roy. Soc. A Math. Phys. Eng. Sci. 359(1), 11–28 (2007) Weng, S., Xia, Y., Xu, Y.L., et al.: Substructuring approach to finite element model updating. Comput. Struct. 89(9–10), 772–782 (2009) Xia, Y., Hao, H., Deeks, A.J., et al.: Condition assessment of shear connectors in slab-girder bridges via vibration measurements. J. Bridg. Eng. 13(1), 43–54 (2008) Zhu, H.P., Li, J.J., Tian, W., et al.: An enhanced substructure-based response sensitivity method for finite element model updating of large-scale structures. Mech. Syst. Signal Process. 154, 107359 (2021)

Part II

Dynamic Condensation Methods

Chapter 10

Dynamic Condensation for Eigensolutions and Eigensensitivities

10.1 Preview In the previous chapters, the substructuring methods are introduced for model reduction of large-scale structures. The methods reduce the global model into a modal space spanned by a few substructural master modes, based on which the dynamic analysis, sensitivity analysis, and model updating are implemented accurately and efficiently. In this chapter, another commonly used model reduction technique, model condensation method, is introduced to compute the eigensolutions and eigensensitivities of large-scale structures. The method selects some DOFs as master DOFs. A transformation matrix is derived to relate other DOFs (slave DOFs) to the master ones. Using the transformation matrix, the global model is reduced into a low-dimensional physical space spanned by the master DOFs (Xia and Lin 2004a, b; Weng et al. 2014). Since the master DOFs are much fewer than the global DOFs, the computational resources and time are saved. Moreover, the computational efficiency can be further improved when combined with the substructuring methods (Choi et al. 2008; Liu and Wu 2011). Model condensation is also widely used in the experimental modal analysis and related fields. In the experimental modal testing, the measured points are usually much fewer than the DOFs of the analytical model. The analytical model is required to be reduced to match the experimental counterparts. The model condensation technique is also useful in determining the sensor position in the experiments (Jeong et al. 2012). This chapter will first give a brief introduction to the traditional static condensation method proposed by Guyan (1965). Afterward, two dynamic condensation approaches are introduced to compute the eigensolutions, namely the iterated improved reduced system (IRS) method (Friswell et al. 1995, 1998) and the iterative order reduction (IOR) method (Xia and Lin 2004a, b). The IOR method is proved to be more effective than other model condensation methods in terms of computational precision and efficiency. The IOR method derives a dynamic transformation matrix linking the master DOFs to the slave DOFs. The frequency-dependent dynamic © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_10

173

174

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

transformation matrix is updated with an iterative process and then used to transform the global eigenequation into a much smaller one. The eigensolutions are then calculated from the reduced eigenequation efficiently. Weng et al. (2014) later developed this method to compute the eigensensitivities. The DOFs associated with the selected elemental parameter are chosen as the additional master DOFs to constrain the change in the elemental parameter within the stiffness and mass matrices of the master DOFs, whereas the derivative matrices associated with the slave DOFs are zeros. The eigensensitivity of the reduced model is obtained by directly performing iterations on the stiffness and mass matrices of the master DOFs, which consumes a small amount of computation time. Finally, the accuracy and efficiency of the condensation methods are compared through the application to the GARTEUR frame and a cantilever plate.

10.2 Static Condensation Approach As before, the free vibration of an undamped structure with N DOFs is described by an eigenequation of (K − λi M)i = 0

(10.1)

If all DOFs of a structure are divided into N m master DOFs and N s slave DOFs (N = N m + N s ), the eigenequation is divided into a block form as (O’Callahan 1989; Friswell et al. 1995, 1998) 

Kmm Kms T Kms Kss



 − λi

Mmm Mms T Mms Mss



m s

 = i

  0 0

(10.2)

Here, subscripts ‘m’ and ‘s’ represent the master and slave DOFs, respectively. It is noteworthy that the normal form subscripts ‘m’ and ‘s’ in previous chapters denote the master and slave modes, which correspond to columns of . In the dynamic condensation approach, however, italic subscripts ‘m’ and ‘s’ are associated with the master DOFs and slave DOFs, corresponding to rows of . For convenience, λ and  represent one mode only, and subscript ‘i’ is omitted in the following analysis: From the second line of Eq. (10.2), the eigenvector of slave DOFs is represented by the master DOFs like  T T m = µm s = −(Kss − λMss )−1 Kms − λMms

(10.3)

10.2 Static Condensation Approach

175

where µ is the transformation matrix relating m and s . In consequence, the complete eigenvector is represented by the master eigenvectors as  =

m s





 Im = m = Vm µ   I V= m µ

(10.4)

(10.5)

where V is the transformation matrix relating m and . µ takes the form of (Xia and Lin 2004a, b)  T T µ = −(Kss − λMss )−1 Kms − λMms

(10.6)

Substituting Eq. (10.4) into Eq. (10.1) and premultiplying Eq. (10.1) by TT , one can obtain a reduced eigenequation containing master DOFs of order N m (K R − λM R )m = 0

(10.7)

where KR = VT KV and MR = VT MV are the reduced stiffness and mass matrices, respectively. They can be written as   

 Mmm Mms Im M R = VT MV = Im µT T Mms Mss µ

 T T = [Mmm + Mms µ] + µ Mms + Mss µ

(10.8)

    Kmm Kms Im K R = V KV = Im µ T Kms Kss µ

 T T = [Kmm + Kms µ] + µ Kms + Kss µ

(10.9)

T



T

The size of the reduced eigenequation (Eq. 10.7) is equal to the number of master DOFs, N m . By using this condensation in the physical space, the reduced eigenequation is much smaller than the original eigenequation of order N. In Eqs. (10.7)–(10.9), Mmm , Mms , Mss , Kmm , Kms , and Kss are reassembled by extracting the specific columns associated with the master and slave DOFs in the system matrix M and K. They are constant given a designated master and slave DOFs. Therefore, the pivot task in the dynamic condensation methods is to compute the dynamic transformation matrix µ. Guyan (1965) first proposed a static condensation method to compute the eigensolutions. The method neglected the inertial effects in the dynamic transformation matrix µ, that is, λ = 0. Equation (10.6) is thus simplified into −1 T µ = µG = −Kss Kms

(10.10)

176

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

where subscript G symbolizes variables associated with the Guyan static condensation. The static transformation matrix µG is constant and frequency-independent. The reduced system matrices are also constant and frequency-independent, which are calculated by 

T 

M R = MG = Mmm + Mms µG + µTG Mms + Mss µG 

T 

K R = KG = Kmm + Kms µG + µTG Kms + Kss µG = Kmm + Kms µG

(10.11)

(10.12)

The eigenvalue λ and eigenvector of mater DOFs Fm are then solved from Eq. (10.7). The eigenvector of the total DOFs F is then calculated by Eq. (10.4). As the inertial effects are ignored and λ = 0, the Guyan static condensation is only accurate for static problems, i.e., the vibration frequency is zero. For dynamic problems, the dynamic condensation method is required to consider the nonzero frequency vibration.

10.3 IOR Method for Eigensolutions The system matrices KR and MR depend on frequency λ, and the eigenproblem cannot be solved directly by the usual eigensolver. Premultiplying Eq. (10.6) by Kss − λMss , one has  T T − λMms (Kss − λMss )µ = − Kms

(10.13)

Rearranging Eq. (5.13) to move the unknown frequency λ on the right-hand side, the transformation matrix µ can be written as  T −1 T −1 Mms + Mss µ = µG + µd µ = −Kss Kms + λKss

(10.14)

where  T  T −1 −1 Mms + Mss µ = Kss Mms + Mss µG + Mss µd µd = λKss

(10.15)

Subscript ‘d’ represents a dynamic item, which is frequency-dependent and will be achieved by an iterative process. Friswell et al. (1998) initialized µ with µG and presented an iterated IRS method to estimate µ accurately with an iterative scheme based on Eqs. (10.11), (10.12), and (10.14).

10.3 IOR Method for Eigensolutions

177

Xia et al. (2004a, b) improved the computational efficiency with the IOR method. They rewrote the left-hand side of Eq. (10.7) as    T T  

+ Kss µG + µd (K R − λM R ) = Kmm + Kms µG + µd + µG + µd Kms

  − λ Mmm + Mms µG + µd  T T   + µG + µd Mms + Mss µG + µd = KG − λMd (10.16) where

T  Md = [Mmm + Mms µ] + µTG Mms + Mss µ

(10.17)

Equations (10.7) and (10.16) lead to (KG − λMd )m = 0. The transformation matrix µ is therefore expressed by  T −1 T −1 µ = −Kss Kms + λKss Mms + Mss µ  T −1 Mms + Mss µ Md−1 KG = µG + Kss

(10.18)

It is noted that, Md−1 KG is used in Eq. (10.18), instead of M−1 R K R used in the iterated IRS method proposed by Friswell et al. (1998). This is because KG is a static item and determined directly, while K R is frequency-dependent and determined through an iterative process. Xia and Lin (2004b) have proved that the new approach was equivalent to the subspace iteration method and converged faster than the method by Friswell et al. (1998). In the dynamic condensation technique, the frequency-dependent transformation matrix µ is estimated by an iterative scheme according to Eq. (10.18). With the iterative updating of the transformation matrix µ, the condensed mass matrix MR and stiffness matrix KR are obtained from Eqs. (10.8) and (10.9), respectively. Consequently, the eigensolutions can be calculated from Eq. (10.7). The dynamic transformation matrix and the condensed system matrices are estimated iteratively with the following procedure: (1) The iteration begins with the Guyan static condensation by neglecting the inertia item (Guyan 1965) µ

[0]

= µG =

−1 T −Kss Kms , V[0]

 =

Im

−1 T −Kss Kms

 (10.19)

T

T Md[0] = MG = V[0] MT[0] = Mmm + Mms µG + µTG Mms + µTG Mss µG (10.20)

[0] T K[0] KV[0] = Kmm + Kms µG (10.21) R = KG = V

178

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

M[0] R = MG

(10.22)

and the eigensolutions with the static condensation are obtained by the eigenfunction KG m = λMG m

(10.23)

(2) The transformation matrix is updated iteratively. In the kth iterations (k = 1, 2, …), the transformation matrix µ[k] is updated by

−1  T −1 µ[k] = µG + Kss KG Mms + Mss µ[k−1] Md[k−1]

(10.24)



T 

Md[k] = Mmm + Mms µ[k] + µTG Mms + Mss µ[k]

(10.25)

The reduced eigenequation is formed as  [k] T T 

[k] + µ Mms + Mss µ[k] M[k] R = Mmm + Mms µ

(10.26)

 [k] T T 

[k] + µ Kms + Kss µ[k] K[k] R = Kmm + Kms µ

(10.27)

[k] [k] [k] K[k] R m = λM R m

(10.28)

(3) The iteration is terminated when the differences of the eigenvalues from two consecutive iterations are less than a predefined tolerance. It is noted that the IOR method introduced here has some similarities to that of the SIS method introduced in Chap. 6. They reduce the size of the global structure to improve its analysis efficiency in different aspects. The dynamic condensation approach reduces the eigenequation of the global structure in the physical coordinates while the SIS method in the modal coordinates.

10.4 IOR Method for Eigensensitivity The first-order derivatives of the ith eigensolutions (i = 1, 2, …, N m ) with respect to a structural design parameter r will be derived in this subsection. The eigenvalue derivative will be discussed, followed by the eigenvector derivative.

10.4 IOR Method for Eigensensitivity

179

10.4.1 Eigenvalue Derivatives Based on Eq. (10.7), the reduced eigenequation for the ith mode is written in the classical form K R m = λM R m

(10.29)

In this equation, m is normalized with respect to the reduced mass matrix MR , which satisfies the orthogonal condition of Tm M R m = Im . Equation (10.29) is differentiated with respect to the elemental parameter r (K R − λM R )

∂m + ∂r



 ∂K R ∂λ ∂M R − MR − λ m = 0 ∂r ∂r ∂r

(10.30)

Premultiplying Eq. (10.30) by Tm leads to Tm (K R

  ∂λ ∂M R ∂m T ∂K R + m − MR − λ m = 0 − λM R ) ∂r ∂r ∂r ∂r

(10.31)

Since (K R − λM R )m = 0, the eigenvalue derivative is obtained as ∂K R ∂M R ∂λ = Tm m − λTm m ∂r ∂r ∂r

(10.32)

The eigensolutions λ and m have been calculated from the eigenequation. R and ∂K are derived from Eqs. (10.8) and (10.9) as ∂r  ∂µ ∂µT T ∂M R ∂Mmm ∂Mms Mms + Mss µ = + µ + Mms + ∂r ∂r ∂r ∂r ∂r   T ∂Mss ∂µ T ∂Mms + µ + Mss +µ ∂r ∂r ∂r  ∂Kmm ∂Kms ∂K R ∂µ ∂µT T = + µ + Kms + Kms + Kss µ ∂r ∂r ∂r ∂r ∂r  T  ∂K ∂µ ∂K ss ms + µT + µ + Kss ∂r ∂r ∂r

∂M R ∂r

(10.33)

(10.34)

The transformation matrix µ is estimated from an iterative scheme according to Eqs. (10.17) and (10.18). Differentiating these two equations with respect to r, one has ∂Mmm ∂Mms ∂µ ∂Md = + µ + Mms ∂r ∂r ∂r ∂r

180

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

  T  ∂µTG T ∂µ ∂Mss T ∂Mms Mms + Mss µ + µG + + µ + Mss ∂r ∂r ∂r ∂r  T ∂µG ∂Mss ∂µ ∂µ −1 ∂Mms = + Kss + µ + Mss ∂r ∂r ∂r ∂r ∂r  −1 ∂Kss −1  T − K Mms + Mss µ Md KG ∂r ss    T ∂KG ∂Md −1 −1 − Md K G Mms + Mss µ Md−1 + Kss ∂r ∂r

(10.35)

(10.36)

where  ∂ Kmm + Kms µG ∂Kmm ∂Kms ∂KG ∂µ = = + µG + Kms G ∂r ∂r ∂r ∂r ∂r  −1 T T Kms ∂ −Kss ∂µG −1 ∂Kss −1 T −1 ∂Kms = = Kss Kss Kms − Kss ∂r ∂r ∂r ∂r

(10.37)

(10.38)

The derivative matrix ∂µ can then be estimated iteratively. ∂r In the full model, the stiffness matrix K and mass matrix M are assembled from the element stiffness matrices and element mass matrices of all elements, according to Eq. (3.3). An element parameter is linearly related to the element stiffness and mass matrices. However, in the reduced matrices, the change of a specific elemental parameter r spreads over the whole reduced system matrices KR and MR . To reduce the computational time, the DOFs associated with the elemental parameter r are included as additional master DOFs, and thus the change of the elemental parameter is constrained within the stiffness and mass matrices of the master DOFs, namely ∂Mmm and ∂K∂rmm are nonzeros. The derivative matrices relating to the slave DOFs are ∂r zeros, i.e., ∂Kss ∂Mms ∂Mss ∂Kms = 0, = 0, = 0, =0 ∂r ∂r ∂r ∂r

(10.39)

G = 0. According to Eq. (10.38), ∂µ ∂r Equations (10.33)–(10.37) are then simplified into

  ∂Mmm ∂M R ∂µ = + Mms + ∂r ∂r ∂r   ∂Kmm ∂K R ∂µ = + Kms + ∂r ∂r ∂r

 ∂µT T ∂µ Mms + Mss µ + µT Mss ∂r ∂r

(10.40)

 ∂µT T ∂µ Kms + Kss µ + µT Kss ∂r ∂r

(10.41)

∂µ ∂µ ∂Md ∂Mmm = + Mms + µTG Mss ∂r ∂r ∂r ∂r

(10.42)

10.4 IOR Method for Eigensensitivity

181

∂µ −1 ∂µ −1 Mss = Kss M KG ∂r ∂r d    T −1 ∂KG ∂Md −1 −1 − Md K G + Kss Mms + Mss µ Md ∂r ∂r ∂Kmm ∂KG = ∂r ∂r

(10.43) (10.44)

R R Therefore, ∂M and ∂K depend on ∂M∂rmm , ∂K∂rmm , and ∂µ . Consequently, the pivot ∂r ∂r ∂r . task of calculating the eigensensitivity of the condensed model is to obtain ∂µ ∂r The eigensensitivity is required together with the eigensolutions. Some interim

T

T   −1 , Mms + Mss µ and Kms + Kss µ , in calcuresults, for example, Md−1 KG , Kss lating the eigensolutions can be re-used in the calculation of the eigensensitivity. , The iterative scheme is performed on the reduced model by iteratively updating ∂µ ∂r which adds a small amount of extra computation time. The eigenvalue derivative of the reduced model is calculated by iteratively according to the following updating the derivative of the transformation matrix ∂µ ∂r procedures. As the DOFs associated with the elemental parameter r are included as the additional master DOFs, the simplified equations (Eqs. 10.40–10.44) are used.

(1) The derivative of the transformation matrix µ is initiated from the static condensation. 

∂µ ∂r

[0] =

∂µG =0 ∂r

(10.45)

Considering that the derivatives relating to the slave DOFs are zeros,

R [0]

R [0] and ∂K are the initial condensed mass and stiffness matrices ∂M ∂r ∂r  [0] calculated on the basis of the initial ∂µ as ∂r 



∂K R ∂r

∂M R ∂r

[0]

[0]

  ∂µTG T ∂µG ∂Kmm + Kms + Kms + Kss µG = ∂r ∂r ∂r ∂Kmm ∂µ (10.46) + µTG Kss G = ∂r ∂r    ∂µTG T ∂µ ∂Mmm + Mms G + Mms + Mss µG = ∂r ∂r ∂r ∂M ∂µ mm (10.47) + µTG Mss G = ∂r ∂r 

182

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

Subsequently, the initial eigenvalue derivatives can be obtained as 

∂λ ∂r



[0] = Tm

∂K R ∂r



[0] m − λTm

∂M R ∂r

[0] m

(10.48)

(2) The derivative of the transformation matrix µ is iteratively updated. In the kth iteration (k = 1, 2, …), the derivative of the transformation matrix µ is  

∂Md ∂r ∂µ ∂r



[k] =

[k] =

 [k−1]   [k−1] ∂Mmm ∂µ ∂µ + µTG Mss + Mms ∂r ∂r ∂r

−1 Kss Mss



∂µ ∂r

 T −1 Mms + Kss

(10.49)

[k−1]

Md−1 KG     −1 ∂KG ∂Md [k−1] −1 − Md K G + Mss µ Md ∂r ∂r (10.50)

The derivatives of the condensed stiffness and mass matrices with respect to r are    [k]   ∂Kmm ∂K R [k] ∂µ + Kms = ∂r ∂r ∂r  [k]  T [k]

T  ∂µ ∂µ T Kms + Kss µ + µ Kss (10.51) + ∂r ∂r   [k]    ∂Mmm ∂M R [k] ∂µ + Mms = ∂r ∂r ∂r  [k]  T [k]

T  ∂µ ∂µ + Mms + Mss µ + Mss µT (10.52) ∂r ∂r The eigenvalue derivative in the kth iteration is 

∂λ ∂r

[k]

 = Tm

∂K R ∂r

[k]

 m − λTm

∂M R ∂r

[k] m

(10.53)

(3) The iterations continue until the relative differences of the eigenvalue derivatives from two consecutive iterations are less than a predefined tolerance,

10.4 IOR Method for Eigensensitivity

183

  [k]  [k−1]   ∂λ  − ∂λ   ∂r e =  ∂r  [k]  ≤ Tol ∂λ  

(10.54)

∂r

It is noted that the eigenvalue derivatives of all required modes can be calculated simultaneously by including those interested modes in λ and m , rather than calculated mode by mode.

10.4.2 Eigenvector Derivatives The eigenvector derivative will be derived in this section based on the reduced eigenequation (Eq. 10.7). The intermediate variables in calculating eigensolutions and eigenvalue derivative will be re-used here directly.  m The eigenvector derivative of the master DOFs ∂ is written as the superpo∂r sition of a particular part and a homogeneous part 

∂m ∂r

 = {vi } + ci {m }

(10.55)

where ci is a participation factor. Once the vector {vi } and participation factor ci are determined, the eigenvector derivative can be achieved. Submitting Eq. (10.55) into Eq. (10.7) gives   ∂λ ∂M R ∂K R − MR − λ m (K R − λM R )({vi } + ci {m }) = − ∂r ∂r ∂r

(10.56)

Since (K R − λM R )m = 0, Eq. (10.56) can be rewritten as   ∂λi ∂K R ∂M R − M R − λi m (K R − λM R ){vi } = − ∂r ∂r ∂r

(10.57)

i R R , ∂M , and ∂λ have been calculated in the previous section, and thus, K R , M R , ∂K ∂r ∂r ∂r {vi } can be solved from Eq. (10.57). The eigenvectors at master DOFs are normalized to the reduced mass matrix, which satisfies the orthogonal condition of

{m }T M R {m } = 1

(10.58)

Differentiating Eq. (10.58) with respect to r gives ∂{m } ∂M R ∂{m }T {m } = 0 M R {m } + {m }T M R + {m }T ∂r ∂r ∂r

(10.59)

184

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

Substituting Eq. (10.55) into Eq. (10.59) leads to 

{vi }T + ci {m }T M R {m } + {m }T M R ({vi } + ci {m }) ∂M R {m } = 0 + {m }T ∂r

(10.60)

The participation factor ci is obtained as   1 T T T ∂M R {m } ci = − {vi } M R {m } + {m } M R {vi } + {m } 2 ∂r

(10.61)

Consequently, the eigenvector derivatives of the master DOFs can be achieved by Eq. (10.55). Since the slave DOFs are represented by the master DOFs in terms of s = µm (Eq. 10.3), the eigenvector derivative of the slave DOFs is calculated from the master DOFs as ∂µ ∂s ∂m = m + µ ∂r ∂r ∂r

(10.62)

The eigenvector derivative of the original full model can be recovered by ∂ = ∂r

 ∂m  ∂r ∂s ∂r

 =

0 I ∂µ µ ∂r



m

∂m ∂r

 (10.63)

All items in Eq. (10.63) have been obtained. The eigenvector derivatives are directly computed based on the reduced model (Eq. 10.7) without iteration performed. The size of the reduced eigenequation is equal to the number of master modes, which is much smaller than the original full eigenequation. Therefore, it is very fast to calculate the eigenvector derivatives. As the calculation of the eigenvector derivatives dominates the computational resources in the traditional method (Fox and Kapoor 1968; Nelson 1976), the present method calculates the eigenvector derivatives on the reduced model and improves the computational efficiency significantly.

10.5 Examples 10.5.1 GARTEUR Frame GARTEUR AG-11 frame structure (Fig. 10.1) serves to demonstrate the accuracy of the dynamic condensation method in the calculation of the eigensensitivity. The frame is modeled by 78 Euler–Bernoulli beam elements and 74 nodes. Each node has 3 DOFs, resulting in 216 DOFs in total. Each element is 1 m long. The Young’s modulus of the material is 75 GPa, and the mass density is 2.80 × 103 kg/m3 . The

10.5 Examples

185

moment of inertia of all members is 0.0756 m4 . The cross-section areas of the vertical, horizontal, and diagonal bars are 0.006 m2 , 0.004 m2 , and 0.003 m2 , respectively. The eigensensitivity of the first 10 modes with respect to the bending rigidities of two elements (r 1 and r 2 in Fig. 10.1b) will be calculated in this example. The master DOFs are selected with a relatively higher ratio of mii /k ii and are preferable to be uniformly distributed around the structure (Xia and Lin 2004a). In this connection, Nodes 21, 22, 45, 46, 69, and 70 are chosen as the master nodes. Nodes 1 and 3 associated with the designed element are additionally included as the master nodes to calculate the eigensensitivity with respect to the bending rigidity of Element 1 (denoted as r 1 in Fig. 10.1b). Consequently, the global model with the size of 216 is reduced to 24. Based on the 24 master DOFs, the eigensensitivity of the first 10 modes with respect to r 1 can be calculated by employing the proposed dynamic condensation method as follows: (1) The initialization values are calculated. Some constant matrices relating to −1 , µTG , the and the values, for example, Kss

Tstatic condensation  T  intermediate −1 −1 Kms + Kss µ , Mms + Mss µ , Md KG , and Kss Mss , have been obtained in the calculation of the eigensolutions. They are re-used here directly. [0] G (2) The initial iteration starts with ∂µ = ∂µ = 0. ∂K∂rmm and ∂M∂rmm are calcu∂r ∂r lated, and then the derivatives of the condensed stiffness and mass matrices are obtained as 

∂K R ∂r

[0] =

    ∂Kmm ∂M R [0] ∂KG ∂Md [0] ∂Mmm = , = = ∂r ∂r ∂r ∂r ∂r

(10.64)

Subsequently, the eigenvalue derivatives in the first iteration are 

∂λ ∂r

[0]



  [0] ∂K R [0] T ∂M R = m − λm m ∂r ∂r ∂Kmm ∂Mmm m − λTm m = Tm ∂r ∂r Tm

(10.65)

(3) In the kth iteration (k = 1, 2, ….), the derivatives of the transformation matrices are updated by  [k−1]  [k−1]  ∂µ ∂µ ∂Md [k] ∂Mmm T + Mms = + µG Mss (10.66) ∂r ∂r ∂r ∂r  [k]  [k−1] ∂µ ∂µ −1 = Kss Mss Md−1 KG ∂r ∂r    [k]  ∂M ∂K d G −1 −1 −1 T − Mms Md K G + Kss + Mss µ Md ∂r ∂r (10.67) 

186

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

Fig. 10.1 GARTEUR frame

10.5 Examples

187

and the derivatives of the condensed mass and stiffness matrices are 

∂M R ∂r



[k]

∂K R ∂r

[k]

 [k] ∂µ ∂Mmm + Mms = ∂r ∂r   T  [k]  ∂µ [k] T ∂µ T Mms + Mss µ + µ Mss + ∂r ∂r  [k] ∂µ ∂Kmm = + Kms ∂r ∂r   T  [k] [k]

T  ∂µ ∂µ Kms + Kss µ + µT Kss + ∂r ∂r

(10.68)

(10.69)

Subsequently, the eigenvalue derivative is achieved by 

∂λ ∂r



[k] = Tm

∂K R ∂r



[k] m − λTm

∂M R ∂r

[k] m

(10.70)

(4) The iteration is terminated until the relative differences of the eigenvalue derivatives from two consecutive iterations are less than a predefined tolerance Tol = 1 × 10−6 .   [k]  [k−1]    ∂λ − ∂λ   ∂r (10.71) err or =  ∂r  [k]  ≤ T ol ∂λ   ∂r

(5) Finally, the eigenvector derivatives of the reduced model are directly calculated following Eqs. (10.55)–(10.62) without iterations performed. Afterward, the eigenvector derivatives of the original full model are recovered by Eq. (10.63). Following the above steps, the eigenvalue and eigenvector derivatives of the first 10 modes with respect to r 1 are calculated. Similarly, Nodes 21, 22, 45, 46, 69, 70, and 24 are selected as the master DOFs to calculate the eigensensitivity with respect to the bending rigidity of Element 8 (denoted as r 2 ). The corresponding size of the reduced model is 21. The calculated eigenvalue and eigenvector derivatives of the first 10 modes are listed in Tables 10.1 and 10.2, in which the results using the global model are treated as the exact value. The results using the traditional Guyan condensation method (Guyan 1965; Lin and Lim 1995) are compared in the tables. They are equivalent to the results of the first iteration of the proposed dynamic condensation method. Guyan condensation method is accurate only at the zero frequency and is acceptable for the low-frequency modes. The relative errors of the eigenvalue derivatives of Modes 1–5 are smaller than 20%, and those of Modes 6–10 are very large. Using the present IOR method, the relative errors of the eigenvalue derivatives are less than 1 × 10−5 for all modes. The proposed method, which calculates the eigensensitivity

188

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

Table 10.1 Eigenvalue and eigenvector derivatives with respect to r 1 Full model

Guyan condensation

Eigenvalue derivative

Eigenvalue derivative

1

5227.958

5241.052

2

8497.763

3

Mode

Difference (%)

IOR method SV

Eigenvalue derivative

Difference

SV

0.25

1.000

5227.955

5.55 × 10−7

1.000

8730.007

2.73

1.000

8497.763

2.93 × 10−8

1.000

62,641.41

73,753.43

17.74

0.964

62,641.38

5.20 × 10−7

1.000

4

87,651.76

98,324.8

12.18

0.938

87,651.67

9.99 × 10−7

0.999

5

92,164.51

86,045.94

−6.64

0.913

92,164.5

8.79 × 10−8

0.999

10−9

1.000

6

27,020.00

132,630.5

390.86

0.122

27,020.00

7.58 ×

7

86,370.03

654,047.6

657.26

0.079

86,370.05

2.32 × 10−7

0.998

8

25,800.43

2,035,768

7790.44

0.497

25,800.37

2.32 × 10−6

0.998

10−8

0.999

9

294,211.8

7,190,623

2344.03

0.005

294,211.9

8.57 ×

10

171,068.2

11,586,267

6672.89

0.300

171,068.6

2.18 × 10−6

0.998

SV

SV: The similarity of eigenvector derivatives

Table 10.2 Eigenvalue and eigenvector derivatives with respect to r 2 Full model

Guyan condensation

Exact eigenvalue derivative

Eigenvalue derivative

1

2754.108

2761.811

2

5516.199

3

Mode

Difference (%)

IOR method SV

Eigenvalue derivative

Difference

0.28

1.000

2754.105

8.93 × 10−7

1.000

5644.271

2.32

0.999

5516.2

1.70 × 10−7

1.000

3502.032

3851.65

9.98

0.989

3502.032

1.31 × 10−7

0.999

4

16,484.25

19,542.57

18.55

0.967

16,484.22

1.68 × 10−6

0.998

5

94,147.79

92,321.55

−1.94

0.921

94,147.8

1.18 × 10−7

0.999

10−8

0.999

6

8721.048

18,683.32

114.23

0.105

8721.048

3.73 ×

7

39,413.64

789,605.9

1903.38

0.574

39,413.65

2.43 × 10−7

0.999

8

109,962.5

3,648,438

3217.89

0.143

109,962.4

1.04 × 10−6

0.999

10−6

0.998 0.998

9

642,616.4

2,405,735

274.37

0.013

642,615.7

1.06 ×

10

161,528.3

3,314,319

1951.85

0.018

161,528.5

1.39 × 10−6

SV: The similarity of eigenvector derivatives

based on the dynamic condensation technique, can achieve high accuracy not only for the lower modes but also for the higher modes. The SV value defined in Eq. (3.51) is used to estimate the accuracy of the eigenvector derivatives. The SV values of the Guyan condensation method and the traditional global method approach 1.0 for Modes 1–5, while those of Modes 6–10 are very small or even approach zero. This indicates that the eigenvector derivatives calculated by the Guyan condensation method are inaccurate for the high modes. In

10.5 Examples

189

: averagely distributed master nodes

Fig. 10.2 Cantilever plate

contrast, the eigenvector derivatives calculated by the present IOR method achieve the predefined accuracy for all modes. The SV values of the eigenvector derivatives for all modes are above 0.998. It again demonstrates that the present IOR technique can accurately calculate the eigenvalue and eigenvector derivatives.

