三维声场边界元高阶单元几乎奇异积分半解析法  

A new semi-analytic algorithm of nearly singular integrals on high order boundary element for 3D acoustic BEM

作　　者：胡宗军[1] 刘翠 牛忠荣[1] 

机构地区：[1]合肥工业大学土木与水利工程学院,合肥230009 [2]安徽省建筑工程质量监督检验站,合肥230088

出　　处：《应用力学学报》2015年第5期743-749,892-893,共7页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金(11272111)

摘　　要：分析了三维边界元法双线性曲面单元几何特征,定义接近度来表征源点与积分单元的接近程度。在单元局部坐标系中构造与三维声场边界元几乎奇异积分核函数具有相同奇异性的近似函数,从奇异积分核函数中扣除其近似函数从而分离出积分核函数中主导的奇异函数部分,并将原奇异积分核函数分解为规则核函数和奇异核函数两项积分。规则核函数积分应用常规Gauss数值积分计算,奇异核函数积分在局部极坐标系ρθ下分离积分变量,对变量ρ积分导出解析计算列式,对变量θ积分应用常规Gauss数值积分计算,从而建立一种三维声场边界元高阶单元几乎奇异积分的新的半解析算法,即双线性单元半解析算法。文中给出了内外声场算例,计算结果表明求解三维声场边界元法几乎奇异积分时基于高阶单元的本文的双线性单元半解析算法比线性单元正则化算法更适合处理距离边界更近的声压问题。By analyzing the geometric feature of bilinear elements in three dimensional boundary element method(3D BEM), the relative distance(named approach degree) from a source point to the surface integral element is defined. In the local coordinate systems on the integral element, the approximate kernel functions are constructed which have the same singularity as the nearly singular kernel functions of 3D acoustic BEM. By subtracting the approximate kernel functions from the nearly singular kernel functions, the leading singular parts are separated. Thus the original nearly singular surface integrals on bilinear elements are transformed into the sum of both the non-singular integrals and new singular integrals. The former can be efficiently computed by the Gaussian quadrature. The integral variables ρ and θ of the later are separated in the local polar coordinates. Firstly the integrals with respect to polar variable ρ are expressed by the analytic formulations, and then the surface integrals are transformed into the line integrals with respect to variableθ, which can be evaluated by the Gaussian quadrature. Consequently, the new semi-analytic algorithm is established to calculate the nearly singular surface integrals on bilinear elements in 3D acoustic BEM. Two internal and external acoustic examples are given in this paper. Both of the results demonstrate that the semi-analytic algorithm on high order element presented in this paper is more effective than linear regularization BEM to solve the problem of nearly singular integrals for 3D acoustic BEM.

关 键 词：三维声场 边界元法 几乎奇异积分 高阶单元 半解析算法 

分 类 号：O348.8[理学—固体力学]