Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Finite element approximation of a phase field model for surface diffusion of voids in a stressed solid

Authors:

John W. Barrett Harald Garcke Robert Nü rnberg.

Affiliation:

Department of Mathematics, Imperial College, London, SW7 2AZ, United Kingdom ; NWF I -- Mathematik, Universität Regensburg, 93040 Regensburg, Germany ; Department of Mathematics, Imperial College, London, SW7 2AZ, United Kingdom

Abstract:

We consider a fully practical finite element approximation of the degenerate Cahn-Hilliard equation with elasticity: Find the conserved order parameter,

, and the displacement field, $underline{u}(x,t) in mathbb{R}^2$ , such that

$displaystyle gamma,textstyle{frac{partial theta}{partial t}}$

subject to an initial condition

and boundary conditions on both equations. Here $gamma in {mathbb{R}}_{>0}$ is the interfacial parameter,

is a non-smooth double well potential,

is the symmetric strain tensor,

is the possibly anisotropic elasticity tensor, $c(s):=c_0+textstylefrac12,(1-c_0),(1+s)$ with $c_0(gamma)in {mathbb{R}}_{>0}$ and

is the degenerate diffusional mobility. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system in two space dimensions. Finally, some numerical experiments are presented.

Keywords:

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