On Orthogonal Collocation Solutions of Partial Differential Equations
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Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ASAC O Oogoa Coocaio Souios o aia ieeia Equaios y ei Suaaa In contrast to the h-version most frequently used, a p-version of the Orthogonal Collocation Method as applied to differential equations in two- dimensional domains is examined. For superior accuracy and convergence, the collocation points are chosen to be the zeros of a Legendre polynomial plus the two endpoints. Hence the method is called the Legendre Collocation Method. the approximate solution in an element is written as a Lagrange interpolation polynomial. This form of the approximate solution makes it possible to fully automate the method on a personal computer using conventional memory. The Legendre Collocation Method provides a formula for the derivatives in terms of the values of the function in matrix form. The governing differential equation and boundary conditions are satisfied by matrix equations at the collocation points. The resulting set of simultaneous equations is then solved for the values of the solution function using LU decomposition and back substitution. The Legendre Collocation Method is applied further to the problems containing singularities. To obtain an accurate approximation in a neighborhood of the singularity, an eigenfunction solution is specifically formulated to the given problem, and its coefficients are determined. by least-squares or .minimax approximation techniques utilizing the results previously obtained by the Legendre Collocation Method. This combined method gives accurate results for the values of the solution function and its derivatives in a neighborhood of the singularity. All results of a selected number of example problems are compared with the available solutions. Numerical experiments confirm the superior accuracy the computed values of the solution function at the collocation points. ON ORTHOGONAL COLLOCATION SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS by Herli Surjanhata A Dissertation Submitted to the Faculty of New Jersey institute of Technology in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Department of Mechanical Engineering January, 1993 Copyright © 1993 by .Herli Stujanhata ALL RIGHTS RESERVED AOA AGE O Oogoa Coocaio Souios o aia ieeia Equaios [ei Suaaa 1-Ian' Hermar!, Dissertation Adviser Associate Chairperson and Professor of Mechanical Engineering, _MIT Dr. Denis Blackmore, Committee Member Professor of Mathematics, NJIT Dr. Zhiming Ji, Committee Member Assistant :Professor oMecaica Egieeig D Bernath Koplik, Committee Member Cflairperson and Professor of Mechanical and Industrial Engineering, NJIT Dr. Nouri Levy, Comliftiee Member Associate Professor of Mechanical Engineering, NJIT BIOGRAPHICAL SKETCH Author: Herli Surjanhata Degree: Doctor of Philosophy Date: January, 1993 Undergraduate and Graduate Education: • Doctor of Philosophy in Mechanical Engineering, New Jersey Institute of -.Technology, Newark, New jersey, 1993 • -Master of Science in Mechanical Engineering, New Jersey Institute of Technology, Newark, New Jersey, 1.984 • Bachelor of of Science in Mechanical Engineering, University of Trisakti, Jakarta, Indonesia, 1976 Major: Mechanical Engineering Presentations and Publications: Spencer, Herbert Harry, and H. Surjanhata. "Plate Buckling Under Partial Edge Loading." Presented in OSU, Columbus, Ohio, September 1985, at 19th Midwestern Mechanics Conference, 1985. Spencer, Herbert Harry, and H. Surjanhata. "On Plate Buckling Coefficients." Journal of Engineering Mechanics, ASCE, Vol. 112, No 3, March 1986. Spencer, Herbert Harry, and H. Surjanhata. "The Simplified Buckling Criterion Applied to Plates with Partial Edge Loading." Journal Applied Scientific Research, Netherlands, Vol. 43, No 2, 1986. iv Dedicated to my parents, and my wife for their love and sacrifice. AC {NOWLEDGMENT The author would like to express his sincere gratitude to his supervisor, Dr. Harty 'Herman, for his valuable guidance and. encouragement during the course of this research. The author also wishes to express his gratitude to the members of his Advisory Committee Professors: Dr. Bernard ..Koplik, Dr. Nonni Levy, Dr. Denis Blackmore and Dr. Zhiming ii for their valuable comments and suggestions. Special thanks to his friend Wen-hua Zhu for his technical suggestions and encouragements during the period of his research. Finally, the author would like to express his deep appreciation and gratitude to his wife 'Limn and his son Brian for their sacrifices and understanding, and also to his parents Lin-Kee Chung and Nyuk-Chun Lian, especially to his mother who is undergoing a chemotherapy treatment at this time, for their love and encouragements during the course of this study. i TABLE OF CONTENTS Cae age 1 INTRODUCTION 2 LEGENDRE COLLOCATION MET:HOD 5 2.1 Introduction. 