An inverse model for a free-boundary problem with a contact line: Steady case

This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to PDE constraints. An advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase. It is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated. We describe an efficient iterative solution method for the Stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional. Performance of the proposed approach is illustrated with sample computations concerning 2D steady solidification phenomena.     

Some remarks on the stochastic model of crystal growth

Conclusions In this paper further possibilities of extending of the stochastic theory of crystal growth as well as possibilities of solving the basic equations are presented. The building of kinetic processes on the solidification front in the stochastic model gives us the possibility of studying the influence of kinetic processes in connection with temperature distribution during crystal growth.The approximations mentioned in chapter 2 considerably accelerate the numerical solution of our problem. Tests in concrete cases show that mentioned approximations do not significantly influence the results if the conditions referred to are fulfilled.     

The Scaling Limit of Polymer Pinning Dynamics and a One Dimensional Stefan Freezing Problem

We consider the stochastic evolution of a 1 + 1-dimensional interface (or polymer) in the presence of a substrate. This stochastic process is a dynamical version of the homogeneous pinning model. We start from a configuration far from equilibrium: a polymer with a non-trivial macroscopic height profile, and look at the evolution of a space-time rescaled interface. In two cases, we prove that this rescaled interface has a scaling limit on the diffusive scale (space rescaled by L in both dimensions and time rescaled by L ² where L denotes the length of the interface) which we describe. When the interaction with the substrate is such that the system is unpinned at equilibrium, then the scaling limit of the height profile is given by the solution of the heat equation with Dirichlet boundary condition ; when the attraction to the substrate is infinite, the scaling limit is given by a free-boundary problem which belongs to the class of Stefan problems with contracting boundary, also referred to as Stefan freezing problems. In addition, we prove the existence and regularity of the solution to this problem until a maximal time, where the boundaries collide. Our result provides a new rigorous link between Stefan problems and Statistical Mechanics.     

Dependence of the growth rate on supercooling and concentration in the framework of the stochastic theory

The analysis of the dependences of the solidification rate on the supercooling and concentration is set forward in the stationary case of the solidification. The stochastic theory of the solidification was used for the theoretical analysis.     

Theoretical analysis of the influence of external conditions upon the solidification of binary metal alloys

In this paper we present theoretical analysis of the influence of cooling rate, kinetic parameters of solidification and transport coefficients upon the solidification rate, the value of supercooling on the solidification front and upon the change of concentration of the solid and liquid phase during the solidification of binary alloys. This analysis is performed within the framework of the stochastic theory of solidification.     

A Schur complement formulation for solving free-boundary,Stefan problems of phase change

A Schur complement formulation that utilizes a linear iterative solver is derived to solve a free-boundary, Stefan problem describing steady-state phase change via the Isotherm–Newton approach, which employs Newton’s method to simultaneously and efficiently solve for both interface and field equations. This formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire Jacobian matrix for a two-dimensional sample problem that discretizes the field equations using a Galerkin finite-element method and employs a deforming-grid approach to represent the melt–solid interface. All methods demonstrate quadratic convergence for sufficiently accurate Newton solves, but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size. Of these two approaches, the Schur formulation proves to be more robust, converging with significantly smaller Krylov subspaces than those required to solve the global system of equations. Further improvement of performance are made through approximations and preconditioning of the Schur complement problem. Hence, the new Schur formulation shows promise as an affordable, robust, and scalable method to solve free-boundary, Stefan problems. Such models are employed to study a wide array of applications, including casting, welding, glass forming, planetary mantle and glacier dynamics, thermal energy storage, food processing, cryosurgery, metallurgical solidification, and crystal growth.     

Pressure effects in the planar solidification of a finite slab: convective cooling and prescribed heat flux boundary conditions

Summary The classical Stefan problem assumes a fixed melting temperature. However, when the solid phase is the one with lower density (e.g., water) the solidification of the system causes an overall volume increase that is often contrasted by the container walls. In that case the growing pressure determines a continuous lowering of the freezing point, and the temperature field as well as the interface motion are strongly affected. This paper is concerned with these aspects of the problem; the planar solidification of a slab of finite thickness, contrasted by an opposing elastic force, is numerically simulated. The effects of two different boundary conditions are analysed. When the solidification is driven by convective cooling, the continuous advancement of the melting front is replaced by an asymptotic behaviour, until thermal equilibrium is attained. When the boundary condition is specified in terms of a prescribed heat flow, the melting front velocity is slowed down by a growing adverse temperature gradient. The influence of various parameters on the process is presented and discussed.     

A Meshless Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem

In this paper, a meshless regularization method of fundamental solutions is proposed for a two-dimensional, two-phase linear inverse Stefan problem. The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions. Furthermore, the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution. Therefore, regularization is necessary in order to obtain a stable solution. Numerical results for several benchmark test examples are presented and discussed.     

Modelling of radiant-conductive heat transfer in the layer of semitransparent medium in approximation of the classical solution to the single-phase Stefan problem

The single-phase Stefan problem was modelled numerically in approximation of the classical solution in application to melting of a flat semitransparent sample by radiant-conductive technique in a wide range of emissivity of the phase transition front.     

