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

Bangti Jin, Jing Li, Yifeng Xu, Shengfeng Zhu

TL;DR

This work develops an adaptive nonconforming finite element method for phase-field dependent topology optimization in Stokes flow, using a conforming linear space for the phase field φ, a nonconforming Crouzeix–Raviart space for velocity u, and piecewise-constant pressure p. It introduces two residual-based a posteriori estimators and an OPTIMIZE–ESTIMATE–MARK–REFINE loop to drive adaptivity, with a Pi_k operator ensuring φ remains within the admissible set. The authors prove convergence: a subsequence of discrete minimizers and associated velocities converges to a solution of the continuous first-order optimality system, and the discrete pressures converge in L^2. Numerical experiments in 2D and 3D show the adaptive method efficiently resolves interfaces, yielding smoother designs and reduced computational effort compared with uniform refinement, thereby highlighting practical benefits for phase-field topology optimization in fluid flows.

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.

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

TL;DR

This work develops an adaptive nonconforming finite element method for phase-field dependent topology optimization in Stokes flow, using a conforming linear space for the phase field φ, a nonconforming Crouzeix–Raviart space for velocity u, and piecewise-constant pressure p. It introduces two residual-based a posteriori estimators and an OPTIMIZE–ESTIMATE–MARK–REFINE loop to drive adaptivity, with a Pi_k operator ensuring φ remains within the admissible set. The authors prove convergence: a subsequence of discrete minimizers and associated velocities converges to a solution of the continuous first-order optimality system, and the discrete pressures converge in L^2. Numerical experiments in 2D and 3D show the adaptive method efficiently resolves interfaces, yielding smoother designs and reduced computational effort compared with uniform refinement, thereby highlighting practical benefits for phase-field topology optimization in fluid flows.

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.

Paper Structure

This paper contains 9 sections, 13 theorems, 129 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Adaptive algorithm
  3. Nonconforming FEM approximation
  4. Adaptive phase-field algorithm
  5. Convergence of adaptive algorithm
  6. Zero limit of estimators
  7. Convergence of the discrete minimizing pair
  8. Numerical results and discussions
  9. Funding

Key Result

Lemma 3.1

There holds for $2\leq p < \infty$ when $d=2$ or $2 \leq p \leq 6$ when $d=3$ with the constant $c_{\mathrm{ds}}$ only depending on the shape regularity of $\mathcal{T}_k$.

Figures (8)

  • Figure 1: Initial phase-field functions for Examples \ref{['exp:leftinflow']}-\ref{['exp:pipe3d']}.
  • Figure 2: Numerical results for Example \ref{['exp:leftinflow']} from top to bottom: mesh, optimized design $\phi_k^*$, estimators $\eta_{k,1}$ and $\eta_{k,2}$. The number of vertices on each mesh is 1441, 2954, 6946 and 17354.
  • Figure 3: The convergence history of the total energy (top) and the volume constraint error (bottom) versus the total number ($K \times N$) of outer iterations performed.
  • Figure 4: Numerical results for Example \ref{['exp:threeinflows']} from top to bottom: mesh, optimized designs $\phi_k^\ast$, and the estimators $\eta_{k,1}$ and $\eta_{k,2}$. The number of vertices on each mesh is 1441, 3070, 7442 and 19364.
  • Figure 5: Numerical results for Example \ref{['exp:bypass']} from top to bottom: mesh, optimized designs $\phi_k^\ast$, and the estimators $\eta_{k,1}$ and $\eta_{k,2}$. The number of vertices on each mesh is 2174, 4973, 11669 and 28468.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.4
  • proof
  • ...and 18 more