On the solution of an ill-posed design solidification problem using minimization techniques in finite- and infinite-dimensional function spaces

This paper provides a comparative study of two alternative methodologies for the solution of an inverse design solidification problem. It is the one-dimensional solidification problem of calculating the boundary heat flux history that achieves a desired freezing front velocity and desired heat fluxes at the freezing front. The front velocity h(t) and flux history q_mS(t) on the solid side of the front control the obtained cast structure. As such, the potential applications of the proposed methods to the control of casting processes are enormous. The first technique utilizes a finite-dimensional approximation of the unknown boundary heat flux function q₀(t). The second technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ‖T_m – T(h(t), t;q₀)‖, that expresses the square error between the calculated T(h(t), t; q₀) and the given freezing front temperature T_m. Both steepest descent (SDM) and conjugate gradient methods (CGM) are examined. A front tracking FEM technique is used for the discretization of the state space. A detailed numerical analysis of the space and time discretization of the ‘parameter’ and state spaces, of the effect of the end condition of the adjoint problem and of other parameters in the solution are examined.