Inverse design of directional solidification processes in the presence of a strong external magnetic field

A computational method for the design of directional alloy solidi cation processes is addressed such that a desired growth velocity vf under stable growth conditions is achieved. An externally imposed magnetic eld is introduced to facilitate the design process and to reduce macrosegregation by the damping of melt ow. The design problem is posed as a functional optimization problem. The unknowns of the design problem are the thermal boundary conditions. The cost functional is taken as the square of the L2 norm of an expression representing the deviation of the freezing interface thermal conditions from the conditions corresponding to local thermodynamic equilibrium. The adjoint method for the inverse design of continuum processes is adopted in this work. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, velocity and electric potential elds such that the gradient of the L2 cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the FE solutions of the continuum direct, sensitivity and adjoint problems. The developed formulation is demonstrated with an example of designing the boundary thermal uxes for the directional growth of a germanium melt with dopant impurities in the presence of an externally applied magnetic eld. The design is shown to achieve a stable interface growth at a prescribed desired growth rate