Numerical Methods for Shape Optimal Design of Fluid-Structure Interaction Problems

TL;DR

This paper develops a numerical framework for shape optimization in unsteady fluid-structure interaction problems, integrating geometric and differential considerations, and validates it through benchmark simulations.

Contribution

It introduces a discretization approach that computes exact discrete gradients, enabling the use of general optimization solvers for FSI shape optimization.

Findings

Validated the numerical implementation on an FSI benchmark problem.

Demonstrated optimization of boundary and interface shapes.

Theoretically showed the adjoint structure in a linear case reverses temporal flow.

Abstract

We consider the method of mappings for performing shape optimization for unsteady fluid-structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on an FSI benchmark problem to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.