An efficient direct BEM numerical scheme for phase change problems using Fourier series

Abstract

Due to the moving boundary nature of the solidification and melting problems, the application of the standard time domain boundary element formulation of the problem presents the difficulty of the numerical evaluation of the history dependence, representing more and more work as time proceeds. In that regard, we present an algorithm based on a Green's function which allows a Fourier representation of the domain integral in the reinitialization scheme of the direct boundary element method formulation of the problem, from which it is possible to carry out the history integration in a recursive manner. To obtain the reinitialization approach, in the integral representation formula, the Green's function corresponding to zero temperature in a box containing the original domain will be used, instead of using the classical free space fundamental solution. This Green's function is given in terms of the original fundamental solution plus a regular solution of the heat equation inside the domain in consideration. It can therefore be used in the integral representation formula of the heat equation (direct formulation) to obtain the solution of a heat problem in such domain. The mentioned Green's function can be obtained by the images method, and the resulting source series can also be rewritten in terms of a double Fourier series, that will be used in the domain integral of the integral representation formula to transform such integral into equivalent surface integrals. In this work a new reinitialization scheme has been applied to solve moving boundary problems, to detect the location of the phase-change interface in solidification and melting problems. The numerical results were compared with analytical solutions (in the case, where they were available) showing a good agreement between the results, giving special emphasis to the interface location and the temperature distribution.