The adjoint method for an inverse design problem in the directional solidification of binary alloys
The authors propose a systematic continuum formulation of the adjoint method to solve an inverse thermo-solutal convection problem. The objective is to control the boundary heating/cooling fluxes such that solidification of dilute binary alloys proceeds with a desired stable interface growth. Such studies identify possible inconsistencies in the previous binary alloy solidification models that a priori assume a macroscopically sharp solid/liquid interface. The stability criterion is here chosen as the absence of constitutional undercooling in the liquid ahead of the interface. Such relation of the interface stability is treated as an overspecified boundary condition in the inverse problem formulation. Based on additional physical arguments of marginal stability and minimum energy input, the constitutional undercooling condition is implemented in the form of an equality. Herein, the cost functional is defined as the thermal deviation of the freezing interface from thermodynamic equilibrium. The inverse design problem is posed as the calculation of the optimum boundary heat flux that leads to the minimization of the cost functional. A close form of the adjoint equation system is derived with coupled thermal, solute, and flow transport mechanism. The definition of the adjoint system is made such that the boundary value of the adjoint thermal variable at the fixed wall is equal to the gradient of the cost functional. The conjugate gradient algorithm is used to solve for the optimum boundary heat flux that minimizes the cost functional. Since such a heat flux is a function varying in space and time, the adjoint method provides an elegant and efficient numerical scheme for the solution of this class of problems. An example case of the solidification of \(\text{NH}_4\text{Cl}\) water solution in a rectangular cavity is performed for moderate strength of thermo-solutal convection. A small amount of `over-stability' is introduced to overcome the problem of loss of stability because of the accuracy level that the numerical optimization scheme can reach. The results are validated to show the achievement of a stable vertical flat freezing interface growth at the desired growth velocity.
inverse thermo-solutal convection problem
scientific article; zbMATH DE number 1283488