快速多极边界元方法在大规模声学问题中的应用  

Application of Fast Multipole Boundary Element Method for Large-scale Acoustic Problems

作　　者：李善德[1] 黄其柏[1] 张潜[2] 

机构地区：[1]华中科技大学数字制造装备与技术国家重点实验室,武汉430074 [2]中船重工701研究所,武汉430064

出　　处：《机械工程学报》2011年第7期82-89,共8页Journal of Mechanical Engineering

基　　金：高等学校博士学科点专项科研基金(20070487403);中央高校基本科研业务费(2010MS080)资助项目

摘　　要：为克服传统边界元方法不适合进行大规模声学问题仿真的困难,将快速多极算法应用到传统边界元方法中,对大规模声学问题进行数值计算。由于在快速多极边界元方法中引入基本解的多极扩展,并应用预处理后的广义极小残差法迭代求解器求解线性方程系统,使得快速多极边界元方法的计算效率与传统边界元方法相比显著提高,计算量和存储量减少到O(N)量级(N为问题的自由度数)。对于传统边界元方法求解外部声学问题时的非唯一解现象,在快速多极边界元方法中采用改进的Burton-Miller方法获得全频段的唯一解。数值算例验证了快速多极边界元方法的准确性,表明快速多极边界元方法的计算效率与传统边界元方法相比有数量级的提高,能够有效求解大规模声学问题。In order to overcome the difficulty of the conventional boundary element method（BEM） is unsuitable for solving large-scale acoustic simulations,the fast multipole method（FMM） is used with the conventional BEM for solving large-scale acoustic problems.The computational efficiency of the fast multipole BEM（FMBEM） is improved significantly compared to the conventional BEM due to the multipole expansion of the fundamental solution and using the preconditioned generalized minimum residual method（GMRES） as an iterative solver to solve system of linear equation.Thus,both the computational complexity and memory requirement of the present FMBEM are drastically reduced to O（N）,where N is the number of degrees of freedom.In order to remove the non-unique problems of the conventional BEM,the FMBEM employs the improved Burton-Miller method to solve the exterior acoustic problems for all frequencies.Numerical examples validate the accuracy of the FMBEM,and show that the present algorithm provides an order of magnitude increase in computational efficiency compared to the conventional BEM.These examples clearly demonstrate that the present FMBEM is effective to solve large-scale acoustic problems.

关 键 词：边界元方法 快速多极算法 HELMHOLTZ方程 声学 大规模 

分 类 号：TB52[理学—物理]