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

Bangti Jin, Jing Li, Yifeng Xu, Shengfeng Zhu

TL;DR

The paper develops a nonconforming Crouzeix-Raviart finite element approach for phase-field topology optimization in Stokes flow, pairing CR velocity discretization with conforming linear phase-field and piecewise-constant pressure. It proves strong convergence of discrete minimizers and associated velocity/pressure to the continuous optimal design as the mesh is refined, utilizing a novel discrete compactness argument for CR spaces and a Brenner-type connection to a conforming FE space. A gradient-flow, augmented-Lagrangian solver is proposed for the discrete problem, and extensive 2D and 3D numerical experiments demonstrate that the CR method achieves comparable optimal designs to Taylor-Hood while significantly reducing degrees of freedom and computational time. The work thereby provides a theoretically sound and practically efficient framework for phase-field topology optimization in fluid-structure interactions with Stokes flow.

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.

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

TL;DR

The paper develops a nonconforming Crouzeix-Raviart finite element approach for phase-field topology optimization in Stokes flow, pairing CR velocity discretization with conforming linear phase-field and piecewise-constant pressure. It proves strong convergence of discrete minimizers and associated velocity/pressure to the continuous optimal design as the mesh is refined, utilizing a novel discrete compactness argument for CR spaces and a Brenner-type connection to a conforming FE space. A gradient-flow, augmented-Lagrangian solver is proposed for the discrete problem, and extensive 2D and 3D numerical experiments demonstrate that the CR method achieves comparable optimal designs to Taylor-Hood while significantly reducing degrees of freedom and computational time. The work thereby provides a theoretically sound and practically efficient framework for phase-field topology optimization in fluid-structure interactions with Stokes flow.

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 -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.
Paper Structure (8 sections, 7 theorems, 79 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 8 sections, 7 theorems, 79 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Crouzeix-Raviart FE approximation of topology optimization in fluid
  3. Problem formulation
  4. Crouzeix-Raviart FE approximation
  5. Numerical solver
  6. Convergence analysis of the FE approximation
  7. Numerical results and discussions
  8. Acknowledgments

Key Result

Lemma 2.1

Let Assumption ass:problem (i) and (iv) hold. Let $\{\phi_n\}_{n\geq 0} \subset \mathcal{U}$ converge to $\phi \in U$ in $L^1(\Omega)$. Then $\{\bold{\mathcal{S}}(\phi_n)\}_{n\geq 0}$ converges to $\bold{\mathcal{S}}(\phi)$ strongly in $H^1(\Omega)$.

Figures (10)

  • Figure 1: The initial phase-field functions for the 2d cases.
  • Figure 2: The optimal designs $\phi_k^\ast$ over each $\mathcal{T}_k$ for case \ref{['exp:pipebend']} by CR-$P_0$ (1st row) and $P_2$-$P_1$ (2nd row).
  • Figure 3: The optimal designs $\phi_k^\ast$ over each $\mathcal{T}_k$ for case \ref{['exp:leftinflow']} by CR-$P_0$ (1st row) and $P_2$-$P_1$ (2nd row).
  • Figure 4: The optimal designs $\phi_k^\ast$ over each $\mathcal{T}_k$ for case \ref{['exp:threeinflows']} by CR-$P_0$ (1st row) and $P_2$-$P_1$ (2nd row).
  • Figure 5: The optimal designs $\phi_k^\ast$ over each $\mathcal{T}_k$ for case \ref{['exp:rugby']} by CR-$P_0$ (1st row) and $P_2$-$P_1$ (2nd row).
  • ...and 5 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Lemma 3.3
  • ...and 5 more