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Adaptive Computation of Elliptic Eigenvalue Topology Optimization with a Phase-Field Approach

Jing Li, Yifeng Xu, Shengfeng Zhu

TL;DR

This work develops and analyzes an adaptive finite element method for elliptic eigenvalue topology optimization in a phase-field framework. By coupling a phase-field design variable with a small-parameter Ginzburg–Landau regularization and solving on adaptively refined meshes, the authors introduce a set of residual-based estimators and a separate-collective marking strategy to drive refinement. Although a guaranteed reliability bound for the estimators is not established due to nonlinearity, they prove vanishing limits for the estimators and show convergence of a subsequence of adaptive solutions to a solution of the continuous optimality system, supported by a limiting problem and convergence lemmas. Numerical experiments in two dimensions demonstrate that the adaptive approach achieves comparable objective values to uniform refinement while markedly reducing computation time and accurately capturing diffuse interfaces and singularities near corners, with additional insights into symmetry phenomena in certain domains.

Abstract

In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional numerical examples for illustration of efficiency and accuracy. Theoretical findings consist in the vanishing limit of a subsequence of estimators and the convergence of the relevant subsequence of adaptively-generated solutions to a solution to the continuous optimality system.

Adaptive Computation of Elliptic Eigenvalue Topology Optimization with a Phase-Field Approach

TL;DR

This work develops and analyzes an adaptive finite element method for elliptic eigenvalue topology optimization in a phase-field framework. By coupling a phase-field design variable with a small-parameter Ginzburg–Landau regularization and solving on adaptively refined meshes, the authors introduce a set of residual-based estimators and a separate-collective marking strategy to drive refinement. Although a guaranteed reliability bound for the estimators is not established due to nonlinearity, they prove vanishing limits for the estimators and show convergence of a subsequence of adaptive solutions to a solution of the continuous optimality system, supported by a limiting problem and convergence lemmas. Numerical experiments in two dimensions demonstrate that the adaptive approach achieves comparable objective values to uniform refinement while markedly reducing computation time and accurately capturing diffuse interfaces and singularities near corners, with additional insights into symmetry phenomena in certain domains.

Abstract

In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional numerical examples for illustration of efficiency and accuracy. Theoretical findings consist in the vanishing limit of a subsequence of estimators and the convergence of the relevant subsequence of adaptively-generated solutions to a solution to the continuous optimality system.
Paper Structure (9 sections, 10 theorems, 145 equations, 14 figures, 2 tables, 2 algorithms)

This paper contains 9 sections, 10 theorems, 145 equations, 14 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Adaptive algorithm
  3. Numerical example
  4. Detail of module SOLVE
  5. Discussion on numerical result
  6. Origin of error estimator
  7. Limiting behavior
  8. Convergence
  9. Acknowledgement

Key Result

Lemma 5.1

With any $\phi_\infty\in \mathcal{U}_\infty$ given, define a bilinear form and the Rayleigh quotient respectively as For $j\in \mathbb{N}$, the corresponding $j$-th eigenvalue $\lambda_{\infty,j}^\varepsilon$ in eigen_vp_phasefield_limit is characterized by the maximum-minimum principle where $S_{j-1}^\perp$ is the orthogonal complement of $S_j\subset L^2(D)$ with respect to the associated scala

Figures (14)

  • Figure 1: Initial phase-field functions for Examples \ref{['example1']} (left), \ref{['example2']} (middle) and \ref{['example7']} (right).
  • Figure 2: Evolution of adaptive mesh levels $k=0,1,3,5$ for Example \ref{['example1']}, with the number of vertices on each mesh being 185, 494, 2702 and 13331. The 2nd, 3rd and 4th rows show optimized designs $\phi_k^\ast$ with a numerical optimal shape highlighted in red over each $\mathcal{T}_k$, two error indicators $\eta_{k,0}$ and $\eta_{k,1}$ respectively.
  • Figure 3: Evolution of adaptive mesh levels $k = 0,1,3,5$ for Example \ref{['example2']}, with the number of vertices on each mesh being 269, 817, 3095 and 10102. The 2nd, 3rd and 4th rows show optimized designs $\phi_k^\ast$ with a numerical optimal shape highlighted in red over each $\mathcal{T}_k$ and two error indicators $\eta_{k,0}$, $\eta_{k,1}$ respectively.
  • Figure 4: Errors in minimizing $\lambda_{1}$ (left) and approximating the associated phase-field function (middle and right) as functions of degrees of freedom for the adaptive refinement and the uniform refinement in Example \ref{['example2']}. $|\lambda_{k,1}^{\varepsilon,\ast}-\lambda_{1}^{\mathrm{ref}}|$ (left), $L^1(D)$ and $L^2(D)$ errors (middle), and $H^1(D)$ error (right) of numerical phase-field functions.
  • Figure 5: Evolution of adaptive mesh levels $k = 0,1,3,5$ for Example \ref{['example4']}, with the number of vertices on each mesh being 259, 663, 2462 and 9083. The 2nd, 3rd and 4th rows show optimized designs $\phi_k^\ast$ with a numerical optimal shape highlighted in red over each $\mathcal{T}_k$ and two error indicators $\eta_{k,0}$, $\eta_{k,1}$ respectively.
  • ...and 9 more figures

Theorems & Definitions (29)

  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 19 more