Tomas Fullana - (d'Alembert) - Adjoint-based optimization of two-dimensional Stefan problems

A range of optimization cases of two-dimensional Stefan problems with a tracking-type cost-functional is presented. We use a level set method for capturing the interface between the liquid and solid phases and discrete Cut Cell operators coupled with an implicit scheme for solving the heat equation. A conservative implicit-explicit scheme is used for solving the level set advection equation. The continuous adjoint is derived formally using tools from shape calculus and transport theorems. Validation of our method with respect to some analytical solutions of the forward Stefan problem is provided. Results show good convergence of the L-BFGS optimization algorithm in this framework.