Development of robust and efficient solution strategies for coupled problems, 2022

There are many applications where the study of coupled physical processes is of great importance. These range from the life sciences with flow in deformable human tissue to structural engineering with fracture propagation in elastic solids. In this doctoral dissertation, there is a twofold focus on coupled problems. Firstly, robust and efficient solution strategies, with a focus on iterative decoupling methods, have been applied to several coupled systems of equations. Secondly, a new thermodynamically consistent coupled system of equations is proposed. Solution strategies are developed for three different coupled problems; the quasi-static linearized Biot equations that couples flow through porous materials and elastic deformation of the solid medium, variational phase-field models for brittle fracture that couple a phase-field equation for fracture evolution with linearized elasticity, and the Cahn-Larché equations that model elastic effects in a two-phase elastic material and couples an extended Cahn-Hilliard phase-field equation and linearized elasticity. Finally, the new system of equations that is proposed models flow through a two-phase deformable porous material where the solid phase evolution is governed by interfacial forces as well as effects from both the fluid and elastic properties of the material.

In the work that concerns the quasi-static linearized Biot equations, the focus is on the fixed-stress splitting scheme, which is a popular method for sequentially solving the flow and elasticity subsystems of the full model. Using such a method is beneficial as it allows for the use of readily available solvers for the subproblems; however, a stabilizing term is required for the scheme to converge. It is well known that the convergence properties of the method strongly depend on how this term is chosen, and here, the optimal choice of it is addressed both theoretically and practically. An interval where the optimal stabilization parameter lies is provided, depending on the material parameters. In addition, two different ways of optimizing the parameter are proposed. The first is a brute-force method that relies on the mesh independence of the scheme's optimal stabilization parameter, and the second is valid for low-permeable media and utilizes an equivalence between the fixed-stress splitting scheme and the modified Richardson iteration.

Regarding the variational phase-field model for brittle fracture propagation, the focus is on improving the convergence properties of the most commonly used solution strategy with an acceleration method. This solution strategy relies on a staggered scheme that alternates between solving the elasticity and phase-field subproblems in an iterative way. This is known to be a robust method compared to the monolithic Newton method. However, the staggered scheme often requires many iterations to converge to satisfactory precision. The contribution of this work is to accelerate the solver through a new acceleration method that combines Anderson acceleration and over-relaxation, dynamically switching back and forth between them depending on a criterion that takes the residual evolution into account. The acceleration scheme takes advantage of the strengths of both Anderson acceleration and over-relaxation, and the fact that they are complementary when applied to this problem, resulting in a significant speed-up of the convergence. Moreover, the method is applied as a post-processing technique to the increments of the solver, and can thus be implemented with minor modifications to readily available software.

The final contribution toward solution strategies for coupled problems focuses on the Cahn-Larché equations. This is a model for linearized elasticity in a medium with two elastic phases that evolve with respect to interfacial forces and elastic effects. The system couples linearized elasticity and an extended Cahn-Hilliard phase-field equation. There are several challenging features with regards to solution strategies for this system including nonlinear coupling terms, and the fourth-order term that comes from the Cahn-Hilliard subsystem. Moreover, the system is nonlinear and non-convex with respect to both the phase-field and the displacement. In this work, a new semi-implicit time discretization that extends the standard convex-concave splitting method applied to the double-well potential from the Cahn-Hilliard subsystem is proposed. The extension includes special treatment for the elastic energy, and it is shown that the resulting discrete system is equivalent to a convex minimization problem. Furthermore, an alternating minimization solver is proposed for the fully discrete system, together with a convergence proof that includes convergence rates. Through numerical experiments, it becomes evident that the newly proposed discretization method leads to a system that is far better conditioned for linearization methods than standard time discretizations.

Finally, a new model for flow through a two-phase deformable porous material is proposed. The two poroelastic phases have distinct material properties, and their interface evolves according to a generalized Ginzburg–Landau energy functional. As a result, a model that extends the Cahn-Larché equations to poroelasticity is proposed, and essential coupling terms for several applications are highlighted. These include solid tumor growth, biogrout, and wood growth. Moreover, the coupled set of equations is shown to be a generalized gradient flow. This implies that the system is thermodynamically consistent and makes a toolbox of analysis and solvers available for further study of the model.

Link to the thesis.