CONTROL OF THE FREEZING INTERFACE MORPHOLOGY IN SOLIDIFICATION PROCESSES IN THE PRESENCE OF NATURAL-CONVECTION

This paper presents a methodology for the solution of an inverse solid ification design problem in the presence of natural convection. In par ticular, the boundary heat flux q(0) in the fixed mold wall, partial d erivative Omega(0), is calculated such that a desired freezing front v elocity and shape are obtained. As the front velocity together with th e flux history q(ms) on the solid side of the freezing front play a de terminant role in the obtained cast structure, the potential applicati ons of the proposed methods to the control of casting processes are en ormous. The proposed technique consists of first solving a direct natu ral convection problem of the liquid phase in an a priori known shrink ing cavity, Omega(L)(t), before solving an ill-posed inverse design co nduction problem in the solid phase in an a priori known growing regio n, Omega(S)(t). The direct convection problem is used to evaluate the flux q(ml) in the liquid side of the freezing Front. A front tracking deforming finite element technique is employed. The flux q(ml) can be used together with the Stefan condition to provide the freezing interf ace flux q(ms) in the solid side of the front. As such, two boundary c onditions (flux q(ms) and freezing temperature theta(m)) are especifie d along the (known) freezing interface partial derivative Omega(l)(t). The developed design technique uses the adjoint method to calculate i n L(2) the derivative of the cost functional, \\theta m - theta(x,t;q( 0))\\(2)(L2), that expresses the square error between the calculated t emperature theta(x,t;q(0)) in the solid phase along partial derivative Omega(I)(t) and the given melting temperature. The minimization of th is cost functional is performed by the conjugate gradient method via t he solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis . Finally, an example is presented for the solidification of a superhe ated incompressible liquid aluminium, where the effects of natural con vection in the moving interface shape are controlled with a proper adj ustment of the cooling boundary conditions.