A Simple Introduction to the SiMPL Method for Density-Based Topology Optimization

TL;DR

The paper presents SiMPL, a simple first-order method for density-based topology optimization that enforces constraints via entropy and uses a latent variable for efficient updates, outperforming existing algorithms.

Contribution

Introduction of SiMPL, a novel, simple, and efficient first-order optimization method for topology optimization using entropy-based constraints and latent variables.

Findings

Outperforms popular first-order algorithms in numerical tests.

Produces feasible iterates even with high-order finite element discretizations.

Applicable to compliance minimization and compliant mechanism problems.

Abstract

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as ``the simple method'') optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi--Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.