This thesis concerns iterative solvers for poromechanics problems. The problems in the studies have involved linear poromechanics, non-linear poromechanics, and poromechanics under large deformation. We included high order discretizations, applied linearization techniques and splitting methods to develop new solvers. We studied the robustness and convergence of these solvers. By studying the fixed stress method as an iterative solver for poromechanics, we developed an optimized version of it. Furthermore, by extending the convergence analysis in the time domain, we developed a new version of the fixed stress method that is partially parallelized. This splitting method was combined with linearization techniques to develop solvers for non-linear poromechanics. By studying the convergence of the linearisation schemes, we developed new solvers and extended the applicability to more complex phenomena, for instance poromechanics with large deformation.
Summary
In this work, we are interested in efficiently solving the quasi‐static, linear Biot model for poroelasticity. We consider the fixed‐stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well known that the convergence properties of the method strongly depend on the applied stabilization/tuning parameter. We show theoretically that, in addition to depending on the mechanical properties of the porous medium and the coupling coefficient, they also depend on the fluid flow and spatial discretization properties. The type of analysis presented in this paper is not restricted to a particular spatial discretization, although it is required to be inf‐sup stable with respect to the displacement‐pressure formulation. Furthermore, we propose a way to optimize this parameter that relies on the mesh independence of the scheme's optimal stabilization parameter. Illustrative numerical examples show that using the optimized stabilization parameter can significantly reduce the number of iterations.
Multiphase poromechanics describes the evolution of multiphase flow in deformable porous media. Mathematical models for such multiphysics system are inheritely nonlinear, potentially degenerate and fully coupled systems of partial differential equations. In this work, we present a thermodynamically consistent multiphase poromechanics model falling into the category of Biot equations and obeying to a generalized gradient flow structure. It involves capillarity effects, degenerate relative permeabilities, and gravity effects. In addition to established models it introduces a Lagrange multiplier associated to a bound constraint on the effective porosity in particular ensuring its positivity. We establish existence of global weak solutions under the assumption of a weak coupling strength, implicitly utilizing the gradient flow structure, as well as regularization, a Faedo-Galerkin approach and compactness arguments. This comprises the first global existence result for multiphase poromechanics accounting for degeneracies that are consistent with the multiphase nature of the flow.
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