10.5.2 A Cantilever Plate The cantilever plate modeled in Sect. 5.4.1 is employed to investigate the computational efficiency of the present method in handling a relatively large-scale structure (Fig. 10.2). The half-bandwidth of the global stiffness and mass matrices is 126. The eigensensitivity of the first 10 modes with respect to the bending rigidities of two elements (denoted as r 1 and r 2 in Fig. 5.2) is calculated. Twenty nodes shown in Fig. 5.2, which are uniformly distributed among the plate structure, are chosen as the master nodes. To calculate the eigensensitivity with respect to an elemental parameter, four nodes associated with the designed element are additionally included as the master nodes. First, the eigensensitivity of the first 10 modes is calculated using the traditional global method with Nelson’s method. The results are regarded as the exact ones. Afterward, the eigensensitivity of the first 10 modes is calculated using the proposed IOR method. The eigenvalue derivatives are recorded in each iteration and compared with the exact values in Fig. 10.3. Only the 1st, 7th, and 10th modes are displayed in the figure for clearness. The iterative process is terminated when the difference of the eigenvalue derivatives from two consecutive iterations is less than the predefined tolerance 10−6 . All modes achieve the predefined tolerance after

190

Relative error (Log)

10 10 10 10 10 10 10

Convergence of Eigenvalue Derivative

2

10

1st mode 7th mode 10th mode

0

10

-2

Relative error (Log)

10

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

-4

-6

-8

-10

10 10 10 10

-12

1

10

4

3

2

5

6

10

Convergence of Eigenvalue Derivative

2

1st mode 7th mode 10th mode

0

-2

-4

-6

-8

-10

-12

1

4

3

2

Iteration

5

6

Iteration

(a) Parameter r1

(b) Parameter r2

Fig. 10.3 Convergence of the eigenvalue derivatives of the 1st, 7th, and 10th modes with respect to elemental parameters

6 iterations. The 1st, 7th, and 10th modes achieve the predefined accuracy after 2, 3, and 5 iterations, respectively. The lower modes converge faster than the higher modes, as expected. The present IOR method can accurately predict the eigenvalue derivatives of the structure with just a few iterations. To compare the computational cost in different schemes, Table 10.3 shows the SV values and running time in calculating the first 10 eigensensitivity with respect to one elemental parameter (r 1 ). The running time is estimated with the MATLAB program in an Intel I5 2.5 GHz PC. The present method requires an iterative scheme to search the eigenvalue derivative, and the iterative scheme is performed for all 10 modes simultaneously. The initialization step consumes about 0.2292 s to compute the interim variables, and each iteration consumes about 0.88 s in the subsequent calculation. Based on the interim results generated in the analysis of the eigenvalue derivative, the eigenvector derivatives are directly calculated based on the reduced model without iteration. The Table 10.3 Computation time in calculating the eigensensitivity with respect to parameter r 1 Scheme

Iteration

Eigenvector derivative

Total

Initialization

Eigenvalue derivative 5 × 0.88

0.1132

4.7824

38.77

12.1619

12.3369

100.00

7.10 × 107

1.15 × 1010

33.25

3.40 × 1010

3.45 × 1010

100.00

Running time (s)

IOR method

0.2292

Full model (traditional method)

0.175

MC

IOR method

7.95 × 108

Full model (traditional method)

5.30 × 108

1.06 × 1010

Relative ratio (%)

References

191

reduced model has a size of 144, much smaller than the full model of 5166. Calculating the eigenvector derivatives costs 0.1132 s only. In sum, the present dynamic condensation method takes about 4.7824 s to calculate the eigensensitivity of the first 10 modes. However, the traditional global method requires about 0.175 s to calculate the eigenvalue derivatives and 12.1619 s to calculate the eigenvector derivative, more than twice the IOR method. In addition, the MC of the IOR method is about 33.25% of the traditional method. It again demonstrates that the IOR method is more efficient than the traditional global method. The IOR is more efficient and thus preferable to calculate the eigensensitivity of a large-scale structure with a large number of DOFs and many design parameters. The nodes should be carefully numbered to reduce the bandwidth of the system matrices and save computation time. The choice of the master DOFs influences the convergence process and the computational efficiency as well. The master DOFs in this example are selected uniformly but not necessarily the best selection.

10.6 Summary This chapter introduces an iterative order reduction method to compute the eigensolutions and eigensensitivities. The method selects some DOFs as master DOFs and derives a dynamic transformation matrix relating the master DOFs to the slave DOFs with an iterative scheme. Using the transformation matrix, the global model is reduced into a much smaller one governed by the master DOFs. The eigensolutions and eigensensitivities are then solved efficiently from the reduced model. The DOFs associated with the specific elemental parameter are included as the master DOFs in calculating eigensensitivities, so that the change of the elemental parameter is constrained within the stiffness and mass matrices of the master DOFs. The proposed IOR method, the traditional global method, and Guyan condensation approach are then applied to two examples. The results indicate that the IOR method can accurately calculate the eigensensitivity of the interested modes in a few iterations. Compared with the traditional global method, the IOR method is more efficient in terms of the multiplication counts and running time.

References Callahan, J.O.: A procedure for an improved reduced system (IRS) model. In: The 7th International Modal Analysis Conference, Orlando, FL, USA (1989) Choi, D., Kim, H., Cho, M.: Iterative method for dynamic condensation combined with substructuring scheme. J. Sound Vib. 317, 199–218 (2008) Fox, R.L., Kapoor, M.P.: Rate of change of eigenvalues and eigenvectors. AIAA J. 6, 2426–2429 (1968) Friswell, M.I., Garvey, S.D., Penny, J.E.T.: Model reduction using dynamic and iterated IRS techniques. J. Sound Vib. 186, 311–323 (1995)

192

10 Dynamic Condensation for Eigensolutions and Eigensensitivities

Friswell, M.I., Garvey, S.D., Penny, J.E.T.: The convergence of the iterated IRS method. J. Sound Vib. 211, 123–132 (1998) Guyan, R.J.: Reduction of stiffness and mass matrices. AIAA J. 3, 380 (1965) Jeong, J., Baek, S., Cho, M.: Dynamic condensation in a damped system through rational selection of primary degrees of freedom. J. Sound Vib. 331, 1655–1668 (2012) Lin, R.M., Lim, M.K.: Structural sensitivity analysis via reduced-order analytical model. Comput. Methods Appl. Mech. Eng. 121, 345–359 (1995) Liu, Z.S., Wu, Z.G.: Iterative-order-reduction substructuring method for dynamic condensation of finite element models. AIAA J. 49, 87–96 (2011) Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14, 1201–1205 (1976) Weng, S., Zhu, A.Z., Zhu, H.P., et al.: Dynamic condensation approach to the calculation of eigensensitivity. Comput. Struct. 132, 55–64 (2014) Xia, Y., Lin, R.: A new iterative order reduction (IOR) method for eigensolutions of large structures. Int. J. Numer. Meth. Eng. 59, 153–172 (2004a) Xia, Y., Lin, R.: Improvement on the iterated IRS method for structural eigensolutions. J. Sound Vib. 270, 713–727 (2004b)

Chapter 11

Dynamic Condensation to the Calculation of Structural Responses and Response Sensitivities

11.1 Preview In the last chapter, the IOR method is employed to compute the eigensolutions and eigensensitivities. This chapter will extend the method for fast computation of the structural responses and response sensitivities of large-scale structures in the time domain (Weng et al., 2017). First, the method derives a dynamic transformation matrix with a simplified iterative scheme to relate the responses of master DOFs to slave DOFs. Then, based on the transformation matrix, the large-scale global vibration equation is reduced into a much smaller condensed one. The response sensitivities are computed efficiently from the derivatives of the master stiffness and mass matrices, by including all the DOFs associated with the concerned element in the master DOFs. Finally, several typical model condensation methods are compared in terms of computational precision and efficiency with an eight-story frame and the cantilever plate used in previous examples.

11.2 IOR Method for Structural Responses As before, the vibration equation of a structure with N DOFs is described as M¨x + C˙x + Kx = f

(11.1)

Dividing the total DOFs into N m master DOFs and N s slave DOFs, the vibration equation is rewritten in a block form as 

Mmm Mms T Mms Mss



x¨ m x¨ s





Cmm Cms + CTms Css



x˙ m x˙ s





Kmm Kms + T Kms Kss



xm xs



 =

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_11

 fm fs (11.2) 193

194

11 Dynamic Condensation to the Calculation of Structural Responses …

Identically, there exists a transformation matrix µ between the slave DOFs and the master DOFs. Therefore, the full response vector can be written as follows (Qu 2002):  x=

xm xs



 =

xm txm





 Im = xm = Vxm µ

(11.3)

Substituting Eq. (11.3) into Eq. (11.1) and premultiplying VT in both sides give M R x¨ m + C R x˙ m + K R xm = f R

(11.4)

 T  M R = VT MV = (Mmm + Mms µ) + µT Mms + Mss µ

(11.5)

 T  K R = VT KV = (Kmm + Kms µ) + µT Kms + Kss µ

(11.6)

C R = VT CV = VT (a1 M + a2 K)V = a1 M R + a2 K R

(11.7)

f R = V T f = f m + µT f s

(11.8)

where

The size of the condensed vibration equation is equal to the number of the master DOFs (N m × N m ), much smaller than that of the original vibration equation (N × N). The pivot task is to calculate the transformation matrix µ. The matrix µ is a constant matrix to relate the master DOFs and slave DOFs, which is irrelevant to the time and external force (including the external excitation and damping force). In consequence, the free vibration system and the excited system share the same matrix µ (Qu 2002). For the sake of clarity, the same free vibration equation of an undamped system is used to derive the dynamic condensation method and is written as       Mmm Mms x¨ m Kmm Kms xm + =0 (11.9) T T x¨ s Mms Mss Kms Kss xs The second line of Eq. (11.9) gives  T  −1 T Mms x¨ m + Mss x¨ s + Kms xs = −Kss xm

(11.10)

Considering Eq. (11.3), Eq. (11.10) can be rewritten as  T  −1 Mms + Mss µ x¨ m + µG xm µxm = −Kss

(11.11)

11.2 IOR Method for Structural Responses

195

−1 T where µG = −Kss Kms is equivalent to the transformation matrix in the Guyan condensation method, as described in Chap. 10. Since the proportional damping does not affect the dynamic condensation of the system matrices (Qu 2002), the free vibration equation of an undamped condensed system is adopted for demonstration

M R x¨ m + K R xm = 0

(11.12)

The accelerations of the master DOFs are written by the displacement as x¨ m = −M−1 R K R xm

(11.13)

Substituting Eq. (11.13) into Eq. (11.11) leads to  T  −1 Mms + Mss µ M−1 µ = Kss R K R + µ G = µ G + µd

(11.14)

The matrix µ is nonlinear and is a function of µ, MR , and KR , which can be calculated with the iterated IRS method based on Eqs. (11.5), (11.6), and (11.14). To improve the computational efficiency, the dynamic condensation matrix KR is simplified as follows.     T   K R = Kmm + Kms µG + µd + µTG + µTd Kms + Kss µG + µd   T    = Kmm + Kms µG + Kms + µTG Kss µd + µTG + µTd Kms + Kss µG + µTd Kss µd

(11.15)

T Noting that Kms + Kss µG = 0 and its transpose Kms + µTG Kss = 0, Eq. (11.15) can be simplified as

K R = KG + µTd Kss µd

(11.16)

Similarly, the condensed mass matrix can be written as  T   T  M R = Mmm + Mms µG + µTG Mms + Mss µG + µTd Mms + Mss µd   + Mms + µTG Mss µd + µTd Mss µd  T    = MG + µTd Mms + Mss µ + Mms + µTG Mss µd + µTd Mss µd

(11.17)

Substituting Eqs. (11.16) and (11.17) into Eq. (11.12) gives   T    M R x¨ m + K R xm = MG + Mms + µTG Mss µd + µTd Mms + Mss µ x¨ m   + KG + µTd Kss µd xm

196

11 Dynamic Condensation to the Calculation of Structural Responses …

   = MG + Mms + µTG Mss µd x¨ m + KG xm  T  + µTd Mms + Mss µ x¨ m + µTd Kss µd xm

(11.18)

As a consequence, Eq. (11.12) can be rewritten as    T   MG + Mms + µTG Mss µd x¨ m + KG xm + µTd Mms + Mss µ x¨ m + µTd Kss µd xm = 0

(11.19)

Substituting Eq. (11.11) into Eq. (11.19), the left-hand side of Eq. (11.19) is simplified as     Md x¨ m + KG xm + µTd −Kss µ − µG + Kss µd xm   = Md x¨ m + KG xm + µTd −Kss µd + Kss µd xm = Md x¨ m + KG xm

(11.20)

Consequently, Eq. (11.19) can be simplified as Md x¨ m + KG xm = 0

(11.21)

x¨ m = −Md−1 KG xm

(11.22)

Equation (11.21) leads to

Substituting Eq. (11.21) into Eq. (11.11) gives  T  −1 Mms + Mss µ Md−1 KG µ = µG + Kss

(11.23)

Equation (11.23) is similar to Eq. (10.18). Accordingly, µ, M R , and K R can be calculated with an iterative process based on Eqs. (10.19)–(10.27). After µ is computed with the iterative process, C R and f R are calculated from Eqs. (11.7) and (11.8). The displacement, velocity, and acceleration of the master DOFs can be solved according to Eq. (11.4) by the Newmark method (Lu and Law 2007; Zhu et al. 2014). The responses of the whole DOFs are finally recovered from the master DOFs by the transformation matrix V as   I V = m , x¨ = V¨xm , x˙ = V˙xm , x = Vxm (11.24) µ Since the computational effort of the structural analysis is approximately proportional to the square of the size of the system, the computational work could be reduced drastically if the size of the vibration equation is reduced significantly.

11.3 IOR Method for Response Sensitivities

197

11.3 IOR Method for Response Sensitivities In this part, the IOR method is used to calculate the structural response sensitivities with respect to r. Equation (11.4) is differentiated with respect to an elemental parameter r on both sides MR

∂f R ∂M R ∂ x˙ m ∂xm ∂ x¨ m ∂C R ∂K R + CR + KR = − x¨ m − x˙ m − xm (11.25) ∂r ∂r ∂r ∂r ∂r ∂r ∂r

As the structural response sensitivities are usually calculated together with the structural responses, variables calculated previously for structural responses can be R R R , ∂K , ∂C , and ∂f∂rR re-used here directly. Therefore, only the derivative matrices ∂M ∂r ∂r ∂r are unknown to be determined. ∂C R and ∂f∂rR are calculated by differentiating Eqs. (11.7) and (11.8) with respect ∂r to r as ∂C R ∂M R ∂K R = a1 + a2 ∂r ∂r ∂r

(11.26)

∂f R ∂µT = fs ∂r ∂r

(11.27)

R and ∂K are calculated from Eqs. (10.33) and (10.34), which are nonlinearly ∂r . Similarly, ∂µ can be achieved iteratively based on Eqs. (10.35) associated with ∂µ ∂r ∂r and (10.36). Again, the elemental design parameter r is located in the master DOFs by selecting the associated DOFs as master DOFs. In this connection, the derivative R , matrices related to the slave DOFs are zeros, as in Eq. (10.39). Consequently, ∂M ∂r ∂µ ∂K R , and are calculated iteratively based on Eqs. (10.45)–(10.54). ∂r ∂r R R R Once ∂M , ∂K , ∂C , and ∂f∂rR are available, the derivatives of the acceleration, ∂r ∂r ∂r   velocity, and displacement of the master DOFs ∂∂rx¨ m , ∂∂rx˙ m , and ∂x∂rm are solved by the Newmark method. The structural response sensitivities of all DOFs are finally recovered from the response sensitivities of the master DOFs by matrix V as

∂M R ∂r

⎡ ⎤  0 Im ∂V = ⎣ ∂µ ⎦ V= , ∂r µ ∂r ∂[V¨xm ] ∂V ∂ x¨ ∂ x¨ m = = x¨ m + V ∂r ∂r ∂r ∂r

(11.28)

(11.29)

∂[V˙xm ] ∂V ∂ x˙ ∂ x˙ m = = x˙ m + V ∂r ∂r ∂r ∂r

(11.30)

∂x ∂xm ∂[Vxm ] ∂V = = xm + V ∂r ∂r ∂r ∂r

(11.31)

198

11 Dynamic Condensation to the Calculation of Structural Responses …

11.4 Examples 11.4.1 A Three-Span Frame The three-span frame introduced in Sect. 2.4.1 is adopted to investigate the procedure and convergence of the proposed IOR method in calculating structural responses and response sensitivities. The Rayleigh damping coefficients with respect to the mass matrix and stiffness matrix are selected as 0.6247 and 0.0039, respectively. The frame structure is excited at Node 92 in the horizontal direction. The excitation force is F = 10 × sin(20t) + 5 × cos(15t) (kN) (0 ≤ t ≤ 5 s). The excitation is discretized into 2501 time steps, each step lasting 0.002 s. The bending rigidities of Elements 104 and 153, denoted r 1 and r 2 in Fig. 11.1b, are chosen as the element parameters. The structural responses and response sensitivities of the frame with respect to the two parameters will be calculated. For comparison, the structural responses and response sensitivities will be calculated by the proposed IOR method, the Guyan condensation method, and the traditional global method. The results from the global method are regarded as the exact results.

(a) Node number

Fig. 11.1 FEM of the frame

(b) Element number

11.4 Examples

199

In the proposed IOR method, the iterative schemes of µ and ∂µ are combined ∂r to calculate the structural responses and response sensitivities simultaneously. The displacements, velocities, and accelerations of the frame and their derivatives with as follows: respect to r1 are calculated by iteratively updating µ and ∂µ ∂r (1) Nodes 40, 41, 45, 49, 92, 93, 97, and 101, which are uniformly distributed among the structure, are selected as the master nodes (Fig. 11.1a). In addition, to calculate the response sensitivities with respect to the bending rigidity of r1 (Element 104), Node 109, attached to Element 104, is additionally chosen as the master node. −1 −1 T −1 (2) Some constant values, for example, Kss , Kss Mms , Kss Mss , ∂M∂rmm , ∂K∂rmm , µG = T −1 T −Kss Kms , µG Mss , and KG = Kmm + Kms tG are calculated to avoid the repeated calculation in later procedures.  (3) The initial values of µ and Md and their derivatives with respect to r ∂µ and ∂r  ∂Md are calculated from the Guyan static condensation as follows: ∂r −1 T µ[0] = −Kss Kms

(11.32)

T  Md[0] = Mmm + Mms µG + Mms µG + µTG Mss µG 

∂µ ∂r

[0]

(11.33)

  −1 T ∂ −Kss Kms ∂µG = =0 = ∂r ∂r   ∂Md [0] ∂Mmm = ∂r ∂r

(11.34)

(11.35)

(4) In the k-th (k = 1, 2, 3, …) iteration, the transformation matrix µ and its are updated iteratively according to derivative ∂µ ∂r −1  T  −1 µ[k] = µG + Kss KG Mms + Mss µ[k−1] Md[k−1]

(11.36)

 T  Md[k] = Mmm + Mms µ[k] + µTG Mms + Mss µ[k] 

∂µ ∂r

[k]





   −1 MT + M µ[k] M[k] −1 ∂KG − ∂Md = Kss ss ms d ∂r ∂r [k−1]   −1 ∂µ −1 M Md[k] KG + Kss ss ∂r



∂Md ∂r

[k]

[k−1] 

 [k]  [k] ∂µ ∂µ ∂Mmm T + Mms = + µG Mss ∂r ∂r ∂r

(11.37) Md[k]

−1

 KG

(11.38)

(11.39)

200

11 Dynamic Condensation to the Calculation of Structural Responses …

The iterative process when the differences of the lowest five eigen  isterminated −1 [k] values λ[k] = eig Md KG between two consecutive iterations are less than 1 × 10−5 . In this frame structure, µ and × 10−5 ) after six iterations.

∂µ ∂r

converge to the predefined tolerance (1

(5) Upon convergence, the condensed matrices (MR , KR , CR , and FR ) are calculated from the updated µ according to Eqs. (11.5)–(11.8). Afterward, the displacements, velocities, and accelerations of the master DOFs are calculated from the condensed vibration equation (Eq. (11.4)) with the Newmark method (Lu and Law 2007; Zhu et al. 2014). The condensed matrices (MR , KR , and CR ) have a size of 24 × 24, much smaller than the original system matrices with a size of 408 × 408. The displacements, velocities, and accelerations of the whole DOFs are recovered from the responses of the master DOFs according to Eq. (11.24). R R (6) Based on the updated derivative ∂µ , ∂M , and ∂K are calculated from Eqs. (6.39) ∂r ∂r ∂r and (6.40). Finally, the derivatives of acceleration, velocity, and displacement of the master DOFs ( ∂∂rx¨ m , ∂∂rx˙ m , and ∂x∂rm ) are calculated from the condensed derivative equation (Eq. (11.25)). The size of the condensed derivative equation is 24 × 24. Therefore, the response sensitivity can be calculated rapidly. In the Guyan condensation method (Guyan 1965; Soheilifard 2015), µ = µG , G = ∂µ , KR = KG , and MR = MG are used to calculate the structural responses. ∂µ ∂r ∂r ∂KG ∂MG ∂K R ∂M R = , and = are used to calculate the response sensitivities. They ∂r ∂r ∂r ∂r are equivalent to the initial step of the proposed condensation method. In the global method, the structural responses and response sensitivities are calculated from the original full model with the Newmark method. The displacement, velocity, and acceleration of Node 40 in the horizontal direction (X direction) by the above three methods are displayed in Fig. 11.2. The structural responses by the Guyan method show an evident discrepancy from the exact responses of the frame. The curves obtained from the proposed IOR method are overlapped with those of the global method, indicating that the proposed method is accurate in calculating the structural responses of a structure. Without losing generality, Table 11.1 lists the average relative errors of the structural responses at randomly selected five nodes, which is estimated by diff(x, x R ) =

average(|x R − x|) average(|x|)

(11.40)

where x represents the exact structural response and x R is the structural response of the condensed model. Among the five nodes, Nodes 40, 41, and 101 are the master nodes, and Nodes 43 and 103 are the slave nodes. The relative errors from the Guyan condensed model are greater than 1% and even larger than 10% sometimes. Therefore, the Guyan condensation method is not accurate enough to estimate the structural responses of a dynamic system. The relative differences in the responses between the proposed condensed model and the original full model are about 10−5 . The relative errors of responses on the two slave

11.4 Examples

Fig. 11.2 Horizontal responses of Node 40 with different methods

201

202

11 Dynamic Condensation to the Calculation of Structural Responses …

Table 11.1 Comparison of structural response with different methods DOFs

Condensation method

Relative error of response Displacement

40(X) 41(X) 101(X) 43(X) 103(X)

10−5

Velocity

Acceleration 9.59 × 10−5

IOR

5.64 ×

Guyan (%)

6.83

7.10

7.18

IOR

5.02 × 10−5

6.59 × 10−5

8.55 × 10−5

Guyan (%)

6.83

7.10

7.18

IOR

4.32 × 10−5

5.58 × 10−5

7.16 × 10−5

Guyan (%)

9.74

11.0

13.1

10−5

7.40 ×

10−5

7.17 ×

10−5

9.31 × 10−5

IOR

5.47 ×

Guyan (%)

6.82

7.08

7.16

IOR

3.84 × 10−5

4.96 × 10−5

6.30 × 10−5

Guyan (%)

9.73

11.0

13.1

nodes are also about 10−5 , indicating that the dynamic transformation matrix µ is a constant matrix irrelevant to the external forces. The response sensitivities of Node 40 with respect to r1 by the three methods are compared in Fig. 11.3. The curves from the proposed IOR model are almost identical to those from the full model, whereas the curves by the Guyan condensed model deviate from the accurate ones. The structural response sensitivities of Nodes 40, 41, 45, 49, 92, 93, 97, and 101 with respect to r 1 are compared. Table 11.2 lists the relative differences between the full model and the condensed model estimated by    average  ∂ x R − ∂ x   ∂r1 ∂r1 ∂x ∂xR   = diff ,  ∂x  ∂r1 ∂r1 average  ∂r1 

(11.41)

∂x where ∂r represents the response derivatives from the full model and ∂∂rx 1R represents 1 those from the condensed model. The relative errors of the response derivatives by the proposed IOR model are about 10−5 . However, the relative errors of the Guyan condensed model almost reach 10%. The Guyan condensed model is insufficiently accurate to calculate the response sensitivities of the dynamic system. Without loss of generality, the response derivatives with respect to r2 (Element 153) are also calculated. Node 137, attached to the element, is chosen as the additional master node. The relative differences of the response derivatives are compared in Table 11.3. The proposed condensed model causes relative errors of about 10−5 , whereas the Guyan condensed model reaches 10% sometimes. It again shows that the proposed IOR method accurately calculates the structural responses and response sensitivities.

