Efficient Boundary Element Method by Multipole Expansion and Generalized Inverse Matrix for Analysing Target Region : Applied for Two-Dementional Elastostatic Problems

The fast multipole boundary element method (FMBEM), which is an efficient BEM that uses the fast multipole method (FMM), is known to suffer from instability at low frequencies when the well-known high-frequency diagonal form is employed. In the present paper, various formulations for a low-frequency FMBEM (LF-FMBEM), which is based on the original multipole expansion theory, are discussed; the LF-FMBEM can be used to prevent the low-frequency instability. Concrete computational procedures for singular, hypersingular, Burton-Miller, indirect (dual BEM), and mixed formulations are described in detail. The computational accuracy and efficiency of the LF-FMBEM are validated by performing numerical experiments and carrying out a formal estimation of the efficiency. Moreover, practically appropriate settings for numerical items such as truncation numbers for multipole/local expansion coefficients and the lowest level of the hierarchical cell structure used in the FMM are investigated; the differences in the efficiency of the LF-FMBEM when different types of formulations are used are also discussed.

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VRP-GMRES(m) Iteration Algorithm for Fast Multipole Boundary Element Method

10.20944/preprints201612.0015.v1 ◽

2016 ◽

Author(s):

Chunxiao Yu ◽

Cuihuan Ren ◽

Xueting Bai

Keyword(s):

Boundary Element Method ◽

Boundary Element ◽

Large Scale ◽

Linear Equations ◽

Iteration Algorithm ◽

Fast Multipole ◽

Minimal Residual Method ◽

Generalized Inverse Matrix ◽

Selection Of ◽

Element Method

To solve large scale linear equations involved in the Fast Multipole Boundary Element Method (FM-BEM) efficiently, an iterative method named the generalized minimal residual method (GMRES)(m)algorithm with Variable Restart Parameter (VRP-GMRES(m) algorithm) is proposed. By properly changing a variable restart parameter for the GMRES(m) algorithm, the iteration stagnation problem resulting from improper selection of the parameter is resolved efficiently. Based on the framework of the VRP-GMRES(m) algorithm and the relevant properties of generalized inverse matrix, the projection of the error vector rm+1 on rm is deduced. The result proves that the proposed algorithm is not only rapidly convergent but also highly accurate. Numerical experiments further show that the new algorithm can significantly improve the computational efficiency and accuracy. Its superiorities will be much more remarkable when it is used to solve larger scale problems. Therefore, it has extensive prospects in the FM-BEM field and other scientific and engineering computing.

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A multipole expansion-based boundary element method for axisymmetric potential problem

Engineering Analysis with Boundary Elements ◽

10.1016/j.enganabound.2008.10.002 ◽

2009 ◽

Vol 33 (5) ◽

pp. 654-660 ◽

Cited By ~ 4

Author(s):

Jitendra Singh ◽

Alain Glière ◽

Jean-Luc Achard

Keyword(s):

Boundary Element Method ◽

Boundary Element ◽

Potential Problem ◽

Multipole Expansion ◽

Element Method

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A Fast Multipole Boundary Element Method Based on Legendre Series for Three-Dimensional Potential Problems

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.468-471.426 ◽

2012 ◽

Vol 468-471 ◽

pp. 426-429

Author(s):

Chun Xiao Yu ◽

Hai Yuan Yu ◽

Yi Ming Chen

Keyword(s):

Boundary Element Method ◽

Boundary Element ◽

Three Dimensional ◽

Multipole Expansion ◽

Fundamental Solutions ◽

Fast Multipole ◽

Legendre Series ◽

Truncation Errors ◽

Potential Problems ◽

Element Method

Vectorization expressions of a Fast Multipole Boundary Element Method (FM-BEM) based on Legendre series are presented for three-dimensional (3-D) potential problems. The formulas are applied to the expression of fundamental solutions for the Boundary Element Method(BEM). Truncation errors of the multipole expansion and local expansion are deduced and analyzed. It shows that the errors can be controlled by truncation terms.

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A Boundary Element Method Based on the Hierarchical Matrices and Multipole Expansion Theory for Acoustic Problems

International Journal of Computational Methods ◽

10.1142/s0219876218500093 ◽

2018 ◽

Vol 15 (03) ◽

pp. 1850009 ◽

Cited By ~ 2

Author(s):

Xiujuan Liu ◽

Haijun Wu ◽

Weikang Jiang

Keyword(s):

Boundary Element Method ◽

Boundary Element ◽

Large Scale ◽

Multipole Expansion ◽

Hierarchical Matrices ◽

Low Rank ◽

Scale Model ◽

Degenerate Kernel ◽

Set Up ◽

Element Method

The coefficient matrices of conventional boundary element method (CBEM) are dense and fully populated. Special techniques such as hierarchical matrices (H-matrices) format are required to extent its ability of handling large-scale problems. Adaptive cross approximation (ACA) algorithm is a widely adopted algorithm to obtain the H-matrices. However, the accuracy of the ACA boundary element method (ACABEM) cannot be adjusted by changing the tolerance [Formula: see text] when it exceeds a certain value. In this paper, the degenerate kernel approximation idea for the low-rank matrices is developed to build a fast BEM for acoustic problems by exploring the multipole expansion of the kernel, which is referred as the multipole expansion H-matrices boundary element method (ME-H-BEM). The newly developed algorithm compresses the far-field submatrices into low rank submatrices with the expansion terms of Green’s function. The obtained H-matrices are applied in conjunction with the generalized minimal residual method (GMRES) to solve acoustic problems. Numerical examples are carefully set up to compare the accuracy, efficiency as well as memory consumption of the CBEM, ACABEM, fast multipole boundary element method (FMBEM) and ME-H-BEM. The results of a pulsating sphere indicate that the ME-H-BEM keeps both storage and operation logarithmic-linear complexity of the H-matrices format as the ACABEM does. Moreover, the ME-H-BEM can achieve better convergence and higher accuracy than the ACABEM. For the analyzed complicated large-scale model, the ME-H-BEM with appropriate number of expansion terms has an advantage in terms of efficiency as compared with the ACABEM. Compared with the FMBEM, the ME-H-BEM is easier to be implemented.

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