THE FMM-MFS SOLUTON TO TWO-DIMENSINAL SCATTERING OF ELASTIC WAVES
This paper presents a new algorithm of the fast multipole fundamental solution method(FMM-MFS)for calculating two-dimensional elastic wave scattering problems. It avoids the singularity of the matrix by placing the line sources of compressional wave and shear wave on a virtual boundary based on single layer potential theory. Additionally, there is no need of elements discretization on the boundary. Combining with the FMM, MFS can solve large-scale problems of wave scattering by greatly reducing the computation and the memory requirement. Taking the two-dimensional scattering of P, SV waves around a cavity in elastic full-space as an example, the implement procedure are presented in detail, and up to millions DOF’s scattering problem are solved successfully on a personal computer. Based on the test of accuracy and efficiency of FMM-MFS, the scattering of plane waves by a cavity as well as random distributed group cavities in full-space are solved. Finally, several important conclusions about scattering of elastic waves around cavity(cavities) are obtained.
1.Department of Civil Engineering, Tianjin Chengjian University, Tianjin 300384, China;
2.Key Laboratory of Soft Soils and Engineering Environmental of Tianjin Province, Tianjin 300384, China
3.Department of Civil Engineering, Tianjin University, Tianjin 300072, China
This paper presents a new algorithm of the fast multipole fundamental solution method(FMM-MFS)for calculating two-dimensional elastic wave scattering problems. It avoids the singularity of the matrix by placing the line sources of compressional wave and shear wave on a virtual boundary based on single layer potential theory. Additionally, there is no need of elements discretization on the boundary. Combining with the FMM, MFS can solve large-scale problems of wave scattering by greatly reducing the computation and the memory requirement. Taking the two-dimensional scattering of P, SV waves around a cavity in elastic full-space as an example, the implement procedure are presented in detail, and up to millions DOF’s scattering problem are solved successfully on a personal computer. Based on the test of accuracy and efficiency of FMM-MFS, the scattering of plane waves by a cavity as well as random distributed group cavities in full-space are solved. Finally, several important conclusions about scattering of elastic waves around cavity(cavities) are obtained.
刘中宪1,2,王冬1,2,梁建文3. 基于快速多极子基本解方法(FMM-MFS)的弹性波二维散射模拟研究[J]. 振动与冲击, 2015, 34(5): 102-109.
LIU Zhong-xian1,2,WANG Dong1,2 Liang Jian-wen3. THE FMM-MFS SOLUTON TO TWO-DIMENSINAL SCATTERING OF ELASTIC WAVES. JOURNAL OF VIBRATION AND SHOCK, 2015, 34(5): 102-109.
[1] Golberg M A, Chen C S. The method of fundamental solutions for potential, Helmholtz and diffusion problems[J]. Boundary Integral Methods-Numerical and Mathematical Aspects, 1998: 103-176.
[2] Fairweather G, Karageorghis A, Martin P A. The method of fundamental solutions for scattering and radiation problems[J]. Engineering Analysis with Boundary Elements, 2003, 27(7): 759-769.
[3] Kupradze V D, Aleksidze M A. The method of functional equations for the approximate solution of certain boundary value problems[J]. USSR Computational Mathematics and Mathematical Physics, 1964, 4(4): 82-126.
[4] Mathon R, Johnston R L. The approximate solution of elliptic boundary-value problems by fundamental solutions[J]. SIAM Journal on Numerical Analysis, 1977, 14(4): 638-650.
[5] Chen J T, Lee Y T, Yu S R, et al. Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept[J]. Engineering Analysis with Boundary Elements, 2009, 33(5): 678-688.
[6] Greengard L, Rokhlin V. A fast algorithm for particle simulations[J]. Journal of Computational Physics, 1997, 135(2): 280-292.
[7] 姚振汉,王海涛. 边界元法,北京:高等教育出版社,2009.
Yao Zhenhan,Wang Haitao. Boundary element methods, Beijing: Higher Education Press, 2009.
[8] 崔晓兵, 季振林. 快速多极子边界元法在吸声材料声场计算中的应用[J]. 振动与冲击, 2011, 30(8): 187-192.
CUI Xiao-bing, JI Zhen-lin. Application of FMBEM to calculation of sound fields in sound-absorbing materials[J]. Journal of Vibration Engineering. 2011, 30(8): 187-192.
[9] Liu Y J, Nishimura N, Yao Z H. A fast multipole accelerated method of fundamental solutions for potential problems[J]. Engineering analysis with boundary elements, 2005, 29(11): 1016-1024.
[10] 许强, 蒋彦涛, 张志佳. 快速多极虚边界元法对含圆孔薄板有效弹性模量的模拟分析[J]. 计算力学学报, 2010, 27(3): 548-555.
XU Qiang JIANG Yan-tao ZHANG Zhi-jia Fast multipole VBEM for analyzing the effective elastic moduli of a sheet containing circular holes. Chinese Journal of Computational Mechanics, 2010, 27(3): 548-555.
[11] Jiang X.R., Chen W., Chen C.S. A fast method of fundamental solutions for solving Helmholtz-type equations, International Journal of Computational Methods, 10(2), 1341008, 2013.
[12] Chen Y H, Chew W C, Zeroug S. Fast multipole method as an efficient solver for 2D elastic wave surface integral equations[J]. Computational mechanics, 1997, 20(6): 495-506.
[13] Chaillat S, Bonnet M, Semblat J F. A new fast multi‐domain BEM to model seismic wave propagation and amplification in 3‐D geological structures[J]. Geophysical Journal International, 2009, 177(2): 509-531.
[14] Saad Y, Schultz M H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems[J]. SIAM Journal on scientific and statistical computing, 1986, 7(3): 856-869.
[15] Yoshida K. Applications of fast multipole method to boundary integral equation method[D]. Dept. of Global Environment Eng., Kyoto Univ., Japan, 2001.
[16] 王海涛. 快速多极边界元法在二维弹性力学中的应用[D]. 清华大学, 2002.
Wang Haitao. Application of Fast Multipole Boundary Element Method for Two Dimensional Elasticity[D]. Tsinghua University, 2002.
[17] Utsunomiya T, Watanabe E, Nishimura N. Fast multipole algorithm for wave diffraction/ radiation problems and its application to VLFS in variable water depth and topography[C]//Proc 20th Int Conf on Offshore Mech and Arcctic Engrg. 2001.