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A New Multidomain Approach and Fast Direct Solver for the Boundary Element Method

Huang, Shuo
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1505125721346283

Abstract Details

2017, PhD, University of Cincinnati, Engineering and Applied Science: Mechanical Engineering.
This dissertation research focuses on the development of new algorithms in the boundary element method (BEM) for solving multidomain problems and a new fast direct linear equation solver for the BEM. The motivation of this research is to further increase the computational efficiencies of the BEM so that BEM models with millions of DOFs (degrees of freedom) can be solved effectively on desktop PCs and those with billions of DOFs on clusters or supercomputers. A simple multidomain fast multipole BEM for solving potential problems is presented first, which can be applied to solve a true multidomain problem or a large-scale single domain problem using the domain decomposition technique. In this multidomain BEM, the coefficient matrix is formed simply by assembling the coefficient matrices of each subdomain and the interface conditions between subdomains without eliminating any unknown variables on the interfaces. Compared with other conventional multidomain BEM approaches, this new approach is more efficient with the fast multipole method, regardless how the subdomains are connected. Instead of solving the linear system of equations directly, the entire coefficient matrix is partitioned and decomposed using Schur complement in this new approach. Numerical results show that the new multidomain fast multipole BEM uses fewer iterations in most cases with the iterative equation solver and less CPU time than the traditional fast multipole BEM in solving large-scale BEM models. A large-scale fuel cell model with more than 6 million elements was solved successfully on a cluster within 3 hours using the new multidomain fast multipole BEM. To accelerate the solution of the multidomain BEM, a new fast direct linear equation solver for the BEM is presented next. The idea of the new fast direct solver stems from the concept of the hierarchical off-diagonal low-rank matrix. The hierarchical off-diagonal low-rank matrix can be decomposed into the multiplication of several diagonal block matrices. The inverse of the hierarchical off-diagonal low-rank matrix can be calculated efficiently with the Sherman-Morrison-Woodbury formula. In this paper, a more general and efficient approach to approximate the coefficient matrix of the BEM with the hierarchical off-diagonal low-rank matrix is proposed. Compared to the current fast direct solver based on the hierarchical off-diagonal low-rank matrix, the proposed method is suitable for solving general 3D boundary element models. Several numerical examples of 3D potential problems with the total number of unknowns up to above 200,000 are presented. The results show that the new fast direct solver can be applied to solve larger 3D BEM models accurately and with better efficiency compared with the conventional direct solvers for the BEM. Discussions and future directions in the research on the fast boundary element method for solving large-scale engineering problems are also presented to end this dissertation.
Yijun Liu, Ph.D. (Committee Chair)
Woo Kyun Kim, Ph.D. (Committee Member)
Gui-Rong Liu, Ph.D. (Committee Member)
Kumar Vemaganti, Ph.D. (Committee Member)
90 p.
Mechanics
Boundary element method; Multidomain problem; Fast direct solver

Recommended Citations

Citations

  • Huang, S. (2017). A New Multidomain Approach and Fast Direct Solver for the Boundary Element Method [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1505125721346283

    APA Style (7th edition)

  • Huang, Shuo. A New Multidomain Approach and Fast Direct Solver for the Boundary Element Method. 2017. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1505125721346283.

    MLA Style (8th edition)

  • Huang, Shuo. "A New Multidomain Approach and Fast Direct Solver for the Boundary Element Method." Doctoral dissertation, University of Cincinnati, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1505125721346283

    Chicago Manual of Style (17th edition)

ucin1505125721346283
190
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This open access ETD is published by University of Cincinnati and OhioLINK.