https://msvlab.hre.ntou.edu.tw/grades/bem(2008)/A%20FAST%203D%20DUAL%20BOUNDARY%20ELEMENT%20METHOD%20BASED%20ON%20HIERARCHICAL%20MATRICES.pdf

A fast 3D dual boundary element method based on hierarchical matrices

It has been previously reported in the open literature that multipole boundary element strategy based on Taylor expansions can result in computer codes which require O(N log N) operations for problems with N degrees of freedom. In this work we present a multipole BEM strategy developed for three-dimensional elasticity problems which is also based on Taylor expansions but requires only O(N) operations and O(N) memory. Results are presented for size of problems in which N=O(104).

An O(N) Taylor series multipole boundary element method for three-dimensional elasticity problems

Fast multipole method (FMM) has been successfully applied to accelerate the numerical solvers of boundary element method (BEM). However, the coefficient matrix implicitly formed by using FMM is sometimes ill-conditioned in cases when mixed boundary conditions exist, resulting in poor rate of convergence for iteration. So preconditioning is a critical part in the development of efficient FMM solver for BEM. In this paper, preconditioners based on sparse approximate inverse type are used for fast multipole BEM to deal with 2D elastostatics. Several sparsity patterns of the preconditioner are considered for single- and multi-domain problems, especially for 2D elastic body with large number of inclusions or cracks. Algorithms and cost analysis of preconditioning under different prescribed sparsity patterns are discussed. GMRES is used as the iterative solver. Numerical results show this type of preconditioner achieves satisfactory rate of convergence for fast multipole BEM and performs well for problems of fairly large sizes.

On the preconditioners for fast multipole boundary element methods for 2D multi-domain elastostatics

The solution of Navier–Stokes equations of time-dependent incompressible viscous fluid flow in planar geometry by the Boundary Domain Integral Method (BDIM) is discussed. The introduction of a subdomain technique to fluid flow problems is considered and improved in order to maintain the stability of BDIM. To avoid problems with flow kinematics computation in the sudomain mesh, a segmentation technique is proposed which combines the original BDIM with its subdomain variant and preserves its numerical stability. In order to reduce the computational cost of BDIM, which greatly depends on the solution of systems of linear equations, iterative methods are used. Conjugate gradient methods, conjugate gradients squared and an improved version of the biconjugate gradient method BiCGSTAB, together with the generalized minimal residual method, are used as iterative solvers. Different types of preconditioning, from simple Jacobi to incomplete LU factorization, are carried out and the performance of chosen iterative methods and preconditioners are reported. Test examples include backward facing step flow and flow through tubular heat exchangers. Test computation results show that BDIM is an accurate approximation technique which, together with the subdomain technique and powerful iterative solvers, can exhibit some significant savings in storage and CPU time requirements.

Iterative methods in solving navier-stokes equations by the boundary element method

In the present paper a fast solver for dual boundary element analysis of 3D anisotropic crack problems is formulated, implemented and tested. The fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix. The admissible low rank blocks are computed by adaptive cross approximation (ACA). The performance of ACA against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated, focusing on the balance between the kernel representation accuracy and the accuracy required for ACA. The system solution is computed by a preconditioned GMRES and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. The effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated, highlighting the accuracy as well as the significant reduction in memory storage and analysis time. In particular, it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom, obtaining up to solution speedups of order 10 for systems of order 104. Moreover, the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed. Copyright © 2009 John Wiley & Sons, Ltd.

A fast dual boundary element method for 3D anisotropic crack problems

The present paper is concerned with the application of iterative techniques to solve boundary element method (BEM) systems of equations.



Initially, a brief description of some algorithms which have been employed for symmetric positive-definite matrices is given (finite element method matrices, for instance). Subsequently, non-symmetric versions of these algorithms are described and their performance is checked for BEM matrices, through various numerical experiments.



Preconditioned algorithms were found to work quite well.

J. E. B. Malaghini

Papers

Solution of BEM systems of equations via iterative techniques