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Analysis of the SiMPL method for density-based topology optimization

Brendan Keith, Dohyun Kim, Boyan S. Lazarov, Thomas M. Surowiec

TL;DR

The paper develops and analyzes the SiMPL method for density-based topology optimization, introducing a latent-variable, sigmoidal representation that preserves density bounds and enables high-order discretizations. Two globalization strategies are proposed: SiMPL-A (Armijo) and SiMPL-B (Bregman), each guaranteeing monotone objective descent and convergence of iterates, with a rigorous KKT-type framework emerging at limit points. Theoretical results establish existence, stability, and convergence properties in non-reflexive function spaces, along with practical guidelines for step-size initialization and volume-correction. Numerical experiments on a 2D cantilever demonstrate mesh- and order-independence, robustness, and rapid convergence, suggesting strong potential for industrial topology-optimization workflows. The framework also supports extensions to broader objective functionals and higher-order discretizations, as discussed alongside the companion application paper.

Abstract

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [31]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. We refer to the new algorithm as the SiMPL method, pronounced like ``simple", which stands for {Si}gmoidal {M}irror descent with a {P}rojected {L}atent variable.

Analysis of the SiMPL method for density-based topology optimization

TL;DR

The paper develops and analyzes the SiMPL method for density-based topology optimization, introducing a latent-variable, sigmoidal representation that preserves density bounds and enables high-order discretizations. Two globalization strategies are proposed: SiMPL-A (Armijo) and SiMPL-B (Bregman), each guaranteeing monotone objective descent and convergence of iterates, with a rigorous KKT-type framework emerging at limit points. Theoretical results establish existence, stability, and convergence properties in non-reflexive function spaces, along with practical guidelines for step-size initialization and volume-correction. Numerical experiments on a 2D cantilever demonstrate mesh- and order-independence, robustness, and rapid convergence, suggesting strong potential for industrial topology-optimization workflows. The framework also supports extensions to broader objective functionals and higher-order discretizations, as discussed alongside the companion application paper.

Abstract

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [31]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. We refer to the new algorithm as the SiMPL method, pronounced like ``simple", which stands for {Si}gmoidal {M}irror descent with a {P}rojected {L}atent variable.
Paper Structure (26 sections, 18 theorems, 104 equations, 4 figures, 2 tables)

This paper contains 26 sections, 18 theorems, 104 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions
  4. Notation
  5. Outline
  6. A general framework for topology optimization
  7. Filtering preserves the admissible set
  8. The gradient is essentially bounded
  9. Deriving SiMPL
  10. Bregman divergences and mirror descent
  11. The Fermi--Dirac entropy on $L^p$-spaces
  12. A primal-dual update rule
  13. SiMPL: A latent variable update rule
  14. Preliminary convergence theory
  15. Convergence of the iterates
  16. ...and 11 more sections

Key Result

Lemma 2.1

\newlabellem:filter-bdd0 Let $\Omega\subset\mathbb{R}^d$ be a Lipschitz domain and $g\in L^q(\Omega)$ with $q>d/2$. Let $w\in H^1(\Omega)$ be the weak solution to the following equation: Then there exists a constant $c>0$, depending only on $\epsilon>0$ and the domain $\Omega$, such that Moreover, $\mathop{\mathrm{ess\,inf}}\limits g\leq w \leq \mathop{\mathrm{ess\,sup}}\limits g\text{ a.e.\ in

Figures (4)

  • Figure 1: Interpolation of a characteristic function $\chi_{\{x>0\}}$ with 3rd (left) and 13th (right) order polynomials. $\chi_h$ is the direct interpolation, and $\chi^*_h$ is the interpolation in the latent space. We set $\chi^*_{h}= \min\{40, \max\{-40, \sigma^{-1}(\chi_{\{x>0\}})\}\}$ to have a finite $\chi^*_{h}$. The $\sigma(\chi_{h})$ used in SiMPL, shows a sharp interface without oscillation. The cut-off interpolation here is used only for demonstration, not in the optimization process. \newlabelfig:gibbs0
  • Figure 1: Experiment 1. Compliance (left), KKT residual (center), and step size (right) for $h=1/64$, $1/128$, $1/256$, and $1/512$. SiMPL-A (top row) and SiMPL-B (bottom row) exhibit mesh-independent behavior.
  • Figure 2: Experiment 1. The discretized design density $\rho_h$ (top) and filtered density $\tilde{\rho}_h$ (bottom) at the final iteration of the SiMPL-B method for $h=1/64$, $1/128$, and $1/256$. SiMPL-A yields visually-identical solutions.
  • Figure 3: Experiment 2. Compliance (left), KKT residual (center), and step size (right) for polynomial degrees $p=1, 2, 3$ and $4$ on a uniform rectangular grid $h=1/64$. SiMPL-A (top row) and SiMPL-B (bottom row) exhibit order-independent behavior.

Theorems & Definitions (41)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1: Properties of Bregman divergences Chen1993
  • Lemma 3.2: Properties of the Fermi--Dirac entropy
  • Proof 1
  • Lemma 3.3
  • Proof 2
  • Theorem 3.4: Primal-dual update rule
  • Proof 3
  • Remark 3.5
  • ...and 31 more