Remark on the influence of gravitation on the solidification of the binary systems

The possible influence of the gravitation on the solidification of the binary systems in the framework of the stochastic theory is discussed. The considerations are based on the change of the kinetic parameters of the phase transformation with the change of the concentration of clusters in the melt.     

Twodimensional stochastic theory of crystal growth

In the paper the twodimensional model of the stochastic theory of crystal growth is presented and used for the study of the morphology of the solidification front under different external conditions and for different rates of kinetic processes on the solidification front. The results show that the stability of the solidification front depends on the kinetic processes, which thus must be taken into account in the stability conditions.     

On a phase transition for semitransparent materials in terms of the Stefan problem

The paper deals with justification of the formula for the latent heat of phase transition of the first kind, taking into account superheating and subcooling of the formed two-phase system, in application to the solution of Stefan problem in semitransparent materials.     

The effect of the evaporation coefficient on the thermophoresis of a volatile single-component drop in a binary gas mixture

A theory of the uniform thermophoretic motion of a liquid volatile spherical drop in a binary gas mixture is developed based on hydrodynamic analysis. One of the components undergoes the phase transition on the surface. The solution of the problem makes it possible to estimate the effect of the evaporation rate on the rate and direction of thermophoresis, as well as on the distributions of the velocity, temperature, and concentration of the volatile component. The thermal diffusion of the gas mixture, together with Stefan and capillary phenomena, is taken into account. The velocity of thermophoretic transport is expressed through the evaporation coefficient of the drop by the formula that generalizes the known results of the conventional theories for the cases of weak and moderately intense diffusive evaporation of a liquid drop.     

Stochastic analog of the quantum theory of Coulomb problem

The path-integral formalism is used to show that the solution to a special stochastic process described by the non-linear Langevin equation can be converted into the quantum theory of the Coulomb problem. The conditional probability density for this stochastic process is obtained by explicit functional integration.The author thanks Professor M. Noga for the proposal of this problem, for helpful suggestions and valuable discussions. The author is also indebted to M. Bedná for critical comments.     

The Stefan problem of solidification of ternary systems in the presence of moving phase transition regions

The process of solidification of ternary systems in the presence of moving phase transition regions has been investigated theoretically in terms of the nonlinear equation of the liquidus surface. A mathematical model is developed and an approximate analytical solution to the Stefan problem is constructed for a linear temperature profile in two-phase zones. The temperature and impurity concentration distributions are determined, the solid-phase fractions in the phase transition regions are obtained, and the laws of motion of their boundaries are established. It is demonstrated that all boundaries move in accordance with the laws of direct proportionality to the square root of time, which is a general property of self-similar processes. It is substantiated that the concentration of an impurity of the substance undergoing a phase transition only in the cotectic zone increases in this zone and decreases in the main two-phase zone in which the other component of the substance undergoes a phase transition. In the process, the concentration reaches a maximum at the interface between the main two-phase zone and the cotectic two-phase zone. The revealed laws of motion of the outer boundaries of the entire phase transition region do not depend on the amount of the components under consideration and hold true for crystallization of a multicomponent system.     

耐腐蚀聚合药物基体上药物的扩散释放

主要对耐腐蚀聚合药物基体上药物的整个释放过程进行解析和数值研究.数学模型为基于Fick第二法则的边值问题,分为状态Ⅰ和Ⅱ.状态Ⅰ为具有移动性扩散波头的Stefan问题,而当此扩散波头消失时,即开始状态Ⅱ.该研究工作可作为数值研究更加复杂系统的基础之一.     

Asymptotical p-moment stability of stochastic impulsive differential system and its application to chaos synchronization

In this paper,the asymptotical p-moment stability of stochastic impulsive differential equations is studied and a comparison theory to ensure the asymptotical p-moment stability of the trivial solution is established,which is important for studying the impulsive control and synchronization in stochastic systems.As an application of this theory,we study the problem of chaos synchronization in the Chen system excited by parameter white-noise excitation,by using the impulsive method.Numerical simulations verify the feasibility of this method.     

A probabilistic study of the influence of parameter uncertainty on solutions of the radiative transfer equation

The influence of uncertainty in the absorption and scattering coefficients on the solution and associated parameters of the radiative transfer equation is studied using polynomial chaos theory. The uncertainty is defined by means of uniform and log-uniform probability distributions. By expanding the radiation intensity in a series of polynomial chaos functions we may reduce the stochastic transfer equation to a set of coupled deterministic equations, analogous to those that arise in multigroup neutron transport theory, with the effective multigroup transfer scattering coefficients containing information about the uncertainty. This procedure enables existing transport theory computer codes to be used, with little modification, to solve the problem. Applications are made to a transmission problem and a constant source problem in a slab. In addition, we also study the rod model for which exact analytical solutions are readily available. In all cases, numerical results in the form of mean, variance and sensitivity are given that illustrate how absorption and scattering coefficient uncertainty influences the solution of the radiative transfer equation.