Phase-field Approximations Research Articles

Homotopy trust-region method for phase-field approximations in perimeter-regularized binary optimal control

We consider optimal control problems that have binary-valued control input functions and a perimeter regularization. We develop and analyze a trust-region algorithm that solves a sequence of subproblems in which the regularization term and the binarity constraint are relaxed by a non-convex energy functional. We show how the parameter that controls the distinctiveness of the resulting phase field can be coupled to the trust-region radius updates and be driven to zero over the course of the iterations in order to obtain convergence to points that satisfy a first-order optimality condition of the limit problem under suitable regularity assumptions. Finally, we highlight and discuss the assumptions and restrictions of our approach and provide the first computational results for a motivating application in the field of control of acoustic waves in dissipative media.

Computing interfacial flows of viscous fluids

We present a novel technique for solving for the flow of a viscous multi-fluid system in which interfaces are present. The Navier-Stokes equations are used in their exact form, without the need for phase-field approximations. A Boussinesq-type approach is invoked, in which the component fluids of differing densities are represented by a single fluid in which the density changes smoothly but rapidly across each narrow interfacial zone. This formulation is illustrated in detail for classical planar Rayleigh-Taylor flow containing two horizontal layers of fluid, in which the upper fluid has greater density. The interface between the fluids is therefore unstable and overturns as time progresses. The results of this new approach are compared against the predictions of classical Boussinesq theory and a recent Extended Boussinesq model proposed by the authors. Finally, a comparison with the predictions of a smoothed-particle-hydrodynamics model gives strong support for this new technique.

Variational fracture: twenty years after

In this work, we propose to succinctly review, in non technical terms, the current mathematical state of brittle fracture first put forth by A. A. Griffith in his seminal work Griffith (Philos Trans R Soc Lond CCXXI-A:163–198, 1920), then re-interpreted under the label “variational fracture” in Francfort and Marigo (J. Mech Phys Solids 46(8):1319–1342, 1998). and subsequent works. We will only address the sharp theory and will limit ourselves to a theoretical exposition, leaving phase-field approximations and other possible implementations to other articles within this volume.

A Phase Field Approach to Trabecular Bone Remodeling

We introduce a continuous modeling approach which combines elastic responds of the trabecular bone structure, the concentration of signaling molecules within the bone and a mechanism how this concentration at the bone surface is used for local bone formation and resorption. In an abstract setting bone can be considered as a shape changing structure. For similar problems in materials science phase field approximations have been established as an efficient computational tool. We adapt such an approach for trabecular bone remodeling. It allows for a smooth representation of the trabecular bone structure and drastically reduces computational costs if compared with traditional micro finite element approaches. We demonstrate the advantage of the approach within a minimal model. We quantitatively compare the results with established micro finite element approaches on simple geometries and consider the bone morphology within a bone segment obtained from $\mu$CT data of a sheep vertebra with realistic parameters.

Phase field approximations of branched transportation problems

In branched transportation problems mass has to be transported from a given initial distribution to a given final distribution, where the cost of the transport is proportional to the transport distance, but subadditive in the transported mass. As a consequence, mass transport is cheaper the more mass is transported together, which leads to the emergence of hierarchically branching transport networks. We here consider transport costs that are piecewise affine in the transported mass with N affine segments, in which case the resulting network can be interpreted as a street network composed of N different types of streets. In two spatial dimensions we propose a phase field approximation of this street network using N phase fields and a function approximating the mass flux through the network. We prove the corresponding Gamma -convergence and show some numerical simulation results.

Numerical approximation of the Steiner problem in dimension $2$ and $3$

The aim of this work is to present some numerical computations of solutions of the Steiner Problem , based on the recent phase field approximations proposed in [12] and analyzed in [5, 4]. Our strategy consists in improving the regularity of the associated phase field solution by use of higher-order derivatives in the Cahn-Hilliard functional as in [6]. We justify the convergence of this slightly modified version of the functional, together with other technics that we employ to improve the numerical experiments. In particular, we are able to consider a large number of points in dimension 2. We finally present and justify an approximation method that is efficient in dimension 3, which is one of the major novelties of the paper.

