A numerical method to compute solidification and melting processes

This paper deals with thermodynamically consistent numerical predictions of solidification and melting processes of pure materials using moving grids. Till date, enthalpy-porosity-based formulations of numerical codes have been generally the popular choice, although because of an artificial numerical smearing of the interface, it is virtually impossible to reproduce a sharp melting/solidification front that is supposed to exist for phase changes of pure substances. Numerical techniques based on moving grid methods have been relatively less used as they rely on complex and time-consuming adaptive grid generations. Using the moving grid approach, the authors present a method to solve solidification and melting problems. A simple linear interpolation is used to slide grid nodes along the interface to handle the otherwise obtained grid skewness near the interface. The numerical approach employed is validated with standard test cases available in the literature. The capability of capturing very complex flow field structures and the superiority of the present approach over enthalpy-porosity-based formulations is discussed. The authors also demonstrate the ability of the set-up computer code to solve complex thermofluid processes such as occur during crystal growth in Czochralski reactors.     

Mathematical modeling and study of the process of solidification in metal casting

The process of melting and solidification in metal casting is considered. The process is modeled by a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the problem and its finite-difference approximation are given. A numerical algorithm is presented for solving the direct problem. The results are described and analyzed in detail. Primary attention is given to the evolution of the solidification front and to how it is affected by the parameters of the problem. Some of the results are illustrated by plots.     

A numerical method for computing singular minimizers

Summary. A numerical method, with truncation methods as a special case, for computing singular minimizers in variational problems is described. It is proved that the method can avoid Lavrentiev phenomenon and detect singular minimizers. The convergence of the method is also established. Numerical results on a 2-D problem are given. Received September 21, 1994     

Numerical Applications of the Scaling Concept

We present a survey of recent developments in the applications of the scaling concept to numerical analysis. In addition, we report on some relevant topics not covered in existing surveys. Therefore, the present work updates and complements the existing surveys on the subject concerned.Applications of the scaling concept are useful in the numerical treatment of both ordinary and partial differential problems. Applications to boundary-value problems governed by ordinary differential equations are mainly related to their transformation into initial-value problems. Within this context, special emphasis is placed on systems of governing equations, eigenvalue, and free boundary-value problems. An error analysis for a truncated boundary formulation of the Blasius problem is also reported. As far as initial-value problems governed by ordinary differential equations are concerned, we discuss the development of adaptive mesh methods. Applications to partial differential problems considered herein are related to the construction of finite-difference schemes for conservations laws, the solution structure of the Riemann problem, rescaling schemes and adaptive schemes for blow-up problems.In writing this paper, our aim was to promote further and more important numerical applications of the scaling concept. Meanwhile, the pertinent bibliography is highlighted and is available on internet as the BIB file sc-gita.bib from the anonymous ftp area at the URL ftp://dipmat.unime.it/pub/papers/fazio/surveys.     

Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type

In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014     

求解椭圆方程柯西问题的修正吉洪诺夫正则化方法

研究了一类变系数椭圆方程的柯西问题,这类问题出现在很多实际问题领域.由于问题的不适定性,不可能通过经典的数值方法来求解上述问题,必须引入正则化手段.采用了一种修正吉洪诺夫正则化方法来求解上述问题.在一种先验和一种后验参数选取准则下,分别获得了问题的误差估计.数值例子进一步显示方法是稳定有效的.     

Cooling of a composite slab in a two-fluid medium

A mixed boundary value problem associated with the diffusion equation that involves the physical problem of cooling of an infinite parallel-sided composite slab in a two-fluid medium, is solved completely by using the Wiener-Hopf technique. An analytical solution is derived for the temperature distribution at the quench fronts being created by two different layers of cold fluids having different cooling abilities moving on the upper surface of the slab at constant speedv. Simple expressions are derived for the values of the sputtering temperatures of the slab at the points of contact with the respective layers, assuming the front layer of the fluid to be of finite width and the back layer of infinite extent. The main problem is solved through a three-part Wiener-Hopf problem of a special type and the numerical results under certain special circumstances are obtained and presented in the form of a table.     

