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.
翻译:在本文中,我们为Cahn-Larch\'e等式提出了一个解决方案战略,这是一个介质线性弹性的模型,介于介质的线性弹性模式,介于介质上,介于两种弹性阶段,演变为Ginzburg-Landau型能源功能。这个系统可以被视为Cahn-Hilliard常规界面方程式和线性弹性的结合,并且不是线性结合,具有来自Cahn-Hilliard子系统的第四个顺序,并且是阶段-实地变量和迁移变量的非线性和非线性。我们提出了一个新的半隐性半隐性离散化,在时间上使用非线性双线性双层能源潜力的标准 convex-concave 分解法,以及作为弹性能源的特异性处理。我们表明,由此形成的离解性系统相当于峰值最小化问题,并提议并证明对其采用的交替最小化最小化和单线性最小化两个变量的趋同性。我们提出了一个数字实验,用以显示相互交替的最小化最小化和单线性新通度分解性新通度的分解法,我们将一个新的离式系统与新通度分解性分解性分解性分解性分解的分解的分解的分解的分解的分解的分式系统比了一种新的分解性分解的分解性分式的分解性系统与新的分解性分解性分解性系统,一个新的分解性分算法与新的分算法。