Björn Stinner

A Diffuse Interface Model for Alloys with Multiple Components and Phases

A nonisothermal phase field model for alloys with multiple phases and components is derived. The model allows for arbitrary phase diagrams. We relate the model to classical sharp interface models by formally matched asymptotic expansions. In addition we discuss several examples and relate our model to the ones already existing.

Modeling and computation of two phase geometric biomembranes using surface finite elements

Biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions. The modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces. This leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints. In this paper we propose a new computational tool for computing equilibria based on an L^2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg-Landau phase field functional. The relaxation dynamics couple a nonlinear fourth order geometric evolution equation of Willmore flow type for the membrane with a surface Allen-Cahn equation describing the lateral decomposition. A novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface, the mean curvature vector and the surface phase field function. The resulting variational formulation uses H^1 spaces, and we employ triangulated surfaces and H^1 conforming quadratic surface finite elements for approximating solutions. Together with a semi-implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only. Numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells, discocytes and starfishes with lateral phase separation.

Allen-Cahn systems with volume constraints

We consider the evolution of a multi-phase system where the motion of the interfaces is driven by anisotropic curvature and some of the phases are subject to volume constraints. The dynamics of the phase boundaries is modeled by a system of Allen–Cahn type equations for phase field variables resulting from a gradient flow of an appropriate Ginzburg–Landau type energy. Several ideas are presented in order to guarantee the additional volume constraints. Numerical algorithms based on explicit finite difference methods are developed, and simulations are performed in order to study local minima of the system energy. Wulff shapes can be recovered, i.e. energy minimizing forms for anisotropic surface energies enclosing a given volume. Further applications range from foam structures or bubble clusters to tessellation problems in two and three space dimensions.

Second order phase field asymptotics for multi-component systems

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

Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

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A Surface Phase Field Model for Two-Phase Biological Membranes

. Since in numerical computations for phase field models the spatial grid size has to be smaller than

ANALYSIS OF A DIFFUSE INTERFACE APPROACH TO AN ADVECTION DIFFUSION EQUATION ON A MOVING SURFACE

\varepsilon

An abstract framework for parabolic PDEs on evolving spaces

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

Parameter identification problems in the modelling of cell motility

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On some linear parabolic PDEs on moving hypersurfaces

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 non-constant 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