Iterative solution of high-order boundary element method for acoustic impedance boundary value problems
Iterative solution of high-order boundary element method for acoustic impedance boundary value problems
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DOI：
10.1016/j.jsv.2005.06.044
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发表时间：
2006-04
影响因子：
4.7
通讯作者：
P. Ylä‐Oijala;S. Järvenpää
中科院分区：
文献类型：
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作者：
P. Ylä‐Oijala;S. Järvenpää
提出了一种求解时谐声阻抗边值问题的高阶边界元方法。该方法是基于Galerkin型制定的Burton-Miller积分方程（BMIE）与高阶多项式的基础和测试功能。该公式具有以下几个重要的优点：它不存在内共振问题，可以避免传统BMIE公式的超奇异积分算子，在未知数方面比低阶方法收敛得更快，并且在迭代求解器中表现出良好的性能。为了避免奇异曲面积分方程的数值计算困难，奇异性提取技术被应用到奇异和近奇异情况下的积分。得到的矩阵方程迭代求解的广义最小残差法（GMRES）和一个简单的预条件的基础上不完全LU分解，以加快收敛。数值结果表明，当采用GMRES方法迭代求解矩阵方程时，基于Galerkin方法和高阶基函数的BMIE公式在宽的频率范围内对各种几何和边界条件都具有良好的收敛性.这反过来又表明，该制剂是非常适合于快速求解程序，如快速多极子方法的有效应用。
A high-order boundary element method for time-harmonic acoustic impedance boundary value problems is presented. The method is based on the Galerkin-type formulation of the Burton–Miller integral equation (BMIE) with high-order polynomial basis and testing functions. This formulation has several important advantages: It is free of the interior resonance problem, the hypersingular integral operator of the traditional BMIE formulation can be avoided, it leads to faster convergence in terms of the number of unknowns than the low-order methods and it shows a good performance with iterative solvers. To avoid the numerical difficulties associated to the implementation of the singular surface integral equations, the singularity extraction technique is applied to evaluate the integrals in the singular and near-singular cases. The resulting matrix equation is solved iteratively with the generalized minimal residual method (GMRES) and a simple preconditioner based on the incomplete LU factorization is applied to expedite the convergence. Numerical results indicate that the BMIE formulation with Galerkin method and high-order basis functions has good convergence properties for various geometries and boundary conditions on a wide frequency range when the GMRES method is applied to solve the matrix equation iteratively. This, in turn, indicates that the formulation is well suited for an efficient application of fast solution procedures, such as the fast multipole method.