Abstract
A general numerical finite element scheme is described for parabolic problems with phase change wherein the elements of the domain are allowed to deform continuously. The scheme is based on the Galerkin approximation in space, and finite difference approximation for the time derivatives. The numerical scheme is applied to the two-phase Stefan problems associated with the melting and solidification of a substance. Basic functions based on Hermite polynomials are used to allow exact specification of flux-latent heat balance conditions at the phase boundary. Numerical results obtained by this scheme indicates that the method is stable and produces an accurate solutions for the heat conduction problems with phase change; even when large time steps used. The method is quite general and applicable for a variety of problems involving transition zones and deforming regions, and can be applied for one multidimensional problems.
Highlights
Application of the finite element method to transport problems is well established, and several valuable texts and compilations are currently available
The use of Galerkin method in conjuction with the finite element technique has significantly extended its scope such that a broad class of transient field problems can be approached confidently with finite elements, for problems allowing continuous mesh deformation, which result that a moving boundary lies on element boundaries, Halabi [i]
The finite element method has proven to be valuable in these types of problems
Summary
Application of the finite element method to transport problems is well established, and several valuable texts and compilations are currently available. The finite element method has proven to be valuable in these types of problems It provides a mechanism for eneratin difference equations on a non-u[iform mesh and allows the use of higher order elements in regions where they are suited to the physics of the problem. The proposed model in this researh, described an internal transition zone ’internal moving boundary’, changing primarily in location, as a function of time. As freezing proceeds with time, the 1ocaion of the transition zone changes, and an important quantities change rapidly across it The complexity in these types of problems arises on the mechanism of generating the governing difference equations on the non-uniform mesh of the domain of definition. S is the internal moving boundary as shown in Figure (I)
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