11.4 Examples

Fig. 11.3 Response sensitivity with different methods

203

204

11 Dynamic Condensation to the Calculation of Structural Responses …

Table 11.2 Comparison of response sensitivities with respect to r 1 DOFs

Condensation method

Relative error of response sensitivity Displacement 1.50 ×

10−5

Velocity 4.81 ×

10−5

Acceleration 1.212 × 10−4

40(X)

IOR Guyan (%)

7.94

7.89

12.60

41(X)

IOR

9.40 × 10−6

3.52 × 10−5

7.80 × 10−5

Guyan (%)

8.14

8.50

11.60

45(X)

IOR

5.43 × 10−6

2.30 × 10−5

4.58 × 10−5

Guyan (%)

8.26

8.89

12.20

3.70 ×

10−6

1.63 ×

10−5

2.96 × 10−5

49(X)

IOR Guyan (%)

8.31

9.06

12.51

92(X)

IOR

1.25 × 10−5

3.26 × 10−5

7.33 × 10−5

Guyan (%)

8.24

10.81

12.31

93(X)

IOR

1.23 × 10−5

3.20 × 10−5

7.45 × 10−5

Guyan (%)

8.71

10.23

11.22

1.10 ×

10−5

3.37 ×

10−5

8.08 × 10−5

97(X)

IOR Guyan (%)

8.05

10.77

11.76

101(X)

IOR

9.03 × 10−6

2.84 × 10−5

7.07 × 10−5

Guyan (%)

8.56

10.29

11.53

Table 11.3 Comparison of response sensitivities with respect to r 2 DOFs 40(X) 41(X) 45(X) 49(X) 92(X) 93(X) 97(X) 101(X)

Condensation

Relative error of response derivative Displacement

Velocity

Acceleration

IOR

1.95 × 10−5

1.15 × 10−4

1.80 × 10−4

Guyan (%)

9.60

6.13

11.87

IOR

2.25 × 10−5

1.24 × 10−4

2.00 × 10−5

Guyan (%)

9.51

6.06

11.55

IOR

2.15 × 10−5

1.03 × 10−4

1.70 × 10−4

Guyan (%)

9.28

5.91

11.13

10−5

1.86 × 10−4

IOR

1.62 ×

Guyan (%)

8.73

5.69

15.10

IOR

4.56 × 10−6

8.21 × 10−6

2.45 × 10−5

Guyan (%)

5.88

9.24

13.62

IOR

6.46 × 10−6

1.16 × 10−5

2.74 × 10−5

Guyan (%)

5.88

9.23

13.41

10−6

5.18 ×

10−5

1.49 ×

10−5

1.32 × 10−5

IOR

8.25 ×

Guyan (%)

5.87

9.21

13.52

IOR

3.40 × 10−5

5.87 × 10−5

2.75 × 10−5

Guyan (%)

5.87

9.20

13.48

11.4 Examples

205

The tolerance is set to be 1 × 10–5 in the proposed IOR method. More iterations are required to achieve a higher precision. It is helpful to perform a trial and error analysis to balance accuracy and efficiency. In this example, the size of the full model of size 408 × 408 is reduced to the condensed model of size 24 × 24. The structural response and response sensitivity are computed based on the vibration equation step by step. Although the proposed condensation method adds a small amount of computation time to calculate µ and ∂µ by an iterative scheme, it is negligible compared to the large number of time steps ∂r in calculating structural responses and response sensitivities. A cantilever plate will be used to investigate the efficiency of the proposed method.

11.4.2 A Cantilever Plate The cantilever plate used in Sect. 5.4.1 is shown in Fig. 11.4. The plate is excited at the free end on Node 861 in the Z direction, and the excitation force is F = –10 × sin(20t) − 5 × cos(15t) (N) (0 ≤ t ≤ 5 s). The excitation is discretized into 2501 time steps, each lasting 0.002 s. The bending rigidity of an element r is randomly selected to calculate the response sensitivity. The bandwidth of the global stiffness matrix is b = 128.

Fig. 11.4 Cantilever plate model

206

11 Dynamic Condensation to the Calculation of Structural Responses …

The responses of the plate are calculated by uniformly selecting Nodes 11, 21, 221, 231, 431, 441, 641, 651, 851, and 861 as the master nodes. Nodes 432, 452, and 453, attached to Element r, are chosen as the additional master nodes to calculate the response sensitivities with respect to r. The structural responses and response sensitivities are compared by the IOR method, the iterated IRS method, Guyan condensed model, and Newmark method on the full model (Lu and Law 2007; Zhu et al. 2014). The Newmark method on the full model is regarded as the exact one. The tolerance of convergence is set to 10−5 for the proposed IOR method and the iterated IRS method. The IOR method converges to the predefined tolerance within seven iterations, while the iterated IRS method takes twelve iterations. The acceleration and acceleration derivatives of Node 21 in the Z direction with respect to r are shown in Figs. 11.5 and 11.6, respectively. The acceleration and acceleration derivative curves calculated by the proposed method overlap with those of the global method. The acceleration and acceleration curves by the Guyan condensation deviate from the exact ones, implying that the Guyan condensation method is not accurate enough. Compared to the iterated IRS method, the acceleration derivative curves calculated by the IOR method are closer to the real ones. The IOR method is more accurate than the iterated IRS method in calculating response sensitivities. Table 11.4 shows the MC and the running time using an ordinary personal computer with a 4.00 GHz CPU and 16 GB memory. ‘Model reduction’ refers to the model reduction process in forming the condensed system, and ‘Newmark’ represents the process of calculating the responses or response sensitivities using the Newmark method.

AZ(21) 0.4

Full model model method method Full Proposed method IOR method IRS method Iterated IRS method Guyan condensation condensation method method Guyan

Acceleration response(m/s 2)

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0

0.5

1

1.5

2

2.5

3

Time (s)

Fig. 11.5 Acceleration response of Node 21 with different methods

3.5

4

4.5

5

11.4 Examples

207 ∂(AZ(21))/∂r

-3

1.5

x 10

Full model model method method Full Proposed method IOR method IRS method Iterated IRS method Guyan condensation condensation method method Guyan

Acceleration response derivative

1

0.5

0

-0.5

-1

-1.5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

Fig. 11.6 Acceleration sensitivity of Node 21 with different methods

The proposed IOS method costs 0.483 s to obtain the condensed model. The size of the vibration equation is reduced from 4920 × 4920 to 60 × 60. The method takes only 0.069 s to calculate the structural responses, about 1% of that consumed by the full model (6.303 s). The proposed IOR method takes only 0.129 s to calculate the response sensitivities, about 2% of that consumed by the full model (6.512 s). As the vibration equation of the condensed model has a much smaller size than that of the full model, the calculation of structural responses using the condensed model performs much faster than the traditional full model. In this example, 60 master nodes are retained in the proposed dynamic condensation, the iterated IRS method, and Guyan static condensation. The sizes of the vibration equation of the three methods are the same. In consequence, the operating time in calculating the responses and response sensitivities is nearly identical, as shown in Table 11.4. However, the precision and efficiency of the three methods in obtaining the condensed model are different. The iterated IRS method takes 0.877 s to form µ with twelve iterations, while the proposed IOR method takes 0.483 s with , while the seven iterations. The iterated IRS method costs 0.773 s to calculate ∂µ ∂r proposed method costs 0.597 s. The proposed IOR method converges to the accurate model faster than the iterated IRS method. Although the Guyan condensation method calculates the condensed model faster than the two dynamic condensation methods, the precision of Guyan static condensed model is not satisfactory, as shown in Figs. 11.5 and 11.6. MC is additionally used to evaluate the efficiency of the proposed method in Table 11.4, to avoid the random influence in running time caused by different operating environments. The MC of the IOR method is approximately 60% of the iterated IRS method. The computational saving can be more significant when applied to a large-scale structure.

Guyan

0.092

Guyan

0.139

0.069

0.115

0.483

IOR

6.303

0.231

0.992

0.552

6.303

3.66

15.74

8.76

100.00

4.52

22.51

3.372 ×

1.904 ×

1.904 × 107 6.767 × 107

108

13.66

107

0.773

0.597

2.604 × 108

1.965 × 108

Total

Relative ratio (%)

2.124 ×

2.816 ×

108

0.102

0.115

0.129

6.512

0.102

0.888

0.726

6.512

1.998 × 107 1.998 × 107

107

2.124 × 107 2.177 × 108

1.57

13.64

11.15

100.00

1.33

18.80

14.53

1.498 × 109 1.498 × 109 100.00

Model reduction Newmark

Relative ratio (%) Sensitivity

1.904 × 107 2.046 × 108

Iterated IRS 0.877

4.863 × 107

108

1.856 × 108

Total

1.498 × 109 1.498 × 109 100.00

Model reduction Newmark

Response

Iterated IRS 3.182 ×

IOR

Full model

Time (s) Full model

MC

Scheme

Table 11.4 Multiplication count and running time of the methods

208 11 Dynamic Condensation to the Calculation of Structural Responses …

11.5 Summary

209

Table 11.5 Running time of the IOR method using different number of master DOFs Case Nm Running time (s) Response Model reduction

Total Newmark

Sensitivity Model reduction

Total Newmark

1

48

0.0571 × 10 = 0.571 0.060

0.631 0.0823 × 10 = 0.823 0.108

0.931

2

60

0.0690 × 7 = 0.483

0.069

0.552 0.0853 × 7 = 0.597

0.129

0.726

3

120 0.126 × 6 = 0.756

0.153

0.909 0.155 × 6 = 0.930

0.210

1.140

The precision and efficiency of the dynamic condensation methods are inevitably influenced by the selection of master DOFs (Bouhaddi and Fillod 1992; Jeong et al. 2012). To investigate the effect of the number of master DOFs on the efficiency of the proposed method, 48, 60, and 120 master DOFs are investigated. The master DOFs are uniformly distributed on the plate, and the convergence criteria are the same. The running time of the three cases is shown in Table 11.5. When the number of master DOFs increases, the IOR method converges to the predefined tolerance within fewer iterations. However, increasing the number of master DOFs will enlarge the size of system matrices and vibration equation. Updating the transformation matrix µ in each iteration takes a longer computational time, and calculating structural responses and response sensitivities using the Newmark method becomes longer. In contrast, using fewer master DOFs will reduce the size of the condensed vibration equation and speed up the calculation, but require more iterations to achieve the condensed model. Therefore, it is suggested to select an appropriate number of master DOFs in practical engineering (Bouhaddi and Fillod 1992; Jeong et al. 2012). The proposed IOR method achieves high accuracy with a few master modes. In this example, using 60 master DOFs is more efficient than using 48 or 120 master DOFs. The efficiency and accuracy of the proposed condensation method have been demonstrated by a relatively large structure of 4920 DOFs and 800 elemental parameters. The excitation lasts 5 s in 2501 time steps. The small amount of computation time in calculating an accurate condensed model is negligible as compared with the large number of time steps. Since model condensation is inherently superior to dealing with large-scale structures, the proposed IOR method will be more promising when applied to practical large-scale structures for calculating responses and response sensitivities.

11.5 Summary This chapter extends the IOR method to calculate the structural responses and response sensitivities. The transformation matrix µ is determined by Md , avoiding the repeated calculation of K R and M R . The method transforms the global vibration equation into a condensed one with a simplified scheme. The DOFs of concerned

210

11 Dynamic Condensation to the Calculation of Structural Responses …

elements are selected as master DOFs to constrain the perturbation of an elemental parameter within the derivatives of master DOFs, and thus, the response sensitivity is quickly calculated by the condensed model. Applications to two numerical examples demonstrate that the IOR method is very accurate and efficient in calculating the responses and response sensitivities. The IOR method is far more accurate than the Guyan static model, although it needs more computation. Compared with the iterated IRS method, the IOR method converges faster with fewer iterations, thus saving more computational resources. In addition, inclusion of more master DOFs increases the size of the vibration equation and thus extends the computation time for calculating structural responses and response sensitivities, whereas it accelerates the convergence of the dynamic condensation. A trial and error analysis may be required to balance the accuracy and efficiency.

References Bouhaddi, N., Fillod, R.: A method for selecting master DOF in dynamic substructuring using the Guyan condensation method. Comput. Struct. 45, 941–946 (1992) Guyan, R.J.: Reduction of stiffness and mass matrices. AIAA J. 3, 380 (1965) Jeong, J., Baek, S., Cho, M.: Dynamic condensation in a damped system through rational selection of primary degrees of freedom. J. Sound Vib. 331, 1655–1668 (2012) Lu, Z.R., Law, S.S.: Features of dynamic response sensitivity and its application in damage detection. J. Sound Vib. 303, 305–329 (2007) Qu, Z.Q.: Model reduction for dynamical systems with local nonlinearities. AIAA J. 40, 327–333 (2002) Soheilifard, R.: A hierarchical non-iterative extension of the Guyan condensation method for damped structures. J. Sound Vib. 344, 434–446 (2015) Weng, S., Tian, W., Zhu, H.P.: Dynamic condensation approach to calculation of structural responses and response sensitivities. Mech. Syst. Signal Process. 88, 302–317 (2017) Zhu, H.P., Mao, L., Weng, S.: A sensitivity-based structural damage identification method with unknown input excitation using transmissibility concept. J. Sound Vib. 333, 7135–7150 (2014)

Chapter 12

Dynamic Condensation Approach to Finite Element Model Updating

12.1 Preview In Chaps. 10 and 11, several model condensation methods, particularly the IOR method, are introduced to compute the eigensolutions, eigensensitivities, structural responses, and response sensitivities. The IOR method is proved to be very accurate and efficient for large-scale structures. This chapter extends the method to sensitivitybased FE model updating using modal data (eigensolutions) and time history data (structural responses). The eigensolutions or structural responses are calculated using the IOR method to construct the objective function. The eigensensitivities or response sensitivities are then computed for each design parameter to serve as a search direction in model updating. The derivative of the transformation matrix is obtained without the additional master DOFs such that the same master DOFs are used in calculating eigensolutions (or structural responses) and eigensensitivities (or response sensitivities), and thus many intermediate variables are shared for efficiency. The method is finally applied to a cable-stayed bridge and a suspension bridge using modal data and time history data, respectively. The results demonstrate that the IOR method is much more efficient than the traditional method.

12.2 Dynamic Condensation-Based FE Model Updating Using Modal Data Based on the eigensolutions and eigensensitivities calculated by the IOR method introduced in Chap. 10, the proposed model updating method using modal data is performed as follows (Zhu et al., 2018):

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_12

211

212

12 Dynamic Condensation Approach to Finite Element Model Updating

(1) The eigensolutions of the FEM are calculated using the IOR method: (a) The total DOFs are divided into master and slave ones. Kms , Kss , Mms , and Mss are extracted from the corresponding system matrices based on the master and slave DOF selection. (b) The transformation matrix µ is initialized with the Guyan condensation −1 T µ[0] = µG = −Kss Kms

 T Md[0] = MG = Mmm + Mms µG + Mms µG + µTG Mss µG KG = Kmm + Kms µG

(12.1) (12.2) (12.3)

(c) µ is calculated iteratively. In the k-th (k = 1, 2, 3 …) iteration, −1  T  −1 Mms + Mss µ[k−1] Md[k−1] µ[k] = µG + Kss KG  T  Md[k] = Mmm + Mms µ[k] + µTG Mms + Mss µ[k]

(12.4) (12.5)

The iteration stops when it reaches the predefined number. The eigensolutions of the master DOFs are gained by solving the eigenequation KG m = λMd m

(12.6)

The mode shapes of all DOFs are recovered by those of the master DOFs as  V=

 Im and  = Vm µ

(12.7)

(2) The objective function J(r) is constructed using the measured modal data and analytical counterparts according to Eq. (9.1). (3) The eigensensitivities are computed for each design parameter r using the IOR method. In this process, many variables generated in the previous steps are reused here directly for efficiency. The detailed process is as follows: ss ms ss , ∂M , and ∂M are extracted from the (a) The derivative matrices ∂K∂rms , ∂K ∂r ∂r ∂r corresponding element matrices associated with the master and slave DOFs. (b) The iteration for eigensensitivity starts with the Guyan static condensation



∂µ ∂r

[0] =

  −1 T T Kms ∂ −Kss ∂µG −1 ∂Kss −1 T −1 ∂Kms = = Kss Kss Kms − Kss ∂r ∂r ∂r ∂r

12.2 Dynamic Condensation-Based FE Model Updating Using Modal Data

= 

−1 −Kss



T ∂Kss ∂Kms µG + ∂r ∂r

213

(12.8)

[0]

 ∂µTG  T ∂Mms ∂µ ∂Mmm + µG + Mms G + Mms + Mss µG ∂r ∂r ∂r ∂r  T ∂Mms ∂Mss ∂µ + µG + Mss G (12.9) + µTG ∂r ∂r ∂r   ∂ Kmm + Kms µG ∂Kmm ∂Kms ∂µ ∂KG = = + µG + Kms G (12.10) ∂r ∂r ∂r ∂r ∂r

(c)

∂Md ∂r

∂µ ∂r



=

is calculated iteratively. In the p-th (p = 1, 2, 3, …) iteration,

   T ∂µ [ p] ∂µ [ p−1] ∂µG −1 ∂Mms + ∂Mss µ[ p] + M = + Kss ss ∂r ∂r ∂r ∂r ∂r     ∂Kss −1 [ p] −1 T + M µ[ p] KG − Md Kss Mms ss ∂r

      ∂Md [ p−1]  [ p] −1 [ p] −1 ∂KG p] −1 T [ Md Md − + Kss Mms + Mss µ KG ∂r ∂r 

(12.11)

     ∂Mms [ p] ∂Md [ p] ∂µ [ p] ∂µT ∂Mmm G MT + M µ[ p] + µ + Mms = + ss ms ∂r ∂r ∂r ∂r ∂r

   T ∂Mms ∂µ [ p] ∂Mss [ p] + µ + M (12.12) + µT ss G ∂r ∂r ∂r

The iteration stops when the predefined tolerance is reached. Usually, this process can be completed with two or three iterations. (d) The derivatives of the condensed stiffness and mass matrices are calculated by the updated µ and ∂µ ∂r  ∂Kmm ∂Kms ∂K R ∂µ ∂µT  T = + µ + Kms + Kms + Kss µ ∂r ∂r ∂r ∂r ∂r  T ∂K ∂K ∂µ ss ms + µ + Kss (12.13) + µT ∂r ∂r ∂r

214

12 Dynamic Condensation Approach to Finite Element Model Updating

∂Mmm ∂Mms ∂µ ∂M R = + µ + Mms ∂r ∂r ∂r ∂r  T  ∂Mss ∂µT  T ∂µ T ∂Mms Mms + Mss µ + µ + µ + Mss + ∂r ∂r ∂r ∂r (12.14) Consequently, the eigenvalue derivative is calculated by  ∂M R ∂λ T ∂K R = m −λ m ∂r ∂r ∂r

(12.15)

Based on Eq. (12.15), the eigenvalue derivatives of all eigenmodes can be calculated simultaneously by including them in λ and m . (4) The eigenvector derivatives are calculated based on the condensed stiffness and mass matrices (KR and MR ) based on Sect. 10.4.2.3 (5) The sensitivity-based FE model updating is conducted by iteratively adjusting all design parameters r through an optimization algorithm. The elemental parameters are adjusted according to the sensitivity matrix. The optimization process stops when the objective function J(r) reaches the predefined tolerance. The flowchart of the processes is illustrated in Fig. 12.1. In each iteration of the model updating process, the eigensolutions are calculated to form objective function once solely, while the eigensensitivities are calculated with respect to numerous designed parameters repeatedly. The calculation of the eigensolutions and eigensensitivities shares the same master DOFs, since the additional master DOFs are avoided in the present method. In consequence, many intermediate variables calculated for eigensolutions are reused directly to compute the eigensensitivities, for example, µ, µG , Md , KG , λ, and m . This significantly improves the efficiency of the FE model updating with numerous updating parameters.

12.3 Dynamic Condensation-Based FE Model Updating Using Time History Data Based on the structural responses and response sensitivities calculated from the IOR method in Chap. 11, the dynamic condensation-based model updating using time history data is implemented as follows (Tian et al., 2021): (1) The structural responses of the FE model are calculated using the dynamic condensation approach: (a) The transformation matrix µ is calculated with an iterative scheme following the same process in Sect. 12.2, Eqs. (12.1)–(12.5).

12.3 Dynamic Condensation-Based FE Model Updating Using Time History …

215

Fig. 12.1 Dynamic condensation-based FE model updating using modal data

(b) With the updated µ, the reduced system matrices MR , KR , CR , and f R are formed from Eqs. (11.5)–(11.8). By usual numerical time integration methods, the structural responses at master DOFs (¨xm , x˙ m , and xm ) are calculated from Eq. (11.4), from which the structural responses of the global model (¨x, x˙ , and x) are finally recovered from those at master DOFs by Eq. (11.24). (2) The objective function is constructed by the residuals of the analytical and experimental responses at the measured DOFs according to Eq. (9.6). (3) The response sensitivities are computed for each design parameter r using the IOR method. Again, the response sensitivities are calculated without additional master DOFs so that the variables obtained in the former steps in calculating structural responses are reused here directly for efficiency. The detailed process is as follows:

216

12 Dynamic Condensation Approach to Finite Element Model Updating

(a) The derivative matrix ∂µ is first estimated iteratively with the same process ∂r in steps (3a)-(3c) in Sect. 12.2. R R R (b) On the basis of the converged ∂µ , the derivative matrices ∂M , ∂K , ∂C , ∂r ∂r ∂r ∂r ∂f R and ∂r are calculated from Eqs. (10.33), (10.34), (11.26), and (11.27),   respectively. The response sensitivities at master DOFs ∂∂rx¨ m , ∂∂rx˙ m , and ∂x∂rm are calculated from Eq. (11.25) using numerical time integration methods. (c) The response sensitivities at all DOFs are recovered from those at master DOFs according to Eqs. (11.28)–(11.31). (4) The sensitivity matrix of the objective function S(r) with respect to each design parameter at the measured DOFs is formed from Eq. (9.6). The design parameters are then iteratively adjusted to minimize the objective function with an optimization algorithm to achieve an accurate FE model. The flowchart of the above processes is illustrated in Fig. 12.2. The proposed model updating method is preferable to the traditional globalbased model updating method in terms of computational efficiency in terms of two aspects. The traditional global-based model updating method calculates the structural responses and response sensitivities from the large-size vibration equation of global structure (Eq. (11.1) directly, whereas the proposed method reduces Eq. (11.1) into a small-sized vibration equation by the dynamic condensation approach. The structural responses and response sensitivities are then calculated from the small-size reduced vibration equation, reducing computational time and resources considerably. In addition, the analytical responses and response sensitivities at the measured DOFs are usually utilized in response-based model updating methods. The structural responses and response sensitivities at all DOFs need to be calculated and stored in the global method, whereas the dynamic condensation approach allows the structural responses and response sensitivities at the measured DOFs to be calculated solely by Eqs. (11.24) and (11.28)–(11.31).

12.4 Examples 12.4.1 Junshan Yangtze River Bridge A practical cable-stayed bridge, Junshan Yangtze River Bridge in China, is employed here to investigate the accuracy and efficiency of dynamic condensation-based model updating using modal data. The bridge is 4881.178 m long and 33.5 m wide. The bridge is modeled by 611 nodes and 758 elements with 3634 DOFs, as shown in Fig. 12.3. Six uniformly distributed nodes (denoted with red dots in Fig. 12.3) on the girder are selected as the master nodes. Each master node has six DOFs, totaling 36 master DOFs. The experimental structure is simulated by the original FEM with the bending rigidity of eight elements in the mid-span of the girder reduced by 30%, denoted

12.4 Examples

217

Fig. 12.2 Flowchart of FE model updating using dynamic condensation-based time history data

with “D” in Fig. 12.3. The model updating method is then conducted to identify the reduced elements. The bending rigidity of all girder elements is selected as the updating parameters, totaling 310 updating parameters. The first ten lowest experimental eigenvalues and mass-normalized mode shapes are utilized to construct the objective function in the model updating process. The model updating process stops when the objective function reaches 1 × 10−9 . For comparison, the traditional global model updating is also employed, in which the eigensolutions and eigensensitivities are calculated based on the global eigenequation, Eq. (10.1), with Lanczos method (Lanczos 2008) presented in Sect. 2.2.2 and Nelson’s method (Nelson 1976) presented in Sect. 3.2.2, respectively. In modal data-based model updating, the convergence of the objective function is sensitive to the accuracy of eigensensitivities. More accurate eigensensitivities are needed when the objective function approaches the optimum, around which minor

218

12 Dynamic Condensation Approach to Finite Element Model Updating

Fig. 12.3 FEM of Junshan Yangtze River Bridge

errors can cause an incorrect searching direction. Therefore, an adaptive scheme is . In the beginning, ∂µ is adopted with a varying number of iterations to estimate ∂µ ∂r ∂r ∂µG = . approximated by the Guyan static condensation without iterations, i.e., ∂µ ∂r ∂r The number of iterations is then gradually increased to obtain more accurate eigensensitivities. Finally, two iterations are used when the objective function approaches the predefined tolerance (1 × 10−9 ). The convergences of the objective function by the proposed IOR method and global method are displayed in Fig. 12.4. It takes 21.01 min for the IOR method to converge, whereas 39.12 min are required in the global method. The IOR method is much more efficient than the global method. Table 12.1 shows the accuracies of frequencies and mode shapes of the proposed IOR method before and after model updating. The IOR method can achieve similar high-accuracy results as the global method. Specifically, the relative errors of frequencies before updating exceed 1 × 10.3 mostly, which drop greatly to about 1 × 10.6 after updating. In addition, the SV values of all mode shapes reach 1.0000. Figure 12.5 shows the location and severity of the identified elemental bending rigidity reductions on the girder. The bending rigidity of the elements in “D” is reduced by 30%, while no obvious reductions are seen in other elements, which matches the real reductions. Therefore, the proposed model updating method is more efficient than the traditional one.

