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Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems

机译:快速多极加速奇异边界方法求解大型三维潜在问题

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摘要

The singular boundary method (SBM) is a relatively new meshless boundary collocation method for the numerical solution of certain elliptic boundary value problems. The method involves a coupling between the boundary element method (BEM) and the method of fundamental solutions (MFS). The main idea is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration-free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. However, similar to the traditional BEM, the SBM produces dense and unsymmetrical coefficient matrices, which requires O(N~2) memory and another O(N~3) operations to solve the system with direct solvers. This paper documents the first attempt to apply the fast multipole method (FMM) to accelerate the solutions of the SBM for the solution of large-scale problems. The FMM formulations for the SBM are presented for three-dimensional (3D) potential problems. Numerical examples with up to 900,000 unknowns are solved successfully on a desktop computer using the developed FMM-SBM code. These results clearly demonstrate the efficiency, accuracy and potentials of the FMM-SBM for solving large-scale problems.
机译:奇异边界法(SBM)是一种相对较新的无网格边界配置方法,用于某些椭圆形边值问题的数值解。该方法涉及边界元素方法(BEM)与基本解方法(MFS)之间的耦合。主要思想是充分继承前者的尺寸和稳定性优势以及后者的无网格和无积分属性。这使得它对于复杂的几何形状和三维问题特别有吸引力。然而,类似于传统的边界元法,SBM产生密集且不对称的系数矩阵,这需要O(N〜2)内存和另一个O(N〜3)运算才能使用直接求解器求解系统。本文记录了首次尝试使用快速多极方法(FMM)来加速SBM解决大型问题的方法。针对SBM的FMM公式针对三维(3D)潜在问题进行了介绍。使用开发的FMM-SBM代码在台式计算机上成功解决了多达900,000个未知数的数值示例。这些结果清楚地证明了FMM-SBM解决大规模问题的效率,准确性和潜力。

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