三维静电场线性插值边界元中的解析积分方法

提出求解三维静电场的三角形线性插值边界元解析积分方法.针对含1/R和1/R2的积分项,将单元形状函数分解为常数项、含x的线性项和含y的线性项,从而将边界单元积分简化为6个基本积分组合,并导出其解析计算公式,避免了因形状函数改变而导致的重复计算.该方法不仅可以准确计算远离奇异情况下的边界元积分,而且可以准确计算一阶和二阶接近奇异积分以及一阶奇异积分.计算结果表明,在接近奇异积分和奇异积分比较突出的问题中,当数值积分方法不能给出正确结果时,用同样的边界元网格,解析积分方法可以给出正确的结果,提高了三维静电场线性插值边界元法的计算精度.

Analytical Integrals in the Linear Interpolation BEM for 3-D Electrostatic Fields

3-D electrostatic fields are calculated by the linear interpolation triangular boundary element method(BEM) with analytical integrals.For integrals with(1/R) and 1/R~2,shape functions are disintegrated into constant,x linear and y linear terms.The boundary element integral is simplified to 6 basic integral assemblies.Analytical integral formulas for them are introduced.Repeat calculations resulting from different shape functions are avoid.The method calculates exactly integrals far from singularity, 1 and 2 order nearly singular integrals and 1 order singular integrals as well.It shows that for problems with nearly integrals and singular integrals the analytical integral method gives correct results while the numerical integral method with the same boundary meshes can not.The precision in linear interpolation BEM for 3-D electrostatic fields is improved.