Search

A fast, accurate, and efficient multilevel boundary-element method (MLBEM) is developed to solve general boundary-value problems arising in computational mechanics. Here we concentrate on problems of two-dimensional steady potential flow and present a fast, direct boundary-element formulation. This novel method extends the pioneering work of Brandt and Lubrecht on multilevel multi-integration (MLMI) in several important ways to address problems with mixed boundary conditions. We utilize bi-conjugate gradient methods (BCGMs) and implement the MLMI approach for fast matrix and matrix transpose multiplication for every iteration loop. After introducing a C-cycle multigrid algorithm, we find that the number of iterations for the bi-conjugate gradient methods is independent of the boundary-element mesh discretization for a broad range of steady-state heat diffusion problems. Here, for a model problem in an L-shaped domain, we demonstrate that the computational complexity of the proposed method approaches the d...

Higher-order boundary-element methods (BEM) are presented for steady-state convective diffusion problems in two dimensions. The free-space steady convective diffusion fundamental solutions considered in this article provide an analytical upwinding for the entire Peclet number range, from zero to infinity. However, integration of the kernels over the boundary elements requires considerable attention, especially at higher Peclet numbers. We define an influence domain due to these convective kernels and then localize the surface integrations only within the domain of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. This, in turn, leads to increasing sparsity and improved conditioning of the global matrix. Consequently, iterative solvers become the primary choice. We consider an example problem with an exact solution, and investigate the accuracy and efficiency of the higher-order BEM formulations for Peclet numbers in the range from 200 to 200,000....