12.4.2 Jiangyin Yangtze River Bridge A practical suspension bridge, Jiangyin Yangtze River Bridge in Jiangsu Province, China, is used to investigate the accuracy and efficiency of the model updating method introduced in Sect. 12.3. The bridge has a main span of 1385 m. The two pylons of

12.4 Examples 10

219

-2

Traditional Full modelmethod method Proposed method iteration) IOR method (No(No iteration) Proposed method (1 iteration) IOR method (1 iteration) Proposed method (2 iterations) IOR method (2 iterations)

-3

Objective function (Log)

10

10

-4

-5

10

-6

10

10

-7

-8

10

-9

10

0

5

10

15

20

25

30

35

40

Time (min)

Fig. 12.4 Convergence of model updating

184 and 187 m high (Xia et al., 2020), and the steel box girder is 32.5 m wide and 3 m high. A simplified FEM is constructed from the design drawings of the bridge, as shown in Fig. 12.6. The model is composed of 653 nodes, 826 elements, and 3178 DOFs. The side spans are independent of the main span and thus excluded in the FEM. The Rayleigh damping is adopted, and the coefficients associated with the mass and stiffness matrices are set to a1 = 0.3309 s−1 and a2 = 5.219 × 10−3 s, respectively. The El Centro earthquake wave with a peak ground acceleration of 0.1 g is applied to the Y direction of the structure. The earthquake wave lasts 15 s with a sampling rate of 200 Hz, as shown in Fig. 12.7. By using the IOR method, nine uniformly distributed nodes on the girder (Fig. 12.6) are selected as the master nodes. Each node has three master DOFs, i.e., X-, Y-, and Z-translations, resulting in 27 master DOFs in total. Three iterations are required to ensure the convergence of µ and ∂µ . ∂r The damage is simulated by intentionally changing the bending rigidity of some elements of the FEM. The dynamic responses are then calculated from the FEM. The bending rigidity of all girder elements is assumed to be randomly changed in range of − 50% to 50%. The displacements of the master nodes in the Y direction are assumed to be the measured data, resulting in nine measured DOFs in total. Since the measured DOFs are included in the master DOFs, only the responses and response sensitivities at the 27 master DOFs need to be computed and stored. The objective function is the residuals between the model predictions and the measured displacement. The model updating process stops when the norm of the objective function reaches the predefined tolerance of 1 × 10−3 . The SRFs of all 260 girder elements are selected as the updating parameters.

0.099286

10

0.099494

0.083299

0.096291

0.082514

0.095127

8

0.076111

0.069690

0.032800

0.018744

9

0.076091

7

0.055333

0.055270

0.069274

0.045195

4

5

0.032720

3

6

0.045280

0.018497

2

0.0095642

Frequency

0.0095641

1 1.0000 0.9999 1.0000

1.334 × 10−2 10−3

1.888 × 10−3 1.0000 0.9979 1.0000

6.010 × 10−3 2.615 × 10−4 0.9968 0.9985 0.9999

1.224 × 10−2 2.101 × 10−3

9.509 ×

10−3

1.133 ×

10−3

2.446 ×

SV 1.0000

Error 1.903 × 10−5

Before updating

Experiment frequency

Mode

0.099285

0.095126

0.082515

0.076091

0.069274

0.055270

0.045195

0.032720

0.018497

0.0095640

Frequency

−6.584 ×

2.903 ×

−6.434 × 10−6

−6.687 × 10−6

10−6

−5.605 × 10−6

−1.371 × 10−7

10−6

−7.605 × 10−6

−2.116 ×

10−6

−8.096 × 10−6

− 4.045 × 10−6

Error

Full model method

After updating MAC

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.099285

0.095126

0.082515

0.076091

0.069274

0.055270

0.045195

0.032720

0.018497

0.0095640

Frequency

IOR method

Table. 12.1 Frequencies and mode shapes of the cable-stayed bridge before and after model updating (unit of frequency: Hz)

Error

−8.224 ×

4.341 ×

−4.568 × 10−6

−8.074 × 10−6

10−6

−6.773 × 10−6

2.236 × 10−7

10−6

−7.038 × 10−6

−2.639 × 10−6

−7.931 × 10−6

−6.088 × 10−6

SV

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

220 12 Dynamic Condensation Approach to Finite Element Model Updating

12.4 Examples

221

1200 -0.4

1100 1000

-0.3 900 -0.2

800 700

-0.1 600 0

500 200 -20

X (m)

400

Y (m)

Fig. 12.5 Location and severity of reductions of bending rigidity on the girders identified using the proposed method

Fig. 12.6 FEM of Jiangyin Yangtze River Bridge

Fig. 12.7 El Centro earthquake wave with peak ground acceleration of 0.1 g

222

12 Dynamic Condensation Approach to Finite Element Model Updating

Fig. 12.8 Convergence of the two model updating methods

Table. 12.2 Computational time and the number of iterations Methods

Time for each iteration (h)

Number of iterations

Total time (h)

Relative ratio (%)

IOR method

0.0297

39

1.16

5.43

Full model method

1.942

11

21.36

100.00

Model updating is performed on the MATLAB Optimization Toolbox in an ordinary desktop computer with a 3.60 GHz CPU and 20 GB RAM. Model updating is performed by both the global model updating method and the dynamic condensation method, while the global model updating method is used as a reference. The convergence process of the two model updating methods in terms of the norm of the objective function is shown in Fig. 12.8. The computational time and number of iterations of the two methods are compared in Table 12.2. The size of vibration equation of the global model is 3178 × 3178, based on which the global method costs 1.942 h for each iteration and 21.36 h in total for the entire model updating process. On the other hand, the dynamic condensation approach requires 0.0297 h per iteration only. The model updating process stops after 39 iterations within 1.942 h, accounting for 5.43% of the global method. The dynamic condensation approach reduces the global system matrices of size 3178 × 3178 to the reduced system matrices of size 27 × 27. Although additional iterations are required by the proposed method, the total computational time is saved remarkably. The computational efficiency of the proposed model updating is improved greatly. Figure 12.9 compares the displacement of a randomly selected measured DOF (Ytranslation of master Node 5). The displacement deviates from the real one before model updating. The two curves are overlapped after the proposed model updating is implemented. Figure 12.10 compares the relative errors of the analytical responses at all measured DOFs before and after model updating. The relative errors decrease

12.5 Summary

223

Fig. 12.9 Displacement of Node 5 in Y direction before and after model updating

Fig. 12.10 Relative errors of the analytical responses before and after model updating

significantly from approximately 10−1 to 10−3 after the model updating. Therefore, the proposed method is very accurate and efficient in updating the FE model of largescale structures with only a few master DOFs and measured time history responses.

12.5 Summary This chapter extends the previously introduced IOR method to updating large-scale structures using modal data and time history data. The IOR method is employed for accurate and efficient computation of the eigensolutions (or structural responses) and eigensensitivities (or response sensitivities) of the global structure, which are used to form the objective function and sensitivity matrix in model updating. The method

224

12 Dynamic Condensation Approach to Finite Element Model Updating

derives the eigensensitivities (or response sensitivities) directly from the condensed system without the additional master DOFs, which boosts the reuse of intermediate variables obtained in calculating eigensolutions (or structural responses). The accuracy and efficiency of the method are finally investigated with a cable-stayed bridge and a suspension bridge using the modal data and time history data, respectively.

References Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bureau Standard 45, 255–282 (2008) Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14, 1201–1205 (1976) Tian, W., Weng, S., Xia, Q., et al.: Dynamic condensation approach for response-based finite element model updating of large-scale structures. J. Sound Vib. 506, 116176 (2021) Xia, Q., Xia, Y., Wan, H.P., et al.: Condition analysis of expansion joints of a long-span suspension bridge through metamodel-based model updating considering thermal effect. Structural Control and Health Monitoring, vol. 27 (2020) Zhu, T.Y., Tian, W., Weng, S., et al.: Sensitivity-based finite element model updating using dynamic condensation approach. Int. J. Struct. Stab. Dyn. 18, 1840004 (2018)

Part III

Nonlinear Substructuring Methods

Chapter 13

Substructuring Method for Responses and Response Sensitivities of Nonlinear Systems

13.1 Preview In the previous chapters, several substructuring methods and dynamic condensation approaches developed by the authors are introduced for dynamic analysis, sensitivity analysis, and model updating of linear systems. Considering the fact that nonlinearity is common in practical structures while linearity is an idealized assumption, the substructuring method is extended to the nonlinear systems in the following chapters. Though the practical structures usually exhibit nonlinearities in a few local regions only, the whole structure behaves nonlinearly. Calculating structural responses and response sensitivities for a nonlinear system demands much more computational resources than a linear system, as the former requires the nonlinear internal forces to be evaluated iteratively within each time step. Linear substructuring methods developed on the basis of mode superposition techniques are, however, inapplicable to nonlinear systems. In this chapter, Kron’s substructuring method presented in Chap. 2 is extended to compute the structural responses and response sensitivities of nonlinear systems (Tian et al. 2021). The ordinary coherence function is firstly used to detect the presence and location of nonlinearity. The global structure is then partitioned into linear and nonlinear substructures. After division, the linear substructures are treated as independent linear structures, whereas the nonlinearities are restricted within a few specific nonlinear substructures. For the linear substructures, the linear substructural responses are decoupled by the combination of a few master modal responses. The slave modal responses are discarded and compensated by the corresponding transformation matrices contributed by the master modal responses, nonlinear substructural responses, and external excitation on the linear substructures. A reduced vibration equation is thus constructed with the transformation matrices. Since the linear substructures are significantly reduced and the nonlinear substructures are localized, the size of the reduced vibration equation is much smaller than that of the original global structure, and the structural responses and response sensitivities are © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_13

227

228

13 Substructuring Method for Responses and Response Sensitivities …

computed efficiently from the reduced vibration equation. The precision and efficiency of the proposed substructuring method are finally verified by a nonlinear spring–mass system and a relatively large-scale nonlinear frame.

13.2 Substructuring Method for Structural Responses of Nonlinear Systems The vibration equation of a nonlinear system with N DOFs is expressed as M¨x + C(x)˙x + K(x)x = f

(13.1)

where C(x) and K(x), being the nonlinear functions of x, are the time-variant damping and stiffness matrices of the nonlinear system, respectively. If the Rayleigh damping is assumed, C(x) = a1 M + a2 K(x), where a1 and a2 are respectively the damping coefficients associated with the mass and stiffness matrices. The nonlinearities usually exist in a few local areas in practical nonlinear systems. The entire nonlinear system can be divided into linear and nonlinear substructures according to the linear and nonlinear location. The global system is assumed to be divided into N S substructures, including the 1st − N L -th linear substructures and the (N L + 1)-th − N S -th nonlinear substructures. In general, the vibration equation of the j-th ( j = 1, 2, …, N L ) linear substructure is expressed as M( j) x¨ ( j) + C( j ) x˙ ( j) + K( j) x( j) = f ( j ) + g( j)

(13.2)

The vibration equation of the k-th (k = N L + 1, N L + 2, …, N S ) nonlinear substructure is expressed as ( ) ( ) M(k) x¨ (k) + C(k) x(k) x˙ (k) + K(k) x(k) x(k) = f (k) + g(k)

(13.3)

After division, only the damping and stiffness matrices of the nonlinear substructures are time-variant, whereas the nonlinearities are restricted within a few nonlinear substructures. The system matrices of the linear and nonlinear substructures are assembled as ) ) ( ( M L = Diag M(1) , · · · , M(N L ) , M N = Diag M(N L +1) , · · · , M(N S ) ) ( ) ( C L = Diag C(1) , · · · , C(N L ) , C N x N ( ) ( )) ( = Diag C(N L +1) x(N L +1) , · · · , C(N S ) x(N S )

(13.4)

(13.5)

13.2 Substructuring Method for Structural Responses of Nonlinear Systems

) ( ) ( K L = Diag K(1) , · · · , K(N L ) , K N x N ( ) ( )) ( = Diag K(N L +1) x(N L +1) , · · · , K(N S ) x(N S ) ⎧ (1) ⎫ ⎧ (1) ⎫ ⎧ (1) ⎫ ⎪ ⎪ ⎪ ⎨ x¨ ⎪ ⎨ x˙ ⎪ ⎨ x ⎪ ⎬ ⎬ ⎬ . . .. L L L .. .. x¨ = , x˙ = ,x = , . ⎪ ⎪ ⎪ ⎩ (N L ) ⎪ ⎩ (N L ) ⎪ ⎩ (N L ) ⎪ ⎭ ⎭ ⎭ x¨ x˙ x ⎧ (N +1) ⎫ ⎧ (N +1) ⎫ ⎧ (N +1) ⎫ L L L ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x¨ ⎨ x˙ ⎨x ⎬ ⎬ ⎬ . . . N N N .. .. .. , x˙ = ,x = x¨ = ⎪ ⎪ ⎪ ⎩ (N S ) ⎪ ⎩ (N S ) ⎪ ⎩ (N S ) ⎪ ⎭ ⎭ ⎭ x¨ x˙ x ⎧ (1) ⎫ ⎧ (N +1) ⎫ L ⎪ ⎪ ⎪ ⎨ f ⎪ ⎨f ⎬ ⎬ . . L N .. .. ,f = f = ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ f (N L ) f (N S ) ⎧ (1) ⎫ ⎧ (N +1) ⎫ L ⎪ ⎪ ⎪ ⎨ g ⎪ ⎨g ⎬ ⎬ . . L N .. .. g = ,g = ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ g(N L ) g(N S )

229

(13.6)

(13.7)

(13.8)

(13.9)

where superscripts L and N respectively denote the assembled vectors/matrices from the (linear ) and nonlinear ( ) substructures. For brevity, the time-variant system matrices C N x N and K N x N are simplified as C N and K N . The vibration equations of the linear and nonlinear substructures are assembled as M L x¨ L + C L x˙ L + K L x L = f L + g L

(13.10)

M N x¨ N + C N x˙ N + K N x N = f N + g N

(13.11)

The substructures satisfy the displacement compatibility and force equilibrium conditions in the interfaces (Klerk et al. 2008; Weng et al. 2020) { { | xL = DL x L + DN x N = 0 Dx = D D xN }T } | |T | |T }T { gp = DT τ = g L g N = DL τ DN τ p

|

L

N

(13.12) (13.13)

where DL and DN are respectively the corresponding component matrices of the connection matrix D associated with the linear and nonlinear substructural DOFs; }T { xp = x L x N is the assembled displacement vector of the linear and nonlinear

230

13 Substructuring Method for Responses and Response Sensitivities …

}T { substructures; gp = g L g N encloses the interface connection forces of the linear and nonlinear substructures; τ is the Lagrange multiplier to imply the interface intensities, as explained before. Equations (13.10)–(13.13) are the primal formulation of the substructural vibration equation, which can be rewritten in a dual form as ⎤⎧ ⎫ ⎡ L ⎤⎧ ⎫ C 0 0 ⎨ x˙ L ⎬ M L 0 0 ⎨ x¨ L ⎬ ⎣ 0 M N 0 ⎦ x¨ N + ⎣ 0 C N 0 ⎦ x˙ N ⎩ ⎭ ⎩ ⎭ τ¨ τ˙ 0 0 0 0 0 0 ⎡ ⎤⎧ ⎫ ⎧ ⎫ | | T KL 0 − DL xL fL | N |T ⎥⎨ N ⎬ ⎨ N ⎬ ⎢ N +⎣ 0 K − D = f ⎦ x ⎩ ⎭ ⎩ ⎭ τ 0 −D L −D N 0 ⎡

(13.14)

Each linear substructure is treated as an independent linear structure. The assembled linear substructures satisfy the mode superposition as x L = o L z L , x˙ L = o L z˙ L , x¨ L = o L z¨ L

(13.15)

where z L , z˙ L and z¨ L are respectively the coordinates of x L , x˙ L and x¨ L in modal space o L . o L is the eigenvector of the assembled linear substructures, which is expressed as ) ( o L = Diag o(1) o(2) · · · o(N L )

(13.16)

Accordingly, the assembled eigenmatrices o L satisfy the orthogonal conditions of | L |T L L o M o = IL |

|T o L K L o L = /\ L

| L |T L L | L |T ( ) o C o = o a1 M L + a2 K L o L = a1 I L + a2 /\ L

(13.17) (13.18) (13.19)

) ( /\ L = Diag /\(1) /\(2) · · · /\(N L ) is the diagonally assembled eigenvalues of all linear substructures. | |T Premultiplying the first line of Eq. (13.14) with o L , and substituting Eq. (13.15) and the orthogonal equations in Eqs. (13.17)–(13.19), Eq. (13.14) is rewritten as ⎤⎧ ⎫ ⎡ L ⎤⎧ ⎫ a1 I + a2 /\ L 0 0 ⎨ z˙ L ⎬ I L 0 0 ⎨ z¨ L ⎬ ⎣ 0 M N 0 ⎦ x¨ N + ⎣ 0 C N 0 ⎦ x˙ N ⎩ ⎭ ⎩ ⎭ τ¨ τ˙ 0 0 0 0 0 0 ⎡

13.2 Substructuring Method for Structural Responses of Nonlinear Systems

231

|T ⎤⎧ L ⎫ ⎧ | L |T L ⎫ | ⎪ o f ⎪ z 0 − DL oL /\ L ⎬ | N |T ⎥⎨ N ⎬ ⎨ ⎢ N +⎣ = ⎦ x fN 0 K −D ⎪ ⎩ ⎭ ⎪ ⎩ ⎭ τ 0 −D L o L −D N 0 ⎡

(13.20)

Equation (13.20) includes the substructural interface DOFs. Therefore, it is larger than the original global vibration equation Eq. (13.1) and is inefficient to solve directly. A reduced vibration equation of a much smaller size will be derived. For brevity, the formulas of the undamped nonlinear systems are derived here. The case with Rayleigh damping can be derived similarly and will be given directly later. The undamped vibration equation (Eq. 13.20) is expressed as |T ⎤⎧ L ⎫ ⎧ | L |T L ⎫ | ⎤⎧ ⎫ ⎡ ⎪ o f ⎪ I L 0 0 ⎨ z¨ L ⎬ z 0 − DL oL /\ L ⎬ | N |T ⎥⎨ N ⎬ ⎨ N ⎣ 0 M N 0 ⎦ x¨ N + ⎢ = ⎣ ⎦ x fN 0 K − D ⎪ ⎩ ⎭ ⎩ ⎭ ⎪ ⎩ ⎭ τ¨ τ 0 0 0 0 −D L o L −D N 0 (13.21) ⎡

The complete eigenmodes of each linear substructure are divided into master and slave modes. Equation (13.21) is thus rewritten as | L L |T ⎤⎧ L ⎫ ⎤⎧ L ⎫ ⎡ L /\ 0 0 − D o ¨ z z ⎪ ImL 0 0 0 ⎪ ⎪ m ⎪ m⎪ ⎪ | L mL |T ⎥⎪ ⎨ mL ⎪ ⎬ L ⎢ 0 I L 0 0 ⎥⎨ z¨ L ⎬ ⎢ zs ⎢ ⎥ 0 /\ 0 − D o s s s ⎢ ⎥ s + ⎢ ⎥ | | T N N N ⎣ 0 0 M 0 ⎦⎪ x¨ ⎪ ⎣ x ⎪ ⎦⎪ 0 0 K N − DN ⎪ ⎪ ⎩ ⎪ ⎩ ⎪ ⎭ ⎭ L L L L N τ¨ τ 0 0 0 0 −D om −D os −D 0 ⎧ | |T ⎫ L L⎪ ⎪ ⎪ ⎪ |om|T f ⎪ ⎪ ⎨ L L ⎬ os f (13.22) = N ⎪ ⎪ ⎪ f ⎪ ⎪ ⎪ ⎩ ⎭ 0 ⎡

L L In Eq. (13.22), /\m , /\sL , om , and osL are assembled from the substructural master modes of linear substructures as ) ( ) ( j) ( ( j ) ( j) ( j) L ( j) (N L ) λ · · · λ λ (13.23) , /\ /\m = Diag /\(1) = Diag · · · /\ · · · /\ j ( ) 1 2 m m m m N m

) ( ) ( j) ( ( j) ( j) ( j) ( j) (N L ) , /\s = Diag λ N ( j ) +1 λ N ( j) +2 · · · λ N ( j ) +N ( j ) /\sL = Diag /\(1) s · · · /\s · · · /\s m

m

m

s

(13.24)

| ) ( j) | ( j) ( j ) ( ( j) L ( j) (N L ) φ · · · φ φ , o om = Diag o(1) = · · · o · · · o j ( ) 1 2 m m m m N

(13.25)

m

| ) ( j) | ( j) ( ( j) ( j) ( j) (N L ) , os = φ N ( j ) +1 φ N ( j ) +2 · · · φ N ( j ) +N ( j ) osL = Diag o(1) s · · · os · · · os m

m

m

s

(13.26)

232

13 Substructuring Method for Responses and Response Sensitivities …

The second line of Eq. (13.22) gives ( )−1 ( )−1 | L L |T ( )−1 | L |T L D os τ + /\sL os f zsL = − /\sL z¨ sL + /\sL

(13.27)

Substituting Eq. (13.27) into the fourth line of Eq. (13.22) and considering the | |T orthogonal condition of osL M L osL = IsL , one can obtain ( | |T )−1 ( L L L L L ) L L D F M os z¨ s − D L om τ = DL FL DL zm − D N x N − D L F L f L (13.28) ( )−1 | L |T os where F L = osL /\sL is the residual flexibility of the assembled linear substructures and can be calculated from the stiffness and master modes of each linear substructure by (| |−1 | (1) |−1 | (1) |T |−1 | om · · · K(N L ) F L = Diag K(1) − o(1) m /\m | (N L ) |−1 | (N L ) |T ) L) /\m om −o(N m

(13.29)

Premultiplying Eq. (13.27) with osL and considering Eq. (13.28), one can obtain | |T ( | |T )−1 ( ) L zL − DN x N − DL FL f L DL FL DL osL zsL = F L D L D L F L M L osL z¨ sL − D L om m − F L M L osL z¨ sL + F L f L L − T N x N − SM L o L z¨ L + Sf L = −T L zm s s

(13.30)

where | |T ( L L | L |T )−1 L L D F D TL = FL DL D om

(13.31)

| |T ( L L | L |T )−1 N D F D TN = FL DL D

(13.32)

| |T ( L L | L |T )−1 L L D F D S = FL − FL DL D F

(13.33)

SM L osL z¨ sL in Eq. (13.30) is associated with the slave modes. It has little contribution to the structural vibration in terms of energy and is thus neglected (Weng et al. 2013, 2020), which will be verified in the numerical example in Sect. 13.4. Equation (13.30) is therefore simplified into osL zsL = −T L zmL − T N x N + Sf L

(13.34)

13.2 Substructuring Method for Structural Responses of Nonlinear Systems

233

Accordingly, one can obtain L osL z¨ sL = −T L z¨ mL − T N x¨ N + Sf¨

(13.35)

where f¨ L is the second-order derivative of f L with respect to t. For a practical structure, f¨ L can be derived directly if f L is an explicit function of t, or calculated by the finite difference method. Equations (13.34) and (13.35) show that the structural responses contributed by the slave modes osL zsL can be transformed into the summation of master modal responses of the linear substructures zmL , responses of the nonlinear substructures x N , and external excitation at the linear substructures f L . TL , TN , and S act as the corresponding transformation matrices. Substituting Eqs. (13.28) and (13.35) into the first and third lines of Eq. (13.22) and considering F L K L F L = F L (Tian et al. 2019), one can obtain |

|{ { | L |T L N | |T T M T IL + TL ML TL z¨ mL | N |T L L | N |T L N N x¨ N T M T M + T M T | |{ { | |T | L |T L N L /\m + TL K L TL T K T zmL | N |T L L | | + T xN T K T K N + TN KL TN {( } )T | |T L L om − T L f L + T L M L Sf¨ = | |T | |T L f N − T N f L + T N M L Sf¨

(13.36)

Equation (13.36) is the reduced vibration equation of the undamped nonlinear system in terms of zmL and x N . The Rayleigh damping is used in this chapter, and the damping coefficient is linearly associated with the structural mass and stiffness matrix. The reduced vibration equation for the damped system can thus be derived with the same procedure directly as ˜ x¨˜ + C ˜ x˙˜ + K ˜ x˜ = f˜ M

(13.37)

˜ is the equivalent time-variant stiffness matrices of the reduced system; where K ˜ is the equivalent mass matrix of the reduced system; C ˜ is the equivalent timeM variant damping matrices of the reduced system; f˜ is the equivalent force vector of the reduced system; and x¨˜ , x˙˜ , and x˜ are respectively the equivalent acceleration, velocity, and displacement vectors of the reduced system. These equivalent variables are expressed as |

| | L |T L N | L |T L L L I T + T M T M T ˜ = m | |T | |T M TN ML TL M N + TN ML TN | | | L |T L N | |T /\mL + T L K L T L T K T ˜ | N |T L L | |T K= T K T K N + TN KL TN

(13.38)

(13.39)

234

13 Substructuring Method for Responses and Response Sensitivities …

˜ = a1 M ˜ + a2 K ˜ C f˜ =

(13.40)

{(

)T )} | |T ( L om − T L f L + T L M L S a1 f˙ L + f¨ L | |T | |T ( ) f N − T N f L + T N M L S a1 f˙ L + f¨ L { L{ { L{ { L{ zm ˙ z˙ m ¨ z¨ m x˜ = , x˜ = , x˜ = x˙ N x¨ N xN

(13.41)

(13.42)

where f˙ L is the first-order derivative of f L with respect to t and can be derived directly or calculated by the finite difference method. In consequence, the responses can be computed from the reduced vibration equation (Eq. (13.37)) with an iterative scheme by classical numerical integration methods like the Newmark-β method or Wilson-θ method. In each time step, ˜ and an iterative process is implemented to determine the time-variant matrices K ˜ C. In Eq. (13.37), the size of the linear substructures is reduced to the total number of the master modes of linear substructures. Consequently, the system matrices are composed)of the linear and nonlinear parts, which have a size of ( NS NL E E ( j) Nm + N (k) and much smaller than that of the global system of N. j=1

k=N L +1

Calculation of structural responses from the reduced vibration equation can be quite efficient. The structural responses of the nonlinear substructures x N , x˙ N , and x¨ N are extracted directly from Eq. (13.42). According to Eqs. (13.15) and (13.34), the responses of the linear substructures x L , x˙ L , and x¨ L are calculated by ) ( L L L x L = o L z L = om zm + osL zsL = om − T L zmL − T N x N + Sf L

(13.43)

) ( L L L L x˙ L = om z˙ m + osL z˙ sL = om − T L z˙ mL − T N x˙ N + Sf˙

(13.44)

) ( L L L L x¨ L = om z¨ m + osL z¨ sL = om − T L z¨ mL − T N x¨ N + Sf¨

(13.45)

Finally, the responses of the global structure are obtained directly by merging the identical values at the interface DOFs.

