Adjoint-based optimization of two-dimensional Stefan problems

TL;DR

This paper develops an adjoint-based optimization framework for two-dimensional Stefan problems, integrating level set and immersed boundary methods, validated against analytical solutions, to optimize interface control strategies.

Contribution

It introduces a novel adjoint-based optimization approach combined with advanced numerical methods for efficient control of Stefan problems.

Findings

Effective control strategies suppress interfacial instabilities.

Parameterised boundary actuation maintains desired crystal shapes.

Validated numerical framework matches analytical solutions.

Abstract

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 optimisation 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 parameterised boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape.