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

Abstract

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.

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References

Brussino, G. , and Sonnad, V. 1989. A comparison of direct and preconditioned iterative techniques for sparse, unsymmetric systems of linear equations, Internat. J. Numer. Methods Engrg. 28:801-815.

Chronopoulos, A.T. , and Ma, S. 1989a. Implementation of iterative methods for large sparse nonsymmetric linear systems on a parallel vector machine. Internat. J. Supercomput. Appl. 4(4):19-24.

Chronopoulos, A.T. , and Ma, S. 1989b. On squaring Krylov subspace iterative methods for nonsymmetric linear systems. UM CSci. TR-89-67. Minneapolis : University of Minnesota.

Comini, G. , Del Giudice, S. , Lewis, R.W. , and Zienkiewicz, O.C. 1974. Finite element solution of nonlinear heat conduction problems with special reference to phase change . Internat. J. Numer. Methods Engrg. 8:613-624.

Concus, P. , Golub, G.H. , and O'Leary, D.P. 1976. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In Sparse matrix computations, edited by J. R. Bunch and D. Rose. New York: Academic Press.

Crank, J. 1984. Free and moving boundary problems. Oxford: Clarendon Press.

Dalhuijsen, A.J. , and Segal, A. 1986. Comparison of finite element techniques for solidification problems. Internat. J. Numer. Methods Engrg. 23:1807-1829.

Eisenstat, S.C. , Elman, H.C. , and Schultz, M.H. 1983. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20:345-357.

Fletcher, R. 1976. Conjugate gradient methods for indefinite systems . Lecture Notes in Math. 509:73-89.

10.

Hestenes, M. , and Steifel, E. 1952. Method of conjugate gradients for solving linear systems . J. Res. Nat. Bur. Standards 49:409-439.

11.

Jennings, A. , and Malik, G.M. 1978. The solution of sparse linear equations by the conjugate gradient method. Internat. J. Numer. Methods Engrg. 12:141-158.

12.

Meijerink, J.A. , and Van der Vorst, H.A. 1977. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix . Math. Comp. 31:148-162.

13.

Rappaz, M. 1989. Modeling of microstructure formation in solidification processes. Internat. Mat. Rev. 34:93-123.

14.

Saad, Y. , and Schultz, M. 1986. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7:856-859.

15.

Shamsundar, N. , and Sparrow, E.M. 1975. Analysis of multidimensional conduction phase change via the enthalpy model. Trans. ASME, J. Heat Trans. 97:333-340.

16.

Sonneveld, P. 1989. CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 10:36-52.

17.

Van der Vorst, H.A. 1982. A vectorizable variant of some ICCG methods. SIAM J. Sci. Statist. Comput. 3:350-356.

18.

Vinsome, P. 1976. An iterative method for solving sparse sets of simultaneous linear equations. J. Res. Nat. Bur. Standards 49:409-439.

19.

Voller, V.R. 1990. A fast implicit finite difference method for the analysis of phase change problems. Num. Heat Trans. 17B:115-129.

20.

Voller, V.R. , and Prakash, C. 1987. A fixed grid numerical modeling methodology for convection/diffusion mushy region phase change problems. Internat. J. Heat Mass Trans. 30:1709-1719.

21.

Voller, V.R. , and Swaminathan, C.R. 1991. A general source-based method for solidification phase change. Num. Heat Trans. 19B:175-190.

22.

Voller, V.R. , Swaminathan, C.R. , and Thomas, B.G. 1990. Fixed grid techniques for phase change problems: a review . Internat. J. Numer. Methods Engrg. 30:875-898.

23.

White, R.E. 1986. A nonlinear parallel algorithm with application to the Stefan problem. SIAM J. Numer. Anal. 23:639-653.

24.

Williams, M.A. 1987. Iterative solution of a nonlinear system arising in phase change problems. Technical report ORNL-6398. Oak Ridge, Tenn.: Oak Ridge National Laboratory.

25.

Young, D.M. , and Lea, K.C. 1980. Generalized conjugate gradient accelerations of nonsymmetric iterative methods. Linear Algebra Appl. 34:159-194.

26.

Zienkiewicz, O.C. , and Taylor, C. 1989. The finite element method. New York : McGraw Hill.