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Applications of Symbolic Computation to the Calculus of Moving Surfaces a Thesis Submitted to the Faculty of Drexel University B

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Applications of Symbolic Computation to the Calculus of Moving Surfaces a Thesis Submitted to the Faculty of Drexel University B Applications of Symbolic Computation to the Calculus of Moving Surfaces A Thesis Submitted to the Faculty of Drexel University by Mark W. Boady in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 2016 c Copyright 2016 Mark W. Boady. All Rights Reserved. i Dedications To my lovely wife, Petrina. Without your endless kindness, love, and patience, this thesis never would have been completed. To my parents, Bill and Charlene, you have loved and supported me in every endeavor. I am eternally grateful to be your son. To my in-laws, Jerry and Annelie, your endless optimism and assistance helped keep me motivated throughout this project. ii Acknowledgments I would like to express my sincere gratitude to my supervisors Dr. Jeremy Johnson and Dr. Pavel Grinfeld. Dr. Johnson's patience and insight provided a safe harbor during the seemingly endless days of debugging and testing code. I cannot thank him enough for improvements in my writing, coding, and presenting skills. This thesis could not have existed without Dr. Grinfeld's pioneering research. His vast knowledge and endless enthusiasm propelled us forward at every roadblock. A special thanks also goes out to the members of my thesis committee: Dr. Bruce Char, Dr. Dario Salvucci, and Dr. Hoon Hong. Without their valuable input, this work would not have been fully realized. They consistently helped to keep this project focused while I was buried in the details of development. All the members of the Drexel Computer Science Department have provided endless help in my development as a teacher and researcher. I am especially grateful to the other members of the High Performance Computing Lab: Lingchuan Meng and Gavin Harrison. Two members of the Drexel Faculty, Dr. Werner Krandick and Dr. Krzysztof Nowak, need to be highlighted for providing valuable sources for discussion and insight. I am very grateful for their interest in both this project and my development as a student. All the members of the Computer Science faculty have been a great help to me, and I am thankful to them all. iii Contents List of Figures .......................................................................................... vi List of Tables ........................................................................................... vii ABSTRACT............................................................................................ xi 1. Introduction......................................................................................... 1 2. The Calculus of Moving Surfaces ................................................................. 6 2.1 Introduction................................................................................... 6 2.2 Elements of Tensor Calculus ................................................................. 6 2.3 Example: Mean Curvature................................................................... 10 2.4 The Calculus of Moving Surfaces............................................................ 14 2.5 Applications of the CMS .................................................................... 16 2.6 Previous Work Automating Tensor Calculus .............................................. 18 3. Term Rewriting Systems .......................................................................... 20 3.1 Introduction................................................................................... 20 3.2 A TRS for the Calculus of Moving Surfaces ............................................... 21 3.3 Implementation ............................................................................... 25 3.3.1 Signature .............................................................................. 25 3.3.2 Reduction ............................................................................. 27 3.3.3 Combining Like Terms .............................................................. 29 3.3.4 Complexity of Reduction ............................................................ 31 3.3.5 Relationship between Equivalence and Graph Isomorphism ..................... 32 3.3.6 Output Methods...................................................................... 34 3.4 Example: Contour Length ................................................................... 35 4. Evaluation of CMS Expressions .................................................................. 38 4.1 Package Overview ............................................................................ 38 4.2 Global Settings ............................................................................... 38 iv 4.3 CMS Object................................................................................... 39 4.4 Primary Functions............................................................................ 40 4.4.1 Multiplication......................................................................... 40 4.4.2 Contraction ........................................................................... 41 4.4.3 Addition............................................................................... 42 4.4.4 Additional Methods .................................................................. 42 4.5 Object Constructors .......................................................................... 42 4.6 Advanced Functions .......................................................................... 47 4.7 Example: Poisson's Equation ............................................................... 50 5. Example: Contour Length ........................................................................ 54 6. Laplace-Dirichlet Eigenvalues ..................................................................... 59 6.1 Introduction................................................................................... 59 6.2 Eigenvalues on the Polygon .................................................................. 60 6.3 Solution on Unit Circle....................................................................... 62 6.4 Regular Polygon ............................................................................. 65 6.5 Second Variation of λ2 ....................................................................... 69 7. Determining Partial Derivatives of u ............................................................. 71 7.1 Introduction................................................................................... 71 7.2 Solving u1 ..................................................................................... 71 7.2.1 Boundary Condition ................................................................. 72 7.2.2 Kernel Condition ..................................................................... 73 7.2.3 Partial Solution to u1 ................................................................ 74 7.2.4 Normalization Condition............................................................. 76 7.2.5 Justification of u1 .................................................................... 77 7.3 General Solution for um ...................................................................... 80 8. Second Variation of λ ............................................................................. 82 8.1 First Convolution ............................................................................. 82 8.2 Second Convolution .......................................................................... 84 8.3 Final Value for λ2 ............................................................................ 86 9. Fourier Library for Maple ......................................................................... 87 9.1 Introduction................................................................................... 87 9.2 Fourier Series Manipulation ................................................................. 87 v 9.3 Numerical Evaluation ........................................................................ 90 9.4 Symbolic Evaluation.......................................................................... 91 9.4.1 Simplification Rules ................................................................. 93 9.4.2 Summation Patterns ................................................................. 94 9.5 Order of Evaluation .......................................................................... 97 9.6 Example: Poisson's Equation Revisited..................................................... 98 10. Level Set Surface ..................................................................................102 10.1 Introduction................................................................................... 102 10.2 Computation of Surface Velocity ............................................................ 102 10.3 Derivatives of the Surface Velocity .......................................................... 104 10.4 Computation of λ............................................................................. 107 11. Results and Analysis ..............................................................................110 11.1 Introduction................................................................................... 110 11.2 Contour Length ............................................................................... 110 11.3 Level Set Problem ............................................................................ 113 11.4 Ellipse Eigenvalues ........................................................................... 115 11.5 N-sided Regular Polygon ................................................................... 118 12. Conclusions .........................................................................................126
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