三维边界元法高阶元几乎奇异积分半解析法

分析了三维边界元法高阶曲面单元几何特征，定义接近度来表征源点与积分单元的接近程度.利用源点在积分单元上的垂足点建立局部极坐标系，构造与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，分离出积分核中主导的奇异函数部分，将奇异积分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ρ和θ，对ρ积分建立解析计算列式，对θ积分应用常规Gauss数值积分计算，从而对三维位势问题高阶边界单元几乎强奇异和几乎超奇异积分建立一种新的半解析算法.给出了若干温度场算例，采用边界元法高阶单元几乎奇异积分半解析法计算了近边界内点位势和位势梯度，并与线性单元正则化算法计算结果对比，结果证明提出的半解析法计算几乎奇异面积分和薄壁结构更加高效. 

A NEW SEMI-ANALYTIC ALGORITHM OF NEARLY SINGULAR INTEGRALS IN HIGH ORDER BOUNDARY ELEMENT ANALYSIS OF 3D POTENTIAL

By analyzing the geometric feature of 8-noded quadrilateral element in three dimensional boundary element method (3D BEM), the relative distance is first defined as the approach degree from a source point to the high order surface element. And then a local polar coordinate ρθ is built which origin point is the project point of the source point on the element surface. The approximate singular kernel function is constructed corresponding to the nearly singular integral on high order surface elements in 3D potential BEM by a series of deduction, which has the same singularity as the nearly singular kernel function. The leading singular part is separated by subtracting the approximate kernel function from the original kernel function. Thus the nearly singular surface integrals on high order elements are transformed into the sum of both the non-singular integrals and 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 coordinate. The singular surface integrals with respect to polar variable ρ are firstly expressed by the analytic formulations. Then the surface integrals are transformed into the line integrals with respect to variable θ, which can be evaluated by the Gaussian quadrature. Consequently, the new semi-analytic algorithm is established to calculate the nearly strongly and hyper-singular surface integrals on high order element in 3D potential BEM. Some numerical examples about the high order BE analysis for 3D heat conduction problems are given to demonstrate the efficiency and accuracy of the present semi-analytic algorithm. In comparison with the published regularization algorithm which is applied to calculating the nearly singular integrals on 3-noded triangular element, the present semi-analytic algorithm with 8-noded quadrilateral element can evaluate the potentials and potential gradients of inner points more close to the boundary. Moreover, the semi-analytic algorithm can be applied to more efficiently analyze thin structures in 3D potentials.