Vanessa Styles

Finite element approximation of a Cahn−Hilliard−Navier−Stokes system

We consider a semi-discrete and a practical fully discrete finite element approximation of a Cahn-Hilliard-Navier-Stokes system. This system arises in the modelling of multiphase fluid systems. We show order h error estimate between the solution of the system and the solution of the semi-discrete approximation. We also show the convergence of the fully discrete approximation. Finally, we present an efficient implementation of the fully discrete scheme together with some numerical simulations.

Phase-field Approaches to Structural Topology Optimization

The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L2- and H-1-gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches.

Numerical diffusion-induced grain boundary motion

In this paper we consider the numerical approximation of phase field and sharp interface models for diffusion-induced grain boundary motion. The phase field model consists of a double-obstacle Allen–Cahn equation with a forcing obtained from the solution of a degenerate diffusion equation. On the other hand the sharp interface model consists of forced mean curvature flow coupled to a diffusion equation holding on the interface itself. Formal asymptotics yield the sharp interface model as the limit of the phase field equations as the width of the associated diffuse interface tends to zero. A finite-element approximation of the phase field model is presented and is shown to be convergent to a weak solution. Numerical simulations of both models are described and compared. It is shown that the two models are consistent.

A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport

We propose and investigate a model for lipid raft formation and dynamics in biological membranes. The model describes the lipid composition of the membrane and an interaction with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple a bulk-diffusion to an evolution equation on the membrane. The latter describes a relaxation dynamics for an energy taking lipid-phase separation and lipid-cholesterol interaction energy into account. It takes the form of an (extended) Cahn--Hilliard equation. Different laws for the exchange term represent equilibrium and non-equilibrium models. We present a thermodynamic justification, analyze the respective qualitative behavior and derive asymptotic reductions of the model. In particular we present a formal asymptotic expansion near the sharp interface limit, where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively. Finally we perform numerical simulations and investigate the long-time behavior of the model and its parameter dependence. Both the mathematical analysis and the numerical simulations show the emergence of raft-like structures in the non-equilibrium case whereas in the equilibrium case only macrodomains survive in the long-time evolution.Using basic thermodynamic principles we derive a Cahn–Hilliard–Darcy model for tumour growth including nutrient diffusion, chemotaxis, active transport, adhesion, apoptosis and proliferation. In contrast to earlier works, the model is based on a volume-averaged velocity and in particular includes active transport mechanisms which ensure thermodynamic consistency. We perform a formally matched asymptotic expansion and develop several sharp interface models. Some of them are classical and some are new which for example include a jump in the nutrient density at the interface. A linear stability analysis for a growing nucleus is performed and in particular the role of the new active transport term is analysed. Numerical computations are performed to study the influence of the active transport term for specific growth scenarios.

Analysis of a mean field model of superconducting vortices

We study a mean-held model of superconducting vortices in one and two dimensions. The existence of a weak solution and a steady-state solution of the model are proved. A special case of the steady-state problem is shown to be of the form of a free boundary problem. The solutions of this free boundary problem are investigated. It is also shown that the weak solution of the one-dimensional model is unique and satisfies an entropy inequality.

Bi-directional diffusion induced grain boundary motion with triple junctions

We propose a multi-order parameter phase field system and a sharp interface model to describe bidirectional diffusion induced grain boundary motion in the presence of triple junctions. Numerical approximations of the models are presented together with some computational results.

Allen-Cahn and Cahn-Hilliard Variational Inequalities Solved with Optimization Techniques

Parabolic variational inequalities of Allen-Cahn and Cahn-Hilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active set approach. We show several numerical computations also involving systems of parabolic variational inequalities.

Stress and diffusion induced interface motion: Modelling and numerical simulations

We propose a phase field model for stress and diffusion-induced interface motion. This model, in particular, can be used to describe diffusion-induced grain boundary motion and generalizes a model of Cahn, Fife and Penrose as it more accurately incorporates stress effects. In this paper we will demonstrate that the model can also be used to describe other stress-driven interface motion. As an example, interface motion resulting from interactions of interfaces with dislocations is studied.

Multi-material Phase Field Approach to Structural Topology Optimization

Multi-material structural topology and shape optimization problems are formulated within a phase field approach. First-order conditions are stated and the relation of the necessary conditions to classical shape derivatives are discussed. An efficient numerical method based on an H 1–gradient projection method is introduced and finally several numerical results demonstrate the applicability of the approach.

Computations of bidirectional grain boundary dynamics in thin metallic films

We propose sharp interface, phase field and level set models for bidirectional diffusion induced grain boundary motion in thin films. Numerical approximations of these models are presented together with computational results comparing the approximate solutions.