In boundary element methods (BEM) in $\mathbb{R}^3$, matrix elements and right hand sides are typically computed via analytical or numerical quadrature of the layer potential multiplied by some function over line, triangle and tetrahedral volume elements. When the problem size gets large, the resulting linear systems are often solved iteratively via Krylov subspace methods, with fast multipole methods (FMM) used to accelerate the matrix vector products needed. When FMM acceleration is used, most entries of the matrix never need be computed explicitly - {\em they are only needed in terms of their contribution to the multipole expansion coefficients.} We propose a new fast method for the analytical generation of the multipole expansion coefficients produced by the integral expressions for single and double layers on surface triangles; charge distributions over line segments and over tetrahedra in the volume; so that the overall method is well integrated into the FMM, with controlled error. The method is based on the $O(1)$ per moment cost recursive computation of the moments. The method is developed for boundary element methods involving the Laplace Green's function in ${\mathbb R}^3$. The derived recursions are first compared against classical quadrature algorithms, and then integrated into FMM accelerated boundary element and vortex element methods. Numerical tests are presented and discussed.
翻译:在 $\ mathb{R ⁇ 3$ 的边界要素方法( BEM) 中, 矩阵元素和右手边通常通过对层潜力的分析或数字梯度乘以线、 三角和四面体体积元素的某些函数来计算。 当问题规模大时, 产生的线性系统往往通过 Krylov 子空间方法迭接解决, 使用快速多极方法加速所需的矩阵矢量产品。 当使用 FMM 加速时, 大多数矩阵条目都无需明确计算 - 只需要对多极扩张系数的贡献来计算 。 } 我们提出一种新的快速方法, 用于分析生成由地表三角的单层和双层整体表达生成的多极扩张系数; 将分布在线段和四面体积上, 使用快速多极法方法来加速所需的矩阵矢量产品。 该方法以每时成本的美元进行递归计算 。 方法是用于将Laplace Greek Green's 的边系方法, 然后是用于正态的正态 和正态 方程3 矩阵 的加速 的矩阵测试 。