On the Kernel-independent Fast Boundary Element Method for the 3-D Laplace Equation
(0) A Kernel-independent fast multipole boundary element method (FMBEM) is developed for the numerical analysis of three-dimensional (3-D) Laplace equations. In this method, the conventional multipole moments of a collection of boundary elements inside a leaf box of the FMM tree are replaced by an equivalent density continuously distributed on a cubic surface enclosing the box. The far-field evaluations are achieved by the integration of potentials of the density. This equivalent density is discretised at the cubic surface and approximated by use of linear or quadratic boundary elements, and the calculation of its value is based on direct kernel evaluations. Therefore, the analytic expansions of the Laplacian kernels are avoided. Compared with the original fast multipole method first introduced by Greengard and Rokhlin in 1987 (J. Comput. Phys. 1987; 73: 325-348), this kernel-independent method has the same data structures and performance procedures except that the equivalent densities are translated instead of both the multipole and local moments in the M2M, M2L and L2L translation operators. The translation matrices are precomputed, stored and repeatedly used at different tree levels. Several control parameters of this method are discussed briefly and their impact on the computational accuracies are tested. The computational efficiencies of this method are studied numerically and compared with several other algorithms including the conventional preconditioned GMRES solver, the original fast multipole boundary element method, the new version of the fast multipole boundary element method and other kernel-independent algorithms like the fully-and partially-pivoted adaptive cross approximation methods (ACA). The numerical results clearly show that the kernel-independent fast multipole boundary element method has O(N) complexity in both the computational time and data storage, and that for the requirement of engineering accuracy, this method has comparable computational efficiencies with the new version of the fast multipole boundary element method. Furthermore, according to the predictable trends in the memory requirement, this method can deal with large scale problems with number of degrees of freedoms (DOFs) orders of magnitude higher than that using the kernel-independent ACA method. In summary, the advantage of the kernel-independent fast multipole boundary element method is that it requires direct kernel evaluations instead of analytic expansions of the kernels, can be extended immediately to the large scale boundary element solutions of other physical problems and remains the same computational complexity of the fast multipole method as well.
H.T.Wang Z.H.Yao
Institute of Nuclear and New Energy Technology,Tsinghua University,Beijing,100084 China Department of Engineering Mechanics,Tsinghua University,Beijing,100084 China
国际会议
南京
英文
1-2
2009-10-18（万方平台首次上网日期，不代表论文的发表时间）