Analysis of the SiMPL method for density-based topology optimization

TL;DR

This paper provides a rigorous convergence analysis of the SiMPL method for density-based topology optimization, demonstrating its robustness, efficiency, and mesh-independent convergence through theoretical proofs and numerical experiments.

Contribution

The paper introduces a new SiMPL algorithm with proven convergence properties and robustness, enhancing density-based topology optimization methods.

Findings

The SiMPL method guarantees point-wise bound preservation.

The algorithm exhibits mesh-independent convergence.

Numerical results confirm strong convergence and robustness.

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.