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International Journal for Multiscale Computational Engineering
Импакт фактор: 1.016 5-летний Импакт фактор: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Печать: 1543-1649
ISSN Онлайн: 1940-4352

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International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2013006012
pages 633-654


Haim Waisman
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA
Luc Berger-Vergiat
Department of Civil Engineering & Engineering Mechanics, Columbia University, New York 10027, USA

Краткое описание

Application of an algebraic multigrid (AMG) solver to linear systems arising from fracture problems modeled by extended finite elements (XFEM) will often result in poor convergence. This is due to coarsening operators in AMG that disregard the discontinuous enrichment functions and automatically coarsen across cracks. To overcome the AMG coarsening limitation, we propose a multiplicative-Schwarz domain decomposition preconditioner to the generalized minimum residual method. In this approach the domain is decomposed into one uncracked subdomain and multiple cracked subdomains. A cracked subdomain is the domain containing the crack and its enrichment functions and the uncracked subdomain contains the rest of the domain with a one-element-layer overlap between the two. Within the preconditioning scheme, one AMG V-cycle is applied to the uncracked subdomain to obtain an approximate solution while the cracked subdomains (often much smaller compared to the uncracked part) are solved concurrently by a direct solver, thus resolving the error from the discontinuous fields exactly. Hence any black box AMG solver can be used for XFEM, and the need for development of special coarsening procedures that handle enriched degrees of freedom can be avoided. We consider multiple propagating cracks and develop an algorithm that adaptively updates the subdomains, following the cracks. This adaptive scheme can be obtained directly from level set values which are updated with crack growth or from close neighbor search algorithms. The level set update scheme is fast but does not guarantee tight subdomains, while a neighbor search is slower but gives optimal subdomains. The preconditioner is tested on structured and unstructured meshes with multiple propagating cracks and shows convergence rates that are significantly better than a brute force application of AMG to the entire domain.


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