Inverse Stefan Problem Research Articles

Inverse stefan problem: Tracking of the interface position from measurements on the solid phase

AbstractThis study presents a sequential algorithm for the identification of the position s of the moving boundary of a variable one‐dimensional or two‐dimensional domain from discrete measurements, temperatures and fluxes, collected at the fixed boundary. The inverse problem is solved by minimization, with respect to s, of a penalized output least square criterion defined on a sliding time horizon of length τ. At every time step, several iterations are performed to estimate the unknown s. Each iteration consists in a guess of s, a computation of the corresponding value of the output y (direct model) and the criterion J, and a step towards a new estimation of s. The impact of the different parameters, space and time discretization intervals, regularization coefficient, dimension of the unknown parameter s, length of the observation horizon, choice of the observed output (temperature or flux) and choice of the direct model is thoroughly analysed for an analytical test case.

SOLUTION OF INVERSE STEFAN PROBLEMS BY A SOURCE‐AND‐SINK METHOD

A method has been developed for the solution of inverse heat diffusion problems to find the initial condition, boundary condition, and the source and sink function in the heat diffusion equation. The method has been used in the development of a source‐and‐sink method to find the boundary conditions in inverse Stefan problems. Green's functions have been used in the solution, and the problems are solved by using two approaches: a series solution approach, and a time incremental approach. Both can be used to find the boundary conditions without reliance on the flux information to be supplied at both sides of the interface. The methods are efficient in that they require less equations to be solved for the conditions. The numerical results have shown to be accurate, convergent, and stable. Most of all, the results do not degrade with time as in other time marching schemes reported in the literature. Algorithms can also be easily developed for the solution of the conditions.

ENTHALPY METHOD FOR INVERSE STEFAN PROBLEMS

In the context of solidification systems, an inverse Stefan problem involves the determination of boundary conditions such that the phase front moves in a prescribed manner. These problems are commonly treated with algorithms based on the temperature Stefan formulation. This paper presents an inverse algorithm based on an enthalpy phase-change formulation. This approach leads to robust and stable solutions of a number of inverse Stefan problems.

Computer simulation of the inverse Stefan problem in conformity with the liquid phase epitaxy of Hg1−xMnxTe

On effectue une simulation par ordinateur du probleme d'inverse de Stefan, en conformite avec l'epitaxie de la phase liquide de Hg 1-x Mn x Te. Dans ce semiconducteur, la bande interdite d'energie depend de la composition. Ceci permet d'indiquer une correlation mutuelle sans ambiguite x(r)↔T(t). La methode peut etre generalisee aux cas de la croissance des monocristaux et des epitaxies electro-liquides et des phases liquides de systemes a plusieurs composantes

A deforming finite element method analysis of inverse Stefan problems

AbstractA deforming FEM (DFEM) analysis of one‐dimensional inverse Stefan problems is presented. Specifically, the problem of calculating the position and velocity of the moving interface from the temperature measurements of two or more sensors located inside the solid phase is addressed. Since the interface velocity is considered to be the primary variable of the problem, the DFEM formulation is found to have many advantages over other traditional front tracking methods. The present inverse formulation is based on a minimization of the error between the calculated and measured temperatures, utilizing future temperature data to calculate current values of the unknown parameters. Also, the use of regularization is found to be useful in obtaining more accurate results, especially when the interface is located far away from the sensors. The method is illustrated with several examples. The effects of the location of the sensors, of the error in the sensor measurements and of several computational parameters were examined.

Solution of a homogeneous inverse stefan problem

Solution of a homogeneous inverse stefan problem

An Analysis of Inverse Heat Transfer Problems With Phase Changes Using an Integral Method

This paper provides a methodology for the solution of certain inverse heat transfer problems with phase changes. It is aimed particularly at the design of casting processes. The idea is to use the inverse method to calculate the boundary flux history that will achieve the velocities and fluxes at the freezing front that are needed to control liquid feeding to the front, as well as yield the desired cast structure. The proposed method also can be applied to predict the freezing front motion using temperature measurements at internal points. A boundary element analysis with constant elements is used here in conjunction with Beck’s sensitivity analysis. The accuracy of the method is illustrated through one-dimensional numerical examples. It is demonstrated that, by using an integral formulation, one can extend all of the current methods for solving inverse heat conduction problems with stationary boundaries, to inverse Stefan problems. Such problems are of great technological significance.

An inverse stefan's problem in casting multicomponent alloys

A model is considered that incorporates the feature that melting (crystallization) occurs over a certain temperature range, and it is shown that the solution to the internal inverse problem is unique in one such formulation.

The Numerical Solution of the Inverse Stefan Problem in Two Space Variables

A method is given for solving the inverse Stefan problem for the heat equation in two space variables. The approach is based on using a complete family of solutions to the heat equation (thus reducing the dimensionality of the problem by one) and minimizing the maximal defect in the initial-boundary data subject to regularizing constraints (thus making the problem well-posed in the sense that the solution now depends continuously on the data). Two numerical examples are given which make use of a linear semi-infinite programming method recently developed by Hettich and Zencke.

A method for the numerical solution of the one-dimensional inverse Stefan problem

In this paper we suggest the use of complete families of solutions of the heat equation for the numerical solution of the inverse Stefan problem. Our approach leads to linear optimization problems which can be established and solved easily. Convergence results are proved. In a final section the method is applied to some examples.

The numerical solution of the inverse Stefan problem

The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.

The inverse Stefan problem as a problem of nonlinear approximation theory

The inverse Stefan problem as a problem of nonlinear approximation theory

Analytic Solutions to the Heat Equation Involving a Moving Boundary with Applications to the Change of Phase Problem (the Inverse Stefan Problem)

Long time analytic solutions to the inverse Stefan problem are presented in Cartesian and spherical coordinates. This problem deals with the phase change of a substance with the boundary conditions specified at the moving boundary. The solutions, expressed in infinite series, facilitate the calculation of temperature and heat flux variations in the phase changing medium and at the interface with the heat source element. Applications of the present analysis are in aerospace engineering, e.g., controlled ablation of thermal shields; metallurgy, e.g., controlled casting; bioengineering, e.g., cryopreservation of tissues, and numerous applications in which a controlled change of phase of a substance is involved.

Calculating the time of isothermal saturation of a sphere

Results of a numerical solution of the inverse Stefan problem are presented for a sphere with boundary conditions of the third kind. Expressions are derived for calculating the time of total diffusion saturation of a spherical core, taking into account the rate of interaction of the gas phase with the surface of the solid.

The inverse Stefan problem for the heat equation in two space variables

Abstract : In this paper the author outlines an inverse method for constructing analytic solutions to the (single phase) Stefan problem for the heat equation in two space dimensions.

The Noncharacteristic Cauchy Problem for Parabolic Equations in One Space Variable

An integral operator is constructed which maps ordered pairs of analytic functions onto analytic solutions of linear second order parabolic equations in one space variable with analytic coefficients. This operator is then used to construct a solution to the noncharacteristic Cauchy problem for parabolic equations in one space variable. Applications are made to the inverse Stefan problem and the analytic continuation of solutions to parabolic equations.

The solution of the inverse Stefan problem

The solution of the inverse Stefan problem