This paper presents a ÿnite element algorithm based on the adjoint method for the design of a certain class of solidiÿcation processes. In particular, the paper addresses the design of directional solidiÿcation processes for pure materials such that a desired freezing front heat ux and growth velocity are achieved. This is the ÿrst time that an inÿnite-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 overspeciÿed boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidiÿcation 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 deÿned 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 solidiÿcation of an initially superheated liquid aluminum conÿned 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 ÿnding 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. ?