Constructing efficient substructure-based preconditioners for BEM systems of equations

Abstract

Keywords:3D boundary-element models Subregion-by-subregion algorithm Krylov solvers Substructure-based block-diagonal preconditioners a b s t r a c tIn this work, a generic substructuring algorithm is employed to construct global block-diagonal preconditioners for BEM systems of equations. In this strategy, the allowable fill-in positions are those on-diagonal block matrices corresponding to each BE subregion. As these subsystems are independently assembled, the preconditioner for a particular BE model, after the LU decomposition of all subsystem matrices, is easily formed. So as to highlight the efficiency of the preconditioning proposed, the Bi-CG solver, which presents a quite erratic convergence behavior, is considered. In the particular applications of this paper, 3D representative volume elements (RVEs) of carbon-nanotube (CNT) composites are analyzed. The models contain up to several tens of thousands of degrees of freedom. The efficiency and relevance of the preconditioning technique is also discussed in the context of developing general (parallel) BE codes.

A three-dimensional cohesive-frictional grain-boundary micromechanical model for intergranular degradation and failure in polycrystalline materials

“…Moreover, performing the numerical tests, it has been noticed that the memory required by the solver PARDISO, see Appendix C, during the symbolic and numerical factorization phases, constitutes a bottleneck of the numerical method. From this point of view, the development of specific iterative Krylov solvers, for sparse systems having the structure of system (11), would be a remarkable achievement [109,110]. Moreover, being the analysis of polycrystalline microstructures a large-scale problem, the consideration of special techniques, with the aim of enhancing both the memory and time performance of the developed model, would be an interesting subject: an example in this context could be given by the use of Hierarchical Matrices in conjunction with iterative solvers [111,112].…”

The nsBETI method: an extension of the FETI method to non‐symmetrical BEM‐FEM coupled problems

“…For the standard FETI algorithm, reproduced in Table , an optimal Dirichlet (or primal) preconditioner exists, but because of its associated storage and computational overhead, a more economical lumped operator is usually preferred when solving second‐order homogeneous or heterogeneous elasticity problems. For the iterative solution of BEM equations, different preconditioners have been proposed in the literature .…”