The University of Southern Mississippi The Aquila Digital Community Dissertations Summer 8-2018 Adaptive Meshfree Methods for Partial Differential Equations Jaeyoun Oh University of Southern Mississippi Follow this and additional works at: https://aquila.usm.edu/dissertations Part of the Numerical Analysis and Computation Commons, and the Partial Differential Equations Commons Recommended Citation Oh, Jaeyoun, "Adaptive Meshfree Methods for Partial Differential Equations" (2018). Dissertations. 1544. https://aquila.usm.edu/dissertations/1544 This Dissertation is brought to you for free and open access by The Aquila Digital Community. It has been accepted for inclusion in Dissertations by an authorized administrator of The Aquila Digital Community. For more information, please contact [email protected]. Adaptive Meshfree Methods for Partial Differential Equations by Jaeyoun Oh A Dissertation Submitted to the Graduate School, the College of Science and Technology, and the Department of Mathematics of The University of Southern Mississippi in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Approved by: Dr. Huiqing Zhu, Committee Chair Dr. CS Chen Dr. Haiyan Tian Dr. Zhifu Xie Dr. Huiqing Zhu Dr. Bernd Schroeder Dr. Karen S. Coats Committee Chair Department Chair Dean of the Graduate School August 2018 COPYRIGHT BY JAEYOUN OH 2018 ABSTRACT There are many types of adaptive methods that have been developed with different algorithm schemes and definitions for solving Partial Differential Equations (PDE). Adaptive methods have been developed in mesh-based methods, and in recent years, they have been extended by using meshfree methods, such as the Radial Basis Function (RBF) collocation method and the Method of Fundamental Solutions (MFS). The purpose of this dissertation is to introduce an adaptive algorithm with a residual type of error estimator which has not been found in the literature for the adaptive MFS. Some modifications have been made in developing the algorithm schemes depending on the governing equations, the domains, and the boundary conditions. The MFS is used as the main meshfree method to solve the Laplace equation in this dissertation, and we propose adaptive algorithms in different versions based on the residual type of an error estimator in 2D and 3D domains. Popular techniques for handling parameters and different approaches are considered in each example to obtain satisfactory results. Dirichlet boundary conditions are carefully chosen to validate the efficiency of the adaptive method. The RBF collocation method and the Method of Approximate Particular Solutions (MAPS) are used for solving the Poisson equation. Due to the type of the PDE, different strategies for constructing the adaptive method had to be followed, and proper error estima- tors are considered for this part. This results in having a new point of view when observing the numerical results. Methodologies of meshfree methods that are employed in this dissertation are introduced, and numerical examples are presented with various boundary conditions to show how the adaptive method performs. We can observe the benefit of using the adaptive method and the improved error estimators provide better results in the experiments. ii ACKNOWLEDGMENTS I would like to thank Dr. CS Chen, Dr. Haiyan Tian, and Dr. Zhifu Xie for their support during my journey to complete my Ph.D. I would also like to thank my advisor, Dr. Huiqing Zhu, for his assistance during this process. I sincerely appreciate his support in preparing this dissertation and all the inspiration he has given me to accomplish my work. iii DEDICATION I would like to thank my husband, Jason, and all my family in South Korea and Mississippi for their love and encouragement; this dissertation would not have been possible without them. iv TABLE OF CONTENTS ABSTRACT :::::::::::::::::::::::::::::::::: ii ACKNOWLEDGMENTS :::::::::::::::::::::::::::: iii DEDICATION :::::::::::::::::::::::::::::::::: iv LIST OF ILLUSTRATIONS ::::::::::::::::::::::::: vii LIST OF TABLES ::::::::::::::::::::::::::::::: ix LIST OF ABBREVIATIONS :::::::::::::::::::::::::: x NOTATION AND GLOSSARY :::::::::::::::::::::::: xi 1 INTRODUCTION TO MESHLESS METHODS :::::::::::::: 1 1.1 RBF Collocation Method 1 1.2 Method of Fundamental Solutions (MFS) 3 1.3 Method of Approximate Particular Solutions (MAPS) 6 1.4 Outline 7 2 ADAPTIVE METHODS :::::::::::::::::::::::::: 9 2.1 Adaptive Algorithm 9 2.2 Adaptive Methods