13.3 Substructuring Method for Response Sensitivities of Nonlinear Systems The response sensitivities can be calculated using the finite difference method, perturbation method, adjoint method, and direct differentiation method. The direct differentiation method is efficient, accurate, and general for nonlinear problems by directly

13.3 Substructuring Method for Response Sensitivities of Nonlinear Systems

235

deriving the response sensitivity from the vibration equation (Li et al. 2017). In this section, first-order derivative of the structural responses with respect to a design parameter r is derived by the direct differentiation method based on the reduced system vibration equation Eq. (13.37). The design parameter can be in either a linear or nonlinear substructure. Differentiating Eq. (13.37) with respect to r leads to ˙ ¨ ˜ ˜ ˜ ˜ ˜ ∂ x˜ + K ˜ ∂ x˜ = ∂ f − ∂ M x¨˜ − ∂ C x˙˜ − ∂ K x˜ ˜ ∂ x˜ + C M ∂r ∂r ∂r ∂r ∂r ∂r ∂r

(13.46)

where ∂ x˜ = ∂r

{

L ∂zm ∂rN ∂x r

}

∂ x˙˜ = , ∂r

{

L ∂ z˙ m ∂rN ∂ x˙ ∂r

}

∂ x¨˜ = , ∂r

{

L ∂ z¨ m ∂rN ∂ x¨ ∂r

} (13.47)

Since the response sensitivities are usually calculated together with the structural ˜ C, ˜ K, ˜ x¨˜ , x˙˜ , and responses, the variables obtained in the previous section, such as M, x˜ , can be reused here directly. ˜ ˜ ˜ ∂K ∂ f˜ ∂M , ∂r , ∂∂rC , and ∂r are calculated by differentiating Eqs. (13.38)–(13.41) with ∂r respect to r as ⎡ | ⎢ ⎢ ⎢ ⎢ ⎢ ˜ ⎢ ∂M =⎢ ⎢ ∂r ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ˜ ⎢ ∂K =⎢ ⎢ ∂r ⎢ ⎢ ⎢ ⎢ ⎣

|T |T | | |T ∂M L | |T ∂M L ∂T L ∂T L L L L L T TN M T + T ML TN + TL ∂r ∂r ∂r ∂r | |T | |T ∂T L ∂T N + TL ML + TL ML ∂r |T | |T ∂r | | |T ∂M L | |T ∂M L ∂T N ∂T N ∂M N L L N L T + TN M T + T ML TN + TN ∂r ∂r ∂r ∂r ∂r | |T | |T ∂T L ∂T N + TN ML + TN ML ∂r ∂r |

|T

|

|T

∂T L

∂T L

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(13.48)

| |T ∂K L K L TL K L TN + TL + TN ∂r ∂r ∂r ∂r | |T | |T ∂K L | |T ∂T L ∂T N TL + TL K L + TL + TL K L ∂r |T ∂r | |T ∂r | | |T ∂K L | |T ∂K L ∂T N ∂T N ∂K N L L N L T + TN K T + T KL TN + TN ∂r ∂r ∂r ∂r ∂r | |T | |T ∂T L ∂T N + TN KL + TN KL ∂r ∂r

L ∂/\m

⎤

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(13.49)

˜ ˜ ˜ ∂C ∂M ∂K = a1 + a2 ∂r ∂r ∂r

(13.50)

236

13 Substructuring Method for Responses and Response Sensitivities …

⎫ ⎧( (| )T |T L ⎪ | L |T ∂M L ⎪ ∂om ∂T L ∂T L ⎪ ⎪ L L ⎪ f + M S+ T − S⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂r ∂r ∂r ∂r ⎪ ⎪ ⎪ ⎪ ) ⎪ ⎪ ⎪ ⎪ ) | L |T L ∂S ( L ⎪ ⎪ L ⎪ ⎪ ˙ ¨ a f + f M + T ⎬ ⎨ 1 ∂! f ∂r ( = | N |T | N |T ⎪ ⎪ | |T ∂M L ∂r ∂T ∂T ⎪ ⎪ ⎪ ⎪ − fL+ ML S + TN S ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂r ∂r ∂r ⎪ ⎪ ⎪ ⎪ ) ⎪ ⎪ ⎪ ⎪ ( | ) | T L ∂S ⎪ ⎪ N L L ⎪ ⎪ a1 f˙ + f¨ M +T ⎭ ⎩ ∂r , ∂T , and In Eqs. (13.48)–(13.51), ∂T ∂r ∂r (13.31)–(13.33) with respect to r as L

N

∂S ∂r

(13.51)

are calculated by differentiating Eqs.

∂T L ∂F L | L |T ( L L | L |T )−1 L L D D F D D om = ∂r ∂r | |T ( L L | L |T )−1 L ∂F L | L |T ( L L | L |T )−1 L L D D F D D F D D om − FL DL D ∂r ( ) L | |T L L | L |T −1 L ∂om (13.52) D F D + FL DL D ∂r ∂F L | L |T ( L L | L |T )−1 N ∂T N = D D F D D ∂r ∂r ( ) | |T ( L L | L |T )−1 L ∂F L | L |T ( L L | L |T )−1 N D D F D D D F D D − FL DL ∂r (13.53) ∂F L ∂F L | L |T ( L L | L |T )−1 L L ∂S = − D D F D D F ∂r ∂r ∂r | |T ( L L | L |T )−1 L ∂F L | L |T ( L L | L |T )−1 L L D D F D D F D + FL DL D D F ∂r | |T ( L L | L |T )−1 L ∂F L (13.54) D F D − FL DL D ∂r As each substructure is treated as an independent structure, the time-invariant N L L L ∂/\ L ∂o L derivative matrices ∂M , ∂K , ∂M , ∂F , ∂rm , and ∂rm are zeros except for the ∂r ∂r ∂r ∂r specific substructure containing r. If r is located in a linear substructure (e.g., the QN L L L ∂/\ L ∂o L = 0. ∂K , ∂M , ∂F , ∂rm , and ∂rm are related to the derivative th substructure), ∂M ∂r ∂r ∂r ∂r matrices of the Q-th substructure solely and expressed as

13.3 Substructuring Method for Response Sensitivities of Nonlinear Systems

( ( ) ∂M L ) ∂K L (Q) (Q) = Diag 0 · · · ∂K∂r · · · 0 , = Diag 0 · · · ∂M∂r · · · 0 , ∂r ∂r L ) ( ( ) L ∂F ∂/\m (Q) (Q) = Diag 0 · · · ∂F∂r · · · 0 , = Diag 0 · · · ∂/\∂rm · · · 0 ∂r ∂r ) ( L ∂om (Q) = Diag 0 · · · ∂o∂rm · · · 0 ∂r

237

(13.55)

(Q)

where the derivative of the residual flexibility matrix of the Q-th substructure ∂F∂r is (Q) (Q) calculated by Eq. (3.39). ∂K∂r and ∂M∂r can be calculated directly from the elemental ∂/\(Q)

∂o(Q)

stiffness and mass matrices associated with r. ∂rm and ∂rm can be calculated within the Q-th substructure by employing the traditional methods like Rogers’ method (Rogers 1970) or Nelson’s method (Nelson 1976). On the other hand, if r is located in a nonlinear substructure (e.g., the A-th substructure), the time-invariant derivative matrices associated with the linear substructures L L L ∂/\ L ∂o L , ∂M , ∂F , ∂rm , and ∂rm are zeros. The derivative matrices are zeros, that is ∂K ∂r ∂r ∂r N ∂T L ∂T N , ∂r , and ∂S are zeros as a consequence. ∂M are zeros except for the A-th ∂r ∂r ∂r substructure, that is ( ∂M N = Diag 0 · · · ∂r The derivative matrix matrix containing r.

˜ ∂M , ∂r

∂M( A) ∂r

··· 0

)

∂M( A) can be derived directly from ∂r ˜ ∂K ∂ f˜ , and ∂r are then simplified into ∂r

(13.56) the specific elemental

| ˜ | ˜ | | ˜ ∂K ∂f ∂M 0 0 0 0 , , = = = {0} N N ∂M ∂K ∂r ∂r ∂r 0 ∂r 0 ∂r

(13.57)

Therefore, when the design parameter is located in a nonlinear substructure, the derivative matrices are calculated in the nonlinear substructure directly, which is simpler than when the parameter is in the linear substructure. N The final unknown variable in Eqs. (13.46)–(13.51) is ∂K . As KN is the function ∂r N N of xN (see Eq. 13.6), ∂K is the function of xN and ∂x∂r . xN is available in the calcu∂r N lation of structural responses. ∂x∂r can be extracted directly from ∂∂rx˜ in Eq. (13.47). Equation (13.46) can thus be calculated with numerical integration methods in an iterative manner. In each time step, an iterative process is performed to search for N . the accurate time-variant matrix ∂K ∂r ¨

˙

After ∂∂rx˜ , ∂∂rx˜ , and ∂∂rx˜ are solved from Eq. (13.46), the response sensitivities of the nonlinear substructures can be extracted directly from Eq. (13.47). The response sensitivities of the linear substructures are calculated by differentiating Eqs. (13.43)– (13.45) with respect to r

238

13 Substructuring Method for Responses and Response Sensitivities …

) L ) ∂zmL ∂om ∂T L L ( L ∂T N N ∂S L ∂x N − zm + om − T L − x − TN + f ∂r ∂r ∂r ∂r ∂r ∂r (13.58) ( L ) ) ∂ z˙ mL ∂T L L ( L ∂T N N ∂S ˙ L ∂om ∂ x˙ N = − z˙ m + om − T L − x˙ − T N + f ∂r ∂r ∂r ∂r ∂r ∂r (13.59) ( L ) ) ∂ z¨ mL ∂om ∂T L L ( L ∂T N N ∂S ¨ L ∂ x¨ N = − z¨ m + om − T L − x¨ − T N + f ∂r ∂r ∂r ∂r ∂r ∂r (13.60)

∂x L = ∂r ∂ x˙ L ∂r ∂ x¨ L ∂r

(

Finally, the response sensitivities of the global structure are achieved by deleting the identical values at the interface DOFs of adjacent substructures. The traditional method calculates the response sensitivities from the large-size global vibration equation. The proposed substructuring method derives the response sensitivities from the reduced vibration equation Eq. (13.37) of a much smaller size. Since the linear substructures are significantly reduced and the nonlinear substructures are localized, the proposed method is expected to be more efficient than the global one, especially for large-scale structures.

13.4 Examples 13.4.1 A Nonlinear Spring–Mass System A simple nonlinear spring–mass system (Fig. 13.1) of 5 DOFs is first employed to illustrate procedure and accuracy of the proposed substructuring method in detail. Five masses are connected by five linear springs and a nonlinear spring. The five masses are m1 = 100 kg, m2 = 200 kg, m3 = 150 kg, m4 = 200 kg, and m5 = 200 kg. The stiffness parameters of the five linear springs are k 1 = k 2 = 1000 N/m, k 3 = k 4 = 2000 N/m, k 5 = 500 N/m. m4 and m5 are also connected with a nonlinear spring with a cubic restoring force F s = −k s (/\x)3 = −k non (x)/\x, where k s = 2 × 105 N/ m3 , /\x is the relative displacements of m5 and m4 (/\x = x 5 − x 4 ), and k non (x) = k s (/\x)2 is the time-variant equivalent stiffness of the nonlinear spring. The Rayleigh damping is used with the damping coefficients of a1 = 0.0508 s−1 and a2 = 0.0396 s. The system is subject to a harmonic acceleration excitation x¨ g = 0.5 sin(2π t) m/s2 (0 ≤ t ≤ 20 s). The excitation is discretized into 10, 000 time steps, each of 0.002 s. The change ratio of k 2 is first selected as the design parameter, denoted as r 1 . The change ratio of k s (denoted as r 2 ) will be studied later. The former is located in the linear substructure and the latter in the nonlinear one.

13.4 Examples

239

Sub1 k1

k2

m1

m2

r1

Sub2 k3

m3

k4

m4 r2

k5 ks

m5

x

ẍg Fig. 13.1 A nonlinear spring–mass system

Since the nonlinearity is usually unknown in advance for a practical structure, the presence and location of the nonlinearity are identified firstly. Several methods have been developed to detect the presence and location of nonlinearities (Worden and Tomlinson 2001; Kerschen et al. 2006; Noel and Kerschen 2017). Here the ordinary coherence function is used (Kerschen et al. 2006; Newland 1993) γ (ω) = 2

| | | S yx (ω)|2 Sx x (ω)S yy (ω)

=

H1 (ω) H2 (ω)

(13.61)

S (ω)

S (ω)

where H1 (ω) = Sxyxx (ω) and H2 (ω) = Syxyy (ω) are respectively the so-called H 1 and H 2 frequency response function estimators. S yy (ω), Sx x (ω), and S yx (ω) denote the auto-power spectral density (PSD) of the response, auto-PSD of the imposed force, and the cross-PSD between the response and the imposed force, respectively. The ordinary coherence function is unity at all accessible frequencies if and only if the system is linear and noise-free. It is a fast and effective tool for detecting nonlinear behavior in a specific frequency band (Allemang and Brown 1987). The structural responses are firstly calculated to detect the presence and location of nonlinearity, using the Newmark method and Newton–Raphson iteration method on the global structure (Newland 1993). The ordinary coherence functions of the imposed force and the measured acceleration of m1 ~ m5 are displayed in Fig. 13.2. The coherence functions for m4 and m5 deviate from 1 significantly, while those of m1 ~ m3 are close to 1, indicating strong nonlinearities at m4 and m5 . The structural responses and response sensitivities of this nonlinear system are then calculated by proposed substructuring method as follows: (1) The system is separated into two substructures based on the nonlinearity location as shown in Fig. 13.1. The first substructure is linear and consists of four DOFs. The second substructure is nonlinear and consists of two DOFs. After division, the assembled mass and stiffness matrices of the linear substructure are ⎡

100 ⎢ 0 ML = ⎢ ⎣ 0 0

0 200 0 0

0 0 150 0

⎡ ⎤ ⎤ 0 2000 −1000 0 0 ⎢ ⎥ 0 ⎥ ⎥, K L = ⎢ −1000 3000 −2000 0 ⎥ ⎣ 0 −2000 4000 −2000 ⎦ 0 ⎦ 100 0 0 −2000 2000

240

13 Substructuring Method for Responses and Response Sensitivities …

Fig. 13.2 Ordinary coherence functions of m1 ~ m5

The assembled mass and stiffness matrices of the nonlinear substructure are | | | | 100 0 500 + knon (x) −500 − knon (x) MN = , KN = −500 − knon (x) 500 + knon (x) 0 200 The derivative matrices associated with the linear substructure are ⎡

⎤ 1000 −1000 0 0 ⎢ −1000 1000 0 0 ⎥ ∂M L ∂K L ⎥ = 0, =⎢ ⎣ 0 0 0 0⎦ ∂r1 ∂r1 0 0 00 The derivative matrices associated with the nonlinear substructure are | | ∂knon (x) (x) − ∂knon ∂M N ∂K N ∂r1 ∂r1 = 0, = (x) ∂knon (x) − ∂knon ∂r1 ∂r1 ∂r1 ∂r1 (x) where ∂knon = 2ks (x5 − x4 ) ∂r1 is constructed as

(

∂ x5 ∂r1

−

∂ x4 ∂r1

) is time-variant. The connection matrix D

| | D = 0 0 0 1 −1 0 The component matrices of D associated with the linear and nonlinear substructures are | | | | D L = 0 0 0 1 , D N = −1 0

13.4 Examples

241

The external excitations of the linear and nonlinear substructures are f L = −M L I4×4 x¨ g , f N = −M N I2×2 x¨ g (2) The first three modes are selected as the master modes of the linear substructure. (1) The master eigenvalues /\(1) m and eigenvectors om of the linear substructure are (1) (1) (1) (1) (1) solved from the eigenequation K om = /\m M om . r 1 is located in the first substructure. The derivatives of the master eigenvalues ∂o(1) m ∂r1 L ∂/\m , ∂r1

∂/\(1) m ∂r1

and eigenvectors

L L are calculated by employing Nelson’s method (Nelson 1976). /\m , om , ∂o L

and ∂r1m are diagonally assembled from the corresponding matrices of all linear substructures. As this example has one linear substructure only, it has L L ∂/\(1) ∂om ∂o(1) L L (1) ∂/\m m m /\m = /\(1) m , om = om , ∂r1 = ∂r1 , and ∂r1 = ∂r1 . ⎡

⎤

0.3977 0 0 L L ∂/\m ⎦, ∂om = ⎣ 0 0.2901 0 ∂r1 ∂r1 0 0 13.1394 ⎡ ⎤ 0.3977 0 0 L L ∂/\m ⎦, ∂om = ⎣ 0 0.2901 0 ∂r1 ∂r1 0 0 13.1394

⎡

−0.009509 ⎢ 0.002722 =⎢ ⎣ 0.0002738 −0.0007645 ⎡ −0.009509 ⎢ 0.002722 ⎢ =⎣ 0.0002738 −0.0007645

−0.014645 0.005317 −0.001284 −0.006183 −0.014645 0.005317 −0.001284 −0.006183

⎤ 0.005248 0.007467 ⎥ ⎥ −0.021185 ⎦ −0.003919 ⎤ 0.005248 0.007467 ⎥ ⎥ −0.021185 ⎦ −0.003919

(3) The residual flexibility of the linear substructure F L is calculated from Eq. (13.29). The intermediate variables T L , T N , and S are calculated by Eqs. L L N ∂S (13.31)–(13.33). The derivative matrices ∂F , ∂T , ∂T , and ∂r are calculated ∂r1 ∂r1 ∂r1 1 ˜ ˜ C, ˜ and f˜ from Eqs. (13.52)–(13.56). Then the reduced system matrices M, K, ˜ ∂K ˜ ∂C ˜ ∂M are calculated by Eqs. (13.38)–(13.41). Their derivative matrices ∂r1 , ∂r1 , ∂r , 1 ˜

and ∂r∂ f1 are calculated by Eqs. (13.48)–(13.51). of the reduced system (˜x, x˙˜ , and x¨˜ ) and their derivatives (4) The ( structural responses ) ∂ x˙˜ ∂ x¨˜ ∂ x˜ , , and ∂r1 are calculated from Eqs. (13.37) and (13.46), respectively, ∂r1 ∂r1 using the Newmark method. In each time step, the Newton–Raphson iteration ˜ C, ˜ ∂ K˜ , and ∂ C˜ . The is employed to obtain the accurate time-variant matrices K, ∂r1 ∂r1 iterations stop when the relative differences of the norm of displacements of all DOFs are less than the predefined tolerance of 1 × 10−5 . (5) Finally, the responses and response sensitivities of the nonlinear substructure are extracted directly according to Eqs. (13.42) and (13.47), respectively. The responses at the linear substructure are recovered from those of the reduced system by Eqs. (13.43)–(13.45). The response sensitivities at the linear substructure are calculated by Eqs. (13.58)–(13.60).

242

13 Substructuring Method for Responses and Response Sensitivities …

In this example, the Euclidean norm of osL zsL and the inertial item SM L osL z¨ sL in Eq. (13.30) are calculated as 0.1561 and 3.7894 × 10–15 , respectively. Therefore, the contribution of SM L osL z¨ sL is negligible. For comparison, the structural responses are also calculated directly from the global nonlinear vibration equation (Eq. (13.1)) with the Newmark method and Newton–Raphson iteration method (Taucer et al. 1991). The response sensitivities are calculated by applying the direct differentiation method to Eq. (13.1) with an iterative scheme (Gu et al. 2019). The process terminates when the relative errors of displacements of all DOFs converge to the same tolerance of 1 × 10−5 . The results are regarded as exact ones. Figure 13.3 compares the displacement responses of m5 by the global method and the proposed substructuring method. Figure 13.4 compares the restoring force of the nonlinear spring. The structural responses and restoring force from the proposed substructuring method are identical to those from the global method, implying that the substructuring method is accurate in calculating structural responses of the nonlinear system. 0.15

Global method

Global method Proposed method Substructuring method

Displacement (m)

0.1 0.05 0 -0.05 -0.1 -0.15 0

2

4

6

8

10 12 Time (s)

14

16

18

20

Fig. 13.3 Displacement response of m5 (x 5 ) 15

Global method

Global method Proposed method Substructuring method

Force (N)

10

5

0

-5

0

2

4

6

8

10 Time (s)

Fig. 13.4 Restoring force of the nonlinear spring (F s )

12

14

16

18

20

13.4 Examples

243

To quantify the precision of the proposed substructuring method, the relative errors of the structural responses at all DOFs and nonlinear restoring force are calculated by Eq. (7.51) and shown in Table 13.1. All errors of the responses are in the order of 10−6 or 10−7 , and that of the restoring force in the order of 10−5 . Therefore, the proposed substructuring method is very accurate in calculating the responses of this nonlinear system. The response sensitivities are also calculated by the substructuring method and the global method. Figure 13.5 compares the derivatives of x 5 and nonlinear restoring force with respect to r 1 . The results obtained from the substructuring method agree well with those from the global method. The proposed substructuring method is also very accurate in calculating the response sensitivities. Without losing generality, the change ratio of k s (denoted as r 2 ) is also chosen as a design parameter. As r 2 is located in the free nonlinear substructure, the indepenL L L L ∂/\ L ∂o L dent linear substructural derivative matrices, such as ∂K , ∂M , ∂F , ∂rm , ∂rm , ∂T , ∂r ∂r ∂r ∂r ∂S ∂T N , and are zeros. The derivatives of the system matrices are obtained directly ∂r ∂r from Eq. (13.57) based on the derivatives of nonlinear substructures. The response sensitivities are then solved directly from Eq. (13.46) with the Newmark method and Newton–Raphson iteration method. The derivatives of x 5 and the nonlinear restoring force with respect to r 2 are calculated by the proposed method and are compared with those by the global method in Fig. 13.6The two curves are almost identical. This indicates that the proposed substructuring method is also very accurate in calculating the response sensitivities with respect to the nonlinear design parameters. Similarly, the error of the calculated response sensitivities is quantified by Eq. (7.52) and listed in Table 13.2. The relative errors of the response and nonlinear restoring force derivatives are in the order of 1 × 10−4 or less. This verifies again that the proposed method accurately calculates response sensitivities with respect to parameters located in the linear or nonlinear substructures. Table 13.1 Relative errors of structural responses and nonlinear restoring force by the proposed method Items

x˙

x

x¨

2.068 ×

10−6

m2

1.368 ×

10−6

m3

8.669 × 10−7

1.692 × 10−6

1.322 × 10−6

m4

1.035 ×

10−6

2.663 ×

10−6

2.894 × 10−6

m5

1.073 ×

10−6

2.143 ×

10−6

2.619 × 10−6

Fs

1.357 × 10−5

m1

2.125 ×

10−6

1.245 × 10−6

2.592 ×

10−6

1.548 × 10−6

244

13 Substructuring Method for Responses and Response Sensitivities … Global method Substructuring method

Displacement derivative (m)

0.2 0.1 0 -0.1 -0.2

2

0

4

8

10 12 Time (s)

14

16

18

20

(a) Derivative of x5 with respect to r1

10

Force derivative (N)

6

Global method Substructuring method

5

0

-5

0

2

4

6

10

8

12

14

16

18

20

Time (s)

(b) Derivative of

Fs

with respect to

r1

Fig. 13.5 Response sensitivity with respect to r 1

13.4.2 A Nonlinear Frame Model A large frame structure with a nonlinear viscous damper is then utilized to investigate the efficiency of the substructuring method for local nonlinear analysis. The frame is modeled with 196 nodes and 216 Euler–Bernoulli beam elements as shown in Fig. 13.7. The cross-sections of the columns and beams are 800 × 800 mm2 and 500 × 800 mm2 , respectively. The mechanical parameters are chosen as: Young’s modulus is 20 GPa; the mass density is 2500 kg/m3 ; and Poisson’s ratio is 0.3. Each node has three in-plane DOFs, and the model has 576 DOFs in total. The model is fixed at Nodes 1, 26, 51, and 76. The Rayleigh damping is assumed with the coefficients of a1 = 1.3255 s−1 and a2 = 1.3791 × 10−3 s. A nonlinear damper is installed on the structure between Nodes 1 and 29. The nonlinear force of the damper is (Li et al. 2018) f d = sign(x˙d )Cd |x˙d |α

(13.62)

13.4 Examples

245

Displacement derivative (m)

0.01

Global method Substructuring method

0.005

0

-0.005

-0.01

0

2

4

6

10

8

12

14

16

18

20

Time (s)

(a) Derivative of x5 with respect to r2

Force derivative (N)

10

Global method Substructuring method

5

0

-5

0

2

4

6

8

10 12 Time (s)

14

16

18

20

(b) Derivative of Fs with respect to r2

Fig. 13.6 Response sensitivity with respect to r 2 Table 13.2 Relative errors of responses and nonlinear restoring force derivatives with respect to r 1 and r 2 by the proposed substructuring method Items

r = r1

r = r2

∂x ∂r

∂ x˙ ∂r

∂ x¨ ∂r

∂x ∂r

∂ x˙ ∂r

∂ x¨ ∂r

m1

4.729 × 10−5

1.040 × 10−4

7.304 × 10−5

6.334 × 10−4

6.424 × 10−4

6.131 × 10−4

m2

3.412 × 10−5

1.052 × 10−4

1.650 × 10−4

6.253 × 10−4

6.842 × 10−4

6.693 × 10−4

m3

2.300 × 10−5

8.543 × 10−5

2.038 × 10−4

5.707 × 10−4

7.176 × 10−4

7.949 × 10−4

m4

2.585 × 10−5

1.154 × 10−4

1.642 × 10−4

4.665 × 10−4

6.740 × 10−4

3.076 × 10−4

m5

3.698 × 10−5

1.440 × 10−4

1.741 × 10−4

5.922 × 10−4

7.418 × 10−4

2.594 × 10−4

∂ Fs ∂r

3.096 × 10−4

2.609 × 10−4

246

13 Substructuring Method for Responses and Response Sensitivities …

where x˙ d is the relative velocity of the two linked nodes along its axial direction, sign(·) is the sign function, C d is the damping coefficient of the nonlinear damper, and α is the exponent of x˙ d . In this example, C d is set to 100 kN/(ms−1 ), and α is 0.3. The frame is excited by the EL Centro earthquake wave in the horizontal

Sub4

Sub3

Sub2

Sub1

24

49

74

99

23

48

73

98

22

173 174 175 176 47

177 178 179 180 72

181 182 183 184 97

21

46

71

96

20

45

70

95

19

161 162 163 164 44

165 166 167 168 69

169 170 171 172 94

18

43

68

93

17

42

67

92

16

149 150 151 152 41

153 154 155 156 66

157 158 159 160 91

15

40

65

90

14

39

64

89

13

137 138 139 140 38

141 142 143 144 63

145 146 147 148 88

12

37

62

87

11

36

61

86

10

125 126 127 128 35

129 130 131 132 60

133 134 135 136 85

9

34

59

84

8

33

58

83

7

113 114 115 116 32

117 118 119 120 57

121 122 123 124 82

6

31

56

81

5

30

55

80

4

101 102 103 104 29

105 106 107 108 54

109 110 111 112 79

28

53

78

27

52

77

26

51

76

3

r

2

damper

1

y x

ẍg 5m×3=15m

Fig. 13.7 A frame with a nonlinear viscous damper

3m×8=24m

185 186 187 188 50 189 190 191 192 75 193 194 195 196 100

25

13.4 Examples

247

10

Acceleration (m/s 2 )

5

0

-5

-10

0

5

10

15

20

25 Time (s)

30

40

35

50

45

Fig. 13.8 Acceleration of the EL Centro earthquake wave

direction, which is displayed in Fig. 13.8. The excitation lasts 50 s and is discretized into 10, 000 time steps of 0.005 s. The structural responses and response sensitivities with respect to the change ratio of the damping coefficient C d (r in Fig. 13.7) will be calculated. The global structure is partitioned into four substructures, as shown in Fig. 13.7. The substructural information is listed in Table 13.3. After partition, the first substructure containing the nonlinear viscous damper is treated as a nonlinear substructure, and the other three are linear substructures. The nonlinear substructure has 39 DOFs. Five master modes of each linear substructure are retained, resulting in 15 master modes in total. As a result, the system matrices of the reduced vibration equation have a size of 54 × 54. Again, the global method is used for comparison. The tolerance of the relative error of the displacement response in the iterative process is set to 1 × 10−5 . In each time step, several iterations are required to calculate the time-variant reduced system matrices (substructuring method) or global system matrices (global method). For the entire 10, 000 time steps, 38, 663 and 43, 701 iterations are required for the proposed substructuring and global methods, respectively. Fewer iterations are required in the substructuring method. The structural responses are then calculated by the substructuring and global methods. Figure 13.9a compares the horizontal displacement of a randomly selected node (Node 4) in the nonlinear substructure calculated by the two methods. A closeup view is displayed in Fig. 13.9b. The response curves obtained from the substructuring method overlap those from the global method. This implies that the proposed Table 13.3 Substructural information of the frame Substructures

