An Adjoint Method for the Inverse Design of Solidification Processes with Natural Convection

This paper presents a nite element algorithm based on the adjoint method for the design of a certain class of solidi cation processes. In particular, the paper addresses the design of directional solidi cation processes for pure materials such that a desired freezing front heat ux and growth velocity are achieved. This is the rst time that an in nite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse=design problems with moving boundaries and Boussinesq incompressible ow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspeci ed boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidi cation design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately de ned continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidi cation of an initially superheated liquid aluminum con ned in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as nding the appropriate spatial-temporal variations of the boundary heat ux on the vertical mold walls that can eliminate or reduce the e ects of convection on the freezing interface heat uxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed. ? 1998 John Wiley & Sons, Ltd.