On an inverse problem arising in continuous casting of steel billets

In continuous casting of steel, the control of the solidification front by means of the amount of water sprayed onto the strand is of great practical interest. We study the thermal history in a continuously cast cylindrical billet. The mathematical model is a two-dimensional nonlinear heat equation div[k(u)gradu] = ut subject to water-cooling and heat radiation boundary conditions. We establish existence, uniqueness and stability results for both the temperature field and the solidification front. We study the monotonicity behaviour of the temperature field and show that certain technically easy-to-realize cooling-strategies may generate double liquid fingers at the final stage of solidification. The inverse problem of determining the cooling strategy is an ill-posed problem. We therefore use Tikhonov regularization as a stable and convergent methodfor treating this problem.     

Problem of the Motion of an Elastic Medium Formed at the Solidification Front

The following self-similar problem is considered. At the initial instant of time, a phase transformation front starts moving at constant velocity from a certain plane (which will be called a wall or a piston, depending on whether it is assumed to be fixed or movable); at this front, an elastic medium is formed as a result of solidification from a medium without tangential stresses. On the wall, boundary conditions are defined for the components of velocity, stress, or strain. Behind the solidification front, plane nonlinear elastic waves can propagate in the medium formed, provided that the velocities of these waves are less than the velocity of the front. The medium formed is assumed to be incompressible, weakly nonlinear, and with low anisotropy. Under these assumptions, the solution of the self-similar problem is described qualitatively for arbitrary parameters appearing in the statement of the problem. The study is based on the authors’ previous investigation of solidification fronts whose structure is described by the Kelvin–Voigt model of a viscoelastic medium.     

Coupling finite difference methods and integral formulas for elliptic problems arising in fluid mechanics

This article is devoted to the numerical analysis of two classes of iterative methods that combine integral formulas with finite‐difference Poisson solvers for the solution of elliptic problems. The first method is in the spirit of the Schwarz domain decomposition method for exterior domains. The second one is motivated by potential calculations in free boundary problems and can be viewed as a numerical analytic continuation algorithm. Numerical tests are presented that confirm the convergence properties predicted by numerical analysis. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 199–229, 2004     

Fast solution of elliptic control problems

Elliptic control problems with a quadratic cost functional require the solution of a system of two elliptic boundary-value problems. We propose a fast iterative process for the numerical solution of this problem. The method can be applied to very special problems (for example, Poisson equation for a rectangle) as well as to general equations (arbitrary dimensions, general region). Also, nonlinear problems can be treated. The work required is proportional to the work taken by the numerical solution of a single elliptic equation.     

AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.     

On numerical implementations of a new iterative method with boundary condition splitting for solving the nonstationary stokes problem in a strip with periodicity condition

Based on finite-difference approximations in time and a bilinear finite-element approximation in spatial variables, numerical implementations of a new iterative method with boundary condition splitting are constructed for solving the Dirichlet initial-boundary value problem for the nonstationary Stokes system. The problem is considered in a strip with a periodicity condition along it. At each iteration step of the method, the original problem splits into two much simpler boundary value problems that can be stably numerically approximated. As a result, this approach can be used to construct new effective and stable numerical methods for solving the nonstationary Stokes problem. The velocity and pressure are approximated by identical bilinear finite elements, and there is no need to satisfy the well-known difficult-to-verify Ladyzhenskaya-Brezzi-Babuska condition, as is usually required when the problem is discretized as a whole. Numerical iterative methods are constructed that are first- and second-order accurate in time and second-order accurate in space in the max norm for both velocity and pressure. The numerical methods have fairly high convergence rates corresponding to those of the original iterative method at the differential level (the error decreases approximately 7 times per iteration step). Numerical results are presented that illustrate the capabilities of the methods developed.     

