DEVELOPMENT AND APPLICATIONS OF MOVING LEAST SQUARE RITZ METHOD IN SCIENCE AND ENGINEERING COMPUTATION LI ZHOU DOCTOR OF PHILOSOPHY 2007 UNIVERSITY OF WESTERN SYDNEY Development and applications of Moving Least Square Ritz Method in Science and Engineering Computation Li Zhou A thesis submitted for the degree of Doctor of Philosophy at University of Western Sydney May 2007 STATEMENT OF AUTHENTICATION I declare that this thesis submitted is, to the best of my knowledge and belief, original except as acknowledged in the text. I certify that this work is not submitted in candidature for any other degrees. Name of Candidate: Li Zhou Signature of Candidate: _________________________ ________/________/________ i ABSTRACT With the rapid development of computing technologies in the past three decades, the numerical methods have become indispensable tools for solving all kinds of science and engineering problems. Extensive researches have been carried out for the development of new numerical methods in the past few decades. The existing numerical methods have achieved great success in finding the numerical solutions for various theoretical and practical problems. A detailed literature review on the development and applications of several numerical methods in solid mechanics and electromagnetic field analysis is presented in the thesis. Despite the great achievements in this research area, there are always the needs to develop new numerical methods or to explore alternative techniques for the purpose of solving the complicate problems and improve the efficiency and accuracy of the existing or new numerical methods. This thesis presents the development of a novel numerical method, the moving least square Ritz (MLS-Ritz) method, and its applications for solving science and engineering problems. The MLS-Ritz method is based on the moving least square (MLS) data interpolation technique and the Ritz minimization principle. The MLS technique is utilized to establish the Ritz trial functions for two-dimensional (2-D) and three-dimensional (3-D) cases. A point substitution approach is developed to enforce boundary conditions. The proposed MLS-Ritz method has the ability to expand the applicability of the conventional Ritz method and meshless method for analysing problems with complex geometries and multiple mediums. The MLS-Ritz method is first applied to solve several solid mechanics problems. The free vibration of square and triangular plates is investigated by the MLS-Ritz ii method. The characteristics of the MLS-Ritz method is examined through the detailed convergence and comparison studies for selected cases. It shows that the MLS-Ritz method is highly stable, accurate and efficient in solving such plate vibration problems. The MLS-Ritz method is also employed to study the challenging problem of free vibration of rhombic plates with large skew angles. The domain decomposition technique is developed and applied in this case to improve the convergence rate of the computations. The study reveals that some of the previous studies on the vibration of rhombic plates with large skew angles did not provide converged results. The MLS-Ritz method is further applied to investigate the 3-D vibration behaviour of isotropic elastic square and triangular plates. The MLS-Ritz method is efficient to generate 3-D vibration frequencies for thick plates with high accuracy. The applications of the MLS-Ritz method are also extended to the analysis of the electromagnetic field problems. Three cases including electrical potential problems in a uniform trough and with dielectric medium and a waveguide eigenvalue problem are analysed and compared with solutions obtained by other methods. Comparison studies show that excellent agreement is achieved for the three cases when comparing with existing results in the open literature. The future directions in the development of the MLS-Ritz method for science and engineering computations are discussed. iii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my principal supervisor, Professor Weixing Zheng, for his great encouragement and expert guidance during my study. Without his support and guidance, I would not be able to complete this work. I would also like to express my sincere appreciation to my co-supervisor, Professor Jianguo Zhu of University of Technology, Sydney, for his constant support, guidance and valuable discussions on my research work during the course of my study. The encouragement and assistance from Dr Jamal Rizk and Associate Professor Mahmood Nagrial of School of Engineering at the initial stage of my study are greatly appreciated. I am very grateful to the University of Western Sydney for the UWS Scholarship, and the School of Computing and Mathematics for the provision of