On the Crouzeix-Raviart Finite Element Approximation of Phase-Field Dependent Topology Optimization in Stokes Flow

TL;DR

This paper explores a nonconforming finite element method for phase-field topology optimization in Stokes flow, demonstrating improved efficiency and convergence properties through theoretical analysis and numerical experiments.

Contribution

It introduces a nonconforming Crouzeix-Raviart finite element approach for phase-field topology optimization, showing convergence and efficiency improvements over conforming methods.

Findings

Nonconforming FEM reduces degrees of freedom compared to conforming FEM.

The numerical scheme converges to a minimizer in relevant norms.

Numerical results compare favorably with Taylor-Hood elements.

Abstract

In this work, we investigate a nonconforming finite element approximation of phase-field parameterized topology optimization governed by the Stokes flow. The phase field, the velocity field and the pressure field are approximated by conforming linear finite elements, nonconforming linear finite elements (Crouzeix-Raviart elements) and piecewise constants, respectively. When compared with the standard conforming counterpart, the nonconforming FEM can provide an approximation with fewer degrees of freedom, leading to improved computational efficiency. We establish the convergence of the resulting numerical scheme in the sense that the sequences of phase-field functions and discrete velocity fields contain subsequences that converge to a minimizing pair of the continuous problem in the $H^1$-norm and a mesh-dependent norm, respectively. We present extensive numerical results to illustrate the performance of the approach, including a comparison with the popular Taylor-Hood elements.