The adjoint method for the inverse design of solidification processes with convection

This investigation presents an adjoint method formulation and computational implementation for certain class of inverse design problems in directional solidification processing. We are interested to find the appropriate heating/cooling thermal conditions at the mold boundaries to achieve a desired growth and other state conditions at the solid-liquid interface throughout the process. Such design objectives have profound implications on the obtained solidification microstructures and thus the quality of the casting product.; The inverse design problems of interest belong to a category of inverse problems with incomplete conditions at a part of the boundary but overspecified conditions at another part of the boundary (here at the solid-liquid interface). The solidification design problem is mathematically posed as a whole time domain optimization problem. An {dollar}Lsb2{dollar} cost functional is defined to measure the freezing interface's deviation from thermodynamic equilibrium under an arbitrary thermal boundary condition. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The process of minimization of the cost functional is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The Petrov-Galerkin streamline-upwinding finite element method with a predictor-corrector time-stepping scheme is chosen to solve these unsteady field equations.; The methodology is developed in three stages. First, the adjoint method is formulated for an inverse natural convection problem in fixed domain with laminar flow induced by temperature dependent density variations according to Boussinesq assumptions. The algorithm is tested by various examples with known solutions. Convergence, accuracy and parametric studies are performed to validate the inversely calculated solutions. Then, the adjoint formulation is applied to an inverse natural convection problem on a variable domain with an explicitly known moving boundary. In particular, we examine the design of solidification processes of pure substances that leads to desired growth and heat flux at the solid-liquid interface. An example case of directional solidification of liquid aluminum in a square cavity is carried out to eliminate the effects of natural convection on the interface morphology and growth velocity. At last, the adjoint formulation is extended to the inverse design of directional solidification processes of binary alloy. Double diffusive and thermal-solutal convection processes are involved as well as the phase change thermodynamics of the binary system. The inverse design problem identifies the thermal flux conditions on the mold walls to achieve a desired stable growth within the regime of the sharp solid-liquid interface model. Possible improvements and extensions of the current algorithm are discussed.