We consider the Cahn-Hilliard equation-a fourth-order, nonlinear parabolic diffusion equation describing phase separation of a binary alloy which is quenched below a critical temperature. The occurrence of two phases is due to a nonconvex double well free energy. The evolution initially leads to a very fine microstructure of regions with different phases which tend to become coarser at later times.The resulting phases might have different elastic properties caused by a different lattice spacing. This effect is not reflected by the standard Cahn-Hilliard model. Here, we discuss an approach which contains anisotropic elastic stresses by coupling the expanded diffusion equation with a corresponding quasistationary linear elasticity problem for the displacements on the microstructure.Convergence and a discrete energy decay property are stated for a finite element discretization. An appropriate timestep scheme based on the strongly A-stable Θ-scheme and a spatial grid adaptation by refining and coarsening improve the algorithms efficiency significantly. Various numerical simulations outline different qualitative effects of the generalized model. Finally, a surprising stabilizing effect of the anisotropic elasticity is observed in the limit case of a vanishing fourth-order term, originally representing interfacial energy.

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Abstract. Spinodal decomposition, i.e., the separation of a homogeneous mixture into different phases, can be modeled by the Cahn-Hilliard equation -a fourth order semilinear parabolic equation. If elastic stresses due to a lattice misfit become important, the Cahn-Hilliard equation has to be coupled to an elasticity system to take this into account. Here, we present a discretization based on finite elements and an implicit Euler scheme. We first show solvability and uniqueness of solutions. Based on an energy decay property we then prove convergence of the scheme. Finally we present numerical experiments showing the impact of elasticity on the morphology of the microstructure.

A new method for the simplification of flow fields is presented. It is based on continuous clustering. A well-known physical clustering model, the Cahn Hilliard model which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional. Here, time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns-the actual clustering-during which the underlying simulation data specifies preferable pattern boundaries. We introduce specific physical quantities in the simulation to control the shape, orientation and distribution of the clusters, as a function of the underlying flow field. In addition the model is expanded involving elastic effects. Thereby in early stages of the evolution shear layer type representation of the flow field can be generated, whereas for later stages the distribution of clusters can be influenced. Furthermore, we incorporate upwind ideas to give the clusters an oriented drop-shaped appearance. Here we discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries. However, the method also carries provisions for other fields as well. The clusters can be displayed directly as a flow texture. Alternatively, the clusters can be visualized by iconic representations, which are positioned by using a skeletonization algorithm.