Abstract
In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes.
Highlights
The coupling of fluid flow and mechanical deformation in unsaturated porous media is relevant for many applications ranging from modeling rainfall-induced land subsidence or levee failure to understanding the swelling and drying-shrinkage of wooden or cement-based materials
The process can be modeled by coupling the Richards equations with quasi-static linear elasticity equations, generalizing the classical Biot equations [1]
We propose three linearization schemes incorporating the fixed-stress splitting scheme, coupled with an Lscheme, Modified Picard and Newton linearization of the flow
Summary
The coupling of fluid flow and mechanical deformation in unsaturated porous media is relevant for many applications ranging from modeling rainfall-induced land subsidence or levee failure to understanding the swelling and drying-shrinkage of wooden or cement-based materials. For the nonlinear Biot equations, as monolithic solver, Newton’s method requires the solver technology to solve saddle point problems coupling mechanics and flow equations. The arising systems are ill-conditioned and require an advanced, Preprint submitted to Elsevier monolithic simulator As the latter might be not available, the goal of this work is to develop a linearization scheme, which is robust and allows the use of decoupled simulators for mechanics and flow equations. For this purpose, we adopt closely related concepts for the linear Biot equations and the Richards equations
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