数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (1): 1-22.doi: 10.1016/S0252-9602(12)60191-X

• 论文 •    下一篇

ON DECOMPOSITION METHOD FOR ACOUSTIC WAVE SCATTERING BY MULTIPLE OBSTACLES

王海兵1,2|刘继军1*   

  1. 1. Department of Mathematics, Southeast University, Nanjing 210096, China;
    2. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China
  • 收稿日期:2011-10-08 出版日期:2013-01-20 发布日期:2013-01-20
  • 通讯作者: 刘继军,jjliu@seu.edu.cn E-mail:wanghb845@gmail.com; jjliu@seu.edu.cn
  • 基金资助:

    This work is supported by NSFC (11071039, 11161130002), and Natural Science Foundation of Jiangsu Province (BK2011584).

ON DECOMPOSITION METHOD FOR ACOUSTIC WAVE SCATTERING BY MULTIPLE OBSTACLES

 WANG Hai-Bing1,2, LIU Ji-Jun1*   

  1. 1. Department of Mathematics, Southeast University, Nanjing 210096, China;
    2. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China
  • Received:2011-10-08 Online:2013-01-20 Published:2013-01-20
  • Contact: LIU Ji-Jun,jliu@seu.edu.cn E-mail:wanghb845@gmail.com; jjliu@seu.edu.cn
  • Supported by:

    This work is supported by NSFC (11071039, 11161130002), and Natural Science Foundation of Jiangsu Province (BK2011584).

摘要:

Consider acoustic wave scattering by multiple obstacles with different sound properties on the boundary, which can be modeled by a mixed boundary value problem for the Helmholtz equation in frequency domain. Compared with the standard scattering problem for one obstacle, the difficulty of such a new problem is the interaction of scattered wave by different obstacles. A decomposition method for solving this multiple scattering problem is developed. Using the boundary integral equation method, we decompose the total scattered field into a sum of contributions by separated obstacles. Each contribution corresponds to scattering problem of single obstacle. However, all the single scattering problems are coupled via the boundary conditions, representing the physical interaction of scattered wave by different obstacles. We prove the feasibility of such a decomposition. To compute these contributions efficiently, an iteration algorithm of Jacobi type is proposed, decoupling the interaction of scattered wave from the numerical points of view. Under the well-separation assumptions on multiple obstacles, we prove the convergence of iteration sequence generated by the Jacobi algorithm, and give the error estimate between exact scattered wave and the iteration solution in terms of the obstacle size and the minimal distance of multiple obstacles. Such a quantitative description reveals the essences of wave scattering by multiple obstacles. Numerical examples showing the accuracy and
convergence of our method are presented.

关键词: multiple obstacles scattering, decomposition method, Jacobi iteration, con-vergence, error estimate, numerics

Abstract:

Consider acoustic wave scattering by multiple obstacles with different sound properties on the boundary, which can be modeled by a mixed boundary value problem for the Helmholtz equation in frequency domain. Compared with the standard scattering problem for one obstacle, the difficulty of such a new problem is the interaction of scattered wave by different obstacles. A decomposition method for solving this multiple scattering problem is developed. Using the boundary integral equation method, we decompose the total scattered field into a sum of contributions by separated obstacles. Each contribution corresponds to scattering problem of single obstacle. However, all the single scattering problems are coupled via the boundary conditions, representing the physical interaction of scattered wave by different obstacles. We prove the feasibility of such a decomposition. To compute these contributions efficiently, an iteration algorithm of Jacobi type is proposed, decoupling the interaction of scattered wave from the numerical points of view. Under the well-separation assumptions on multiple obstacles, we prove the convergence of iteration sequence generated by the Jacobi algorithm, and give the error estimate between exact scattered wave and the iteration solution in terms of the obstacle size and the minimal distance of multiple obstacles. Such a quantitative description reveals the essences of wave scattering by multiple obstacles. Numerical examples showing the accuracy and
convergence of our method are presented.

Key words: multiple obstacles scattering, decomposition method, Jacobi iteration, con-vergence, error estimate, numerics

中图分类号: 

  • 74J20