ISSN:
0029-5981
Keywords:
Engineering
;
Engineering General
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Mathematics
,
Technology
Notes:
In this paper we present a Legendre spectral element method for solution of multi-dimensional unsteady change-of-phase Stefan problems. The spectral element method is a high-order (p-type) finite element technique, in which the computational domain is broken up into general (curved) quadrilateral macroelements, and the solution, data and geometry are expanded within each element in terms of tensor-product Lagrangian interpolants. The discrete equations are generated by a Galerkin formulation followed by Gauss-Lobatto Legendre quadrature, for which it is shown that exponential convergence to smooth solutions is obtained as the polynomial order of fixed elements is increased. The spectral element equations are inverted by conjugate gradient iteration, in which the matrix-vector products are calculated efficiently using tensor-product sum-factorization.To solve the Stefan problem numerically, the heat equations in the liquid and solid phases are transformed to fixed domains applying an interface-local time-dependent immobilization transformation technique. The modified heat equations are discretized using finite differences in time, resulting at each time step in a Helmholtz equation in space that is solved using Legendre spectral element elliptic discretizations. The new interface position is then computed using a variationally consistent flux treatment along the phase boundary, and the solution is projected upon the corresponding updated mesh. The rapid convergence rate and stability of the method are discussed, and numerical results are presented for a one-dimensional Stefan problem using both a semi-implicit and a fully implicit time-stepping scheme. Finally, a two-dimensional Stefan problem with a complex phase boundary is solved using the semi-implicit scheme.
Additional Material:
17 Ill.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/nme.1620241204
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