Penalization of stationary Navier-Stokes equations and applications in topology optimization

TL;DR

This paper investigates a penalization approach for steady Navier-Stokes equations with mixed boundary conditions, providing error estimates, numerical tests, and applications in topology optimization and geometric inverse problems.

Contribution

It introduces a penalization method for the stationary Navier-Stokes system, with new error estimates and convergence results applicable to topology optimization and inverse problems.

Findings

Error estimates for penalized solutions with small viscosity

Numerical tests demonstrating the method's effectiveness

Convergence results for geometric inverse and control problems

Abstract

We consider the steady Navier-Stokes system with mixed boundary conditions, in subdomains of a holdall domain. We study, via the penalization method, its approximation properties. Error estimates, obtained using the extension operator, other evaluations and the uniqueness of the solution, when the viscosity may be arbitrarily small in certain subdomains, are also discussed. Numerical tests, including topological optimization applications, are presented. A general convergence result for the approximation of this type of geometric inverse problems and of the associated optimal control problems, is investigated in the last part of the paper.