Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow

TL;DR

This paper develops and analyzes an adaptive nonconforming finite element method for phase-field topology optimization in Stokes flow, proving convergence and demonstrating effectiveness through numerical examples.

Contribution

It introduces a new convergence analysis for an adaptive nonconforming FEM applied to phase-field topology optimization governed by Stokes equations.

Findings

The adaptive method's minimizers converge to the optimal solution.

The pressure fields from the discrete solutions also converge.

Numerical results show the adaptive method outperforms uniform refinement.

Abstract

In this work, we develop an adaptive nonconforming finite element algorithm for the numerical approximation of phase-field parameterized topology optimization governed by the Stokes system. We employ the conforming linear finite element space to approximate the phase field, and the nonconforming linear finite elements (Crouzeix-Raviart elements) and piecewise constants to approximate the velocity field and the pressure field, respectively. We establish the convergence of the adaptive method, i.e., the sequence of minimizers contains a subsequence that converges to a solution of the first-order optimality system, and the associated subsequence of discrete pressure fields also converges. The analysis relies crucially on a new discrete compactness result of nonconforming linear finite elements over a sequence of adaptively generated meshes. We present numerical results for several examples to illustrate the performance of the algorithm, including a comparison with the uniform refinement strategy.