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. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 3, pp. 564–570, May–June, 1998.     

A method of numerical solution of one-dimensional stefan problems of two types

A one-dimensional conjugate problem of heat transfer with phase transitions of two types (with interfaces of two phases and with a two-phase zone) is solved by a finite difference method based on the general initial heat conduction equation written with the Dirac delta function. A calculating scheme is developed using a nonuniform spatial net with floating nodes and the method of oppositely directed pivots.Nizhnii Novgorod Polytechnic Institute. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 65, No. 3, pp. 332–340, September, 1993.     

Method of numerical solution of one-dimensional multifront stefan problems

A finite-element method with explicit isolation of fronts is proposed for Stefan problems with an arbitrary number of phase transition boundaries.Inzhenerno-Fizicheskii Zhurnal, Vol. 58, No. 4, pp. 681–689, April, 1990.     

Use of classical stefan problem for initializing solution in the numerical prediction of freezing

The exact solution of the classical Stefan problem is examined from the point of view of using it as an initial solution in numerical solutions of appropriate problems.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 43, No. 5, pp. 847–850, November, 1982.     

A direct numerical method for the solution of field problems

A numerical method for the analysis of field problems is described. The algorithm is based upon the generalized Betti-Maxwell theorem. Using a set of known solutions to problems with similar boundary conditions produces a set of ‘integral’ equations for the required solution. Using any convenient numerical integration formula reduces the problem to the solution of a set of simultaneous algebraic equations. The accuracy of the solution depends upon the accuracy of the integration formula as applied to the problem under consideration and is independent of the known auxiliary solutions. The method is described in detail as applied to harmonic problems.     

An efficient numerical method for the solution of sliding contact problems

In this paper, an efficient numerical method to solve sliding contact problems is proposed. Explicit formulae for the Gauss–Jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with Cauchy kernels are derived. The resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a Jacobi polynomial. Gaussian quadratures obtained in this manner involve fixed nodal points and are exact for polynomials of degree 2n ? 1, where n is the number of nodes. From this Gauss–Jacobi quadrature, the existing Gauss–Chebyshev quadrature formulas can be easily derived. Another apparent advantage of this method is its ability to capture correctly the singular or regular behaviour of the tractions at the edge of the region of contact. Also, this analysis shows that once if the total normal load and the friction coefficient are given, the external moment M and contact eccentricity e (for incomplete contact) in fully sliding contact are uniquely determined. Finally, numerical solutions are computed for two typical contact cases, including sliding Hertzian contact and a sliding contact between a flat punch with rounded corners pressed against the flat surface of a semi‐infinite elastic solid. These results provide a demonstration of the validity of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

A numerical method for the solution of two-dimensional inverse heat conduction problems

A numerical method for the solution of inverse heat conduction problems in two-dimensional rectangular domains is established and its performance is demonstrated by computational results. The present method extends Beck's⁸ method to two spatial dimensions and also utilizes future times in order to stabilize the ill-posedness of the underlying problems. The approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one-dimensional problems which can be decoupled.     

A fixed grid method for the numerical solution of phase change problems

A fixed grid method using an updated iterative implicit scheme is developed to solve one-dimensional phase change problems. The temperature field is deduced from the resolution of the governing equations whose discretization takes into account the discontinuous variation of the temperature derivative at the phase change front. At each iteration an updated position of the moving front is found from the resolution of the energy conservation at the solid-liquid interface. The accuracy of the proposed numerical method has been checked on three test problems.     

An explicit unconditionally stable variable time-step method for one-dimensional stefan problems

Modelling of heat conduction processes with phase changes benefits from the application of variable time-step methods when the behaviour of the moving boundary is not known a prioiri. Due to convergence and stability constraints only implicit difference equations have been used with these methods. Implicit methods show a significant loss of accuracy and exhibit convergence difficulties when used for relatively slow or rapid moving-boundary problems. To overcome these problems an improved explicit variable time-step method which combines the explicit exponential difference equation and a variable time-step grid network with virtual subspace increments around the moving boundary is presented and tested for both a solidification and a melting problem. A virtual subinterval time-step elimination technique is incorporated to ensure that stability is automatically maintained for any mesh size. Unlike the implicit variable time-step methods, the accuracy of the resulting method is not affected by the velocity of the moving boundary. For both test problems numerical results are in better agreement with known analytical solutions than results predicted by other numerical methods.     

