Adjoint-based optimization of two-dimensional Stefan problems

A range of optimization cases of two-dimensional Stefan problems, solved using a tracking-type cost-functional, is presented. A level-set method is used to capture the interface between the liquid and solid phases and an immersed boundary (cut cell) method coupled with an implicit time-advancement scheme is employed to solve the heat equation. A conservative implicit-explicit scheme is then used for solving the level-set transport equation. The resulting numerical framework is validated with respect to existing analytical solutions of the forward Stefan problem. An adjoint-based algorithm is then employed to efficiently compute the gradient used in the optimization algorithm (L-BFGS). The algorithm follows a continuous adjoint framework, where adjoint equations are formally derived using shape calculus and transport theorems. A wide range of control objectives are presented, and the results show that using parameterized boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape.