Some new analysis results for a class of interface problems

Interface problems modeled by differential equations have many applications in mathematical biology, fluid mechanics, material sciences, and many other areas. Typically, interface problems are characterized by discontinuities in the coefficients and/or the Dirac delta function singularities in the source term. Because of these irregularities, solutions to the differential equations are not smooth or discontinuous. In this paper, some new results on the jump conditions of the solution across the interface are derived using the distribution theory and the theory of weak solutions. Some theoretical results on the boundary singularity in which the singular delta function is at the boundary are obtained. Finally, the proof of the convergency of the immersed boundary (IB) method is presented. The IB method is shown to be first‐order convergent in L ^∞ norm. Copyright © 2013 John Wiley & Sons, Ltd.     

A phase-field model of grain boundary motion

We consider a phase-field model of grain structure evolution, which appears in materials sciences. In this paper we study the grain boundary motion model of Kobayashi-Warren-Carter type, which contains a singular diffusivity. The main objective of this paper is to show the existence of solutions in a generalized sense. Moreover, we show the uniqueness of solutions for the model in one-dimensional space. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday     

Three-dimensional phase field simulations of grain growth in materials containing finely dispersed second-phase particles

Phase field modelling has proven to be a versatile tool for simulating microstructural evolution phenomena, such as grain growth in polycrystalline materials. However, the computational requirements of a phase field model impose strong limitations on the number of phase field variables employed in a practical implementation. In this paper, a bounding box algorithm is proposed allowing the use of a large number of phase field variables without excessive computational requirements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase-field modeling of the effect of interfacial energy on pyrolytic carbon morphology in chemical vapor deposition

A conserved phase-field model is proposed to investigate the effect of interfacial energy on the morphological evolution of the pyrolytic carbon deposit in chemical vapor deposition. The equilibrium geometry of carbon deposit islands is analytically predicted, of which the contact angle was controlled through the boundary conditions of the phase-field parameter at the substrate surface according to the Young-Laplace equation. Simulations of deposit growth are carried out for single and multi island nucleation. It is clarified that the island morphology depends on the magnitude of the interface energy. It is also observed that high interface energy results in large island size fluctuation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Void dynamics in lead-free Sn-Ag-Cu solder joints

As a result of thermal and mechanical loading of Sn-Ag-Cu solder joints pores, flaws and voids may nucleate and grow. Such void nucleation is studied here by means of a phase-field approach which accounts for the decomposition of the solder into phases described by its concentration c. In this investigation, the void growth results from boundary inward flux only, effects due to grain size and interstitials are not involved, cf. [1]. The free energy functional is approximated by a second order Taylor expansion and thus composed of a bulk free energy density and a Ginzburg-type gradient term. The bulk free energy density ϕ follows a classical Ginzburg-Landau double-well potential. The gradient-energy coefficient κ depends on ϕ and is calculated similar to [2]. Experimental data have been adapted for the modeling of the temporal evolution of the concentration of voids located on a square domain in 2D. The simulation is based on a B-spline finite element analysis. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

STUDY OF CRYSTAL GROWTH AND SOLUTE PRECIPITATION THROUGH FRONT TRACKING METHOD

Crystal growth and solute precipitation is a Stefan problem. It is a free boundary problem for a parabolic partial differential equation with a time-dependent phase interface. The velocity of the moving interface between solute and crystal is a local function. The dendritic structure of the crystal interface, which develops dynamically, requires high resolution of the interface geometry. These facts make the Lagrangian front tracking method well suited for the problem. In this paper, we introduce an upgraded version of the front tracking code and its associated algorithms for the numerical study of crystal formation. We compare our results with the smoothed particle hydrodynamics method （SPH） in terms of the crystal fractal dimension with its dependence on the Damkohler number and density ratio.     

An interface-fitted adaptive mesh method for elliptic problems and its application in free interface problems with surface tension

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting.     

On RF-pairs, Bäcklund transformations and linearization of nonlinear equations

Some classes of nonlinear equations of mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where the unknown function is taken as a new independent variable and an appropriate partial derivative is taken as the new dependent variable. RF-pairs and associated Bäcklund transformations are constructed for evolution equations of general form. The results obtained are used for order reduction of hydrodynamic equations (Navier-Stokes and boundary layer) and constructing exact solutions to these equations. A generalized Calogero equation and a number of other new linearizable nonlinear differential equations of the second, third and forth orders are considered. Some integro-differential equations are analyzed.     

Modelling of recrystallization behavior and austenite grain size evolution during the hot rolling of GCr15 rod

In this paper, four 3-D finite element models are developed to simulate the whole rod rolling process of GCr15 steel. The distribution and evolution of different field-variables, such as effective strain, effective strain rate and temperature, are obtained. Based on the simulated results and the microstructure evolution models of the steel, the paper designs a FORTRAN program to predict the evolution of recrystallization behavior and austenite grain size in rolled piece during the rolling. The surface temperatures of rolled piece calculated by FEM agree well with measured values. Comparison between calculated values and measured ones of grain size shows the validity of the program.     

A Variational Approach to Dynamic Recrystallization Using Probability Distributions

We investigate a model of dynamic recrystallization in polycrystalline materials. A probability distribution function is introduced to characterize the state of individual grains by grain size and dislocation density. Specifying free energy and dissipation within the polycrystalline aggregate we are able to derive an evolution equation for the probability density function via a thermodynamic extremum principle. Once the distribution function is known macroscopic quantities like average strain and stress can be calculated. For distribution functions which are constant in time, describing a state of dynamic equilibrium, we obtain a partial differential equation in parameter space which we solve using a marching algorithm. Numerical results are presented and their physical interpretation is given. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control of time discrete two-phase flow governed by a diffuse interface model

We present results on optimal control of two-phase flows. The fluid is modeled by a thermodynamically consistent diffuse interface model and allows to treat fluids of different densities and viscosities. In earlier work we proposed an energy stable time discretization for this model that we now employ to derive existence of optimal controls for a time discrete optimal control problem. The control aim is to obtain a desired distribution of the two phases in the system. For this we investigate three control actions. We use tangential Dirichlet boundary control and distributed control. We further consider the inverse problem of finding an initial distribution such that the evolution over a given time horizon starting from this value is close to a desired distribution. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase Field Model for Martensitic Transformations

The martensitic transformation is described using a phase field model which is in mathematical terms the regularization of a sharp interface approach. In this work, up to two martensitic orientation variants are considered. The evolution of microstructure is assumed to follow a time dependent Ginzburg-Landau equation. The coupled problem of the mechanical balance equation and the evolution equations is solved using finite elements and an implicit time integration scheme. In this work, the global energy evolution during the martensitic transformation and the influence of external loads on the formation of the different martensitic phases are studied in 2d. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A grain boundary defect model for ZnO ceramic varistors by deep heat treatment

Studies on ZnO ceramic varistors by deep heat treatment at 650–900 C are reported. The current creep time curve exhibits a peak during the continuous action of a dc biasing voltage; the forwardV-l characteristic is improved rather than degraded after the action of the biasing voltage. We assume that the zinc interstitial cations Zn_i are out diffused rapidly and the concentration of Zn_i in the depletion layer is decreased rapidly during deep heat treatment; the oxygen anions O’_o could be accumulated at the grain interface if the out diffusion quantity of Zn_i is not enough to react with the O’_o; the current creep phenomenon above results from the migration of the interface O’_o by the biasing voltage. We suggest an improved grain boundary defect model for the ZnO varistors by deep heat treatment, and examine the model using the experimental data of lifetime positron-annihilation spectroscopy. Project supported by the National Natural Science Foundation of China.     

A new discrete dynamical system of signed integer partitions

In Brylawski (1973) Brylawski described the covering property for the domination order on non-negative integer partitions by means of two rules. Recently, in Bisi et al. (in press), Cattaneo et al. (2014), Cattaneo et al. (2015) the two classical Brylawski covering rules have been generalized in order to obtain a new lattice structure in the more general signed integer partition context. Moreover, in Cattaneo et al. (2014), Cattaneo et al. (2015), the covering rules of the above signed partition lattice have been interpreted as evolution rules of a discrete dynamical model of a two-dimensional p–n semiconductor junction in which each positive number represents a distribution of holes (positive charges) located in a suitable strip at the left semiconductor of the junction and each negative number a distribution of electrons (negative charges) in a corresponding strip at the right semiconductor of the junction. In this paper we introduce and study a new sub-model of the above dynamical model, which is constructed by using a single vertical evolution rule. This evolution rule describes the natural annihilation of a hole–electron pair at the boundary region of the two semiconductors. We prove several mathematical properties of such new discrete dynamical model and we provide a discussion of its physical properties.     

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.     

Modeling Dynamic Recrystallization in Polycrystalline Materials via Probability Distribution Functions

A model of dynamic recrystallization in polycrystalline materials is investigated in this work. Within this model a probability distribution function representing a polycrystalline aggregate is introduced. This function characterizes the state of individual grains by grain size and dislocation density. By specifying free energy and dissipation within the polycrystalline aggregate an evolution equation for the probability density function is derived via a thermodynamic extremum principle. For distribution functions describing a state of dynamic equilibrium we obtain a partial differential equation in parameter space. To facilitate numerical treatment of this equation, the equation is further modified by introducing an appropriately rescaled variable. In this the source term is considered to account for nucleation of grains. Then the differential equation is solved by an implicit time-integration scheme based on a marching algorithm [2]. From the obtained distribution function macroscopic quantities like average strain and stress can be calculated. Numerical results of the theory are subsequently presented. The model is compared to an existing implementation in Abaqus as well. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

q-Shock soliton evolution

By generating function based on Jackson’s q-exponential function and the standard exponential function, we introduce a new q-analogue of Hermite and Kampe-de Feriet polynomials. In contrast to q-Hermite polynomials with triple recurrence relations similar to [1], our polynomials satisfy multiple term recurrence relations, which are derived by the q-logarithmic function. It allows us to introduce the q-Heat equation with standard time evolution and the q-deformed space derivative. We find solution of this equation in terms of q-Kampe-de Feriet polynomials with arbitrary number of moving zeros, and solved the initial value problem in operator form. By q-analog of the Cole–Hopf transformation we obtain a new q-deformed Burgers type nonlinear equation with cubic nonlinearity. Regular everywhere, single and multiple q-shock soliton solutions and their time evolution are studied. A novel, self-similarity property of the q-shock solitons is found. Their evolution shows regular character free of any singularities. The results are extended to the linear time dependent q-Schrödinger equation and its nonlinear q-Madelung fluid type representation.     

On the Geometric Densities of Random Closed Sets

Abstract

In many applications it is of great importance to handle evolution equations about random closed sets of different (even though integer) Hausdorff dimensions, including local information about initial conditions and growth parameters. Following a standard approach in geometric measure theory such sets may be described in terms of suitable measures. For a random closed set of lower dimension with respect to the environment space, the relevant measures induced by its realizations are singular with respect to the Lebesgue measure, and so their usual Radon–Nikodym derivatives are zero almost everywhere. In this paper we suggest to cope with these difficulties by introducing random generalized densities (distributions) á la Dirac–Schwarz, for both the deterministic case and the stochastic case. In this last one we analyze mean generalized densities, and relate them to densities of the expected values of the relevant measures. Many models of interest in material science and in biomedicine are based on time dependent random closed sets, as the ones describing the evolution of (possibly space and time inhomogeneous) growth processes; in such a situation, the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the local relevant kinetic parameters of birth and growth. In this context connections with the concepts of hazard function, and spherical contact distribution function are offered.     

Dynamic simulation on the preparation process of thin films by pulsed laser

An ablation model of targets irradiated by pulsed laser is established. By using the simple energy balance conditions, the relationship between ablation surface location and time is derived. By an adiabatic approximation, the continuous-temperature condition, energy conservation and all boundary conditions can be established. By applying the analytical method and integral-approximation method, the solid and liquid phase temperature distributions are obtained and found to be a function of time and location. The interface of solid and liquid phase is also derived. The results are compared with the other published data. In addition, the dynamics process of pulsed laser deposition of KTN (Kta_0.65Nb_0.35O₃) thin film is simulated in detail by using fluid dynamics theory. By combining the expression of the target ablation ratio and the dynamic equation and by using the experimental data, the effects of laser action parameters on the thickness distribution of thin film and on the thin film component characteristics are discussed. The results are in good agreement with the experimental data.     

Simulation of Surface Wetting by Droplets Using a Phase Field Model

In the proposed phase field model a continuous order parameter indicates the phase distribution (liquid/gas). An energy density functional which is dependent on the surface tensions and defined by three contributions yields the total energy of the system. An equilibrium state is then computed by minimizing this energy of the system using an evolution equation. Details of the algorithmic implementation are discussed by illustrative examples. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)