A robust solution strategy for the Cahn-Larch\'e equations

TL;DR

This paper introduces a new solution approach for the complex, coupled Cahn-Larché equations involving a semi-implicit discretization, convex minimization, and alternating minimization, demonstrating improved robustness and conditioning.

Contribution

It presents a novel semi-implicit discretization and a convergence proof for alternating minimization tailored to the coupled Cahn-Larché system, enhancing numerical stability.

Findings

The proposed discretization leads to a convex minimization problem.

Alternating minimization converges for the discretized system.

Numerical experiments show improved robustness and conditioning.

Abstract

In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment to the elastic energy. We show that the resulting discrete system is equivalent to a convex minimization problem, and propose and prove the convergence of alternating minimization applied to it. Finally, we present numerical experiments that show the robustness and effectiveness of both alternating minimization and the monolithic Newton method applied to the newly proposed discrete system of equations. We compare it to a system of equations that has been discretized with a standard convex-concave splitting of the double-well potential, and implicit evaluations of the elasticity contributions and show that the newly proposed discrete system is better conditioned for linearization techniques.