Multidimensional Stefan Problem Research Articles

A second order accurate fixed-grid method for multi-dimensional Stefan problem with moving phase change materials

This paper proposes a front-tracking fixed grid method for solving Stefan problem with moving phase change materials in any arbitrary (irregular) bounded domains. To achieve this objective, the governing partial differential equations are transformed, using a curvilinear coordinate transformation, in to a fixed rectangular domain, at each time level and an alternate direction implicit (ADI) scheme is used to solve the resultant differential system. The proposed scheme is consistent, unconditionally stable and also produces second-order accurate results. Many numerical experiments are conducted, in both one and two phase environments, to validate the proposed method. All these validations confirmed the theoretical accuracy of the produced results which are presented in the form of error graphs and tables in each case. Importance is given to understand the influence of the physical parameters like Peclet number, Stefan number, material velocity, latent heat, and thermal conductivity, etc., on the rate of change of phase and observed a continuous enhancement of the same with in the chosen range of the parameters. Also noticed a faster movement of the interface with the increase in these parameters in the direction of melting or freezing.

Global-in-time strong solvability of the multi-dimensional one-phase Stefan problem for an incompressible viscous fluid

In the present paper, we consider the multi-dimensional one-phase Stefan problem describing the process of phase transition in an incompressible viscous fluid. The model is described as a free boundary problem consisting of the heat equation with a transport term and the Navier–Stokes equations. We prove the existence of a global-in-time strong solution with small data by introducing Lagrangian coordinates.

DYNAMIC ADAPTATION METHOD FOR NUMERICAL SOLUTION OF AXISYMMETRIC STEFAN PROBLEMS

Application of the dynamic adaptation method for the numerical solution of multidimensional axisymmetric Stefan problems with explicit tracking of interfaces is presented. The dynamic adaptation method is based on the idea of transition of the physical coordinate system to the non‐stationary coordinate system. The results of computational experiments for modelling the action of high energy fluxes on metals are given.

Dynamic adaptation method for numerical solution of axisymmetric Stefan problems

Abstract Application of the dynamic adaptation method for the numerical solution of multidimensional axisymmetric Stefan problems with explicit tracking of interfaces is presented. The dynamic adaptation method is based on the idea of transition of the physical coordinate system to the non‐stationary coordinate system. The results of computational experiments for modelling the action of high energy fluxes on metals are given.

Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators

In this paper we discuss weak solutions of the multi-dimensional two-phase Stefan problem with the boundary condition including the subdifferential operator of the convex function on $\mathbb R$ so that the boundary condition is nonlinear and contains a multi-valued operator, in general. Kenmochi and Pawlow already established the uniqueness and existence of a solution to our problem in the sense of the vanishing viscosity solution. The purpose of this paper is to prove the uniqueness theorem for a solution defined in the usual variational sense. Our proof is due to the standard method in which the dual problem of the original problem plays a very important role.

Solution of the Multi-interface Stefan Problem by the Method of Dynamic Adaptation

AbstractThe application of the dynamic adaptation method to the numerical solution of multidimensional Stefan problems with explicit tracking of interfaces is considered. The dynamic adaptation method is based on the idea of transition to any non-stationary coordinate system. The results of the numerical experiment on modelling the pulsed action of high-energy fluxes on metals are presented.

An implicit method for reconstructing dynamic three-dimensional phase boundaries under low Peclet number conditions

This article develops an implicit inverse method for reconstructing dynamic multidimensional phase boundaries. The technique is suitable for problems having small liquid phase Peclet numbers, Pe r = (V̂ rL̂/ α ̂ ), where V̂ r is the characteristic liquid phase velocity scale evaluated relative to the solid phase velocity scale, L̂ is a characteristic length scale, and α ̂ is a characteristic thermal diffusivity. Under these conditions, a multidimensional Stefan problem emerges. Explicit front-tracking procedures are eliminated by incorporating the latent heat effect in an effective, temperature dependent specific heat. Time-sequential reconstruction is then performed by solving a multidimensional nonlinear inverse heat conduction problem. As an illustration, evolving phase boundaries are reconstructed within moving and stationary plates subject to concentrated, high energy density heat sources. It is found that boundaries can be accurately reconstructed using either exact or noisy temperature measurements.

Existence of the Global Classical Solution for a Two-Phase Stefan Problem

In this work we prove the existence of the global classical solution in a two-phase multidimensional Stefan problem. We apply a method which consists of the following. First, we construct a special system of difference--differential approximating elliptic problems, then we prove some uniform estimates and pass to the limit. We prove that the free boundary is given by the graph of a function from the $ H^{2+\alpha ,1+\frac {\alpha }{2}} $ class.