Sub1

Sub2

Sub3

Sub4

No. nodes

15

48

72

72

No. elements

14

48

77

77

Linear/Nonlinear

Nonlinear

Linear

Linear

No. interface nodes

3

4

Linear 4

248

13 Substructuring Method for Responses and Response Sensitivities …

substructuring method accurately calculates the structural responses of a large-scale locally nonlinear system with only five master modes required in each substructure. The response sensitivities with respect to r are then calculated. As r is located in the nonlinear substructure, the derivatives of the system matrices are formed directly from Eq. (13.57), avoiding the computation of many intermediate derivative matrices associated with the linear substructures. The response sensitivities are also calculated by the global method for comparison. Figure 13.10 compares the derivatives of x 4 with respect to r by the two methods. Again, the results of the substructuring method are consistent with those of the global method. Table 13.4 compares the CPU time in calculating the structural responses and response sensitivities, which is counted on the MATLAB platform in a desktop computer with a 3.60 GHz CPU and 20 GB RAM. The traditional global method takes 3.32 s to calculate the structural responses. On the other hand, the initialization process of the substructuring method takes 0.02 s to form the reduced system vibration equation. The substructuring method consumes 0.58 s for the structural responses with the Newmark method and Newton–Raphson iteration method, totaling 0.60 s for the whole process. In addition, the substructuring method consumes 0.72 s to 10 -3

8

Global method Substructuring method

Displacement (m)

6 4 2 0 -2 -4 -6 -8

0

5

10

15

20

25

30

35

40

45

50

Time (s)

(a) 0-50 s

Displacement (m)

0.01

Global method Substructuring method

0.005 0 -0.005 -0.01

0

1

2

3

4

5 6 Time (s)

(b) 0-10 s Fig. 13.9 Horizontal displacement of Node 4 (x 4 )

7

8

9

10

13.4 Examples

249 10 -4

Displacement derivative (m)

2

Global method Substructuring method

1 0 -1 -2

2 Displacement derivative (m)

5

0

10

15

20

25 30 Time (s)

35

40

45

50

(a) 0-50 s

10 -4

Global method Substructuring method

1 0 -1 -2 0

1

2

3

4

5 6 Time (s)

7

8

9

10

(b) 0-10 s Fig. 13.10 Derivatives of x 4 with respect to r

calculate the response sensitivities, while the global method takes 3.44 s. Therefore, the substructuring method is much more efficient than the global method in calculating structural responses and response sensitivities. This is because the size of the system matrices of global method is 576 × 576, whereas the system matrices of the substructuring method are reduced to 54 × 54. Although the substructuring method spends a small amount of time in the initialization process, it is negligible compared to the computational time consumed in the Newmark method on global structure, which takes numerous time steps. The accuracy and efficiency of the substructuring method are closely related to the number of master modes. The effect of the number of master modes is investigated by examining the relative error and computational time. When 9, 15, 24, 36, and 45 master modes are used, the relative errors of x 4 and ∂∂rx4 are calculated and compared in Fig. 13.11. When only nine master modes are used, the computational time for structural responses and response sensitivities is short but their relative errors are significant. When the master modes increase from 9 to 36, the relative errors drop

250

13 Substructuring Method for Responses and Response Sensitivities …

Table 13.4 Computational time for structural responses and response sensitivities Responses

Methods

Response sensitivities

Relative ratio (%)

One parameter

All parameters

Relative ratio (%)

Proposed method

0.60 s

18.07

0.72 s

2.90 min

22.85

Global method

3.32 s

100.00

3.44 s

12.69 min

100.00

gradually, whereas the computational time increases as well. As the number of master modes continues to increase, the relative errors decrease slightly while the computational time increases significantly. After the number of master modes saturates, the inclusion of more master modes slightly improves the precision. On the other hand, when the number of master modes is comparable to the number of the nonlinear DOFs in substructures, the inclusion of more master modes enlarges the size of the reduced vibration equation (Eq. 13.37) significantly. In this example, 15 to 36 master modes are preferable to balance computational accuracy and efficiency. In model updating or damage detection, the parameters of structural model need to be updated continuously. The structural responses and response sensitivities with respect to all design parameters are calculated in each iteration, and hundreds of iterations may be required to achieve the predefined convergence criterion (Weng et al. 2012, 2013). In this example, there are 218 design parameters in total, where C d and α of the nonlinear damper and the bending rigidities of all elements are selected as the design parameters. The global method takes 12.69 min to compute the response sensitivities with respect to all design parameters, whereas the proposed substructuring method needs 2.90 min only. If hundreds of iterations are performed in model updating or damage detection, the proposed substructuring method can save 1.5

-2

Relative error

Response RE

10

-3

10

-4

10

-5

10

-6

Sensitivity RE

Response CT

Sensitivity CT

1

Computational time (s)

10

0.5 5

10

15

20 25 30 Number of master modes

35

40

45

Fig. 13.11 Relative errors (RE) of x 4 and its sensitivity and the total computational time (CT) versus numbers of master modes

References

251

a large amount of computational time. A practical locally nonlinear system usually has thousands of DOFs and design parameters. The proposed substructuring method is efficient to be used for dynamic analysis and model updating.

13.5 Summary This chapter extends Kron’s substructuring method to the nonlinear systems to compute the structural responses and response sensitivities. The coherence function is used to detect the presence and location of the nonlinearity beforehand. Based on the location of nonlinearity, the global structure is then divided into independent nonlinear and linear substructures. The size of the linear substructures is reduced by the superposition of a small number of master modes and a compensation of the slave modes. The expensive nonlinear analysis is only constrained within the local area other than the global structure. Applications to a nonlinear spring–mass system and a nonlinear frame show that the substructuring method is accurate and efficient in calculating the structural responses and response sensitivities of nonlinear systems. Since the linear parts are reduced to a much smaller size and the nonlinear analysis is localized at the substructure level, the proposed substructuring method significantly improves the computation efficiency.

References Allemang, R.J., Brown, D.L.: Experimental modal analysis and dynamic component analysis−vol. 2: measurement techniques for experimental modal analysis. AFWAL Technical Report TR 87-3069 (1987) de Klerk, D., Rixen, D.J., Voormeeren, S.N.: General framework for dynamic substructuring: history, review, and classification of techniques. AIAA J. 46(5), 1169–1181 (2008) Gu, Q., Wang, L., Li, Y., et al.: Multi-scale response sensitivity analysis based on direct differentiation method for concrete structures. Composites 157, 295–304 (2019) Kerschen, G., Worden, K., Vakakis, A.F., et al.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20(3), 505–592 (2006) Li, Y., Huang, S.R., Lin, C., et al.: Response sensitivity analysis for plastic plane problems based on direct differentiation method. Comput. Struct. 182, 392–403 (2017) Li, F., Zhang, S., Peng, H.J., et al.: The model predictive control for structural vibration with local nonlinearity. Chin. J. Comput. Mech. 35(5), 582–588 (2018). (in Chinese) Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14(9), 1201–1205 (1976) Newland, D.E.: An Introduction to Random Vibrations, Spectral and Wavelet Analysis. PrenticeHall, Harlow (1993) Noel, J.P., Kerschen, G.: Nonlinear system identification in structural dynamics: 10 more years of progress. Mech. Syst. Signal Process. 83, 2–35 (2017) Rogers, L.C.: Derivatives of eigenvalues and eigenvectors. AIAA J. 8(5), 943–944 (1970) Taucer, F.F., Spacone, E., Filippou, F.C.: A fiber beam-column element for seismic response analysis of reinforced concrete structures. Earthquake Engineering Research Center, College of Engineering, University of California Berkekey, California (1991)

252

13 Substructuring Method for Responses and Response Sensitivities …

Tian, W., Weng, S., Xia, Y., et al.: An iterative reduced-order substructuring approach to the calculation of eigensolutions and eigensensitivities. Mech. Syst. Signal Process. 130, 361–377 (2019) Tian, W., Weng, S., Xia, Y.: Kron’s substructuring method to the calculation of structural responses and response sensitivities of nonlinear systems. J. Sound Vib. 502, 116101 (2021) Weng, S., Xia, Y., Zhou, X.Q., et al.: Inverse substructure method for model updating of structures. J. Sound Vib. 331(25), 5449–5468 (2012) Weng, S., Zhu, H.P., Xia, Y., et al.: Damage detection using the eigenparameter decomposition of substructural flexibility matrix. Mech. Syst. Signal Process. 34(1–2), 19–38 (2013) Weng, S., Zhu, H.P., Xia, Y., et al.: A review on dynamic substructuring methods for model updating and damage detection of large-scale structures. Adv. Struct. Eng. 23(3), 584–600 (2020) Worden, K., Tomlinson, G.R.: Nonlinearity in Structural Dynamics: Detection, Identification and Modelling. Institute of Physics Publishing (2001)

Chapter 14

Model Updating of Nonlinear Structures Using Substructuring Method

14.1 Preview In nonlinear model updating, the time-domain responses or their decomposed timevarying properties, such as proper orthogonal modes, frequency responses, nonlinear normal modes, Wiener series, and instantaneous frequencies, are used to capture nonlinear behaviors. The response sensitivities of each design parameter serve as a rapid search direction of the optimization. Calculating structural responses and response sensitivities requires the internal force vector and tangent stiffness to be evaluated with numerous iterations, which demands considerable computational time. Consequently, conventional nonlinear model updating is limited to small structures only. Alternatively, the substructuring method has been developed to efficiently update large-scale nonlinear systems (Tian et al. 2021). The structural responses and response sensitivities are calculated accurately and efficiently with the substructuring method, as introduced in Chap. 13. These will be used for updating the nonlinear system in this Chapter. The proposed nonlinear model updating method will be applied to a frame with nonlinear base isolations.

14.2 Procedure of the Substructure-Based Nonlinear Model Updating Method Based on the structural responses and response sensitivities of nonlinear systems formulated in Chap. 13, the procedure of the substructure-based nonlinear model updating method is summarized as follows (Tian et al. 2022): (1) An initial FEM is built based on the design drawings of the corresponding structure. The elemental design parameters r are selected and initialized as zeros. The input excitation f and responses x E , x˙ E , and x¨ E are measured. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_14

253

254

14 Model Updating of Nonlinear Structures Using Substructuring Method

(2) In the i-th iteration, the system matrices M, C, and K are updated based on r. Based on the updated system matrices, the analytical structural responses (x A , A A xA x˙ A , and x¨ A ) and response sensitivities ( ∂x , ∂∂rx˙ , and ∂R ) in each iteration are ∂r ∂r calculated by the substructuring method developed in Chap. 13. (3) After the response sensitivities of all design parameters are calculated, the Jacobian matrix S(r) is formed from Eq. (9.7). Based on S(r), the design parameters r are adjusted using optimization algorithms. (4) Steps (2)–(3) are iteratively repeated until the relative difference of the objective function between two consecutive iterations reaches the predefined tolerance.

14.3 Example: A Nonlinear Frame 14.3.1 Model Updating Without Measurement Noises A three-bay nine-story reinforced concrete planar frame with nonlinear base isolations in Fig. 14.1 is used to demonstrate the accuracy and efficiency of the substructure-based nonlinear model updating method. The cross-sections of columns and beams are 600 × 600 mm2 and 400 × 700 mm2 , respectively. The density and elastic modulus of concrete are 2500 kg/m3 and 30 GPa, respectively. The frame includes 166 nodes, 189 two-dimensional Euler–Bernoulli beam elements and four base isolators. The Rayleigh damping is adopted with a damping ratio of 0.05. The damping coefficients with respect to the mass and stiffness matrices are then calculated as a1 = 1.3563 s−1 and a2 = 7.8676 × 10−4 s by using the 1st and 10th modes of the initial static system (Chopra 2011). The structure is assumed to be excited by the El Centro earthquake wave with a peak ground acceleration of 0.1 g in the horizontal direction. The wave is recorded at a sampling rate of 200 Hz lasting for 15 s, as shown in Fig. 14.2. Four base isolators are installed underneath the column bottoms. Each isolator has only one horizontal DOF with the movement in all other directions being constrained. Each other node has three DOFs, and the frame has 490 DOFs in total. The base isolators are modeled with the Bouc − Wen hysteretic model (Haukaas 2006; Hui et al. 2021) as f b = αk0 x + (1 − α)k0 z

(14.1)

z˙ = a0 x˙ − β|x||z| ˙ p−1 z − γ x|z| ˙ p,

(14.2)

where f b , x, and z are the restoring force, shear deformation, and normalized hysteresis force of the base isolation, respectively; α is the ratio of post-yield stiffness to the initial elastic stiffness; a0 controls the tangent stiffness; β and γ control the

14.3 Example: A Nonlinear Frame

135 131 117 113 99 95

158 154 155 156 157 150

159 160

140 136 137 138 139 132

141 142

122 118 119 120 121 114

123 124

104 100 101 102 103 96

105 106

143

82

83

84

85

64

65

66

67

46

47

48

49

28

29

30

31

10

11

12

13

87

88

23 D1

69

70

112 108 109 110 111 98

51

52

33

34

15

16

94 90

91

92

93

72

73

74

75

54

55

56

57

36

37

38

39

17 D2 18 7

19

20

21

79

61

43

6 2

25

62

Sub 3

58 44 40 26

Sub 2

22

3 D4 C

B

80 76

35

24

Sub 4

130 126 127 128 129 116

53

42

14

1 D3 A y x

148 144 145 146 147 134

71

60

32

27

97

166 162 163 164 165 152

89

78

50

45 41

115 107

68

63 59

133 125

86

81 77

9 5

161 151

3.6 m ×9=32.4 m

153 149

255

8 4

D Sub 1

ẍg 9m

6m

Accelerometer Base isolation

9m

24 m

(a) Global model

(b) Substructure configuration

Fig. 14.1 Reinforced concrete frame with nonlinear base isolations

Acceleration (m/s 2 )

1 0.5 0 -0.5 -1

0

5

10

15

Time (s)

Fig. 14.2 El Centro earthquake wave with the peak ground acceleration of 0.1 g

shape of the hysteresis loop; p controls the transition from linear to nonlinear range; and k 0 is the initial elastic stiffness. In this example, α = 0.1, a0 = 1, β = γ = 50, p = 1.0, and k 0 = 2.0 × 103 kN/m. The structural responses and response sensitivities are first calculated by the substructuring method. The change ratio of the bending rigidity of Element D1 (see Fig. 14.1a) is selected as the design parameter. The frame model is divided into four substructures, as shown in Fig. 14.1b. The detailed substructural information is listed in Table 14.1. The first substructure containing the nonlinear base isolators is treated as the nonlinear substructure, and the three others are linear substructures.

256

14 Model Updating of Nonlinear Structures Using Substructuring Method

The nonlinear substructure has 16 DOFs. The first 15 modes of each linear substructure are selected as the master modes, totaling 45 for the structure. The traditional global method is also used for comparison, and the results are treated as exact. In the global method, the structural responses are calculated from the global vibration equation (Eq. 13.1) with the Newmark-β method and Newton–Raphson iteration, and the response sensitivities of the structure with the direct differentiation method (Li et al. 2017). The horizontal displacement and hysteretic loop of base isolator A are compared in Fig. 14.3. The derivative of the horizontal displacement of the base isolator with respect to the design parameter is compared in Fig. 14.4. The curves from the substructuring method are overlapped with those from the global method, indicating a high precision of structural responses and response sensitivities by the substructuring method. In numerical model updating, the FEM is usually modified by intentionally changing several system parameters. The model updating is then implemented to identify these changes. The elemental bending rigidity of all 189 elements and the six coefficients (α, k 0 , p, γ , β, and a0 ) of all base isolators are selected as updating parameters. Consequently, there are 213 updating parameters in total, including 189 linear and 24 nonlinear ones. 27 accelerometers are assumed to be uniformly installed on the frame to measure the horizontal accelerations of the columns and vertical accelerations of the beams. The sensor location and directions are shown in Fig. 14.1a. The measured accelerations are used in model updating, and the displacement and velocity are excluded from the objective function, Eq. (9.6). The model updating is then conducted to minimize the objective function optimally using the Levenberg–Marquardt algorithm (Levenberg 1944; Marquardt 1963) on the MATLAB platform. The algorithm constructs the Jacobian matrix S(r) to provide a rapid searching direction for model updating. The model updating process stops when the relative difference of the objective function between two consecutive iterations is less than the predefined tolerance 1 × 10−4 . Five cases listed in Table 14.2 are studied in this example. The l1 regularization parameters for Cases 1, 2, and 3 are set to 6 × 10−4 , 3 × 10−3 , and 6 × 10−4 , respectively. Cases 4 and 5 contain the measurement noise and will be studied in the next section. In Case 1, two linear parameters are changed. Specifically, the bending rigidities of Elements D1 and D2 are reduced by 40% and 30%, respectively, while other parameters (including the nonlinear ones) remain unchanged. The parameter reduction factor (PRF) quantifies the relative change between the updated parameters and Table 14.1 Substructural information of the frame Substructures

Sub 1

Sub 2

Sub 3

Sub 4

No. nodes

8

58

58

54

No. elements

4

63

63

59

Linear/Nonlinear

Nonlinear

Linear

Linear

Linear

No. interface nodes

4

4

4

14.3 Example: A Nonlinear Frame

257

0.02

Global method Substructuring method

Displacement (m)

0.015 0.01 0.005 0 -0.005 -0.01 -0.015 0

5

10

15

Time (s) (a) Displacement

Restoring force (kN)

15 10 5 0 -5 -10 -15 -0.015

Global method Substructuring method

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Displacement (m) (b) Hysteretic loop of restoring force

Fig. 14.3 Displacement and restoring force of base isolator A 10 -5

Displacement derivative (m)

2 1.5 1 0.5 0 -0.5 -1 -1.5 -2

0

Global method Substructuring method

5

Time (s)

Fig. 14.4 Displacement derivative of base isolator A

10

15

258

14 Model Updating of Nonlinear Structures Using Substructuring Method

Table 14.2 Reductions in elemental bending rigidity and nonlinear parameters Cases

Noise level

No. of Measured points

Assumed PRFs

Case 1 0

27

Elements D1 (−40%) and D2 (−30%)

Case 2 0

27

k0A (−20%) and k0C (−10%)

Case 3 0

27

Elements D1 (−40%) and D2 (−30%); k0A (−20%) and k0C (−10%)

Case 4 2%

27

Elements D1 (−40%) and D2 (−30%); k0A (−20%) and k0C (−10%)

Case 5 2%

15

Elements D1 (−40%) and D2 (−30%); k0A (−20%) and k0C (−10%)

(a) Linear parameters

(b) Nonlinear parameters

Fig. 14.5 Identified PRFs for linear and nonlinear parameters (Case 1)

the original ones. Using the proposed substructure-based model updating method, the PRFs of all design parameters are identified and shown in Fig. 14.5. The bending rigidities of Elements D1 and D2 are identified to be reduced by 37.9% and 29.0%, respectively, and some unchanged elements have small values, which are very close to the actual parameter reductions. No apparent PRFs are identified on the nonlinear parameters of the base isolators. Case 2 considers the reductions on nonlinear parameters solely. Specifically, the initial elastic stiffness of base isolations A and C (denoted as k0A and k0C in Table 14.2) is reduced by 20% and 10%, respectively. Figure 14.6 shows the identified PRFs of all design parameters by the proposed method. k0A and k0C are reduced by 18.9% and 8.7%, respectively, whereas other linear and nonlinear parameters show no apparent changes. Again, the identified results agree well with the actual parameter reductions. Case 3, which is the combination of Cases 1 and 2, considers both linear and nonlinear parameters to be reduced. The identified PRFs for all design parameters are shown in Fig. 14.7. Again, the reductions in both the linear elements and nonlinear parameters are accurately identified, and no apparent false identification in other parameters is noted.

14.3 Example: A Nonlinear Frame

(a) Linear parameters

259

(b) Nonlinear parameters

Fig. 14.6 Identified PRFs for linear and nonlinear parameters (Case 2)

(a) Linear parameters

(b) Nonlinear parameters

Fig. 14.7 Identified PRFs for linear and nonlinear parameters (Case 3)

The analytical and measured vertical accelerations of Node 11 before and after updating for Cases 1–3 are shown in Fig. 14.8. The analytical accelerations before model updating considerably deviate from the measured ones. They agree very well after the proposed nonlinear model updating method is conducted. Therefore, the proposed method accurately reproduces the actual accelerations. The computational time of the proposed substructure-based model updating method is compared with that of the global approach. Both are carried out in an ordinary desktop computer with a 3.60 GHz CPU and 20 GB RAM. The convergence processes of Case 3 by the two methods are compared in Fig. 14.9. The computational time and number of iterations are listed in Table 14.3. The global method completes the model updating within 24 iterations and consumes 5.04 h. By contrast, the substructuring method requires 27 iterations and 2.43 h, accounting for 48.1% computational time of the global method to achieve the same accuracy. The computational efficiency is remarkably improved. This is because 16 nonlinear substructural DOFs and 45 linear substructural master modes are used in the substructuring method. The reduced system matrices have a size of 61 × 61, much smaller than the matrices of 490 × 490 in the global method.

260

14 Model Updating of Nonlinear Structures Using Substructuring Method 0.025

Experimental data Analytical data (before updating) Analytical data (after updating)

Acceleration (m/s 2 )

0.02 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02

0

5

Time (s)

10

15

(a) Case 1 0.025

Experimental data Analytical data (before updating) Analytical data (after updating)

Acceleration (m/s 2 )

0.02 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 0

5

Time (s)

10

15

(b) Case 2 0.025

Experimental data Analytical data (before updating) Analytical data (after updating)

Acceleration (m/s 2 )

0.02 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 0

5

Time (s)

10

(c) Case 3

Fig. 14.8 Vertical acceleration of Node 11 before and after updating

15

14.3 Example: A Nonlinear Frame

261

2

Global method Substructuring method

1.8

Objective function

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

1

2

3

5

4

6

Time (hour)

Fig. 14.9 Convergence process of the model updating for Case 3

Table 14.3 Computational time and the number of iterations for the two methods (Case 3) Methods

Size

Time per iteration (hour)

No. iterations

Total time (h)

Substructuring method

61 × 61

0.090

27

2.43

48.1%

Global method

490 × 490

0.210

24

5.04

100.0%

Relative ratio

14.3.2 Model Updating with Measurement Noises The effect of measurement noises on the performance of the proposed nonlinear substructuring method is investigated. Case 4 considers white Gaussian noise in the input excitation and acceleration responses. All other conditions remain the same as those in Case 3. The measured acceleration responses and input earthquake acceleration polluted by noises are simulated by x¨ˆ E = x¨ E (1 + e1 U )

(14.3)

x¨ˆ gE = x¨ gE (1 + e2 U ),

(14.4)

where x¨ˆ E and x¨ˆ gE are the measured acceleration responses and earthquake acceleration, respectively; e1 and e2 are their corresponding noise levels, and U is a normally distributed random number with a zero mean value and a unit standard deviation. 2% noises are considered in this study, that is, e1 = e2 = 2%. The proposed model updating method is then implemented using the noisy data. Again, the bending rigidities of all beams and columns and the six coefficients of all isolation bearings serve as the design parameters. The regularization parameter is 8 × 10−4 . The model updating process stops when the relative differences of the

262

14 Model Updating of Nonlinear Structures Using Substructuring Method

Fig. 14.10 Identified PRFs in Cases 3–5

objective function between two consecutive iterations are smaller than 1 × 10−4 . 30 iterations are required by the proposed method. The identified PRFs are shown in Fig. 14.10. Similarly, actual PRFs are identified for the bending rigidities of Elements D1 and D2 , k0A , and k0C only, with the values of − 0.35, − 0.28, − 0.18, and − 0.10, respectively. Therefore, the proposed method is robust to the measurement noise. The effect of the number of measurement points on the proposed method is then studied by reducing the number of measurement points from 27 in Case 4 to 15 in Case 5. That is, only the accelerations at the 1st, 3rd, 5th, 7th, and 9th floors of the frame are measured. All other conditions remain the same as those in Case 4, including 2% noise in the input excitation and acceleration responses. The updating results are also compared in Fig. 14.10. Again, PRFs are identified for the bending rigidities of Elements D1 and D2 , k0A , and k0C , with the values of −0.29, −0.25, −0.18, and −0.12, respectively. The actual locations of the presumed parameter reductions can still be identified accurately, while the identification accuracy drops slightly. This is because fewer measurement data provide less structural behaviors for model updating.

14.4 Summary This chapter introduces an accurate and efficient substructuring-based nonlinear model updating method. The substructuring method divides the global structure into several linear and nonlinear substructures and then reduces the large-size vibration system into a much smaller one. The analytical responses and response sensitivities are computed from the reduced system for the nonlinear regularized model updating.

References

263

Application to a frame with nonlinear base isolations demonstrates that the substructuring method is accurate, robust to the measurement noise, and more efficient than the global model updating approach.

References Chopra, A.K.: Dynamics of Structures: Theory and Applications to Earthquake Engineering, 4th edn. Prentice Hall (2011) Haukaas, T.: BoucWenMaterial.cpp. https://opensees.berkeley.edu/OpenSees/api/doxygen2/html/ BoucWenMaterial_8cpp-source.html. Accessed 5 Sept 2006 Hui, Y., Law, S.S., Zhu, W.D.: Efficient algorithm for the dynamic analysis of large civil structures with a small number of nonlinear components. Mech. Syst. Signal Process. 152, 107480 (2021) Levenberg, K.: A method for the solution of certain problems in least squares. Q. Appl. Math. 2, 164–168 (1944) Li, Y., Huang, S.R., Lin, C., et al.: Response sensitivity analysis for plastic plane problems based on direct differentiation method. Comput. Struct. 182, 392–403 (2017) Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11, 431–441 (1963) Tian, W., Weng, S., Xia, Y.: Model updating of nonlinear structures using substructuring method. J. Sound Vib. 521, 116719 (2022) Tian, W., Weng, S., Xia, Y.: Kron’s substructuring method to the calculation of structural responses and response sensitivities of nonlinear systems. J. Sound Vib. 502, 116101 (2021)

Chapter 15

A Modal Derivative Enhanced Kron’s Substructuring Method for Response and Response Sensitivities of Geometrically Nonlinear Systems

15.1 Preview Structural nonlinearities majorly include two forms, material nonlinearities and geometric nonlinearities. Chapters 13 and 14 introduce a substructuring method for structural responses, response sensitivities, and model updating of structures with material nonlinearities. This chapter extends the substructuring method to calculate responses and sensitivities of structures with geometric nonlinearities. Geometric nonlinearities may exist in thin-walled and flexible structural components, which characterizes the nonlinearities of stiffness and internal force vectors in the structure due to the changing geometry as it deflects. Different from the material nonlinearities that are usually localized in a few small regions, geometric nonlinearities may exist in the whole structure. Consequently, the global structure cannot be divided into linear and nonlinear substructures, and the substructuring methods introduced in Chaps. 13 and 14 are thus inapplicable. This chapter introduces another substructuring method to deal with geometrically nonlinear systems using modal derivatives (MDs). MDs are the first-order derivatives of the vibration modes with respect to a specific modal coordinate at the initial equilibrium position, which indicates the coupling among vibration modes (VMs) as a consequence of geometric nonlinearities (Idelsohn and Cardona 1985; Weeger et al. 2016; Jain et al. 2017; Wu et al. 2016, 2019). The proposed method first partitions the global structure into several independent substructures. The substructural displacements are assumed to be dominated by a low dimensional nonlinear manifold around the initial equilibrium position, characterizing only a few master modal coordinates of each substructure. The nonlinear manifold is approximated by a quadratic manifold. A time-variant reduction basis is derived from the quadratic manifold, which is augmented by the master modal derivatives to account for the geometric nonlinearities accurately. Using the reduction basis, the global system is projected into a reduced model of a much smaller size. The structural responses and response sensitivities are computed efficiently based on the reduced system with numerical time © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_15

265

266

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

integration methods. The effectiveness of the proposed method is finally investigated by applying it to a hinged plate.

