Dirk Blömker: Stochastic interface motion and slow manifolds

In the four lectures we present the theory of slow motion of interfaces and thus of metastable behavior, in stochastic models of phase separation like Allen-Cahn or Cahn-Hilliard. We mainly limit ourselves to the one-dimensional case and study the motion of kinks described by stochastic ordinary equations (SDEs) for their position. In higher dimension we need to study the theory of free boundary problems for the interfaces, which up to now yields to many unsolved problems. In the first part we give a brief introduction to the theory of stochastic partial differential equations (SPDEs) with additive noise. This is based on the mild formulation using the semigroup approach, path-wise fixed-point arguments, and a-priori estimates. In a second part we discuss the theory of slow manifolds for the motion of interfaces. We discuss briefly how to construct such a manifold for 1D Allen-Cahn and 1D Cahn-Hilliard. A crucial point for the dynamics is the spectral theory for the operator linearized around an element of the slow manifold. In the final part we discuss the dynamics of interfaces, where we need two key ingredients. First we present the derivation of the effective equation for the motion of the interfaces. The second result is the stability of the slow manifold. Both results are discussed in the deterministic as well as in the stochastic case.

Patrick Dondl: Sharp Interface Limits of Phase Field Models

1) Spaces of Bounded Variation and other preliminaries 2) Gamma-Convergence of the Modica-Mortola Functional 3) Convergence of solutions to the Allen-Cahn equation to Mean Curvature Flow (I) 4a) Convergence of solutions to the Allen-Cahn equation to Mean Curvature Flow (II) 4b) An outlook to higher and fractional order models

After discussing some preliminaries, like properties of functions of bounded variation and sets of finite perimeter as well as the basics of the theory of Gamma-convergence, we will derive the classical result by Modica and Mortola. This result shows that, after rescaling, the sharp-interface limit of the the phase field energy by Cahn and Hilliard is given by the perimeter functional. We will then turn our attention to the gradient flow and consider convergence of solutions of Allen-Cahn equation to solutions of mean curvature flow. Finally, we will study both higher order versions of the Modica-Mortola functional (yielding a phase field approximation of Willmore’s energy) as well as lower order versions (yielding non-local perimeter functionals in the limit).

Britta Nestler: Fundamentals of Phase-Field Modelling for HPC Systems and Coupling with Continuum Mechanics

The fundamentals of phase-field modelling, its efficient implementation on HPC systems and the extension of the method to enable coupling with continuum mechanical methods will be presented in two lecture units. In an introductory lecture, the base equations and energy contributions of the phase-field model will be formulated, and the relation to thermodynamic energies will be shown. The equations of motion will be derived and numerical solution methods will be described. For an efficient use of high-performance computers, optimisations of the methods will furthermore be discussed. The first lecture will finish with a selection of applications from fields such as solidification, the formation of grain structures, wetting and cellular structures. The second lecture will focus on the coupling of phase-field modelling with mechanical balance equations. The fundamental balance equations and the fulfilment of the mechanical jump conditions with phase-field modelling will be discussed, and the mechanical driving forces for a phase transformation will be derived. The lecture will finish with examples of stress propagation in microstructures, stress induced phase transformations and crack formations. The course will be accompanied by a computer-assisted internship, which will be taught in blocks. To solve the phase-field equations, a C programme will be provided during the internship. The participants will receive an introduction to the use of the programme and into its modification and extension. The individual contributions of the phase-field model will be tested in an exercise. After initial experiences with the solver, extensions of the basic model such as the consideration of anisotropic patterns of growth are to be implemented by the participants.

Robert Nürnberg: Numerical Approximation of Phasefield Models

In this lecture series we will consider the numerical approximation of phase field models with the help of the finite element method. After a short review of the theory of finite element approximations for linear elliptic and parabolic equations, we introduce and analyse approximations for the Allen--Cahn and the Cahn--Hilliard equations with a smooth potential. Here we will focus on the gradient flow structure of the two equations and derive appropriate stability bounds for the finite element approximations. With a view towards extending the derived results to the case of an obstacle potential, we give a short introduction to variational inequalities and their applications. This will enable us to derive finite element approximations for Allen--Cahn and Cahn--Hilliard with the standard double-well obstacle potential. In the final part of the lecture series we will consider the effect of an anisotropic surface energy. Once again, we will focus on the derivation of (unconditionally) stable finite element approximations. To this end, we will restrict our attention to a suitable class of admissible anisotropic surface energy densities.