On the numerical solution of linear exterior problems via the uncoupling method

The application of the uncoupling of boundary integral and finite element methods to solve exterior boundary value problems in R ² yields a weak formulation that contains only one boundary term. This is the so-called uncoupling term, which is determined by the boundary integral operator of the single-layer potential acting on a circle centered at the origin. The purpose of this paper is to provide a suitable formula, which combines analytical and numerical methods, to approximately integrate the uncoupling term to any exacteness. Our method provides sharper error estimates than the one that uses Truncated Infinite Fourier Series (TIFE). As a model we consider the exterior Dirichlet problem for the Laplacian, and use linear finite elements for the corresponding Galerkin scheme. Some numerical experiments are also presented. © 1998 John Wiley & Sons, Ltd.     

A numerical method for heat transfer problems using collocation and radial basis functions

Simple, mesh/grid free, numerical schemes for the solution of heat transfer problems are developed and validated. Unlike the mesh or grid-based methods, these schemes use well-distributed quasi-random collocation points and approximate the solution using radial basis functions. The schemes work in a similar fashion as finite differences but with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no extra difficulty. © 1998 John Wiley & Sons, Ltd.     

On the selection of coordinate functions for the solution of boundary problems by Galerkin's method

We propose a method for determining the coordinate functions to be employed in Galerkin's method for the solution of boundary problems, which increases significantly the precision of the calculations in a first approximation.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 2, pp. 309–315, February, 1970.     

A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.     

A new family of time integration methods for heat conduction problems using numerical green’s functions

This paper is concerned with the formulation and numerical implementation of a new class of time integration schemes applied to linear heat conduction problems. The temperature field at any time level is calculated in terms of the numerical Green’s function matrix of the model problem by considering an analytical time integral equation. After spatial discretization by the finite element method, the Green’s function matrix which transfers solution from t to t + Δt is explicitly computed in nodal coordinates using efficient implicit and explicit Runge-Kutta methods. It is shown that the stability and the accuracy of the proposed method are highly improved when a sub-step procedure is used to calculate recursively the Green’s function matrix at the end of the first time step. As a result, with a suitable choice of the number of sub-steps, large time steps can be used without degenerating the numerical solution. Finally, the effectiveness of the present methodology is demonstrated by analyzing two numerical examples.     

A meshless method of lines for the numerical solution of KdV equation using radial basis functions

In this paper, a meshless method of lines (MOL) is presented for the numerical solution of the Korteweg–de Vries (KdV) equation. This novel method has an advantage over the traditional method of lines which approximates the spatial derivatives using finite difference method (FDM) or finite element method (FEM), because it does not need the mesh in the domain, and it approximates the solution using the radial basis functions (RBFs) on a set of node scattered in problem domain. A comparison among some RBFs is made in numerical examples. Numerical examples demonstrate the accuracy and easy implementation of this novel method and it is an efficient method for the nonlinear time-dependent partial differential equations (PDEs).     

Time-space method for multidimensional melting and freezing problems

In this paper, the time–space method (TSM) for multidimensional melting and solidification problems is proposed. In the proposed TSM, the timewise co-ordinate is incorporated into one of the spatial co-ordinates, thereby transforming the usual transient 2-D (or 3-D) problems into steady 3D (or 4-D) boundary-value problems. Since time integration is not necessary, the TSM has a feature that eliminates the so-called numerical instability which has been a great concern in the principal numerical methodologies in the past. That is, no error propagation in the timewise direction occurs in the TSM calculation. The TSM is applicable to almost all transient heat transfer and flow problems. The computer running time will be reduced to only 1/100th–1/1000th of the existing schemes for 2-D or 3-D problems. The sample calculations are presented for a 2-D melting problem in a square cavity and the validity of the present method is examined.     

A numerical method for unilateral problems

Variational inequalities connected with Signorini's problem have appeared as a natural generalization of the minimum potential-energy theorem for bodies with unilateral constraints. In this paper, we describe numerical experience on the use of variational inequalities and Pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity, like those describing the equilibrium configuration of an elastic membrane stretched over an elastic obstacle. These problems have the peculiar feature of being alternatively formulated as nonlinear boundary value problems without constraints for which the technique of Pade approximants can be successfully employed. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution.     

Use of singularity capturing functions in the solution of problems with discontinuous boundary conditions

A method is proposed to improve the accuracy of the numerical solution of elliptic problems with discontinuous boundary conditions using both global and local meshless collocation methods with multiquadrics as basis functions. It is based on the use of special functions which capture the singular behavior near discontinuities in boundary conditions. In the case of global collocation, the method consists in enlarging the functional space spanned by the RBF basis functions, while in the case of local collocation, the method consists in modifying appropriately the problem in order to eliminate the singularities from the formulation. Numerical results for benchmark problems such as a stationary heat equation in a box (harmonic) and Stokes flow in a lid-driven square cavity, show significant improvements in accuracy and in compliance with the continuity equation.