Shape optimisation with nearly conformal transformations

TL;DR

This paper introduces a novel shape optimisation algorithm that employs least squares solutions to the Cauchy-Riemann equations, enabling nearly conformal mesh deformations that preserve triangle angles during optimisation.

Contribution

The proposed method is a new approach that allows for mesh deformations in shape optimisation while maintaining mesh quality through nearly conformal transformations.

Findings

Effective preservation of mesh angles during deformation

Successful application to unconstrained shape functions

Demonstrated utility in constrained Stokes shape optimisation

Abstract

In shape optimisation it is desirable to obtain deformations of a given mesh without negative impact on the mesh quality. We propose a new algorithm using least square formulations of the Cauchy-Riemann equations. Our method allows to deform meshes in a nearly conformal way and thus approximately preserves the angles of triangles during the optimisation process. The performance of our methodology is shown by applying our method to some unconstrained shape functions and a constrained Stokes shape optimisation problem.