15.2 Substructuring Method for Responses of Geometrically Nonlinear Systems The governing equation of a geometrically nonlinear system of N DOFs is expressed as M¨x + C˙x + fnl (x) = f

(15.1)

where fnl (x) is the nonlinear internal force vector, a function of system displacement when geometrically nonlinear effects are considered. The global structure is divided into N S independent substructures. The vibration equation of the kth substructure of N (k) DOFs is expressed as ( ) M(k) x¨ (k) + C(k) x˙ (k) + fnl(k) x(k) = f (k) + g(k) ,

(15.2)

where g is the substructural interface connection force introduced in Eq. (6.12). The primitive matrices Mp , Cp , x¨ p , x˙ p , xp , gp , and f p are assembled from the corresponding substructural components according to Eqs. (6.6), (6.7), (6.9), (6.10), p (6.11), (6.12), and (6.13), respectively. fnl (xp ) are assembled by ⎧ (1) ( (1) ) ⎫ ⎪ ⎪ ⎨ fnl x ⎬ p . p . fnl (x ) = . ⎪ ⎩ (N S ) ( (N S ) ) ⎪ ⎭ fnl x

(15.3)

Combining the vibration equations of all substructures gives p

Mp x¨ p + Cp x˙ p + fnl (xp ) = f p + gp

(15.4)

Similarly, the displacement compatibility and force equilibrium conditions are also satisfied at the substructural interfaces (Weng et al. 2020; Zhu et al. 2021) Dxp = 0

(15.5)

gp = DT τ

(15.6)

Equations (15.4)−(15.6) are the primal formulation of Kron’s substructuring method for the geometrically nonlinear system, which can also be written in a dual form as

15.2 Substructuring Method for Responses of Geometrically Nonlinear Systems

|

Mp 0 0 0

267

|| p { | p ( p ) { | p { || p { | p || p { | f x x C 0 f x¨ x˙ 0 −DT + + + nl = τ¨ τ˙ τ 0 0 0 −D 0 0 p

(15.7)

p

When the geometric nonlinearities are insignificant, fnl (xp ) = Keq xp . Then, Eq. (15.7) is equivalent to the linear elastic case as |

Mp 0 0 0

||

x¨ p τ¨

{

| +

Cp 0 0 0

||

x˙ p τ˙

{

| +

p

Keq −DT −D 0

||

xp τ

{

| =

fp 0

{ (15.8)

p

where Keq is the stiffness matrix at the initial equilibrium position assembled from the corresponding substructural components ) ( p (1) (2) (N S ) Keq = Diag Keq Keq · · · Keq ( | | ∂fnl(1) (x(1) ) || ∂fnl(2) (x(2) ) || ··· = Diag ∂x(1) | ∂x(2) | 0

( )| (N ) | ∂fnl S x( N S ) |

0

∂x( N S )

| |

) (15.9) 0

In Chap. 8, we develop a substructuring method from Eq. (15.8) with a modal truncation technique, in which the substructural displacement can be approximated with a few master modal coordinates of each substructure via a reduction basis u (see Eq. (8.14)) as ) ( xp ≈ ypm − ws zm = uzm

(15.10)

( )−1 u = ypm − ws = ypm − Fp DT DFDT Dopm

(15.11)

where

zm encloses all substructural master modal coordinates { (1) } (k) (N S ) T z = zm · · · zm · · · zm

(15.12)

}T { (k) (k) (k) (k) where zm = z 1 z 2 · · · z N (k) is composed of the first Nm(k) master modal coordim nates of the k-th substructure, and z is the modal coordinate of a specific substructural mode. In Eq. (15.11), ypm and Fp are formed by the assembled substructural master eigenvectors and flexibility calculated in the initial equilibrium condition, according to Eqs. (2.40) and (2.48). Equation (15.10) demonstrates that the substructural displacement xp is estimated by a few master modal coordinates via the linear reduction basis u. Therefore, when the geometric nonlinearities become significant, we can assume that an analytical manifold around the initial equilibrium position is governed by the master modal coordinates

268

15 A Modal Derivative Enhanced Kron’s Substructuring Method … p

x ≈ X(zm ) = p

Nm E

u i (zm )z i

(15.13)

i=1

where u(zm ) is the instantaneous reduction basis for geometrically nonlinear systems. Assume the nonlinear manifold X is second-order differentiable with respect to zm , Eq. (15.13) is expanded into the Taylor series around the initial state zm = 0 as p p p | | Nm E Nm Nm E ∂X || ∂ 2 X || 1E z + zi z j + · · · x =0+ i ∂z i |0 2 j=1 i=1 ∂z i ∂z j |0 i=1

p

(15.14)

The derivative matrices with respect to specific modal coordinates zi and zj are computed from Eq. (15.13) by ∂u j (zm ) ∂X = u i (zm ) + zj ∂z i ∂ zi

(15.15)

∂ 2X ∂u i (zm ) ∂u j (zm ) ∂u k (zm ) = + + zk ∂z i ∂ z j ∂z j ∂z i ∂ zi ∂ z j

(15.16)

Evaluation at zm = 0 leads to | ∂X || = u i |0 ∂z i |0 | | | ∂u j || ∂ 2 X || ∂u i || = + ∂z i ∂z j |0 ∂ z j |0 ∂z i |0

(15.17)

(15.18)

| i | is its derivative with respect to modal coordinate u i |0 is the reduction basis, and ∂u ∂z j |0 zj at the initial equilibrium position. For brevity, the time dependency is omitted to | ∂u i | specifically represent the initial state, z = 0, thus u i |0 and ∂ z j | are simplified into u i and

∂u i ∂z j

0

.

u i is calculated from Eq. (15.11). Eq. (15.11) with respect to zj

∂u i ∂z j

can be obtained by differentiating

| | p ∂u ∂ym ∂Fp T ( p T )−1 =P − D DF D Dopm ∂z j ∂z j ∂z j

(15.19)

where the intermediate variable P is expressed as ( )−1 P = Ip − Fp DT DFp DT D

(15.20)

15.2 Substructuring Method for Responses of Geometrically Nonlinear Systems ∂yp

269

As each substructure is treated to be independent, ∂ z mj and ∂F are therefore zeros ∂z j except for the specific substructure containing zj . If zj corresponds to a master mode A−1 A E (k) E p ∂yp of the A-th substructure, Nm < j ≤ Nm(k) , ∂ z mj and ∂F are expressed in the ∂z j k=1

p

k=1

form of ( ) ( A) ∂ypm m = Diag 0 · · · ∂y · · · 0 ∂z j ∂z j ) ( ∂Fp ( A) ··· 0 = Diag 0 · · · ∂F ∂z j ∂z j where

∂F(A) ∂z j

(15.21) (15.22)

is derived in a similar manner to Eq. (3.38) as

( A) | ( A) |−1 ∂y(mA) | (A) |−1 | ( A) |T | (A) |−1 ∂Keq ∂F( A) Keq /\m ym = − Keq − ∂z j ∂z j ∂z j

+

∂/\(mA) ∂z j

and

y(A) m

∂y(mA) ∂z j

|T | | ( A) |−1 ∂/\(mA) | ( A) |−1 | ( A) |T | ( A) |−1 ∂ y(mA) (A) /\m /\m ym − ym /\m ∂z j ∂z j (15.23)

are expressed in the form of ) ( ∂/\(A) ∂λ(A) (A) m = Diag ∂λ1 · · · Nm(A) ∂z j ∂z j ∂z j | | ∂y(mA) ∂φ ( A)( A) ( A) ∂φ N 1 = · · · ∂ zmj ∂z j ∂z j

(15.24)

(15.25)

∂φ where ∂∂λz j and ∂z are the eigenvalue derivative and MD of a specific substructural j mode with respect to the modal coordinate zj , respectively, which indicate how λ and φ change when the system is perturbed by a displacement in the direction of vibration mode φ j . They effectively show the coupling among different VMs and the departure of the nonlinear system from the initial state. The eigenvalue derivatives and MDs can ( ) be solved from the corresponding substructural eigenequation, Keq − λM φ = 0.. The expansion in Eq. (15.14) provides a quadratic manifold approximation between the substructural master modal coordinates zm and the physical displacements xp as

1 xp = X(zm ) = uzm + (0 : zm )zm 2 p

p

(15.26)

where 0 is a third-order N p × Nm × Nm -tensor. (· : ·) is used to denote the product between a third-order tensor and a vector, which results in a matrix. For example,

270

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

(0 : zm ) is rewritten by Einstein notation as ( { } { }) (0 : zm ) = 0 I i j z j I ∈ 1 · · · N p ;i, j ∈ 1 · · · Nmp

(15.27)

where 0I i j =

∂u I j ∂u I i + ∂z j ∂z i

(15.28)

and 0 I i j = 0 I ji . It is noted from Eqs. (15.11), (15.19), and (15.28) that Du I i = 0 and D0 I i j = 0. Accordingly, the approximated displacement in Eq. (15.26) also satisfies the displacement compatibility condition in Eq. (15.5). The substructural velocity x˙ p and acceleration x¨ p are obtained by the first- and second-order time derivatives of Eq. (15.26) x˙ p = u z˙ m + (0 : zm )˙zm = [u + (0 : zm )]˙zm = ||˙zm

(15.29)

x¨ p = u z¨ m + (0 : zm )¨zm + (0 : z˙ m )˙zm = ||¨zm + (0 : z˙ m )˙zm

(15.30)

where || = u + (0 : zm )

(15.31)

|| is the time-variant reduction basis of the geometrically nonlinear system. Besides the linear reduction basis u, a time-dependent item comprising the modal derivative matrix 0 and the master modal coordinate zm is added to capture the geometrical nonlinearities accurately. Premultiplying Eq. (15.4) with ||T and considering Eqs. (15.6) and (15.29)−(15.31) and D|| = 0, one can obtain a reduced system vibration equation ˜ zm + C˙ ˜ zm + f˜nl (xp ) = f˜ M¨

(15.32)

˜ = ||T Mp || M

(15.33)

˜ = ||T Cp || C

(15.34)

p f˜nl (xp ) = ||T fnl (xp )

(15.35)

| | f˜ = ||T f p − Mp (0 : z˙ m )˙zm

(15.36)

where

15.3 Substructuring Method for Response Sensitivities of Geometrically …

271

˜ and C ˜ are the reduced mass and damping matrices, respectively; f˜nl (xp ) where M and f˜ are the reduced internal force and external load vectors, respectively. Equation (15.32) can be solved iteratively with numerical time integration methods, such as Newmark-β method and Wilson-θ method. In each time step, iterations are required ˜ C, ˜ f˜nl (xp ), and f. ˜ After zm , z˙ m , and z¨ m are to evaluate the time-variant variables ||, M, solved from Eq. (15.32), the substructural responses xp , x˙ p , and x¨ p are obtained from Eqs. (15.26), (15.29), and (15.30), respectively. The responses of the global system, x, x˙ , and x¨ , are finally extracted from corresponding DOFs of the substructural components. The reduced system comprises the substructural master modes only, which has p an order of Nm , much smaller than N p of the global system. Therefore, calculating structural responses from Eq. (15.32) is expected to be computationally efficient. It is noted that f˜nl (xp ) in Eq. (15.32) is associated with the instantaneous internal p force of the assembled system, fnl (xp ), which requires the assembly of all elemental forces in each substructure within each iteration. This dominates online computation time as the system becomes large. An effective solution is to evaluate the reduced nonlinear force f˜nl (xp ) in the offline stage (before time integration) using hyperreduction techniques (Chaturantabut and Sorensen 2010; Rutzmoser and Rixen 2017; Hernandez et al. 2017; Jain and Tiso 2019; Kim et al. 2022) or reduced tensors (Touzé et al. 2014), such that the computation cost on f˜nl (xp ) does not scale with the system size. Compared with the conventional MD-enhanced Craig–Bampton method (Wu et al. 2016), the proposed method is efficient in two main aspects. First, it is a free interface method, and thus the size of the reduced system is irrelevant to the number of interface DOFs, which is promising to deal with cases with complicated interfaces. Second, the proposed reduced system is constructed with a quadratic manifold method and thereby has an identical size to the number of the reduced unknowns. The conventional method, however, uses a linear manifold, and the size of the reduced system grows quadratically with the increase of reduced unknowns.

15.3 Substructuring Method for Response Sensitivities of Geometrically Nonlinear Systems As explained previously, the direct differentiation method is developed here to derive the first-order derivative of the structural responses with respect to a design parameter r. Equation (15.32) is differentiated with respect to r as p ˜ ˜ ˜ ˜ ˜ ∂ z˙ m + ∂ fnl (x ) = ∂ f − ∂ M z¨ m − ∂ C z˙ m ˜ ∂ z¨ m + C M ∂r ∂r ∂r ∂r ∂r ∂r

(15.37)

Equation (15.37) can be solved with numerical time integration methods once the p ˜ ˜ ˜ ∂ f˜ derivative variables, ∂∂rM , ∂∂rC , ∂ fnl∂r(x ) , and ∂r , are available.

272

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

Differentiating Eqs. (15.33)−(15.36) with respect to r gives ˜ ∂Mp ∂|| ∂M ∂||T p = M || + ||T || + ||T Mp ∂r ∂r ∂r ∂r

(15.38)

˜ ∂||T p ∂Cp ∂|| ∂C = C || + ||T || + ||T Cp ∂r ∂r ∂r ∂r

(15.39)

∂f (xp ) ∂xp ∂||T p p ∂||T p p ∂ f˜nl (xp ) ∂xp p = fnl (x ) + ||T nl p = fnl (x ) + ||T Kt (xp ) ∂r ∂r ∂x ∂r ∂r ∂r (15.40) p

| ∂||T | p ∂~ f = f − Mp (0 : z˙ m )˙zm ∂r ∂r | ( ) | ∂Mp ∂0 ∂ z˙ m : z˙ m z˙ m + 2Mp (0 : z˙ m ) (15.41) − ||T (0 : z˙ m )˙zm + Mp ∂r ∂r ∂r p

p

∂f (xp )

where Kt = nl∂xp is the instantaneous tangent stiffness matrix of the assembled system and formed from the substructural components as p Kt

= Diag

(

Kt(1)

···

Kt(N S )

)

( = Diag

∂fnl (x(1) ) ∂x(1)

( ) ∂fnl x( N S )

···

)

∂x( N S )

(15.42)

As each substructure is treated to be independent, ∂M and ∂C are nonzeros only ∂r ∂r at the specific substructure (for instance, the B-th substructure) containing r as p

( ∂Mp = Diag 0 · · · ∂r ( ∂Cp = Diag 0 · · · ∂r ∂M(B) ∂r

p

∂M(B) ∂r

··· 0

∂C(B) ∂r

··· 0

) (15.43)

) (15.44)

(B)

and ∂C∂r are computed directly from the mass and damping matrices of the specific element containing r. p In Eqs. (15.38)−(15.41), ∂x and ∂|| are derived from Eqs. (15.26) and (15.31) as ∂r ∂r ) ( ∂u 1 ∂0 ∂xp ∂zm ∂zm = zm + u + : zm zm + (0 : zm ) ∂r ∂r ∂r 2 ∂r ∂r ) ( ∂u 1 ∂0 ∂zm zm + : zm zm + || = ∂r 2 ∂r ∂r ) ( ( ) ∂zm ∂u ∂0 ∂|| = + : zm + 0 : ∂r ∂r ∂r ∂r

(15.45) (15.46)

15.3 Substructuring Method for Response Sensitivities of Geometrically …

273

It is noted from Eqs. (15.38)−(15.46) that only the derivative matrices ∂0 are unknown to be determined. ∂u is derived from Eq. (15.11) as ∂r ∂r ∂ypm ∂Fp T ( p T )−1 ∂u = − D DF D Dopm ∂r ∂r ∂r ( )−1 ∂Fp T ( p T )−1 + Fp DT DFp DT D Dopm D DF D ∂r ( )−1 ∂yp − Fp DT DFp DT D m ∂r ∂yp

∂u ∂r

and

(15.47)

are nonzeros As each substructure serves as an independent structure, ∂rm and ∂F ∂r ∂ypm ∂Fp for the substructure containing r (the B-th substructure). ∂r and ∂r are therefore expressed in the form of ) ( ∂ypm (B) = Diag 0 · · · ∂y∂rm · · · 0 ∂r ) ( ∂Fp (B) = Diag 0 · · · ∂F∂r · · · 0 ∂r

p

(15.48) (15.49)

where the residual flexibility derivative of the B-th substructure is derived as follows, similarly to Eq. (15.23), |T | (B) | (B) |−1 ∂Keq | (B) |−1 ∂y(B) ∂ y(B) ∂F(B) ∂/\(B) m m = − Keq Keq Q + QT m Q − QT − ∂r ∂r ∂r ∂r ∂r (15.50) The intermediate variable Q is expressed by | |−1 | (B) |T Q = /\(B) ym m ∂/\(B) m ∂r

(15.51)

∂y(B)

and ∂rm are solved from the B-th substructure using Nelson’s method (Nelson 1976). Based on Eq. (15.28), ∂0 is computed for each modal coordinate as ∂r ∂0 I i j ∂ 2uI j ∂ 2uI i = + ∂r ∂r ∂z j ∂r ∂z i The second-order derivative matrix

∂2u ∂r ∂ z j

(15.52)

is derived from Eqs. (15.19) and (15.20)

274

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

) ( ∂ 2u ∂P ∂ypm ∂Fp T ( p T )−1 p = − D DF D Dom ∂r ∂z j ∂r ∂z j ∂z j ( 2 p ∂ 2 Fp T ( p T )−1 ∂ ym − D DF D Dopm +P ∂r ∂z j ∂r ∂z j )) ( p ∂Fp T ( p T )−1 ∂ypm ∂F T ( p T )−1 p D DF D + D DF D D Dom − ∂z j ∂r ∂r

(15.53)

where ( ( )−1 ) ∂Fp T ( p T )−1 ∂P D = − Ip − Fp DT DFp DT D D DF D ∂r ∂r ∂Fp T ( p T )−1 = −P D DF D D ∂r

(15.54)

As the substructures are treated as independent structures, the second-order deriva2 p ∂ 2 yp tive matrices ∂r ∂ zmj and ∂r∂ ∂Fz j are nonzeros when the master modal coordinate zj corresponds to the same substructure as the design parameter r, i.e., A = B and B−1 B E (k) E Nm < j ≤ Nm(k) . Therefore, they are expressed as k=1

k=1

⎧ ⎨

( ∂ 2 ypm Diag 0 · · · = ⎩ ∂r ∂z j ⎧ ⎨

( ∂ 2 Fp Diag 0 · · · = ⎩ ∂r ∂z j

∂ 2 y(B) m ∂r ∂ z j

) B−1 B E (k) E Nm < j ≤ Nm(k) ··· 0 , k=1

0 ∂ 2 F(B) ∂r ∂ z j

k=1

(15.55)

others ) B−1 B E (k) E ··· 0 , Nm < j ≤ Nm(k) k=1

0

k=1

(15.56)

others

Differentiating Eq. (15.48) with respect to zj gives ( ) (B) (B) (B) (B) (B) ∂ 2 Keq ∂Keq | (B) |−1 ∂Keq | (B) |−1 ∂Keq | (B) |−1 ∂Keq ∂ 2 F(B) − Keq Keq = Keq + ∂r ∂z j ∂z j ∂r ∂r ∂z j ∂r ∂z j | (B) |−1 ∂ 2 /\(B) ∂ 2 y(B) ∂y(B) ∂QT ∂/\(B) m m ∂Q m m Q + QT Q− Q− + Keq ∂r ∂z j ∂r ∂z j ∂r ∂z j ∂ z j ∂r |T | (B) |T | 2 (B) ∂QT ∂ y(B) m T ∂ ym T ∂/\m ∂Q −Q +Q − (15.57) ∂r ∂ z j ∂z j ∂r ∂r ∂z j where

∂Q ∂z j

is derived from Eq. (15.51) as

|T | | (B) |−1 | (B) |T | (B) |−1 ∂ y(B) | (B) |−1 ∂/\(B) ∂Q m m /\m ym = − /\m + /\m ∂z j ∂z j ∂z j

(15.58)

15.4 Computational Operation ∂ 2 /\(B) m ∂r ∂ z j

275

∂ 2 y(B)

and ∂r ∂ mz j are the second-order eigenvalue and eigenvector derivatives of the master modes, respectively. They can be derived from the B-th substructure via second-order eigenvector derivative calculation methods (Tan et al. 1994; Friswell 1995). p ˜ ˜ ˜ ∂ f˜ Once ∂∂rM , ∂∂rC , ∂ fnl∂r(x ) , and ∂r are determined from the above equations, the response derivative of the reduced system, ∂z∂rm , ∂∂rz˙ m , and ∂∂rz¨ m , is then solved from Eq. (15.37) with numerical time integration methods in an iterative manner. The displacement derivative is obtained from Eq. (15.45). The velocity derivative and acceleration derivative are calculated based on the first- and second-order derivatives of Eq. (15.45) as )| | ( ∂u ∂0 ∂ z˙ m ∂ x˙ p ∂zm = + : zm z˙ m + (0 : z˙ m ) + || ∂r ∂r ∂r ∂r ∂r ) )| ( ( | ∂0 ∂0 ∂ x¨ p ∂u = : z˙ m z˙ m + + : zm z¨ m ∂r ∂r ∂r ∂r ∂ z¨ m ∂ z˙ m ∂zm + 2(0 : z˙ m ) + || + (0 : z¨ m ) ∂r ∂r ∂r

(15.59)

(15.60)

The response sensitivities of the global system, ∂x , ∂ x˙ , and ∂∂rx¨ , are finally extracted ∂r ∂r from corresponding DOFs of the substructural components. The proposed method derives the response sensitivities directly from the reduced system of a much smaller size, which is expected to save plenty of computational time.

15.4 Computational Operation The computational procedures of the proposed substructuring method can be divided into the offline and online stages. The former stage calculates the constants used in the proposed method, and the latter completes the numerical time integration process. The Newmark-β method is used here. For completeness, this section combines the procedures to calculate the structural responses and response sensitivities simultaneously as follows. Offline stage: compute constants used in the online stage (k) (k = Inputs: Connection matrix D, substructural matrices M(k) , C(k) , g(k) , and Keq 1, 2, …, N S ), and element design parameter r. Outputs: Linear reduction basis u, modal derivative matrix 0, and their derivatives with respect to r, ∂u and ∂0 . ∂r ∂r

1. for k = 1:N S (k) 2. Solve /\(k) m and ym from the eigenequation of the kth substructure. 3. end p 4. Assemble Mp , Cp , f p , and Keq from each substructure.

276

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

5. Obtain ypm and Fp from Eqs. (2.40) and (2.48), respectively. 6. Compute u from Eq. (15.11). 7. For the substructure B containing r, calculate ∂/\(B)

∂y(B)

∂M(B) ∂C(B) , ∂r , ∂r

(B) ∂Keq . ∂r

and

8. Calculate ∂rm and ∂rm from the B-th substructure by Nelson’s method in Sect. 3.2.2. (B) 9. Calculate ∂F∂r using Eqs. (15.50) and (15.51). p p and ∂C as Eqs. (15.43) and (15.44). 10. Assemble ∂M ∂rp ∂r p ∂ym ∂F 11. Assemble ∂r and ∂r as Eqs. (15.48) and (15.49). 12. Calculate ∂u from Eq. (15.47). ∂r p p p 13. j = 0. Generate empty tensors 0 and ∂0 ∈ R N ×Nm ×Nm . ∂r 14. for A = 1:N S 15. for j(A) = 1:Nm( A) 16. 17. 18.

Calculate j = j + 1.

Calculate respectively. Assemble 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

(A) ∂λ(A) (A) ∂Keq , j ∂ z j (A) ∂z j (A) ( A) ∂Keq ∂z j

=

, and

(A) ∂Keq , ∂ z j ( A)

∂/\(mA) ∂y(mA) , ∂z j , ∂z j

∂φ (A) (A) j

∂z j ( A)

∂λ(A) j (A)

and

∂z j ( A)

∂F ∂z j

=

based on Lin et al. (2020). ∂λ(A) (A) j

∂φ (A) ( A)

j

∂z j

, and ∂ z ( A)

j

=

∂φ (A) ( A) j

∂z j ( A)

.

using Eqs. (15.24), (15.25), and (15.23),

p ∂ypm and ∂F from Eqs. (15.21) and ∂z j ∂z j ∂u using Eqs. (15.19) and (15.20). ∂z j

(15.22), respectively.

Calculate if A = B ∂ 2 K(B) ∂ 2 /\(B) ∂ 2 y(B) Calculate ∂r ∂ eqz j , ∂r ∂ mz j , and ∂r ∂ mz j based on Tan et al. (1994) or Friswell (1995). 2 p ∂ 2 yp F based on Eqs. (15.55)−(15.58). Calculate ∂r ∂ zmj and ∂r∂ ∂z j else 2 p ∂ 2 ypm = 0 and ∂r∂ ∂Fz j = 0. ∂r ∂z j end end end for I = 1:N p p for i = 1:Nm p for j = 1:Nm Calculate 0(I, i, j) from Eq. (15.28). Calculate ∂0 i, j ) from Eq. (15.52). ∂r (I, end end end

Online stage: using implicit Newmark time integration ∂z[0]

∂ z˙ [0]

p

[0] [0] , z˙ m , ∂rm , and ∂rm ∈ R Nm ; Inputs: Initial conditions for Eq. (15.37), namely zm integration parameters β and γ ; time step /\t; number of time steps Nt; error tolerance of residual force for iteration convergence ε.

