三维常数势边界元中的精确积分

对三维问题边界元方法中应用最广泛的常数边界元的积分提出一种精确积分方法。借助于一个假想的闭合曲面,将特定的势场应用于边界积分方程,发现对于三维问题,常数势项的积分可以化作球面三角型的面积计算,而导数项的积分则可在平面域用极坐标进行。本文方法结果精确,公式简单,同一计算公式可以用来计算非奇异、几乎奇异和奇异积分,统一了积分算法。

Accurate Integration of 3D Constant Potential Boundary Element

In boundary element methods,the common integrals are divided into three types,regular,nearly singular and singular.A new method to get the accurate integrals for 3D constant potential elements is proposed.With the aid of an assumed close surface,the certain potential field is taken into the boundary integration equations.The boundary integration is transformed to an assumed closed-surface.It is proved that the integral related to the potential term is equal to the area of a spherical triangle on a unit ball,which leads to a further application to the integral of constant derivative term,thus the above mentioned three types of integration can be evaluated by a unified formula.