The Adjoint Method for an Inverse Design Problem in the Directional Solidification of Binary Alloys

This paper presents a systematic procedure based on the adjoint method for solving a class of inverse directional alloy solidification design problems in which a desired growth velocityv_fis achieved under stable growth conditions. To the best of our knowledge, this is the first time that a continuum adjoint formulation is proposed for the solution of an inverse problem with simultaneous heat and mass transfer, thermo-solutal convection, and phase change. In this paper, the interfacial stability is considered to imply a sharp solid–liquid freezing interface. This condition is enforced using the constitutional undercooling criterion in the form of an inequality constraint between the thermal and solute concentration gradients,GandG_c, respectively, at the freezing front. The main unknowns of the design problem are the heating and/or cooling boundary conditions on the mold walls. The inverse design problem is formulated as a functional optimization problem. The cost functional is defined by the square of theL₂norm of the deviation of the freezing interface temperature from the temperature corresponding to thermodynamic equilibrium. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, and velocity fields such that the gradient of the cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the finite element method solutions of the continuum direct, sensitivity, and adjoint problems. The developed formulation is demonstrated with an example of designing the directional solidification of a binary aqueous solution in a rectangular mold such that a stable vertical interface advances from left to right with a desired growth velocity.