Iterative solution techniques in boundary element analysis

Abstract

SUMMARYIterative techniques for the solution of the algebraic equations associated with the direct boundary element analysis (BEA) method are discussed. Continuum structural response analysis problems are considered, employing single-and multi-zone boundary element models with and without zone condensation. The impact on convergence rate and computer resource requirements associated with the sparse and blocked matrices, resulting in multi-zone BEA, is studied. Both conjugate gradient and generalized minimum residual preconditioned iterative solvers are applied for these problems and the performance of these algorithms is reported. Included is a quantification of the impact of the preconditioning utilized to render the boundary element matrices solvable by the respective iterative methods in a time competitive with direct methods. To characterize the potential of these iterative techniques, we discuss accuracy, storage and timing statistics in comparison with analogous information from direct, sparse blocked matrix factorization procedures. Matrix populations that experience block fill-in during the direct decomposition process are included. With different degrees of preconditioning, iterative equation solving is shown to be competitive with direct methods for the problems considered.

Fast multipole method applied to 3-D frequency domain elastodynamics

“…Numerical tests [20,21,22] establish that the rate of convergence of the iterative solver is strongly dependent on the spectral properties of the matrix system. The preconditioning aims at transforming the original system of equations into another, formally equivalent, one with improved spectral properties.…”

On the use of partial re-analysis in conceptual mechanical design

“…At present, the re-analysis performed in the current work solves the complete revised matrix system using a diagonal preconditioned GMRES scheme, 6 which has been shown to give good performance for many BEM solutions. 7,8 The previous solution is used as a good ®rst approximation.…”

Multiresolution green's function methods for interactive simulation of large-scale elastostatic objects

“…Multizone substructuring methods are commonly used in boundary element analysis, e.g., [KKP91], to avoid the construction of large dense BEM matrices (H and G) such as for direct solvers and also for use with Krylov iterative methods. A common strategy is to use condensation on the system of multizone equations to arrive at a reduced system of equations relating only boundary values belonging to interzonal interfaces.…”

“…An interesting practical point is that iterative solvers for BEM discretizations on our Loop subdivision surfaces may be exaggerating iteration counts due to problems introduced by the steep mesh grading at extraordinary vertices [KKP91], e.g., valence 3. This problem might be reduced by the preconditioner used, but we plan to investigate this issue in the future.…”