Block-partitioned solvers for coupled poromechanics: A unified framework
By Joshua A. White, Nicola Castelletto, Hamdi A. Tchelepi. 2016
155 citations recorded by the source corpus.
Abstract 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.
Subfields: Biomedical Engineering, Computational Mechanics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/eb20756a498b2e0024e715389d70bd4a5d4e9c50
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In Computational Mechanics
In Computational Theory and Mathematics
Citations in the corpus
- On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics (2017, background): “drained-split, undrained-split, fixed-stress and fixed-strain [5], can be seen in this way [13].”
- On the optimization of the fixed‐stress splitting for Biot's equations (2018, unknown): “…as mentioned already above, the number of iterations can differ considerably depending on the choice of the stabilization parameter [15, 3, 23, 2] (unless one uses the fixed-stress scheme as a preconditioner for a monolithic solver, as done in [26, 27] or as a smoother for a multigrid solver [28]).”