Adaptive pseudospectral solution of a diffuse interface model

A diffuse interface type model, using an energy-based variational formulation with a free energy that is a function of the density and its gradients is presented. All of the boundary terms are retained and related to external surface forces, which can be of particular interest when considering the fluid–fluid–solid region. The numerical solution of these types of problems can be troublesome if a thin transition layer is desired. Here, Chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied. A mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived. In order to achieve high precision for problems exhibiting thin transition layers, a modified version of the arc-length monitor function is proposed which yields a sufficiently smooth coordinate transformation. At every step of the continuation process, a fixed number of iterations is implemented, so that the equidistribution equations are not solved completely at each step, which saves a considerable amount of computational effort. Numerical results for the static phase field model exhibiting thin transition layers are presented.     

On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grün model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).     

Kinks in two-phase lipid bilayer membranes

Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least C ¹ across phase boundaries. We consider the spontaneous curvature model for closed rotationally symmetric two-phase membranes without excluding tangent discontinuities at interfaces a priorily. We introduce a family of energies for smooth surfaces and phase fields for the lipid phases and derive a sharp interface limit that coincides with the Γ-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to tangent discontinuities. The theoretical result is illustrated by numerical examples.     

A phase field model for martensitic transformations with a temperature-dependent separation potential

Metallic materials often exhibit a complex microstructure with varying material properties in the different phases. Of major importance in mechanical engineering is the evolution of the austenitic and martensitic phases in steel. The martensitic transformation can be induced by heat treatment or by plastic surface deformation at low temperatures. A two dimensional elastic phase field model for martensitic transformations considering several martensitic orientation variants to simulate the phase change at the surface is introduced in [1]. However here, only one martensitic orientation variant is considered for the sake of simplicity. The separation potential is temperature dependent. Therefore, the coefficients of the Landau polynomial are identified by results of molecular dynamics (MD) simulations for pure iron [1]. The resulting separation potential is applied to analyse the mean interface velocity with respect to temperature and load. The interface velocity is computed by use of the dissipative part to the configurational forces balance as suggested in [3]. The model is implemented in the finite element code FEAP using standard 4-node elements with bi-linear shape functions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global sharp interface limit of the Hele–Shaw–Cahn–Hilliard system

In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two‐phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by ? , and the mobility coefficient (the diffusional Peclet number) is ? ^α . We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as ? →0 in the case of 0≤α  < 1. Copyright © 2016 John Wiley & Sons, Ltd.     

Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system

A well-known diffuse interface model consists of the Navier–Stokes equations nonlinearly coupled with a convective Cahn–Hilliard type equation. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a variant of this model where the standard Cahn–Hilliard equation is replaced by its nonlocal version. More precisely, the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter φ, while the potential F may have any polynomial growth. Therefore the coupling with the Navier–Stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of φ. We establish the global existence of a weak solution. In the two-dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate, provided that F fulfills a suitable coercivity condition.     

Diffuse Interface Methods for Multiple Phase Materials: An Energetic Variational Approach

In this paper, we introduce a diffuse interface model for describing the dynamics of mixtures involving multiple (two or more) phases. The coupled hydrodynamical system is derived through an energetic variational approach. The total energy of the system includes the kinetic energy and the mixing (interfacial) energies. The least action principle (or the principle of virtual work) is applied to derive the conservative part of the dynamics, with a focus on the reversible part of the stress tensor arising from the mixing energies. The dissipative part of the dynamics is then introduced through a dissipation function in the energy law, in line with Onsager's principle of maximum dissipation. The final system, formed by a set of coupled time-dependent partial differential equations, reflects a balance among various conservative and dissipative forces and governs the evolution of velocity and phase fields. To demonstrate the applicability of the proposed model, a few two-dimensional simulations have been carried out, including (1) the force balance at the three-phase contact line in equilibrium, (2) a rising bubble penetrating a fluid-fluid interface, and (3) a solid particle falling in a binary fluid. The effects of slip at solid surface have been examined in connection with contact line motion and a pinch-off phenomenon.     

A Regularized Sharp Interface Model for Phase Transformation Accounting for Prescribed Sharp Interface Kinetics

The macroscopic mechanical behavior of multi-phasic materials depends on the formation and evolution of their microstructure by means of phase transformation. In case of martensitic transformations, the resulting phase boundaries are sharp interfaces. We carry out a geometrically motivated discussion of the regularization of such sharp interfaces by use of an order parameter/phase-field and exploit the results for a regularized sharp interface model for two-phase elastic materials with evolving phase boundaries. To account for the dissipative effects during phase transition, we model the material as a generalized standard medium with energy storage and a dissipation function that determines the evolution of the regularized interface. Making use of the level-set equation, we are thereby able to directly translate prescribed sharp interface kinetic relations to the constitutive model in the regularized setting. We develop a suitable incremental variational three-field framework for the dissipative phase transformation problem. Finally, the modeling capability and the associated numerical solution techniques are demonstrated by means of a representative numerical example. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exponential Finite Elements for a Phase Field Fracture Model

Sharp interface material models can be related to phase field models by introducing an order parameter, whose value is assigned to the different phases of a material. The elastic material law is coupled to the evolution equation of the order parameter and cracking is addressed as a phase transition problem instead of a moving boundary value problem. A regularization parameter ϵ controls the width of the diffuse cracks represented by the order parameter and the underlying sharp interface model can be recovered from the phase field model by the limit ϵ → 0. However, in numerical simulations using standard finite elements with linear shape functions, the minimum value of ϵ is restricted by the grid size and therefore the discretization of the crack field requires extensive mesh refinement for small values of ϵ. In this work, we construct special 2d shape functions which take into account the exponential character of the crack field and its dependence on the parameter ϵ. Especially in simulations with small values of ϵ and a rather coarse mesh, the elements with exponential shape functions perform significantly better than standard linear elements. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of Multiphase Flows with Strong Shocks and Density Variations

The system of extended Euler type hyperbolic equations is considered to describe a two-phase compressible flow. A numerical scheme for computing multi-component flows is then examined. The numerical approach is based on the mathematical model that considers interfaces between fluids as numerically diffused zones. The hyperbolic problem is tackled using a high resolution HLLC scheme on a fixed Eulerian mesh. The global set of conservative equations (mass, momentum and energy) for each phase is closed with a general two parameters equation of state for each constituent. The performance of various variants of a diffuse interface method is carefully verified against a comprehensive suite of numerical benchmark test cases in one and two space dimensions. The studied benchmark cases are divided into two categories: idealized tests for which exact solutions can be generated and tests for which the equivalent numerical results could be obtained using different approaches. The ability to simulate the Richtmyer-Meshkov instabilities, which are generated when a shock wave impacts an interface between two different fluids, is considered as a major challenge for the present numerical techniques. The study presents the effect of density ratio of constituent fluids on the resolution of an interface and the ability to simulate Richtmyer-Meshkov instabilities by various variants of diffuse interface methods. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous,incompressible fluids

We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.     

Single phase limit for melting nanoparticles

The melting of a spherical or cylindrical nanoparticle is modelled as a Stefan problem by including the effects of surface tension through the Gibbs–Thomson condition. A one-phase moving boundary problem is derived from the general two-phase formulation in the singular limit of slow conduction in the solid phase, and the resulting equations are studied analytically in the limit of small time and large Stefan number. Further analytical approximations for the temperature distribution and the position of the solid–melt interface are found by applying an integral formulation together with an iterative scheme. All these analytical results are compared with numerical solutions obtained using a front-fixing method, and are shown to provide good approximations in various regimes. The inclusion of surface tension, which acts to decrease the melting temperature as the particle melts, is shown to accelerate the melting process. Unlike the classical one-phase Stefan problem without surface tension, the solid–melt interface exhibits blow-up at some critical radius of the particle (which for metals is of the order of a few nanometres), a phenomenon that has been observed experimentally. An interesting feature of the model is the prediction that surface tension drives superheating in the solid particle before blow-up occurs.     

Strong solutions to 1D compressible Navier‐Stokes/Allen‐Cahn system with free boundary

This paper is concerned with a diffuse interface model for two‐phase flow of compressible fluids with a type of free boundary. We establish the existence and uniqueness of global strong solutions of a coupled Navier‐Stokes/Allen‐Cahn system in 1D.     

On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain by allowing for a degenerate mobility. The model has been developed by Abels, Garcke and Grün for fluids with different densities and leads to a solenoidal velocity field. It is given by a non-homogeneous Navier–Stokes system with a modified convective term coupled to a Cahn–Hilliard system, such that an energy estimate is fulfilled which follows from the fact that the model is thermodynamically consistent.     

