期刊
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
卷 303, 期 -, 页码 55-74出版社
ELSEVIER SCIENCE SA
DOI: 10.1016/j.cma.2016.01.008
关键词
Poromechanics; Iterative methods; Preconditioning; Algebraic multigrid; Fixed-stress split
资金
- U.S. Department of Energy by Lawrence Livermore National Laboratory [DE-AC52-07NA27344]
- Petroleum Institute (PI) of Abu Dhabi
- Reservoir Simulation Industrial Affiliates Consortium at Stanford University (SUPRI-B)
Coupled poromechanical problems appear in a variety of disciplines, from reservoir engineering to biomedical applications. This work focuses on efficient strategies for solving the matrix systems that result from discretization and linearization of the governing equations. These systems have an inherent block structure due to the coupled nature of the mass and momentum balance equations. Recently, several iterative solution schemes have been proposed that exhibit stable and rapid convergence to the coupled solution. These schemes appear distinct, but a unifying feature is that they exploit the block-partitioned nature of the problem to accelerate convergence. This paper analyzes several of these schemes and highlights the fundamental connections that underlie their effectiveness. We begin by focusing on two specific methods: a fully-implicit and a sequential-implicit scheme. In the first approach, the system matrix is treated monolithically, and a Krylov iteration is used to update pressure and displacement unknowns simultaneously. To accelerate convergence, a preconditioning operator is introduced based on an approximate block-factorization of the linear system. Next, we analyze a sequential-implicit scheme based on the fixed-stress split. In this method, one iterates back and forth between updating displacement and pressure unknowns separately until convergence to the coupled solution is reached. We re-interpret this scheme as a block-preconditioned Richardson iteration, and we show that the preconditioning operator is identical to that used within the fully-implicit approach. Rapid convergence in both the Richardson-and Krylov-based methods results from a particular choice for a sparse Schur complement approximation. This analysis leads to a unified framework for developing solution schemes based on approximate block factorizations. Many classic fully-implicit and sequential-implicit schemes are simple sub-cases. The analysis also highlights several new approaches that have not been previously explored. For illustration, we directly compare the performance and robustness of several variants on a benchmark problem. (C) 2016 The Authors. Published by Elsevier B.V.
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