The Stefan Problem Solved Via Conjugate Gradient—Like Iterative Methods On a Parallel Vector Machine

The aim of this paper is to illustrate the validity and efficiency of iterative methods for solving large linear systems arising from the finite element discretizations of the equation governing conduction-controlled solidi fication processes. Starting from the basic enthalpy equation, two alternative formulations are obtained and fixed-grid finite element discretizations are devel oped. These discretizations yield a set of nonlinear equations that are linearized using the Newton-Raph son scheme. The linearized equations are used as a basis for evaluating different iterative methods of the conjugate gradient type. Symmetric scaling and in complete factorization preconditioning of the linear equations are used to improve the convergence prop erties of the iterative methods. Vectorization and paral lelization are also employed to make full use of the CRAY-2 supercomputer. The results indicate that the implementation of currently available iterative solvers leads to efficient solution methodologies for phase change problems.

[1]  P. K. W. Vinsome,et al.  Orthomin, an Iterative Method for Solving Sparse Sets of Simultaneous Linear Equations , 1976 .

[2]  Ephraim M Sparrow,et al.  ANALYSIS OF MULTIDIMENSIONAL CONDUCTION PHASE CHANGE VIA THE ENTHALPY MODEL. , 1975 .

[3]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[4]  R. White.  A nonlinear parallel algorithm with application to the Stefan problem , 1986 .

[5]  M. Rappaz,et al.  Modelling of microstructure formation in solidification processes , 1989 .

[6]  Henk A. van der Vorst,et al.  A Vectorizable Variant of some ICCG Methods , 1982 .

[7]  R. Fletcher.  Conjugate gradient methods for indefinite systems , 1976 .

[8]  Anthony T. Chronopoulos,et al.  Implementation of Iterative Methods for Large Sparse Nonsymmetric Linear Systems On a Parallel Vector Machine , 1990, Int. J. High Perform. Comput. Appl..

[9]  Anthony T. Chronopoulos,et al.  On Squaring Krylov Subspace Iterative Methods for Nonsymmetric Linear Systems on Squaring Krylov Subspace Iterative Methods for Nonsymmetric Linear Systems , 2022 .

[10]  Brian G. Thomas,et al.  Fixed grid techniques for phase change problems: A review , 1990 .

[11]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[12]  J. Crank.  Free and moving boundary problems , 1984 .

[13]  Kang C. Jea,et al.  Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods , 1980 .

[14]  A. Segal,et al.  Comparison of finite element techniques for solidification problems , 1986 .

[15]  Vijay Sonnad,et al.  A comparison of direct and preconditioned iterative techniques for sparse, unsymmetric systems of linear equations , 1989 .

[16]  V. Voller,et al.  A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems , 1987 .

[17]  J. Z. Zhu,et al.  The finite element method , 1977 .

[18]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[19]  Paul T. Boggs,et al.  Moving boundary problems , 1978 .

[20]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[21]  Roland W. Lewis,et al.  Finite element solution of non‐linear heat conduction problems with special reference to phase change , 1974 .

[22]  A. Jennings,et al.  The solution of sparse linear equations by the conjugate gradient method , 1978 .

[23]  V. Voller,et al.  ERAL SOURCE-BASED METHOD FOR SOLIDIFICATION PHASE CHANGE , 1991 .

[24]  V. Voller.  FAST IMPLICIT FINITE-DIFFERENCE METHOD FOR THE ANALYSIS OF PHASE CHANGE PROBLEMS , 1990 .

[25]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .