Abstract:

The computational cost of conventional BEM is relatively high when solving large scale acoustic problems.A multipole Galerkin BEM based on Burdon-Miller formulation was established.Based on the former two dimensional fast multipole BEM code,diagonal expansion form of kernel function,the adaptive tree
structure and the approximate inverse pre-conditioner were  adopted to improve the numerical efficiency.Finally,two numerical examples including a rectangular tube and a rigid cylinder scattering problem were solved by multipole Galerkin BEM.The numerical results show that, the non-unique problem associated with infinite domain acoustic problems is solved by the Burdon-Miller formulation.Compared with conventional BEM,the computational complexity of the multipole Galerkin BEM will be reduced from
O(n2) to O(nlog2 n).These two examples demonstrate clearly  that the present multipole Galerkin BEM is effective to solve large-scale acoustic problems.

Key words: Galerkin boundary element method(BEM), fast multipole(FM) method, approximate inverse pre-conditioner, acoustics, numerical computation

摘要：

针对传统边界元法在数值求解大规模声学问题时的超大计算量问题,将快速多极算法与伽辽金边界元法相结合，提出了基于Burton-Miller方程的伽辽金多极边界元法。在已有二维快速多极算法的基础上，引入核函数的对角展开形式及自适应树结构算法，同时使用经过近似求逆预处理的广义极小残差法求解系统线性方程组，最后将该方法应用于二维矩形管道与刚性圆柱面声散射问题的求解。数值计算结果表明：在求解无限域声学问题时，Burton-Miller方程保证了全频率段解的唯一性，特别是在特征频率处解的稳定性。与传统边界元方法相比，伽辽金多极边界元法的计算量由原来的O(n2)降到了O(nlog2n)量级,该方法非常适合用于求解大规模声学问题。

关键词: 伽辽金边界元法, 快速多极方法, 近似求逆预处理, 声学, 数值计算