A Regularized Sharp Interface Model for Martensitic Phase Transformations at Large Strains

The macroscopic mechanical behavior of many functional materials crucially depends on the formation and evolution of their microstructure. When considering martensitic shape memory alloys, this microstructure typically consists of laminates with coherent twin boundaries. We suggest a variational-based phase field model for the dissipative evolution of microstructure with coherence-dependent interface energy and construct a suitable gradient-extended incremental variational framework for the proposed dissipative material. We use our model to predict laminate microstructure in martensitic CuAlNi. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Calculation of deformations in nanocomposites using the block multipole method with the analytical-numerical account of the scale effects

Local scale effects for linear continuous media are investigated as applied to the composites reinforced by nanoparticles. A mathematical model of the interphase layer is proposed that describes the specific nature of deformations in the neighborhood of the interface between different phases in an inhomogeneous material. The characteristic length of the interphase layer is determined formally in terms of the parameters of the mathematical model. The local stress state in the neighborhood of the phase boundaries in the interphase layer is examined. This stress can cause a significant change of the integral macromechanical characteristics of the material as a whole if the interphase boundaries are long. Such a situation is observed in composite materials reinforced by microparticles and nanoparticles even when the volume concentration of the inclusions is small. A numerical simulation of the stress state is performed on the basis of the block analytical-numerical multipole method with regard for the local effects related to the special nature of the deformation of the interphase layer in the vicinity of the interface.     

Modeling the Oscillations of the CO + O2 Reaction Rate in a Granular Catalyst Layer

A new point model is proposed for the oxidation reaction of CO on the surface of a Pd cluster. The model reflects the oxidation–reduction mechanism on the palladium surface and incorporates all the stages of the kinetic scheme from the previous three-component model. A new assumption based on a series of experimental facts claims that the adsorbed oxygen atoms may diffuse into deeper-lying layers of the crystal lattice and thus influence the processes in the adsorption layer due to the micro size of the palladium clusters. The new point system has been built into the distributed general model for a granular catalyst developed previously. As a result, for parameter values corresponding to experimental conditions we have managed for the first time to obtain a wide region of chaos and complex mixed modes, close to those observed in real experiments.     

Instability results related to compressible Korteweg system

This paper presents the study of surface tension effects in compressible mixtures in the framework of diffuse interface models. In the first part, we describe results previously obtained on the so-called compressible Korteweg and shallow water models and we present nonlinear stability using energy estimates and a new entropy equality recently discovered. These diffuse interface models also allow to take account of capillarity effects in turbulent mixtures and plasma flows subject to Rayleigh–Taylor instabilities. The aim of the last part is to study the influence of surface tension on this instability phenomena. More precisely we look at the expression of the growth rate under a small perturbation of wave number k. We prove that for an appropriate choice of the capillary number σ in terms on the surface tension coefficient T _s (that means particular pressure laws), we find the same expression as for the two incompressible fluids model with surface tension coefficient on a sharp interface studied for instance by Chandrasekhar (Hydrodynamic and hydromagnetic stability. Dover Publications, Inc. New York, 1981). The first author would like to thank the CEA/DAM (Bruyères le Chatel, France) for its financial support through the contract no. 4600052302/P6H28. He is also partially supported by the IDOPT project in Grenoble and the “ACI jeunes chercheurs 2004” du ministère de la Recherche “études mathématiques de paramétrisations en océanographie”.     

Two-layer muti-parameterized Schwarz Alternating Method

The convergence rate of a numerical procedure based on Schwarz Alternating Method (SAM) for solving elliptic boundary value problems (BVP’s) depends on the selection of the interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that theRobin condition(mixed interface condition), controlled by a parameter, can optimize SAM’s convergence rate. Since the convergence rate is very sensitive to the parameter, Tang[17] suggested another interface condition calledover- determined interface condition. Based on the over-determined interface condition, we formulate thetwo-layer multi-parameterized SAM. For the SAM and the one-dimensional elliptic model BVP’s, we determine analytically the optimal values of the parameters. For the two-dimensional elliptic BVP’s, we also formulate the two-layer multiparameterized SAM and suggest a choice of multi-parameter to produce good convergence rate.