15.4 Computational Operation

277

Outputs: Structural responses (x, x˙ , x¨ ), and the response derivatives ) ∂ x¨ . ∂r

( ∂x ∂r

,

∂ x˙ , ∂r

and

37. Calculate {xp }[0] and ||[0]( using Eqs. (15.26) and (15.31), respectively. ) ˜ [0] , C ˜ [0] , f˜ [0] {xp }[0] , and f˜ [0] using Eqs. (15.33)−(15.36). 38. :Calculate M nl |−1 { || ( )} [0] [0] [0] [0] ˜ ˜ ˜ [0] z˙ m f −C 39. z¨ m = M − f˜nl[0] {xp }[0] . ( ) [0] [0] ˜ [0] z¨ m ˜ [0] z˙ m 40. Calculate initial system resisting force f˜r es = M +C + f˜nl[0] {xp }[0] . | |T p ( ) 1 ˜ [k+1] + 41. Estimate initial Jacobian matrix J = ||[0] Kt {xp }[0] ||[0] + β(/\t) 2M γ ˜ [k+1] C β/\t { p }[0] [0] and ∂|| using Eqs. (15.45) and (15.46), respectively. 42. Calculate ∂x ∂r ∂r [0] ˜ [0] ∂ f˜nl ({xp }[0] ) ˜ [0] ∂ C f˜ [0] ∂M 43. Calculate ∂r , ∂r , , and ∂∂r using Eqs. (15.38)−(15.41). ∂r | ( ) || −1 p [0] [0] [0] ˜ [0] [0] [0] ˜ ˜ ˙ ∂ z ∂ z¨ m ∂f ∂C [0] [0] ˜ [0] m − ∂ fnl ({x } ) − ∂ M˜ [0] z¨ m ˜ [0] ˙ z − − C . 44. ∂r = M m ∂r ∂r ∂r ∂r ∂r [0] p }[0] [0] [0] ˜ [0] [0] {x ˜ ∂ f ˜ ˜ ¨ ˙ ∂ z ∂ z ( ). fr es [0] [0] ˜ [0] m + ∂ C z˙ m ˜ [0] m + nl z¨ m = ∂M +M +C 45. Calculate ∂∂r ∂r ∂r ∂r ∂r ∂r T | ) |T ∂Ktp ({xp }[0] ) [0] ∂ [||[0] ] p( 46. Calculate ∂J = Kt {xp }[0] ||[0] + ||[0] || + ∂r ∂r ∂r | [0] |T p ( p [0] ) ∂||[0] ˜ [k+1] ˜ [k+1] γ ∂C 1 ∂M || + β(/\t)2 ∂r + β/\t ∂r Kt {x } ∂r 47. for k = 0:Nt − 1 [k+1] [k] [k+1] [k] [k+1] [k] 48. Initialize zm = zm , z˙ m = z˙ m , z¨ m = z¨ m and ||[k+1] = ||[k] . 49. Calculate f˜ [k+1] using Eq. (15.36). 50. Calculate system unbalanced force f˜U = f˜ [k+1] − f˜r es . [k+1] [k+1] [k+1] [k+1] [k] ∂zm ∂z[k] ∂ z˙ m ∂ z˙ [k] ∂ z¨ m ∂ z¨ [k] = ∂rm , ∂r = ∂rm , ∂r = ∂rm , and ∂||∂r = ∂|| . 51. Initialize ∂r ∂r 52. Calculate

∂ f˜ [k+1] using Eq. (15.41). ∂r ∂ f˜U ∂ f˜r es ∂ f˜ [k+1] ||∂r =|| ∂r − || ∂r . || ˜ [k+1] || ||

53. Calculate || || 54. while ||f˜U || > ε||f

||

55. Modal coordinate increment /\zm = J−1 f˜U . γ [k+1] [k+1] [k+1] [k+1] [k+1] [k+1] = zm + /\zm , z˙ m = z˙ m + β/\t /\zm , and z¨ m = z¨ m + 56. Update zm 1 /\z m β(/\t)2 [k+1] using Eqs.)(15.26) and (15.31), respectively. 57. Update {x(p }[k+1] and ) || p p( p [k+1] 58. Form Kt {x } and fnl {xp }[k+1] according to structural constitutive relation. ( ) ˜ [k+1] , C ˜ [k+1] , f˜ [k+1] {xp }[k+1] , and f˜ [k+1] using Eqs. 59. Calculate M nl (15.33)−(15.36). ( ) [k+1] ˜ [k+1] z¨ m[k+1] + C ˜ [k+1] z˙ m 60. f˜r es = M + f˜nl[k+1] {xp }[k+1] . 61. f˜U = f˜ [k+1] − f˜r es . |T p ( ) | 1 ˜ [k+1] + γ C ˜ [k+1] . 62. J = ||[k+1] Kt {xp }[k+1] ||[k+1] + β(/\t) 2M β/\t ( ) ˜ m 63. Calculate ∂/\z = J−1 ∂∂rfU − ∂J /\zm . ∂r ∂r [k+1] ∂zm ∂r ∂/\zm 1 . β(/\t)2 ∂r

64. Update

=

[k+1] ∂zm ∂r

m + ∂/\z , ∂r

[k+1] ∂ z˙ m ∂r

=

[k+1] ∂ z˙ m ∂r

γ + β/\t

[k+1] ∂ z¨ m ∂/\zm , and ∂r ∂r

=

[k+1] ∂ z¨ m ∂r

+

278

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

66. 67. 68. 69. 70.

71. 72. 73. 74. 75.

{ ∂xp }[k+1]

[k+1] and ∂||∂r using Eqs. (15.45) and (15.46), respectively. ∂r p p [k+1] p ∂K {x } ) and ∂fnl ({xp }[k+1] ) according to structural constitutive relation. Derive t ( ∂r ∂r [k+1] ˜ [k+1] ∂ f˜nl ({xp }[k+1] ) ˜ [k+1] ˜ [k+1] ∂ C ∂M Calculate ∂r , ∂r , , and ∂ f∂r using Eqs. (15.38)−(15.41). ∂r [k+1] [k+1] [k+1] ˜ [k+1] [k+1] ∂ f˜r es [k+1] ˜ [k+1] ∂ z¨ m + ∂ C˜ [k+1] z˙ m ˜ [k+1] ∂ z˙ m + ∂ f˜nl . = ∂ M∂r z¨ m +M +C ∂r ∂r ∂r ∂r ∂r ˜ ˜ [k+1] ∂ f˜U = ∂ f∂r − ∂∂rf R . ∂r T ) |T ∂Ktp ({xp }[k+1] ) [k+1] | ∂ ||[k+1] p( Calculate ∂J = [ ∂r ] Kt {xp }[k+1] ||[k+1] + ||[k+1] || + ∂r ∂r | [k+1] |T p ( p [k+1] ) ∂||[k+1] ˜ [k+1] ˜ [k+1] γ ∂C 1 ∂M || Kt {x } + β(/\t)2 ∂r + β/\t ∂r ∂r

65. Update

end Calculate {˙xp }[k+1] and {¨xp }[k+1] using Eqs. (15.29) and (15.30), respectively. { p }[k+1] { p }[k+1] and ∂∂rx¨ using Eqs. (15.59) and (15.60), respectively. Calculate ∂∂rx˙ end Extract global structural responses and response sensitivities ( ) ∂ x˙ ∂ x¨ x, x˙ , x¨ , ∂x , , and from the corresponding substructural components ( p p ∂rp ∂x∂rp ∂ x˙ p ∂r ∂ x¨ p ) x , x˙ , x¨ , ∂r , ∂r , and ∂r .

15.5 Example: A Hinged Plate Model A thin plate, as shown in Fig. 15.1, is used here to investigate the accuracy of the proposed substructuring method. The longitudinal boundaries of the plate are hinged and immovable, whereas the transverse boundaries are free. The plate is 4 m long, 2 m wide, and 0.1 m thick. The structural material properties are: Young’s modus is 206 GPa, the mass density is 7800 kg/m3 , and Poisson’s ratio is 0.3. The plate is subjected to a uniformly distributed pressure on the top surface, q = 2 × [sin(5t) + sin(5π t)] kPa (0 ≤ t ≤ 5 s), as displayed in Fig. 15.2. The structure is uniformly discretized into an FEM of 861 (41 × 21) nodes and 800 (40 × 20) elements, as shown in Fig. 15.1b. Mindlin plate elements are used to account for the geometric nonlinearities (Patil 2016). Each node has five DOFs, x-translation, y-translation, z-translation, x-rotation, and y-rotation, totaling 4179 DOFs for the global structure. The Rayleigh damping is adopted at the initial equilibrium state, C = a1 M + a2 Keq . The damping coefficients are selected as a1 = 4.9586 s−1 and a2 = 1.1167 × 10−4 s. The change ratio of the bending rigidity of an element in the mid-span (highlighted in Fig. 15.1b) is selected as the design parameter r to calculate the response sensitivities. The global structure is partitioned into three substructures, as shown in Fig. 15.1b. Table 15.1 lists the detailed substructural information. The first 12 modes of each substructure are selected as master modes, resulting in 36 master modes total. ( in ∂/\pm ∂ypm p p The master eigenmatrices (/\m and ym ) and their derivative matrices ∂z j , ∂ z j , ) ∂yp ∂/\pm , and ∂rm are calculated with Lanczos’s method (Lanczos 1950) and Nelson’s ∂r ∂ 2 /\p

method (Nelson 1976), respectively. The second-order derivatives ( ∂r ∂ zmj and

∂ 2 ypm ) ∂r ∂ z j

15.5 Example: A Hinged Plate Model

z

y

279

0.1 m x

4m

(a) Global structure

P

Sub 1

Sub 2

Sub 3

(b) Substructural model

Fig. 15.1 Hinged rectangular plate

Pressure (Pa)

5000

0

-5000 0

0.5

1

1.5

2

3 2.5 Time (s)

3.5

4

4.5

5

Fig. 15.2 Uniform surface pressure

are calculated with Friswell’s method (Friswell 1995), which is a direct differentiation ∂/\p ∂yp ∂ 2 /\p ∂ 2 yp method based on Nelson’s method. The derivative matrices ∂rm , ∂rm , ∂r ∂ zmj , and ∂r ∂ zmj are computed only within the second substructure that contains r. Table 15.1 Substructural information of the plate Substructures

Sub1

Sub2

Sub3

No. nodes

231

441

231

No. elements

200

400

200

No. interface nodes

21

42

21

280

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

The linear reduction basis u and its modal derivative ∂u associated with the ∂z master modes are first calculated offline at the initial equilibrium position. Figure 15.3 illustrates the first three modes of u associated with each substructure. The first three modes of Sub 1 and Sub 3 are bending modes, whereas those of the Sub 2 are stretching modes. Figure 15.4 displays the modal derivatives corresponding to the . It shows that the MDs inherently capture the first mode of each substructure, ∂u ∂z bending-stretching coupling associated with structural geometric nonlinearities. The responses and response sensitivities are solved with the Newmark-β method and Newton–Raphson iteration method for time-variant matrix determination in each Ψ1( ) 1

Ψ (2 )

Ψ (3 )

Ψ1(

2)

Ψ (2 ) 2

Ψ 3( )

Ψ1( )

Ψ (2 )

Ψ (3 )

3

1

1

2

3

3

Fig. 15.3 Linear reduction basis (u) corresponds to the first three modes of the substructures ∂Ψ1( ) 1 ∂z1( )

∂Ψ1( ) 2 ∂z1( )

∂Ψ1( ) 2 ∂z1( )

∂Ψ1( ) 1 ∂z1( )

∂Ψ1( ) 2 ∂z1( )

∂Ψ1( ) 3 ∂z1( )

∂Ψ1( ) 2 ∂z1( )

∂Ψ1( ) 3 ∂z1( )

1

2

∂Ψ1( ) 1 ∂z1( ) 3

Fig. 15.4 Linear reduction derivative

1

2

3

1

2

3

15.5 Example: A Hinged Plate Model

281

time step. Other three methods, namely the global nonlinear, global linear, and linear manifold methods, are also used for comparison. In the global nonlinear method, the structural responses are solved directly from the global nonlinear vibration equation, Eq. (15.1), with the Newmark-β method and Newton–Raphson iteration method. The response sensitivities are computed using the direct differentiation method to Eq. (15.1). The global nonlinear solutions are treated as exact. The global linear method treats the system to be linear, and the responses and response sensitivities are calculated without iteration. In the linear manifold method, the nonlinear mapping in Eq. (15.4) is approximated by the linear manifold rather than the quadratic manifold in the proposed method, resulting in a constant reduction basis of the geometrically nonlinear system in Eq. (15.31), namely || = u. The method uses u only for model order reduction, which is equivalent to the proposed quadratic manifold method when ∂u are assumed to be zeros for all master modal coordinates. ∂z Figure 15.5 compares the vertical displacement response of a node (denoted with P in Fig. 15.1b) in the mid-span by the four methods. The global linear solution deviates remarkably from the actual value, indicating strong geometric nonlinearities in the system. The response curve from the present quadratic manifold method is overlapped with the exact one, whereas the results from the linear manifold method deviate from the true value moderately. It implies that the quadratic manifold method is accurate in calculating the structural enhanced by the modal derivative matrix ∂u ∂z responses of the geometrically nonlinear system. Figure 15.6 compares the derivatives of xP with respect to r. Again the global linear solution deviates significantly from the real value. The linear manifold method deviates moderately from the exact result, whereas the difference is more significant than that in the response calculation (Fig. 15.4). This is because the response sensitivities are computed on the basis of the structural responses. The inaccuracy in structural responses exaggerates the errors in the response sensitivities. Therefore, the is significant in capturing the geometric nonlinearities. modal derivative matrix ∂u ∂z Global nonlinear solution Global linear solution Linear manifold method Quadratic manifold method

0.03

Displacement (m)

0.02 0.01 0 -0.01 -0.02 -0.03

0

0.5

1

1.5

2

2.5

Time (s)

Fig. 15.5 Vertical displacement at node P (xP )

3

3.5

4

4.5

5

282

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

In contrast, the results obtained from the quadratic manifold method agree very well with those from the global nonlinear method. The quadratic manifold method is very accurate in calculating response sensitivities of geometrically nonlinear systems. The relative errors of the structural responses and response sensitivities are quantified by Eqs. (7.51) and (7.52), respectively. Table 15.2 shows the relative errors of the vertical responses and response sensitivities at P by the above methods. The proposed quadratic manifold method has a calculation error of 10−3 or less, whereas the global linear solution and linear manifold method lead to errors higher than 5% and 10%, respectively. This is consistent with the results in Figs. 15.4 and 15.5. The number of master modes affects the accuracy and efficiency of the linear substructuring method significantly. 5, 8, 12, and 15 master modes for each substructure are investigated, totaling 15, 24, 36, and 45 master modes, respectively. Figure 15.7 compares the relative errors of the response and response sensitivity at P in the four cases using the proposed method. Both errors exceed 1% when 15 master modes are used only. The errors decrease rapidly as the number of master modes increases from 15 to 36. When 45 master modes are used, the errors are almost unchanged. This implies that 36 master modes are sufficient in this situation 10

Displacement derivative (m)

1.5

-4

Global nonlinear solution Global linear solution Linear manifold method Quadratic manifold method

1 0.5 0 -0.5 -1 -1.5

0

1

0.5

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

Fig. 15.6 Derivative of xP with respect to r

Table 15.2 Relative errors of response and response sensitivity at the selected DOF Methods

Responses

Response derivative

xP

x˙P

x¨P

∂ xP ∂r

∂ x˙P ∂r

∂ x¨P ∂r

Global linear (%)

20.23

11.56

24.34

45.51

28.02

34.48

Linear manifold (%)

6.17

5.10

5.66

14.15

19.49

16.51

Quadratic manifold

1.18 × 10−3

6.02 × 10−4

1.35 × 10−3

5.66 × 10−3

3.41 × 10−3

6.48 × 10−3

15.6 Summary

Relative errors

10

10

10

283 -1

Response Response sensitivity

-2

-3

15

20

25

30

35

40

45

Number of master modes

Fig. 15.7 Relative errors of the response and response sensitivity at P versus the number of master modes

and including more master modes improves the precision little. However, including more master modes requires more VMs and MDs to be computed in the offline stage and enlarges the size of the reduced system equation (Eq. 15.32) significantly, thereby resulting in low computational efficiency. 24 to 36 master modes are appropriate in this example to balance accuracy and efficiency. ( ) p In nonlinear dynamics, the tangent stiffness Kt {xp }[k+1] and internal force vector ( ) p fnl {xp }[k+1] are constructed by iteratively assembling all elements within each time step, which adds much computational effort for calculating the reduced nonlinear ( ) force vector f˜nl[k+1] {xp }[k+1] , as shown in Steps 58 and 59 in Sect. 15.4. These steps consume most of the computational time for a large-scale system with complex nonlinearities. In this case, the computational efficiency can be further improved with hyper-reduction techniques (Chaturantabut and Sorensen 2010; Rutzmoser and Rixen 2017; Hernandez et al. 2017; Jain and Tiso 2019; Kim et al. 2022) once the reduced order model is constructed. efforts ( )The methods reduce the computational [k+1] rather by directly evaluating f˜nl[k+1] {xp }[k+1] with the master modal coordinate zm than assembling each element with {xp }[k+1] . These techniques have been used in order reduction problems. The proposed nonlinear substructuring method is expected to be more efficient after utilizing the hyper-reduction techniques.

15.6 Summary This chapter introduces an MD-enhanced Kron’s substructuring method to calculate structural responses and response sensitivities of geometrically nonlinear systems. The method approximates the substructural displacement as a quadratic manifold governed by a few substructural master modes around the initial equilibrium position. A time-variant reduction basis is constructed and augmented by the MDs of master

284

15 A Modal Derivative Enhanced Kron’s Substructuring Method …

modes to capture the geometric nonlinearities accurately. A reduced model of a much smaller size is built via the reduction basis, which is later developed to compute the structural responses and response sensitivities of geometrically nonlinear systems. The accuracy of the method is finally demonstrated using a thin plate structure with strong geometric nonlinearities.

References Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010) Friswell, M.I.: Calculation of second and higher order eigenvector derivatives. J. Guid. Control. Dyn. 18, 919–921 (1995) Hernandez, J.A., Caicedo, M.A., Ferrer, A.: Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Comput. Methods Appl. Mech. Eng. 313, 687–722 (2017) Idelsohn, S.R., Cardona, A.: A reduction method for nonlinear structural dynamic analysis. Comput. Methods Appl. Mech. Eng. 49(3), 253–279 (1985) Jain, S., Tiso, P.: Hyper-reduction over nonlinear manifolds for large nonlinear mechanical systems. J. Comput. Nonlinear Dyn. 14(8), 081008 (2019) Jain, S., Tiso, P., Rutzmoser, J.B., et al.: A quadratic manifold for model order reduction of nonlinear structural dynamics. Comput. Struct. 188, 80–94 (2017) Kim, Y., Kang, S.H., Cho, H.S., et al.: Improved nonlinear analysis of a propeller blade based on hyper-reduction. AIAA J. 60, 1909–1922 (2022) Lanczos, C.: An iterative method for solution of eigenvalue problem of linear differential and integral operations. J. Res. Natl. Bureau Standards 45(4), 255–282 (1950) Lin, R.M., Mottershead, J.E., Ng, T.Y.: A state-of-the-art review on theory and engineering applications of eigenvalue and eigenvector derivatives. Mech. Syst. Signal Process. 138, 106536 (2020) Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA J. 14, 1201–1205 (1976) Patil, A.: FEM MATLAB Code for Linear and Nonlinear Bending Analysis of Plates. https://www. mathworks.com/matlabcentral/fileexchange/54226-fem-matlab-code-for-linear-and-nonlinearbending-analysis-of-plates. MATLAB Central File Exchange (accessed by 17 April 2016) Rutzmoser, J.B., Rixen, D.J.: A lean and efficient snapshot generation technique for the hyperreduction of nonlinear structural dynamics. Comput. Methods Appl. Mech. Eng. 325, 330–349 (2017) Tan, R.C.E., Andrew, A.L., Hong, F.M.L.: Iterative computation of 2nd-order derivatives of eigenvalues and eigenvectors. Commun. Numer. Methods Eng. 10, 1–9 (1994) Touzé, C., Vidrascu, M., Chapelle, D.: Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models. Comput. Mech. 54(2), 567–580 (2014) Weeger, O., Wever, U., Simeon, B.: On the use of modal derivatives for nonlinear model order reduction. Int. J. Numer. Meth. Eng. 108, 1579–1602 (2016) Weng, S., Zhu, H.P., Xia, Y., et al.: A review on dynamic substructuring methods for model updating and damage detection of large-scale structures. Adv. Struct. Eng. 23, 584–600 (2020) Wu, L., Tiso, P., van Keulen, F.: A modal derivatives enhanced Craig-Bampton method for geometrically nonlinear structural dynamics. In: Sas, P., Moens, D., Van De Walle, A. (eds.) Proceedings of ISMA2016 International Conference on Noise and Vibration Engineering and USD2016 International Conference on Uncertainty in Structural Dynamics, pp. 3615–3624 (2016) Wu, L., Tiso, P., Tatsis, K., et al.: A modal derivatives enhanced Rubin substructuring method for geometrically nonlinear multibody systems. Multibody Sys. Dyn. 45, 57–85 (2019)

References

285

Zhu, H.P., Li, J.J., Tian, W., et al.: An enhanced substructure-based response sensitivity method for finite element model updating of large-scale structures. Mech. Syst. Signal Process. 154, 107359 (2021)

Chapter 16

Epilogue

16.1 Conclusions This book has developed the substructuring methods for SHM comprehensively and systematically, covering methods in frequency and time domains, linear and nonlinear structures, eigensolutions/responses and their sensitivities, and FE model updating. It consists of three parts, linear substructuring methods, model condensation techniques, and nonlinear substructuring methods. The first part includes Chaps. 2–9, covering eigensolutions/responses, their sensitivities, and FE model updating of linear structures. Chapters 2–6 develop a modal truncation technique to calculate the eigensolutions and eigensensitivities of the global structure in the frequency domain. The contribution of the complete eigenmodes of substructures to the global structure is replaced by a few master modes and a residual flexibility matrix using the first-order approach (FRFS) in Chap. 2, second-order (SRFS) approach in Chap. 3, and highorder approach in Chap. 4. The second-order approach is more accurate than the first-order approach, taking more computational sources. The high-order one applies to cases of significant variations in the design parameters or the closely spaced natural frequencies. The substructuring methods have two benefits: (1) the computational work required in extracting the eigenmodes of the substructures is reduced, and (2) the assembled eigenequation is much smaller than that of the original substructuring method. Chapters 5 and 6 develop the iterative schemes to compensate for the discarded higher modes in calculating eigensolutions and eigensensitivities of the original structure. In particular, Chap. 5 considers the eigensolutions and eigensensitivities of each mode separately, whereas Chap. 6 considers all modes simultaneously in an elegant manner more efficiently. The iterative schemes require fewer master modes than the FRFS and SRFS methods. Consequently, the size of the reduced eigenequation is smaller. Chapters 7 and 8 develop the substructuring method to calculate the dynamic responses and response sensitivities of the global structure in the time domain. The © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Weng et al., Substructuring Method for Civil Structural Health Monitoring, Engineering Applications of Computational Methods 15, https://doi.org/10.1007/978-981-99-1369-5_16

287

288

16 Epilogue

time-domain vibration equation includes the elastic, inertial, and damping forces. The time-domain substructuring method considers the elastic effect from substructural slave modes in Chap. 7 and considers both elastic and inertial effects in Chap. 8. The former is simpler and easier to understand than the latter from a practical point of view. Nevertheless, including both elastic and inertial effects involve more calculations but requires fewer master modes. As a result, the latter leads to more accurate results but takes longer computation time. Chapter 9 develops the substructuring-based FE model updating methods in frequency and time domains. The substructuring methods in previous chapters are used to construct the objective function and the sensitivity matrix. An adaptive scheme is designed to ensure the efficiency and convergence of the model updating process simultaneously. The second part of the book includes Chaps. 10–12, which present the dynamic condensation method in the frequency domain, time domain, and model updating, respectively. The substructuring methods reduce the global model into a modal space spanned by a few master modes of the substructures, while the model condensation methods reduce the global model into a low-dimensional physical space spanned by the master DOFs. The model condensation is promising to be used together with substructuring methods. The third part of the book includes Chaps. 13–15, covering the responses, response sensitivities, and model updating of nonlinear structures. Material and geometrical nonlinearities are studied. In particular, Chaps. 13 and 14 develop a substructuring method for responses, response sensitivities, and model updating of structures with local material nonlinearities. The global structure is divided into nonlinear and linear substructures. The linear substructures are simplified by the mode superposition and modal truncation, while the time-consuming nonlinear analysis is constrained in local nonlinear substructures. Since the linear substructures are significantly reduced and the nonlinearities are restrained locally, the nonlinear dynamic analysis and model updating are conducted efficiently from the reduced vibration equation. Chapter 15 finally develops a substructuring method for geometrically nonlinear systems. The method approximates the substructural displacements as a quadratic manifold governed by a few substructural master modes around the initial equilibrium position. A time-variant reduction basis augmented by the master MDs is constructed to capture the geometric nonlinearities accurately. A reduced model of a much smaller size is built via the reduction basis, which is later developed to compute the structural responses and response sensitivities of geometrically nonlinear systems.

16.2 Prospects

289

16.2 Prospects This book has demonstrated that the substructuring methods are advantagous and promising in SHM. It is authors’ viewpoint that the substructuring method can be further developed in the following aspects. (1) Automatic multi-scale substructuring method The structural failure of a complex civil structure is a multi-scale process, from microscale to mesoscale and macroscale, corresponding to the material, component, and structural levels in the structural system. This book presents a linear and nonlinear substructuring scheme for dynamic analysis and model updating of large and complex models. The nonlinear substructuring method can model the details of civil structures on a microscale. The multi-scale substructuring method recursively divides the large FE model into many substructures on several levels based on the details of the system matrices. It not only benefits grasping details in the local area but also does not reduce the computation efficiency. An automatic partitioning procedure will reduce the manual operation of the analyst. Automatic algorithms with rapid convergence are required to implement the multi-level substructuring methods. In addition, the influences of master modes and the divisional formation of substructures on computational efficiency and accuracy are investigated with extensive examples in this book. The automatic selection of master modes and the divisional formation of substructures require further study to make the multi-scale substructuring method feasible for the analyst. With the development of automatic substructuring algorithms and the robust substructure division and modes selection scheme, the multi-scale substructuring method will become a commercially viable application. (2) Parallel/cloud computing for substructuring method Parallel computing/cloud computing provides abound opportunities for substructuring methods and vice versa. Since the substructures are independent, the substructuring method can be more efficient to be combined with parallel computing and cloud computing. Parallel computing is easy to be performed in software like MATLAB or Python. At the time of this writing, the authors have begun the parallel implementation of substructuring methods on the MATLAB platform, which significantly improves computation efficiency. Cloud computing utilizes clusters of distributed computers that provide on-demand resources and services over a network. Cloud computing is a promising paradigm for information delivery and fast computation. Moreover, cloud computing is recurrently used for storing and processing big data from the SHM system. It provides a potential way to analyze the independent substructures on distributed computers by different individuals to accomplish the dynamic analysis of giant models. (3) Substructuring methods and machine learning The revolution of computation capacity creates the opportunity to analyze much larger and more accurate models for structural design, structural control, and damage assessment. It allows analysts to access more accurate computer simulations of

290

16 Epilogue

greater complexity with less dependence on expensive testing of physical prototypes. Machine learning acquires information via tagged data to identify damage. It is difficult to collect the training data and labels of damaged structures, which makes it challenging to implement machine learning in civil engineering damage identification. An alternative way is to collect the training data and labels from simulations on numerical models. However, it is laborious and time-consuming to collect massive training data and perform machine learning on the global structure as a whole. The substructuring method is promising to be used in machine learning and transfer learning. As compared with the global structure, the substructure has a smaller size and fewer damage labels. Therefore, substructuring-based machine learning/transfer learning can improve the precision and efficiency of the analysis. On the other hand, the idea of substructuring methods can be extended to machine learning of large-scale systems, not for SHM only. Big data and machine learning networks have become huge in many scenarios. For example, ImageNet contains over 15 million images with storage of around 1 TB. Training and application of such a huge dataset are very time-consuming or even prohibited. The substructuring method provides a new idea to model training. One option may be, for example, dividing the entire network into several smaller ones, training the subnetworks separately, and assembling the subnetworks into the global one. As we have found in the proposed substructuring method, the pivot work is to assemble the substructural results into the global ones by exerting the connecting matrix for realizing the displacement and force compatibility conditions on the interfaces of the substructures. Similarly, when the substructuring method is applied to the network model, the pivot task may be the connection between the subnetworks and the assembly. Moreover, parallel/cloud computing can be integrated into the process. This book presents our work on substructuring methods in the past fifteen years. The substructuring method, which divides a global structure into smaller substructures for analysis, has shown advantages in analyzing a large-scale civil engineering structure. The book does not cover all substructuring methods. Most likely, some work introduced in this book will be improved and surpassed in the future. Its extensive and commercial application deserves further improvement by the younger generation. There is always room for improved performance. The substructuring method is no exception.