5 2.2 One-dimensional Lagrangian Trial Solution. 9 Differentiation of a Lagrange Interpolation Polynomial 2.4 Coefficient Matrices for First and Second Derivatives 15 2.5 Zeros of Legendre Polynomials 18 2.6 Legendre Collocation Method Applied to Differential Equation in Two-dimensional Domain 2.7 Linear Transformation of Two-dimensional Domains 22 2.8 The Legendre Collocation Element Formulation in Two-dimensional Domains 24 2.9 Treatment of Corners )_) 3 TREATMENT OF BOUNDARY SINGULARITIES 37 3.1 Introduction 37 3.2 Eigenfunctions Solution for Torsion of an L-shaped Bar ... 41 3.3 Discrete Least-Squares Approximation for Torsion of L-shaped Bar 47 3.4 Mini . ax Fit at Chebyshev Zeros 56 3.5 Series Solution for the Problem of Motz 59 vii 4 APPLICATIONS 67 4.1 Introduction 67 4.2 Torsion of A Square Bar 68 4.3 The Problem of Motz 4.4 Solution of Laplace's Equation in L-shaped Domain 107 5 CONCLUDING REMARKS 1.127 APPENDIX A Solution of Dirichlet Problem for Laplace's Equation by Separation of Variables 130 B Solution of Mixed-Boundary Conditions for Laplace's Equation by Separation of Variables -136 C Weights of Gauss-Legendre Q.adrature 140 D The Position of the Collocation Points 145 REFERENCES 146 iii LIST OF TABLES Table Page 3.1 Interpolation points 0A along the sector arc from 9 ----- 0 to 0 1.5 ;I- 59 4.1 Maximum Stress Function P - Torsion of Square Bar 4.2 k value for Maximum Shearing Stress - Torsion of Square Bar 4.3 ki - value for Torque - Torsion of Square Bar 73 4.4 Comparison, of torsion function 1,1 at the colloction points ............... 79 5 Seaig sess at the collocation points 80 4.6 Pointwise convergence of a solution u on a square grid points of 3.5 x 3.5. The results shown are based on a mesh size on each element: 3 x 3, 5 x 5, 7 x 7, 9 x 9 92 4.7 Comparison results of solution u of the problem of Motz on a unit mesh 93 4.8 The coefficients c, 1 computed from interpolated results obtained through the Legendre Collocation Method with 9 x 9 mesh size 94 4.9 Comparison the solution u or on the sector arc of the problem of Motz ( see Figure 4.10) 95 4.10 Comparison results of the solution it of the problem of Motz on a 0.25 unit mesh in the neighborhood of the singular point 96 4.11 The derivative (;1:. of the problem of Motz on a 0.25 unit mesh in the neighborhood of the singular point 0 97 4.12 The derivative 7(72,;,-- of the problem of Motz oil a 0.25 unit mesh in the neighborhood of the singular point 0 98 ix 4. 13 Pointwise convergence of least-squares approximation on the eigensolution u in the neighborhood of the singular point O. The results are on a 0.25 unit mesh and based on 9 x 9 mesh size for each element in the Legendre Collocation Method solution 99 4.14 Pointwise convergence of minitnax approximation on the eigensolution u in the neighborhood of the singular point O. The results are on a 0.25 unit mesh and based on 9 x 9 mesh size fOr each element in the Legendre Collocation Method solution ..... 100 4.1.5 Pointwise convergence of;-,(71 .7.` of least-squares approximation on the eigensolution 1/ in the neighborhood of the singular point O. The results are on a 0.25 unit mesh and based on 9 x 9 mesh size for each element in the Legendre Collocation Method solution ....... 101 4.16 Pointwise convergence of (,); ofot least-squares approximation on the eigensolutionu in the neighborhood of the singular point O. The results are on. a 0.25 unit .mesh. and based on 9 x 9 mesh size for each element in the Legendre Collocation Method solution ..., 102 4.1.7 Pointwise convergence of of minimax approximation on the eigensolution u in the neighborhood of the singular point O.