Preconditioned Krylov solvers for BEA

Abstract

The performance of a number of preconditioned Krylov methods is analysed for a large variety of boundary element formulations. Low-and high-order element, two-dimensional (2-D) and three-dimensional 3-D, regular, singular and hypersingular, collocation and symmetric Galerkin, single-and multi-zone, thermal and elastic, continuous and discontinuous boundary formulations with and without condensation are considered. Preconditioned Conjugate Gradient (CG) solvers in standard form and a form effectively operating on the normal equations (CGN), Generalized Minimal Residual (GMRES), Conjugate Gradient Squared (CGS) and Stabilized Bi-conjugate Gradient (Bi-CGSTAB) Krylov solvers are employed in this study. Both the primitive and preconditioned matrix operators are depicted graphically to illustrate the relative amenability of the alternative formulations to solution via Krylov methods, and to contrast and explain their computational performances. A notable difference between 2-D and 3-D BEA operators is readily visualized in this manner. Numerical examples are presented and the relative conditioning of the various discrete BEA operators is reflected in the performance of the Krylov equation solvers. A preconditioning scheme which was found to be uncompetitive in the collocation BEA context is shown to make iterative solution of symmetric Galerkin BEA problems more economical than employing direct solution techniques. We conclude that the preconditioned Krylov techniques are competitive with or superior to direct methods in a wide range of boundary formulated problems, and that their performance can be partially correlated with certain problem characteristics.

Rapid re-meshing and re-solution of three-dimensional boundary element problems for interactive stress analysis

“…Such iterative methods have already been applied to the BEM [15][16][17] and have produced positive results. However, they do not achieve the speed necessary for real-time re-analysis.…”

Performance of GMRES for the MFS

“…We shall experiment by taking the pre-conditioner P to be a diagonal matrix consisting of the diagonal elements of G d . Such pre-conditioners were found to be effective when using GMRES to solve systems resulting from Boundary Element Method (BEM) discretizations [7][8][9].…”

Application of a generic domain‐decomposition strategy to solve shell‐like problems through 3D BE models

“…Note that the strategies above can be promptly employed for parallelizing FE and BE codes. Among the Krylov solvers applicable to the solution of the BE systems of equations, PetrovGalerkin-type schemes, such as Lanczos and bi-conjugate gradient (Bi-CG), minimal residual approaches as GMRES, and hybrid procedures like CGS and Bi-CGSTAB, have all been considered before [23,[31][32][33][34][35][36][37][38][39][40]. Here, the Jacobi-preconditioned biconjugate gradient solver (J-BiCG) [31], which in a number of numerical experiments has proven to be reliable and efficient, is the one selected.…”