Adjoint-Based Optimal Control of Time-Dependent Free Boundary Problems

TL;DR

This paper presents a simplified adjoint-based optimization method for free boundary problems using an extended operator splitting technique, enabling efficient shape optimization in complex, time-dependent PDE scenarios.

Contribution

It introduces a novel operator splitting approach that decouples domain deformation from PDE solving, simplifying the application of adjoint methods in free boundary problems.

Findings

Successfully applied to Navier-Stokes flow optimization.

Effective in shape optimization of Stefan-type problems.

Numerical verification confirms method's efficiency and accuracy.

Abstract

In this paper we show a simplified optimisation approach for free boundary problems in arbitrary space dimensions. This approach is mainly based on an extended operator splitting which allows a decoupling of the domain deformation and solving the remaining partial differential equation. First we give a short introduction to free boundary problems and the problems occurring in optimisation. Then we introduce the extended operator splitting and apply it to a general minimisation subject to a time-dependent scalar-valued partial differential equation. This yields a time-discretised optimisation problem which allows us a quite simple application of adjoint-based optimisation methods. Finally, we verify this approach numerically by the optimisation of a flow problem (Navier-Stokes equation) and the final shape of a Stefan-type problem.