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

TL;DR

SiMPL addresses density-based topology optimization by delivering a first-order method that preserves feasibility at every discretization step through a latent-variable representation and a Bregman divergence derived from the negative Fermi–Dirac entropy. The method yields a simple two-stage update in the latent space and can be formulated in both discretize-then-optimize and optimize-then-discretize paradigms, with a practical emphasis on high-order discretizations that retain pointwise feasibility. Numerical experiments across 2D and 3D TO problems—including MBB, multiple loads, self-weight, and compliant mechanisms—show that SiMPL converges faster and is more robust than classic first-order methods such as OC and MMA, while exhibiting mesh- and degree-independence. An open-source MFEM implementation accompanies the work, and extensions to augmented Lagrangian formulations are proposed to handle additional constraints.

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.

Figures (8)

• Figure 1: Schematics of the SiMPL method in primal (left) and latent (right) spaces, respectively, in $\mathbb{R}^2$. $\bm{\rho}_{k+1/2}=\sigma(\bm{\psi}_{k+1/2})=\sigma(\bm{\psi}_k-\alpha_k\nabla F(\bm{\rho}_k))$ is an auxiliary step before the volume correction. Black curves represent the feasible set $K$ with volume constraint; cf. \ref{['eq:full-update']}. Both the gradient step and volume correction are linear operations in the latent space (right), but are nonlinear in the primal space (left).
• Figure 2: Problem 1. Filtered density, $\tilde{\rho}$, for selected iterations $k$. From left to right: SiMPL-A, SiMPL-B, OC, and MMA. The final number of iterations are 50 (SiMPL-A), 46 (SiMPL-B), 300 (OC), and 300 (MMA).
• Figure 3: Problem 1. Compliance (top left), successive difference of compliance (top right), relative stationarity error (bottom left), and volume (bottom right) for the MBB beam with mesh size $h=1/256$.
• Figure 4: Problem 1. Compliance (left), relative stationarity error (center), and step size (right) with SiMPL-A and SiMPL-B methods for the MBB beam with various mesh sizes $h=1/64,\;1/128$, and $1/256$.
• Figure 5: Problem 3. Optimized frame designs with SiMPL-A (27 iterations) and SiMPL-B (25 iterations) for this multiple load problem. The final objective function values are $2.8092\times 10^{-3}$ and $2.8185\times10^{-3}$, respectively.

Theorems & Definitions (7)

• Remark 1: Computing the gradient
• Remark 2: Quickly reaching binary designs
• Remark 3: Solving the volume projection equation
• Remark 4: Selecting the step sizes $\alpha_k$
• Remark 5: Convergence analysis
• Remark 6: Taming the overflow
• Proposition 1