Second order phase field asymptotics for multi-component systems

Abstract

Abstract. We derive a phase field model which approximates a sharp interface model for solidification of a multicomponent alloy to second order in the interfacial thickness ε. Since in numerical computations for phase field models the spatial grid size has to be smaller than ε the new approach allows for considerably more accurate phase field computations than have been possible so far.In the classical approach of matched asymptotic expansions the equations to lowest order in ε lead to the sharp interface problem. Considering the equations to the next order, a correction problem is derived. It turns out that, when taking a possibly nonconstant correction term to a kinetic coefficient in the phase field model into account, the correction problem becomes trivial and the approximation of the sharp interface problem is of second order in ε. By numerical experiments, the better approximation property is well supported. The computational effort to obtain an error smaller than a given value is investigated revealing an enormous efficiency gain.

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

“…Thus the gradient ∇d points from Ω i to Ω j and we may use ∇d on Γ as a unit normal ⃗ ν. Let g(t, s) denote a parameterisation of Γ by arclength s, and in a tubular neighbourhood of Γ, for smooth functions f (⃗ x), we have In this new (t, s, z)-coordinate system, the following change of variables apply (see [1,27]):…”

“…In order to match the inner expansions valid in the interfacial region to the outer expansions of Section 3.1, we employ the matching conditions, see [27]:…”

Stress- and diffusion-induced interface motion: Modelling and numerical simulations

“…We refer to the papers [14,13,17,21,2,27] for more details about this method. Obtaining the equations in the bulk is straightforward and we only discuss the derivation of the equations on the interface.…”

Numerical schemes for a three component Cahn-Hilliard model

“…For two phase situations, there has been much algorithm development and many simulations of the Cahn-Hilliard equations [2,3,5,7,12,13,16,19,20,24]. Generalizations of diffuse-interface models to any number of components have been recently introduced and studied as well as associated numerical methods and simulations, see for instance [1,4,6,8,14,15,[21][22][23]25,27,28].…”