Image Segmentation Using Parametric Contours With Free Endpoints.

In this paper, we introduce a novel approach for active contours with free endpoints. A scheme for image segmentation is presented based on a discrete version of the Mumford-Shah functional where the contours can be both closed and open curves. Additional to a flow of the curves in normal direction, evolution laws for the tangential flow of the endpoints are derived. Using a parametric approach to describe the evolving contours together with an edge-preserving denoising, we obtain a fast method for image segmentation and restoration. The analytical and numerical schemes are presented followed by numerical experiments with artificial test images and with a real medical image.

Erratum: Numerical computations of the dynamics of fluidic membranes and vesicles [Phys. Rev. E 92, 052704 (2015)].

This corrects the article DOI: 10.1103/PhysRevE.92.052704.

A stable numerical method for the dynamics of fluidic membranes.

We develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in Navier-Stokes flow. Local inextensibility of the membrane is ensured by solving a tangential Navier-Stokes equation, taking surface viscosity effects of Boussinesq-Scriven type into account. In our approach the bulk and surface degrees of freedom are discretized independently, which leads to an unfitted finite element approximation of the underlying free boundary problem. Bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by Dziuk (Numer Math 111:55-80, 2008). The obtained numerical scheme can be shown to be stable and to have good mesh properties. Finally, the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions. The numerical results demonstrate the robustness of the method, and it is observed that the conservation properties are fulfilled to a high precision.

Numerical computations of the dynamics of fluidic membranes and vesicles.

Vesicles and many biological membranes are made of two monolayers of lipid molecules and form closed lipid bilayers. The dynamical behavior of vesicles is very complex and a variety of forms and shapes appear. Lipid bilayers can be considered as a surface fluid and hence the governing equations for the evolution include the surface (Navier-)Stokes equations, which in particular take the membrane viscosity into account. The evolution is driven by forces stemming from the curvature elasticity of the membrane. In addition, the surface fluid equations are coupled to bulk (Navier-)Stokes equations. We introduce a parametric finite-element method to solve this complex free boundary problem and present the first three-dimensional numerical computations based on the full (Navier-)Stokes system for several different scenarios. For example, the effects of the membrane viscosity, spontaneous curvature, and area difference elasticity (ADE) are studied. In particular, it turns out, that even in the case of no viscosity contrast between the bulk fluids, the tank treading to tumbling transition can be obtained by increasing the membrane viscosity. Besides the classical tank treading and tumbling motions, another mode (called the transition mode in this paper, but originally called the vacillating-breathing mode and subsequently also called trembling, transition, and swinging mode) separating these classical modes appears and is studied by us numerically. We also study how features of equilibrium shapes in the ADE and spontaneous curvature models, like budding behavior or starfish forms, behave in a shear flow.

A phase-field approach for wetting phenomena of multiphase droplets on solid surfaces.

We study the equilibrium wetting behavior of immiscible multiphase systems on a flat, solid substrate. We present numerical computations which are based on a vector-valued multiphase-field model of Allen-Cahn type, with a new boundary condition, based on appropriately designed surface energy contributions in order to ensure the right contact angles at multiphase junctions. Experimental investigations are carried out to validate the method and to support the numerical results.

Numerical computations of faceted pattern formation in snow crystal growth.

Faceted growth of snow crystals leads to a rich diversity of forms with remarkable sixfold symmetry. Snow crystal structures result from diffusion-limited crystal growth in the presence of anisotropic surface energy and anisotropic attachment kinetics. It is by now well understood that the morphological stability of ice crystals strongly depends on supersaturation, crystal size, and temperature. Until very recently it was very difficult to perform numerical simulations of this highly anisotropic crystal growth. In particular, obtaining facet growth in combination with dendritic branching is a challenging task. We present numerical simulations of snow crystal growth in two and three spacial dimensions using a computational method recently introduced by the present authors. We present both qualitative and quantitative computations. In particular, a linear relationship between tip velocity and supersaturation is observed. In our computations, surface energy effects, although small, have a pronounced effect on crystal growth. We compute solid plates, solid prisms, hollow columns, needles, dendrites, capped columns, and scrolls on plates. Although all these forms appear in nature, it is a significant challenge to reproduce them with the help of numerical simulations for a continuum model.

Phase-field model for multiphase systems with preserved volume fractions.

We report on an interesting formulation of a phase-field model which incorporates a description of individual phases and particles with preserved volume evolving in a system of multiple phases such that the interfacial energy decreases. In our model, an antiforcing free energy density is defined to fulfill constraints on selected volume fractions by counterbalancing phase changes. Phases are defined as regions with energy bearing boundaries that may differ in their physical states, i.e., the regions may be distinguished in structure (crystal transformations), in composition (alloys, mixtures of fluids), or in the orientation of the crystal lattice (grains). The method allows one to simulate the formation of equilibrium crystal shapes and of the migration of inert particles and phases in microstructures. We show two- and three-dimensional simulations of bubble ensembles and foam textures and demonstrate the excellent agreement of crystal morphology configurations with analytical results.

Multicomponent alloy solidification: phase-field modeling and simulations.

A general formulation of phase-field models for nonisothermal solidification in multicomponent and multiphase alloy systems is derived from an entropy functional in a thermodynamically consistent way. General expressions for the free energy densities, for multicomponent diffusion coefficients, and for both weak and faceted types of surface energy and kinetic anisotropy are possible. A three-dimensional simulator is developed to show the capability of the model to describe phase transitions, complex microstructure formation, and grain growth in polycrystalline textures.