The Cahn-Hilliard equation with elasticity -- finite element approximation and qualitative studies
type: Item
instance of: scholarly article
title: The Cahn-Hilliard equation with elasticity -- finite element approximation and qualitative studies
zbMATH Open document ID: 0972.35164
DOI: 10.4171/IFB/34
author: Harald Garcke
author: Martin Rumpf
author: Ulrich Weikard
published in: Interfaces and Free Boundaries
publication date: 2001-03-28T00:00:00Z
review text: Summary: 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.
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 74N15
Mathematics Subject Classification ID: 74S05
zbMATH DE Number: 1582146
zbMATH Keywords: Cahn-Hilliard equation
zbMATH Keywords: fourth-order parabolic equation
zbMATH Keywords: finite element approximation
zbMATH Keywords: elasticity
zbMATH Keywords: adaptive methods
zbMATH Keywords: second-order time discretization
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/ifb/34
OpenAlex ID: W2055481609
description: scientific article; zbMATH DE number 1582146