by using the MFS 9 2.3 Adaptive Methods by using the RBF collocation method 10 2.4 Adaptive Procedure 12 3 ADAPTIVE METHOD BY USING THE MFS IN 2D :::::::::::: 13 3.1 A Residual-Type Adaptive Algorithm for 2D 13 3.2 The Choice of Parameter d 14 3.3 The Initial Distribution of Collocation points and Source points 15 3.4 Adaptive Algorithms 16 3.5 Numerical Experiments 17 4 ADAPTIVE METHOD BY USING THE MAPS AND RBF IN 2D ::::: 42 4.1 Method of Approximate Particular Solutions (MAPS) 42 4.2 Marking Strategies 44 4.3 Error Estimators 46 v 4.4 Numerical Experiments 49 5 ADAPTIVE METHOD BY USING THE MFS IN 3D :::::::::::: 59 5.1 Scheme for Adaptive MFS in 3D 59 5.2 The Initial Distribution of Collocation points 61 5.3 Numerical Experiments 63 6 CONCLUSIONS AND FUTURE WORK :::::::::::::::::: 79 6.1 Conclusions 79 6.2 Future work 80 BIBLIOGRAPHY ::::::::::::::::::::::::::::::: 82 vi LIST OF ILLUSTRATIONS Figure 3.1 General Adaptive Procedure . 14 3.2 W1 (Circle), W2 (Amoeba-like), and W3 (Gear-shaped, k = 7) . 16 −5 −9 3.3 W1, V1, Non-harmonic BC, e1 = 10 , and e2 = 10 .............. 21 −4 −9 3.4 W2, V1, Non-harmonic BC, Approach 4, e1 = 10 , and e2 = 10 ....... 21 −4 −9 3.5 W2, V2, Non-harmonic BC, Approach 4, e1 = 0:4, tol=2 · 10 , and e2 = 10 . 22 −4 −9 3.6 W2, V3, Non-harmonic BC, Approach 4, tol=10 , and e2 = 10 ....... 22 −2 −6 3.7 W3, V1, Non-harmonic BC, Approach 3, e1 = 10 , and e2 = 10 ....... 23 −3 −6 3.8 W3, V2, Non-harmonic BC, Approach 3, e1 = 1:1, tol=5 · 10 , and e2 = 10 . 23 −3 −6 3.9 W3, V3, Non-harmonic BC, Approach 3, tol=2 · 10 , and e2 = 10 ...... 24 −2 −6 3.10 W1, V1, Nonsmooth BC, h = 0:7, e1 = 10 , and e2 = 10 .......... 28 −3 −6 3.11 W1, V2, Nonsmooth BC, h = 0:7, e1 = 0:1, tol=5 · 10 , and e2 = 10 .... 28 −3 −6 3.12 W1, V3, Nonsmooth BC, h = 0:7, tol=10 , and e2 = 10 ........... 29 −2 −6 3.13 W1, V1, Nonsmooth BC, h = 0:1, e1 = 10 , and e2 = 10 .......... 29 −2 −6 3.14 W1, V2, Nonsmooth BC, h = 0:1, e1 = 0:8, tol=2 · 10 , and e2 = 10 .... 30 −2 −6 3.15 W1, V3, Nonsmooth BC, h = 0:1, tol=1:5 · 10 , and e2 = 10 ......... 30 −2 −6 3.16 W1, V1, Nonsmooth BC, h = 0:01, e1 = 10 , and e2 = 10 .......... 31 −1 −6 3.17 W1, V2, Nonsmooth BC, h = 0:01, e1 = 1:1, tol=10 , and e2 = 10 ..... 31 −2 −6 3.18 W1, V3, Nonsmooth BC, h = 0:01, tol=5 · 10 , and e2 = 10 ......... 32 −2 −6 3.19 W2, V1, Nonsmooth BC, h = 0:7, Approach 4, e1 = 10 , and e2 = 10 .... 32 −3 3.20 W2, V2, Nonsmooth BC, h = 0:7, Approach 4, e1 = 0:6, tol=2 · 10 , and −6 e2 = 10 ..................................... 33 −3 −6 3.21 W2, V3, Nonsmooth BC, h = 0:7, Approach 4, tol=2 · 10 , and e2 = 10 ... 33 −2 −6 3.22 W2, V1, Nonsmooth BC, h = 0:1, Approach 4, e1 = 10 , and e2 = 10 .... 34 −2 3.23 W2, V2, Nonsmooth BC, h = 0:1, Approach 4, e1 = 0:9, tol=2 · 10 , and −6 e2 = 10 ..................................... 34 −2 −6 3.24 W2, V3, Nonsmooth BC, h = 0:1, Approach 4, tol=2 · 10 , and e2 = 10 ... 35 −2 −6 3.25 W2, V1, Nonsmooth BC, h = 0:01, Approach 4, e1 = 10 , and e2 = 10 ... 35 −2 −6 3.26 W2, V2, Nonsmooth BC, h = 0:01, Approach 4, e1 = 1, tol=4·10 , and e2 = 10 36 −2 −6 3.27 W2, V3, Nonsmooth BC, h = 0:01, Approach 4, tol=4 · 10 , and e2 = 10 .. 36 −2 −6 3.28 W3, V1, Nonsmooth BC, h = 0:7, Approach 3, e1 = 10 , and e2 = 10 .... 37 −4 −6 3.29 W3, V2, Nonsmooth BC, h = 0:7, Approach 3, e1 = 1, tol=5·10 , and e2 = 10 37 −4 −6 3.30 W3, V3, Nonsmooth BC, h = 0:7, Approach 3, tol=2 · 10 , and e2 = 10 ... 38 −2 −6 3.31 W3, V1, Nonsmooth BC, h = 0:1, Approach 3, e1 = 10 , and e2 = 10 .... 38 −3 −6 3.32 W3, V2, Nonsmooth BC, h = 0:1, Approach 3, e1 = 1:1, tol=10 , and e2 = 10 39 −4 −6 3.33 W3, V3, Nonsmooth BC, h = 0:1, Approach 3, tol=5 · 10 , and e2 = 10 ... 39 −2 −6 3.34 W3, V1, Nonsmooth BC, h = 0:01, Approach 3, e1 = 10 , and e2 = 10 ... 40 vii −3 3.35 W3, V2, Nonsmooth BC, h = 0:01, Approach 3, e1 = 0:4, tol=4 · 10 , and −6 e2 = 10 ..................................... 40 −3 −6 3.36 W3, V3, Nonsmooth BC, h = 0:01, Approach 3, tol=4 · 10 , and e2 = 10 .. 41 4.1 Steps for Strategy 1 . 46 4.2 Steps for Strategy 2 . 47 4.3 Initial Distribution of Collocation Points . 50 4.4 Example 4.4.1, MAPS, Distributions at the final step and RMSEs . 51 −2 4.5 Example 4.4.1, MAPS, Polynomial order of 15, e1 = 10 , Strategy 1, h1 ... 52 4.6 Example 4.4.2, TPS-RBF, Distributions at the final step and RMSEs . 53 4.7 Example 4.4.2, TPS-RBF, e1 = 0:3 · max(h2), Strategy 2, h2 .......... 54 4.8 Example 4.4.2, Distributions of collocation points for each step .