Vanishing specific heat for the classical solutions of a multidimensional Stefan problem with kinetic condition

In this paper we prove that the multidimensional Hele-Shaw problem with kinetic condition at the free boundary is the limit case of the Stefan problem with kinetic condition at the free boundary in the classical sense when the specific heat ε \varepsilon goes to zero. The method is the use of a fixed point theorem; the key step is to construct a suitable function space in which we can get the existence and uniform estimates with respect to ε > 0 \varepsilon > 0 at the same time as for classical solutions of the multidimensional Stefan problem with kinetic condition at the free boundary. For the sake of simplicity, we only consider one-phase problems in three space dimensions, although the method used here is also applicable for two-phase problems and any space dimensions.

Use of the method of green functions for numerical solution of multidimensional stefan problems

It is shown that use of an auxiliary Green function, namely, the Green function of the boundary-value problem for the Laplace operator with a condition of the third kind artificially introduced on part of the boundary, makes it possible to find numerical solutions of multidimensional Stefan problems with any boundary conditions. The efficiency of the method is verified for a Stefan problem that has an exact solution.

Numerical solution of stefan problems by the method of green functions

A method for reducing a multidimensional Stefan problem to a system of Hammerstein integral equations is proposed. Application of the proposed method to numerical solution of one-dimensional nonstationary Stefan problems formulated for the cases of an internal phase front, coincidence of the phase front with the external boundary, and a movable external boundary is considered. The efficiency of the method is tested on an exactly solvable Stefan problem.

Multi-dimensional stefan problems with dynamic boundary conditions

We consider multi-phase Stefan problems for a class of nonlinear parabolic equations with dynamic boundary conditions formulated in bounded domains in RN, N≥2. Our dynamic boundary condition is described by a non-linear parabolic (pseudo) differential equation for the boundary temperature. The existence and uniqueness for a weak solution as well as the monotone dependence of the solution upon data will be shown. Our approach to this problem is based on the abstract theory of nonlinear evolution equations governed by time-dependent subdifferentials in Hilbert spaces.

Existence of classical solutions of the multidimensional Stefan problem on a finite interval

The existence of classical solutions of the two-phase Stefan problem on any finite interval is shown.

Error estimates for multidimensional singular parabolic problems

We consider a rather general class of singular parabolic problems; multidimensional two-phase Stefan problem and porous-medium equation involving additional nonlinearities are included. We analyze an approximation scheme consisting in a regularization procedure and discretization by means ofC °-piecewise linear finite elements in space and backward-differences in time. We prove severalL β-error estimates for the various unknowns, wherep depends on the problem. As a by-product of our technique, we obtain an optimalL 2-error estimate for enthalpy.

Optimal control of a Stefan problem with state-space constraints

An optimal control problem with state-space constraints for a multidimensional Stefan problem in an enthalpy formulation with nonlinear boundary conditions is investigated. The existence and stability of a solution of the optimal control problem is shown providing a set of admissible controls is compact. Also, a boundary observation is admitted. The original problem is then approximated by using a penalty-function method, finite-element method in space and backward Euler formula in time. The convergence of the approximate optimal controls is proved. Moreover, an example for non-stability of the penalty-function method inL p spaces is constructed. Finally, numerical results of an illustrative example are presented.

Convergence of the approximate free boundary for the multidimensional one-phase Stefan problem

The variational inequality arising from the one-phase multidimensional Stefan problem is discretized by piecewise-linear finite elements in space and by backward-differences in time. Error estimates for the discrete free boundary at each time-step are proved.

Approximate analytical methods for multi-dimensional stefan problems

Two approximate analytical methods are presented for solving two-dimensional Stefan problems. One of them selects one-dimensional quadratic temperature profiles in one variable at the preselected values of the other variable. The second one is based on piecewise linear profiles. The first method is extended for solving a three-dimensional problem also. The numerical results for some selected problems are obtained by using the present methods as well as some methods due to other authors. The merits of the proposed methods are discussed.

On the numerical approach to a two-phase stefan problem with non-linear flux

This paper is devoted to the numerical analysis of a multidimensional two-phase Stefan problem, with a non-linear flux condition on the fixed boundary; the enthalpy formulation is used. A numerical approach suggested by the theory of non-linear semigroup of contractions in L1 (Ω) is introduced; some converging algorithms based on the Crandall-Liggett formula and on the non-linear Chernoff formula are studied. The algebraic non-linear equations are solved by a modified Gauss-Seidel method. The results of several numerical tests are exhibited and discussed.

Numerical solution of a multidimensional two-phase stefan problem

This paper treats a multidimensional two-phase Stefan problem with variable coefficients and mixed type boundary conditions. A numerical method for solving the problem is of fixed domain type, based on a variational inequality formulation of the problem. Numerical solutions are obtained by using piecewise linear finite elements in space and finite difference in time, and by solving a strictly convex minimization problem at each time step. Some computational results are presented.

Regularity of the boundary data and the convergence of the finite element discretization in two-phase stefan problems

Motivated by some problems from the control of free boundaries the convergence of a finite element discretization of multidimensional Stefan problems is discussed in connection with the regularity properties of the boundary data. The numerical results concern two-dimensional problems.