Abstract: ompositional profiles of binary crystals, growth by directional solidification, can be modelated by a diffusive problem in the presence of a moving solid-liquid interface [1]. Unless very specific cases, there are not analytical solutions and the one-dimensional direct problem is solved by an implicit finite difference scheme that requires the knowledge of the diffusion coefficient, that is itself a function of the composition. The associated implicit inverse solver reconstructs this coefficient from synthetic data related to the solid compositional profile. The objective function is given by the sum of the square differences between model and synthetic compositional profiles. A former work [2] proposed the use of an Ant Colony System (ACS) implementation to reconstruct the diffusion coefficient. The ACS [3] was originally developed as a computational method for stochastic combinatorial optimization. A particular scheme for coding real into integer numbers was used for the discrete values of the diffusion coefficient. In the ACS paradigm, the best ant of its generation, proceeding along a path, sheds a substance called pheromone. Each path is associated to a possible solution and paths that are marked by several ants in sucessive generations have more probability to be chosen. This implies that ants can learn from the experience of previous generations. However, no regularization technique was used in that work and results were poor, even for noiseless data. The current work proposes a new regularization scheme, specific for the ACS inverse solver in crystal growth problems. The use of extra information can be imbedded in the ACS in order to influence the choice of the paths. In this particular case, the additional information is related to the smoothness of the solution curve, the diffusion coefficient versus composition. This approach is different from the traditional formulation that includes the regularization term in the objective function. Results are presented for noisy data.

References:

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

[2] N.N. Arai, R.P. Souto, A.J. Preto, H.F. de Campos Velho, J.C. Becceneri, M. Fabbri, S. Stephany (2002): An inverse formulation for diffusive Bridgman growth using ant-colony optimization in a high performance environment, Brazilian Congresso on Applied and Computational Mathematics, Brazil.

[3] M. Dorigo, V. Maniezzo, A. Colorni (1996): The ant system: optimization by a colony of cooperating agents, IEEE T. Syst. Man Cy. B, 26(2), 29-41.