Research On The Fast Multipole Boundary Element Method For 3D Potential And Elastic Problems

In this paper, we study 3D fast multipole boundary element method (FM-BEM) for potential problems and 3D FM-BEM for elastic problems, prove the existence and uniqueness of 3D FM-BEM for potential problems, the singular integrals in 3D Fast Multipole BEM for potential problems can be dealt with, also, the FMM expansion of the kernels for 3D elastic problems can be derived. Due to its high efficiency and low computing memory demanding, the above method makes the large-scale computing possible.The Paper includes five chapters. Chapter 1 is introduction, which summarizes the development of BEM, potential problems, FMM, and Fast Multipole-BEM, then presented the source, significance of the task and the major work of the paper.In chapter 2, we introduce the multipole expansion and the fast multipole boundary element method basic theory knowledge, and it establishes the foundation of the multipole expansion method fuses with BEM form and the followlling to studied multipole expansion BEM; also we discuss the multipole expansion method's suitable scope, this method renews the traditional BEM theory and calculation structure, it is suitable for big scale operate project problem.In chapter 3, the basic theory of Krylov Subspace Method is introduced, and the detailed iterative steps and the utilization disposal of GMRES algorithm are presented. Then the GMRES algorithm based on FMM is presented.In chapter 4, the FM-BEM for 3D potential problems is introduced, the proof of existence and uniqueness of 3D FM-BEM for potential problems is given, the mathematical theory of the FM-BEM for 3D potential problems can be completed. Also, the method of solving the singularity in 3D Fast Multipole BEM for potential problems can be obtained.In chapter 5, on basis of FMM and traditional BEM, we can give the mathematical theory of the surface FM-BEM. Then we introduce the BEM formulation for 3D elastic problems, derive the FMM expansion of the kernels for 3D elastic problems, which can adapt to FM-BEM. Furthmore, the mathematical theory of the FM-BEM can be completed, the computing efficiency and precision can be improved.