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 [Theta]-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.
Harald Garcke
1
;
Martin Rumpf
2
;
Ulrich Weikard
2
1
Universität Regensburg, Germany
2
Universität Bonn, Germany
Harald Garcke; Martin Rumpf; Ulrich Weikard. The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies. Interfaces and free boundaries, Tome 3 (2001) no. 1, pp. 101-118. doi: 10.4171/ifb/34
@article{10_4171_ifb_34,
author = {Harald Garcke and Martin Rumpf and Ulrich Weikard},
title = {The {Cahn-Hilliard} equation with elasticity-finite element approximation and qualitative studies},
journal = {Interfaces and free boundaries},
pages = {101--118},
year = {2001},
volume = {3},
number = {1},
doi = {10.4171/ifb/34},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/34/}
}
TY - JOUR
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AU - Martin Rumpf
AU - Ulrich Weikard
TI - The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies
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PY - 2001
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