Energy stable gradient flow schemes for shape and topology optimization in Navier-Stokes flows

TL;DR

This paper introduces energy-stable gradient flow schemes for topology optimization in Navier-Stokes flows, utilizing phase field models and demonstrating their stability and effectiveness through numerical experiments.

Contribution

It proposes novel stabilized semi-implicit schemes for Allen-Cahn and Cahn-Hilliard gradient flows with proven energy stability for Navier-Stokes based topology optimization.

Findings

Unconditional energy stability of the schemes in continuous and discrete settings.

Effective and robust optimization algorithms demonstrated in 2D and 3D CFD simulations.

Abstract

We study topology optimization governed by the incompressible Navier-Stokes flows using a phase field model. Novel stabilized semi-implicit schemes for the gradient flows of Allen-Cahn and Cahn-Hilliard types are proposed for solving the resulting optimal control problem. Unconditional energy stability is shown for the gradient flow schemes in continuous and discrete spaces. Numerical experiments of computational fluid dynamics in 2d and 3d show the effectiveness and robustness of the optimization algorithms proposed.