Isogeometric analysis for phase-field models of geometric PDEs and high-order PDEs on stationary and evolving surfaces

In this paper, we present an isogeometric analysis (IGA) for phase-field models of three different yet closely related classes of partial differential equations (PDEs): geometric PDEs, high-order PDEs on stationary surfaces, and high-order PDEs on evolving surfaces; the latter can be a coupling of the former two classes. In the context of geometric PDEs, we consider mean curvature flow and Willmore flow problems and their corresponding phase-field approximations which yield second-order and fourth-order nonlinear parabolic PDEs. Through some numerical examples, we study the convergence behavior of isogeometric analysis for these equations using the method of manufactured solutions. Moreover, we study numerically the convergence of these phase-field approximations to the sharp interface solutions. As for the high-order PDEs on stationary surfaces, we consider a model problem which is the Cahn–Hilliard equation on a unit sphere, where the surface is modeled using a diffuse-interface approach. Finally, as a model problem for high-order PDEs on evolving surfaces, we consider a phase-field model of a deforming multicomponent vesicle which couples the vesicle shape changes with the phase separation process on the vesicle surface. The model consists of two fourth-order nonlinear PDEs which their direct finite element formulation in a Galerkin framework necessitates smooth basis functions with at least global C1 continuity; a condition that can be easily satisfied using spline bases in IGA. We solve the coupled equations both in two dimensions, where the vesicle is a curve, and in three dimensions, where the vesicle is a surface. The simulation results agree with the numerical and experimental results from the literature.

Phase‐field boundary conditions for the voxel finite cell method: Surface‐free stress analysis of CT‐based bone structures

The voxel finite cell method uses unfitted finite element meshes and voxel quadrature rules to seamlessly transfer computed tomography data into patient-specific bone discretizations. The method, however, still requires the explicit parametrization of boundary surfaces to impose traction and displacement boundary conditions, which constitutes a potential roadblock to automation. We explore a phase-field-based formulation for imposing traction and displacement constraints in a diffuse sense. Its essential component is a diffuse geometry model generated from metastable phase-field solutions of the Allen-Cahn problem that assumes the imaging data as initial condition. Phase-field approximations of the boundary and its gradient are then used to transfer all boundary terms in the variational formulation into volumetric terms. We show that in the context of the voxel finite cell method, diffuse boundary conditions achieve the same accuracy as boundary conditions defined over explicit sharp surfaces, if the inherent length scales, ie, the interface width of the phase field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human femur and a vertebral body.

A numerical convergence study regarding homogenization assumptions in phase field modeling

SummaryFrom a mathematical point of view, phase field theory can be understood as a smooth approximation of an underlying sharp interface problem. However, the smooth phase field approximation is not uniquely defined. Different phase field approximations are known to converge to the same sharp interface problem in the limiting case—if the thickness of the diffuse interface converges to zero. In this respect and focusing on numerics, a question that naturally arises is as follows: What are the convergence rates of the different phase field models? The paper deals precisely with this question for a certain family of phase field models. The focus is on an Allen–Cahn‐type phase field model coupled to continuum mechanics. This model is rewritten into a unified variational phase field framework that covers different homogenization assumptions in the diffuse interfaces: Voigt/Taylor, Reuss/Sachs and more sound homogenization approaches falling into the range of rank‐one convexification. It is shown by means of numerical experiments that the underlying phase field model—that is, the homogenization assumption in the diffuse interface—indeed influences the convergence rate. Copyright © 2017 John Wiley & Sons, Ltd.

A diffuse interface method for the Navier–Stokes/Darcy equations: Perfusion profile for a patient-specific human liver based on MRI scans

We present a diffuse interface method for coupling free and porous-medium-type flows modeled by the Navier–Stokes and Darcy equations. Its essential component is a diffuse geometry model generated from the phase-field solution of a separate initial boundary value problem that is based on the Allen–Cahn equation. Phase-field approximations of the interface and its gradient are then employed to transfer all interface terms in the coupled variational flow formulation into volumetric terms. This eliminates the need for an explicit interface parametrization between the two flow regimes. We illustrate accuracy and convergence for a series of benchmark examples, using standard low-order stabilized finite element discretizations. Our diffuse interface method is particularly attractive for coupled flow analysis on imaging data with complex implicit interfaces, where procedures for deriving explicit surface parametrizations constitute a significant bottleneck. We demonstrate the potential of our method to establish seamless imaging-through-analysis workflows by computing a perfusion profile for a full-scale 3D human liver based on MRI scans.

On the Gamma-limit of joint image segmentation and registration functionals based on phase fields

A classical task in image processing is the following: Given two images, identify the structures inside (for instance detect all image edges or all homogeneous regions; this is called segmentation) and find a deformation which maps the structures in one image onto the corresponding ones in the other image (called registration). In medical imaging, for instance, one might segment the organs in two patient images and then identify corresponding organs in both images for automated comparison purposes. The image segmentation is classically performed variationally using the Mumford–Shah model, and the obtained structures are then mapped onto each other by minimizing a registration energy in which the deformation is regularized via elasticity. Experimentally it often seems more robust to perform segmentation and registration simultaneously so that both can benefit from each other. The question to be examined here is how phase field approximations of the Mumford–Shah model behave if used for the joint segmentation and registration problem.We mathematically analyze corresponding generic phase field models and reveal interesting phenomena that rule out some of the models. These phenomena are characteristic of coupling phase fields with deformations and thus are interesting in their own right. In essence, region-based segmentation and registration problems can well be approximated using phase fields, while edge-based approaches typically suffer from different types of vanishing or newly appearing edges. We conjecture how the introduction of a different scaling could remedy those shortcomings.

Diffuse-Interface Approximations of Osmosis Free Boundary Problems

Free boundary problems based on mass conservation and surface tension with application in osmotic swelling are the topic of this contribution. Phase-field approximations of such models are introduced, in order to numerically investigate properties of the solutions. Formal justification of the proposed approximations is provided by matched asymptotic expansions supported by numerical tests reproducing the convergence for shrinking interface thickness.

Phase-field approximations of the Willmore functional and flow

We discuss in this paper phase-field approximations of the Willmore functional and the associated \({\mathrm L}^{2}\)-flow. After recollecting known results on the approximation of the Willmore energy and its \({\mathrm L}^{1}\) relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence is proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.

Phase-field models for the approximation of the willmore functional and flow

This paper is a short account on phase-field approximations of the Willmore functional and the associated L 2 -flows.

Stable phase field approximations of anisotropic solidification

We introduce unconditionally stable finite element approximations for a phase field model for solidification, which take highly anisotropic surface energy and kinetic effects into account. We hence approximate Stefan problems with anisotropic Gibbs--Thomson law with kinetic undercooling, and quasi-static variants thereof. The phase field model is given by {align*} \vartheta\,w_t + \lambda\,\varrho(\varphi)\,\varphi_t & = \nabla \,.\, (b(\varphi)\,\nabla\, w) \,, \cPsi\,\tfrac{a}\alpha\,\varrho(\varphi)\,w & = \epsilon\,\tfrac\rho\alpha\,\mu(\nabla\,\varphi)\,\varphi_t -\epsilon\,\nabla \,.\, A'(\nabla\, \varphi) + \epsilon^{-1}\,\Psi'(\varphi) {align*} subject to initial and boundary conditions for the phase variable $\varphi$ and the temperature approximation $w$. Here $\epsilon > 0$ is the interfacial parameter, $\Psi$ is a double well potential, $\cPsi = \int_{-1}^1 \sqrt{2\,\Psi(s)}\;{\rm d}s$, $\varrho$ is a shape function and $A(\nabla\,\varphi) = \tfrac12\,|\gamma(\nabla\,\varphi)|^2$, where $\gamma$ is the anisotropic density function. Moreover, $\vartheta \geq 0$, $\lambda > 0$, $a > 0$, $\alpha > 0$ and $\rho \geq 0$ are physical parameters from the Stefan problem, while $b$ and $\mu$ are coefficient functions which also relate to the sharp interface problem. On introducing the novel fully practical finite element approximations for the anisotropic phase field model, we prove their stability and demonstrate their applicability with some numerical results.

Gamma-convergence results for phase-field approximations of the 2D-Euler Elastica Functional

We establish some new results about the Γ -limit, with respect to the L 1 -topology, of two different (but related) phase-field approximations {Ee}e, { � Ee}e of the so-called Euler's Elastica Bending Energy for curves in the plane. In particular we characterize the Γ -limit as e → 0o fEe ,a nd show that in general the Γ -limits of Ee and � Ee do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of the Γ -limit of � Ee strictly contains the domain of the Γ -limit of Ee.

