Abstract: A method of reconstructing the diffusion coefficient with dependence on the composition is presented. It is considered the liquid phase of a binary alloy, under Bridgman growth. The proposed method uses Ant Colony Optimization [2]. The directional solification of totally miscible alloy under a constant growing rate, inside of a closed ampoule, can be modelled as a diffusion problem with a moving solid-liquid boundary. Its solution yields the axial profile of the final solid composition, and depends on the diffusion coefficient which is itself a function of the composition [1]. The model was numerically implemented by a finite difference method. In the case of Bridgman growth in microgravity conditions, or reduced diameter ampoules, and in the abscence of constitutional supercooling, the experimental profile is well decribed by a purely diffusive model under quasi-equilibrium [4]. The associated inverse problem consists of recovering the diffusion coefficient from the solid composition profile. This optimization problem is solved by a meta-heuristic based on the behaviour of ants over a trajectory between two places in terrain. Each candidate solution is assessed by one ant and, since the ants are totally independently each other, the method can be efficiently parallelized. The inversion was executed in a distributed memory machine, a 17-node multicomputer based on IA-32 architecture, using Fortran 90 code and the MPI (Message Passing Interface) communication library [3]. The inversion was performed from noiseless synthetic data and the reconstructed diffusion coefficient curve was close to the exact solution, but presented some spikes. Future work points out to the use of some regularization technique in order to get smoother curve [5]. As it would be expected, increasing the number of ants per generation caused better efficiency: up to 6 processors this figure was above 0.7 for 30 ants and above 0.9 for 300 ants. This was expected once, for each generation, every processor selects its best ant and the best-of-the-bests ant is chosen among all processors. Thus an increasing number of ants per generation uses more processing time for the same amount of communication time.

References:

[1] V. Alexiades, A.D. Solomon (1993): Mathematical modeling of melting and freezing processes, Hemisphere Publishing Corporation Washington.

[2] M. Dorigo, V. Maniezzo, A. Colorni (1999): The ant system: optimization by a colony of cooperating agent, IEEE Trans. Syst. Man Cy. B, 26 (2), 29-41.

[3] W. Gropp, E. Lusk, A. Skjellum (1999): Using MPI: Portable Parallel Programming with the Message-Passing Interface, Massachusetts: MIT.

[4] K. Kinoshita, T. Yamada (1990): Conditions for diffusion-controlled steady-state growth of Pb_1-xSn_xTe under microgravity, J. Cryst. Growth, 99, 1276-1280.

[5] W.B. Muniz, F.M. Ramos, H.F. Campos Velho (2000): Entropy- and Tikhonov-based Regularization Techniques Applied to the Backwards Heat Equation, Comput. Math. Appl., 40, 1071-1084.