Regularization of optimal design problems for bars and plates,part 2

For the problems of optimal design considered in Ref. 1, contradictions arising in the necessary conditions of optimality are eliminated by suitable extension of the initially given class of admissible materials. The extended class includes composites of some special (layered) microstructure. Elastic properties of such composites are described, and alternative (regularized) formulations of the optimal design problems are given. Necessary conditions of Weierstrass are shown to be satisfied, both for the case in which the strip of variations is small compared with the width of the layers and for the opposite case. Numerical results are given for the regularized problem of a bar of extremal torsional rigidity.The authors are indebted to Dr. N. A. Lavrov for performing numerical calculations.     

Solving some problems for systems of linear ordinary differential equations with redundant conditions

Numerical methods are proposed for solving some problems for a system of linear ordinary differential equations in which the basic conditions (which are generally nonlocal ones specified by a Stieltjes integral) are supplemented with redundant (possibly nonlocal) conditions. The system of equations is considered on a finite or infinite interval. The problem of solving the inhomogeneous system of equations and a nonlinear eigenvalue problem are considered. Additionally, the special case of a self-adjoint eigenvalue problem for a Hamiltonian system is addressed. In the general case, these problems have no solutions. A principle for constructing an auxiliary system that replaces the original one and is normally consistent with all specified conditions is proposed. For each problem, a numerical method for solving the corresponding auxiliary problem is described. The method is numerically stable if so is the constructed auxiliary problem.     

On transient heat conduction in the walls of a rectangular gas channel

In this paper a problem on transient heat conduction in the walls of a gas channel with a rectangular cross-section is solved. The temperature of the gas flow in the channel rises linearly while the temperature of the surrounding open air is constant.The differential equation and its auxiliary conditions are Laplace transformed, the subsidiary equations are solved by a method resembling the two-dimensional relaxation method for steady state heat conduction problems, and the resulting temperatures are obtained by numerical inversion.Numerical results are presented at the end of the paper.     

Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems

A pseudospectral method for generating optimal trajectories of linear and nonlinear constrained dynamic systems is proposed. The method consists of representing the solution of the optimal control problem by an mth degree interpolating polynomial, using Chebyshev nodes, and then discretizing the problem using a cell-averaging technique. The optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the proposed method avoids many of the numerical difficulties typically encountered in solving standard optimal control problems. Furthermore, for discontinuous optimal control problems, we develop and implement a Chebyshev smoothing procedure which extracts the piecewise smooth solution from the oscillatory solution near the points of discontinuities. Numerical examples are provided, which confirm the convergence of the proposed method. Moreover, a comparison is made with optimal solutions obtained by closed-form analysis and/or other numerical methods in the literature.     

Numerical solutions of the thermistor equations

This paper examines heat conduction in a thermistor used as a current surge regulator. The problem consists of coupled nonlinear, nonlocal parabolic initial boundary value problems. Simplifying assumptions are made which lead to two different problems each of which consists of a one (space) dimensional nonlocal parabolic initial boundary value problem.Numerical methods for the approximate solutions of both the steady state and the transient problems are described and the results of the numerical experiments are presented.     

Asymptotic analysis of stationary propagation of the front of a two-stage exothermic reaction in a gas : PMM vol. 37, n6, 1973, pp. 1049–1058

We construct an approximate solution of the problem concerning the propagation of a planar. front of a two-stage exothermic sequential chemical reaction in a gas, by the method of matched asymptotic expansions. As the parameter in the expansion we use the ratio of the adiabatic combustion temperature to the sum of the activation temperatures of both reactions. Depending on the values of the characteristic parameters of the problem, we consider several solutions, each with a different asymptotic behavior, corresponding to the various flame front propagation modes. The analytical results obtained are compared with numerical data available in the literature.     

Inverse linear multistep methods for the numerical solution of initial value problems of second order differential equations

In this paper inverse linear multistep methods for the numerical solution of second order differential equations are presented. Local accuracy and stability of the methods are defined and discussed. The methods are applicable to a class of special second order initial value problems, not explicitly involving the first derivative. The methods are not convergent, but yield good numerical results if applied to problems they are designed for. Numerical results are presented for both the linear and nonlinear initial value problems.