期刊
JOURNAL OF COMPUTATIONAL PHYSICS
卷 452, 期 -, 页码 -出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2021.110910
关键词
Extended finite element method; Domain decomposition preconditioners; Sequence of linear systems; Crack propagation; Auxiliary tip subspace
资金
- National Natural Science Foundation of China [G12071469]
In this paper, a recycling preconditioning method with auxiliary tip subspace is proposed for solving ill-conditioned linear systems arising from elastic crack propagation problems. The method utilizes a Schwarz type preconditioner and a finite element mesh decomposition to handle crack tip and regular subdomains separately, resulting in significant computational cost savings.
In this paper, we propose a recycling preconditioning method with auxiliary tip subspace for solving a sequence of highly ill-conditioned linear systems of equations of different sizes arising from elastic crack propagation problems discretized by the extended finite element method. To construct a Schwarz type preconditioner, the finite element mesh is decomposed into crack tip subdomains, which contain all the degrees of freedom (DOFs) of the branch enrichment functions, and regular subdomains, which contain the standard DOFs and the DOFs of the Heaviside and the Junction enrichment functions. As cracks propagate these subdomains are modified accordingly, and the subdomain matrices are constructed as the restriction of the global matrix to the subdomains. In the overlapping Schwarz preconditioners, the crack tip subproblems are solved exactly and the regular subproblems are solved by some inexact solvers, such as ILU. We consider problems with and without crack intersections and develop a simple scheme to update, instead of re-computing, the subdomain problems as cracks propagate, in which only crack tip subdomains are updated around the new crack tips and all the regular subdomains remain unchanged. Therefore, no extra search is required, and the sizes of crack tip subproblems do not increase as cracks propagate, which greatly saves the computational cost. Moreover, starting from the second system, the Krylov subspace method uses a nontrivial initial guess constructed using the solution of the previous system with a modification around the new crack tips. The strategy accelerates the convergence remarkably. Numerical experiments demonstrate the efficiency of the proposed algorithms applied to problems with several types of cracks. (C) 2021 Elsevier Inc. All rights reserved.
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