The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies

Abstract

We consider the Cahn-Hilliard equation-a fourth-order, nonlinear parabolic diffusion equation describing phase separation of a binary alloy which is quenched below a critical temperature. The occurrence of two phases is due to a nonconvex double well free energy. The evolution initially leads to a very fine microstructure of regions with different phases which tend to become coarser at later times.The resulting phases might have different elastic properties caused by a different lattice spacing. This effect is not reflected by the standard Cahn-Hilliard model. Here, we discuss an approach which contains anisotropic elastic stresses by coupling the expanded diffusion equation with a corresponding quasistationary linear elasticity problem for the displacements on the microstructure.Convergence and a discrete energy decay property are stated for a finite element discretization. An appropriate timestep scheme based on the strongly A-stable Θ-scheme and a spatial grid adaptation by refining and coarsening improve the algorithms efficiency significantly. Various numerical simulations outline different qualitative effects of the generalized model. Finally, a surprising stabilizing effect of the anisotropic elasticity is observed in the limit case of a vanishing fourth-order term, originally representing interfacial energy.

Cahn–Hilliard equations incorporating elasticity: analysis and comparison to experiments

“…In this article, we concentrate on extending the Cahn-Larché model. The system with linear elasticity was mainly studied in [48][49][50][51]. Finally, up-to-date numerics and references to computational methods for the CH model can be found in [52].…”

Multiple Scales in Phase Separating Systems with Elastic Misfit

“…The discrete solution at time t n is denoted by (c h n , w h n , u h n ). The resulting numerical scheme has been analyzed in [GRW01,GW05]. In [GRW01] optimal error estimates have been shown in the case that C does not depend on the concentration (homogeneous elasticity).…”

“…This process happens on a very short time scale and the regions with different phases have sizes which are given by a small length scale. If the elasticity tensor or the eigenstrains are anisotropic, one will observe that the phase regions orientate themself in certain directions (see [GMW03], [GRW01]) for numerical simulations). We will now describe how one can make these observations quantitative.…”

“…The fully discrete scheme has the properties that mass is conserved and that the total discrete free energy decreases (see [GRW01,GW05]). The last observation is a consequence of the fact that the discrete problem reflects the gradient flow property of the continuous problem and this is an important fact in the analysis of the scheme (see [GW05]).…”

“…All the computations presented in the following are with the help of the Θ-scheme, but we made sure that computations with the implicit Euler scheme lead to qualitatively similar results although with higher computational effort. We consider adaptive triangular grids in space and a corresponding a posteriori error control [GRW01,GW05]. The discrete linear systems were solved with the help of the BICG and GM-RES algorithms and for the nonlinear discrete problem we used Newton's method (see [GRW01,GW05] for more details).…”