Ulrich Weikard

A phase field model for continuous clustering on vector fields

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 (1958) 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, where time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns, 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. In the early stages of the evolution, a shear-layer-type representation of the flow field can thereby 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. 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.

The Cahn-Hilliard equation with elasticity—finite element approximation and qualitative studies

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 –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.

A continuous clustering method for vector fields

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 (J. Cahn and J. Hilliard, 1958), 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, and meanwhile the underlying simulation data specifies preferable pattern boundaries. The authors discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries, but the method also carries provisions in other fields as well. The clusters are visualized via iconic representations. A skeletonization algorithm is used to find suitable positions for the icons.

Transient coarsening behaviour in the Cahn–Hilliard model

Abstract We study two-dimensional coarsening by simulations for the Cahn–Hilliard model. A scale invariance of the sharp interface limit of this model suggests that the characteristic length scale grows proportional to t1/3, respectively the energy density decreases as t−1/3. We compare the coarsening dynamics for different choices of data for different volume fractions. We observe that, depending on the specific data, the coarsening process can, over a large time window, be much slower than expected by dimensional analysis.

Error indicators for multilevel visualization and computing on nested grids

Abstract Nowadays computing and post processing of simulation data is often based on efficient hierarchical methods. While multigrid methods are already established standards for fast simulation codes, multiresolution visualization methods have only recently become an important ingredient of real–time interactive post processing. Both methodologies use local error indicators which serve as criteria where to refine the data representation on the physical domain. In this article we give an overview on different types of error measurement on nested grids and compare them for selected applications in 2D as well as in 3D. Furthermore, it is pointed out that a certain saturation of the considered error indicator plays an important role in multilevel visualization and computing on implicitly defined adaptive grids.

Numerical approximation of the Cahn-Larché equation

SummarySpinodal 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.

On Level Set Formulations for Anisotropic Mean Curvature Flow and Surface Diffusion

Anisotropic mean curvature motion and in particular anisotropic surface diffusion play a crucial role in the evolution of material interfaces. This evolution interacts with conservations laws in the adjacent phases on both sides of the interface and are frequently expected to undergo topological chances. Thus, a level set formulation is an appropriate way to describe the propagation. Here we recall a general approach for the integration of geometric gradient flows over level set ensembles and apply it to derive a variational formulation for the level set representation of anisotropic mean curvature motion and anisotropic surface flow. The variational formulation leads to a semi-implicit discretization and enables the use of linear finite elements.

Spinodal Decomposition in the Presence of Elastic Interactions

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.

A Comparison of Error Indicators for Multilevel Visualization on Nested Grids

Multiresolution visualization methods have recently become an indispensable ingredient of real time interactive post processing. Here local error indicators serve as criteria where to refine the data representation on the physical domain. In this article we give an overview on different types of error measurement on nested grids and compare them for selected applications in 2D as well as in 3D. Furthermore, it is pointed out that a certain saturation of the considered error indicator plays an important role in multilevel visualization and can be reused for the evaluation of data bounds in hierarchical searching or for a multilevel backface culling of isosurfaces.

Multiple scales in phase separating systems with elastic misfit

harald.garcke@mathematik.uni-regensburg.de 2 Institut fur Numerische Simulation, Rheinische Friedrich-Wilhelms-Universitat Bonn, Nusallee 15, 53115 Bonn. martin.lenz@ins.uni-bonn.de, martin.rumpf@ins.uni-bonn.de, 3 Institut fur Mathematik, Humboldt-Universitat zu Berlin, Unter den Linden 6, 10099 Berlin. niethamm@mathematik.hu-berlin.de 4 Fachbereich Mathematik, Gerhard-Mercator-Universitat Duisburg, Lotharstr. 63/65, 47048 Duisburg. weikard@math.uni-duisburg.de