Mini-Workshop: Mathematics of Biological Membranes

From the early 1970's on, the understanding of the variety of equilibrium shapes of cell membranes has been a very active field in biophysics. Relaxation dynamics, phase separation on membranes, the coupling to inner/outer flows, and the contribution of domains on membranes to the biological function of cells are still a major focus of research. The mathematical analysis of the shape and evolution of cell membranes is challenging. Over the last few years we have seen an increasing number of contributions from modelling, analysis and numerics. This workshop reports on recent progress made by the different communities – biophysics, numerics, analysis – and aims at a more complete picture of the relevant questions and adequate techniques. Variational approaches for the description of equilibrium shapes of membranes are based on phenomenological bending energies, such as proposed by Canham [4] and Helfrich [7]. The Willmore functional is the most important reduction of the Helfrich energy, and understanding its properties is key in the analysis. Recent progress on the Willmore functional and its L^2 -gradient flow by Kuwert–Schätzle [10, 9, 11] and Rivière [12] [Rivière], such as a new formulation of the Euler–Lagrange equation, provide promising techniques to investigate (constrained) minimization problems for different bending energies. (Here and below, names between square brackets refer to workshop presentations). In many minimization or evolution problems approximations by phase-field models are convenient, since they can deal with topological changes of membranes and phases on the membrane respectively [2, 16, 8]. The connection between the Willmore energy and its phase field approximations are meanwhile better understood [13]. Emphasis in talks has been on thermodynamically consistent phase-field models [Misbah] as well as on their capability to recover singularities in the membrane shapes [Stinner,Rätz]. To find equilibrium shapes a L^2 gradient flow of an appropriate bending energy is the most practical approach. A discussion of more physical models for the relaxation towards an equilibrium has started. One choice is to take the flow of the lipid molecules on the membranes into account [Arroyo]. Other effects have to be considered as well, such as the orientation of molecules in ordered phases [Dolzmann]. Much progress in the numerical simulation of the Willmore flow, of partial differential equations on surfaces, and the coupling of reaction-diffusion processes or inner flows on evovling surfaces with their motion laws has been made [6, 5, 1]. For example, new weak formulations of the gradient flow for the Willmore functional avoid the evaluation of higher derivatives [Dziuk], and the inclusion of tangential motion in the numerical scheme leads to a convenient redistribution of the mesh [Garcke]. It has become feasible to compare equilibrium shapes and phase transition diagrams that have been determined either experimentally or by minimization within some restricted classes of shapes (for instance rotational symmetry) [14, 3] [Svetina]. Cell membranes consist of many different lipid molecules (and other components, such as rafts) and phase separation processes and evolution of phase boundaries are important for the function of a biological cell [Ben Amar]. Variational models account for different physical properties of the distinct phases (for instance different bending or spontaneous curvature constants) and a line tension energy at the boundary between the phases. First simulations of equilibrium configurations and relaxations have been obtained and are becoming more and more efficient [16, 15] [Stinner, Rätz]. The participants of this workshop came from 8 different countries and their expertise was evenly spread over the areas biophysics, analysis, numerics and modeling. On the first day there were three introductory talks on modeling, analysis and numerics respecvtively. On the following days there were more specialized talks, two in the morning and one in the afternoon. The schedule was kept flexible and allowed to have room for extensive and lively discussions which arose after virtually every talk. This was very much appreciated by all participants and was repeatedly reported as a significant advantage of the concept of a miniworkshop in comparison to the regular Oberwolfach meetings.

First Principle Materials Design of Half-Metallic Ferromagnetic Half-Heusler Alloys

We present a first principle study of the half-metallicity and ferromagnetism in half-Heusler alloys XYSi (X= Ni, Pd and Pt; Y= Cr and Mn). The equilibrium lattice constants are predicted by means of the ultrasoft pseudo-potential method. The electronic structure and magnetic exchange interaction of various alloys are investigated within KKR-LSDA. The Curie temperature is predicted by using the Monte Carlo simulation, the random phase and mean field approximations, as well. The role of elements X, Y components and influence of the lattice parameters in stabilizing half-metallicity and ferromagnetism are discussed. Except PdMnSi, alloys are half-metallic at equilibrium lattice constant. The Curie temperatures of 634 K, 785 K, and 1076 K for NiCrSi, PtMnSi and NiMnSi, respectively, are predicted by Monte Carlo simulation. The structural stability of C1 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> is also considered.

Convergence of phase-field approximations to the Gibbs–Thomson law

We prove the convergence of phase-field approximations of the Gibbs–Thomson law. This establishes a relation between the first variation of the Van der Waals–Cahn–Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs–Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn–Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta–Kawasaki as a model for micro-phase separation in block-copolymers.

Mini-Workshop: Interface Problems in Computational Fluid Dynamics

This workshop was concurrent with the workshop on Analytical and Numerical Methods in Image and Surface Processing (see: Report No. 10/2005), and since the participants interests overlapped several combined talks were organized. In particular, we thank P. Schroder, U. Reif, G. Dziuk, C. Elliot, and F. Cirak, for their contributions. Currently there are three major techniques for tracking or characterizing interfaces present in a CFD computation: (i) The level set method, (ii) Phase field approximations, and (iii) Explicit parameterization using Lagrangian variables. Each of these techniques, and various hybrids, were presented by the participants to solve a variety of challenging CFD problems involving interfaces. Several trends emerged during the workshop.