On the competition of elastic energy and surface energy in discrete numerical schemes
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On the competition of elastic energy and surface energy in discrete numerical schemes T. Blesgen
Received: 25 February 2004 / Accepted: 9 May 2005 / Published online: 7 March 2007 © Springer Science + Business Media B.V. 2007
Abstract The -limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the -limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions. Keywords Discrete schemes · Surface energy · Elastic energy Mathematics Subject Classifications (2000) 82C26 · 74N20 · 74S20
1 Introduction This article is concerned with the behaviour of certain discrete schemes where there is competition between surface energy and elastic energy. Often, the surface energy will not appear in the limit, but sometimes if the ratio between surface energy and elastic energy is suitable this may be the case. We will compare two discrete approximation schemes that are related to discrete energy functionals H1h , H2h . These schemes compute the free energy of double well potentials and discretise the deformation gradient ∇u with a step size h > 0. In the first example, ∇u is
Communicated by A. Zhou. T. Blesgen (B) Max-Planck-Institute for Mathematics in the Sciences, Inselstraße 22-26, D-04103 Leipzig, Germany e-mail: [email protected]
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approximated by a two point stencil, and no surface energy appears for h 0. If three or more points are used in the approximation of the deformation gradient, this may be different. The second functional H2h is one simple example which produces a surface energy in the limit. The choice on H1h , H2h is motivated by the approximation of the free energy in physical systems where phase transitions take place and are especially related to phenomena like crystal growth, [2], segregation processes, [7], polymers and particular effects in fluid mechanics as cavitation, [4, 12]. As a practical example, we will consider the Cahn–Hilliard model with linear elasticity where two phases are assumed to occupy a bounded domain ⊂ R D √ with Lipschitz boundary. If γ is a given constant where γ is the thickness of the transition layer between two phases, the corresponding free energy has the form
F (, u) :=
W((x)) +
γ |∇(x)|2 + Q((x), u(x)) dx. 2
(1)
Here : × (0, T0 ) → R+ defines the density of one component of the alloy, u : → R D the deformation applied to the solid, T0 > 0 is a chosen stop time, Q(, u) ≥ 0 the elastic energy density and W ≥ 0 a double well potential. W has two spatially separated minima 1 , 2 , for instance W() := ( − 1 )2 ( − 2 )2 .
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