Numerical approximation of phase field based shape and topology optimization for fluids

TL;DR

This paper develops a phase field based numerical method for shape and topology optimization of fluid domains governed by Navier-Stokes equations, employing diffuse interfaces, gradient flows, and adaptive finite element techniques.

Contribution

It introduces a well-posed phase field formulation for fluid shape optimization with a novel numerical scheme using mass conserving gradient flows and adaptive mesh refinement.

Findings

The formulation is mathematically well-posed with proven existence of minimizers.

The numerical scheme effectively solves the optimization problem with adaptive mesh refinement.

Comparison studies demonstrate the method's accuracy and efficiency.

Abstract

We consider the problem of finding optimal shapes of fluid domains. The fluid obeys the Navier--Stokes equations. Inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow. We formulate a corresponding optimization problem where flow outside the fluid domain is penalized. The resulting formulation of the shape optimization problem is shown to be well-posed, hence there exists a minimizer, and first order optimality conditions are derived. For the numerical realization we introduce a mass conserving gradient flow and obtain a Cahn--Hilliard type system, which is integrated numerically using the finite element method. An adaptive concept using reliable, residual based error estimation is exploited for the resolution of the spatial mesh. The overall concept is numerically investigated and comparison values are provided.