essential facilities. I would like to thank the School staff, in particular, Professor Yan Zhang, Associate Professor Richard L. Ollerton, Dr Carmel Coady, Ms Lyn Dormer and Ms Judith Kearns for their kind assistance during my study at UWS. My sincere thanks also go to Dr Keith Mitchell of the School of Engineering, UWS, Dr Yun Bai of the School of Computing and Mathematics, UWS, and Dr Mike Zhong and Dr Youguang Guo of the University of Technology, Sydney, for their friendship and support. Finally, I would like to express my heartfelt gratitude to my family. My parents always supported me in the study and everyday life. My father is of a great passion on acquiring knowledge and contributing to the society, which has inspired me to iv pursuit this research work. My sons Hao and Kieren are wonderful children in so many ways, which gives me the best support. Specially, I wish to thank my husband Yang for always being here for me and sharing his ideas and experience in study and research. Without his unconditional support, I would not be able to complete the long journey of my PhD study. v TABLE OF CONTENTS STATEMENT OF AUTHENTICATION i ABSTRACT ii ACKNOWLEDGMENTS iv TABLE OF CONTENTS vi LIST OF TABLES x LIST OF FIGURES xv CHAPTER 1. INTRODUCTION 1 1.1 Significance and Background 1 1.2 Scope and Objectives 3 1.3 Thesis Outline 5 CHAPTER 2 LITERATURE REVIEW 7 2.1 Introduction 7 2.2 Finite Element Method 7 2.3 Finite Difference Method 11 2.4 Boundary Element Method 13 2.5 Ritz Method 15 2.5.1 General background of Ritz method 15 2.5.2 Application of Ritz method for analysis of solid mechanics problems in particular plate analysis 17 2.5.3 Application of Ritz method for analysis of electromagnetic field problems 21 2.6 Meshless Method 22 vi 2.7 Other Emerging Methods 26 2.7.1 Differential quadrature (Collocation-interpolation) method 26 2.7.2 Discrete singular convolution method 28 2.8 Conclusions 29 CHAPTER 3 MOVING LEAST SQUARE INTERPOLATION 31 3.1 Introduction 31 3.2 Moving Least Square Method 32 3.2.1 Least square method 33 3.2.2 Moving least square method 33 3.2.3 Moving least square weight functions 34 3.3 Moving Least Square (MLS) Interpolation Scheme for 2-D Function 36 3.3.1 2-D MLS shape function 36 3.3.2 Differentiation of 2-D MLS shape function 38 3.3.3 Evaluation of 2-D MLS shape function and its derivatives 39 3.4 3-D Moving Least Square Formulation 41 3.4.1 3-D MLS shape function 43 3.4.2 Differentiation of 3-D MLS shape function 44 3.4.3 Evaluation of 3-D MLS shape function and its derivatives 44 3.5 Conclusion 45 CHAPTER 4 APPLICATION OF MLS-RITZ METHOD FOR VIBRATION ANALYSIS OF RECTANGULAR AND TRIANGULAR PLATES 46 4.1 Introduction 46 4.2 Kirchhoff Plate Theory 48 4.2.1 The Kirchhoff hypothesis 49 4.2.2 Plate energy functiona 52 4.2.3 Boundary conditions 56 4.3 MLS-Ritz Modelling of 2-D Kirchhoff Plates 57 vii 4.4 Implementation of Boundary Conditions 58 4.5 Eigenvalue Equation 60 4.6 Results and Discussion 61 4.6.1 Square plates 62 4.6.2 Right-angled isosceles triangular plates 76 4.7 Conclusion 88 CHAPTER 5 APPLICATION OF MLS-RITZ METHOD FOR VIBRATION ANALYSIS OF SKEW PLATES 89 5.1 Introduction 89 5.2 Problem Definition and Modelling of Skew Plates 92 5.2.1 Problem definition 92 5.2.2 One calculation domain 93 5.2.3 Multiple calculation domains 94 5.3 Results and Discussion 98 5.3.1 Plate with a single computational domain 98 5.3.2 SSFF plate with two and more computational domains 112 5.4 Conclusion 117 CHAPTER 6 APPLICATION OF MLS-RITZ METHOD FOR 3-D SOLID MECHANICS 118 6.1 Introduction 118 6.2 Mathematical Formulation 120 6.2.1 Problem definition 120 6.2.2 Three-dimensional energy functional for a plate 121 6.2.3 MLS-Ritz formulations 123 6.2.4 Implementation of boundary conditions 126 6.3 Results and Discussion 128 6.3.1 Vibration of thick square plates 129 6.3.2 Vibration of right-angled isosceles triangular plates 143 viii 6.4 Conclusion 149 CHAPTER 7 APPLICATION OF THE MLS-RITZ METHOD FOR ANALYSIS OF 2-D ELECTROMAGNETIC FIELDS 150 7.1 Introduction 150 7.2 Theory of Electromagnetic Fields 153 7.2.1 Maxwell’s equations 153 7.2.2 Electric and magnetic potentials and wave equations 155 7.2.3 Boundary conditions 157 7.3 Variational Formulation 158 7.4 MLS-Ritz Method for Analysis of Static Electrical Field Problem 160 7.4.1 Problem definition 161 7.4.2 MLS-Ritz formulation for 2-D static electric field problem 163 7.4.3 Boundary conditions 164 7.4.4 Numerical results 166 7.5 MLS-Ritz Method for Analysis of Waveguide Problems 173 7.5.1 Problem definition 173 7.5.2 MLS-Ritz formulation for a double ridged waveguide 175 7.5.3 Numerical results 176 7.6 Conclusion 180 CHAPTER 8 CONCLUSIONS AND FUTURE WORK 181 8.1 Conclusions 181 8.2 Future Work 184 APPENDIX LIST OF PUBLICATIONS FROM THIS WORK 185