Frank Wendler

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.

Simulations of Complex Microstructure Formations

In this article, we review the progress on phase-field modelling, sharp interface asymptotics, numerical simulations and applications to microstructure evolution and pattern formation in materials science. Model formulations and computations of pure substances and of binary alloys are discussed. Furthermore, a thermodynamically consistent class of non-isothermal phase-field models for crystal growth and solidification in complex alloy systems is presented. Explicit expressions for the different energy density contributions are proposed. Multicomponent diffusion in the bulk phases including interdiffusion coefficients as well as diffusion in the interfacial regions are discussed. Anisotropy of both, the surface energies and the kinetic coefficients is incorporated in the model formulation. A 3D parallel simulator based on a finite difference discretization is introduced illustrating the capability of the model to simultaneously describe the diffusion processes of multiple components, the phase transitions between multiple phases and the development of the temperature field. The numerical solving method contains parallelization and adaptive strategies for optimization of memory usage and computing time. Applying the computational methods, we show a variety of simulated microstructure formations in complex multicomponent alloy systems occuring on different time and length scales. In particular, we present 2D and 3D simulation results of dendritic and eutectic solidification in pure substances and binary and ternary alloys. Another field of application is the modelling of competing polycrystalline grain structure formation and grain growth.