Gradient-free neural topology optimization: Towards effective fracture-resistant designs

TL;DR

The proposed gradient-free neural topology optimization method using a pre-trained neural reparameterization strategy can optimize toughness of a structure undergoing brittle fracture more effectively than a traditional gradient-based optimizer, delivering an objective improvement in the order of 30% for all tested configurations.

Abstract

Gradient-free optimizers allow for tackling problems regardless of the smoothness or differentiability of their objective function, but they require many more iterations to converge when compared to gradient-based algorithms. This has made them unviable for topology optimization due to the high computational cost per iteration and the high dimensionality of these problems. We propose a gradient-free neural topology optimization method using a pre-trained neural reparameterization strategy that addresses two key challenges in the literature. First, the method leads to at least one order of magnitude decrease in iteration count to reach minimum compliance when optimizing designs in latent space, as opposed to the conventional gradient-free approach without latent parameterization. This helps to bridge the large performance gap between gradient-free and gradient-based topology optimization for smooth and differentiable problems like compliance optimization, as demonstrated via extensive computational experiments in- and out-of-distribution with the training data. Second, we also show that the proposed method can optimize toughness of a structure undergoing brittle fracture more effectively than a traditional gradient-based optimizer, delivering an objective improvement in the order of 30% for all tested configurations. Although gradient-based topology optimization is more efficient for problems that are differentiable and well-behaved, such as compliance optimization, we believe that this work opens up a new path for problems where gradient-based algorithms have limitations.

Figures (26)

• Figure 1: Schematics of our framework. a) Schematic of a variational autoencoder (VAE). In the first step of our method, we train the generative model -- a VAE to reparameterize topology designs into latent space. Here $\mu$ and $\sigma$ are mean and variance vectors in latent space, $\epsilon$ is the noise vector sampled from a multivariate Gaussian, and $z$ is a latent variable. b) Schematic of latent space optimization process with gradient-free optimizer. In the second step of our method, we use a trained generative model, to optimize the designs using latent space representation of the generative model. Note that the latent vector $z$ is no longer stochastic, and it is fully controlled and updated ($z_{new}$) by the optimizer, based on the objective values obtained from the simulator.
• Figure 2: Performance of unprocessed designs: cumulative probability (fraction of problems) vs relative error w.r.t. the MMA solution, for two variants of latent parameterization models -- Variational Autoencoder (VAE, latent space dimensionality: 64), and Latent Bernoulli Autoencoder (LBAE, latent space dimensionality: 256) -- compared against conventional pixel parameterization. The designs were optimized with a BIPOP-CMA-ES optimizer. Note that the shown distributions do not add up to 1 due to the cut-off at 300%. Best, typical, and worst designs correspond to the best, median, and worst random initializations for each problem.
• Figure 3: Performance of thresholded designs: cumulative probability (fraction of problems) vs relative error w.r.t. the MMA solution, calculated for thresholded designs for two variants of latent parameterization models -- Variational Autoencoder (VAE, latent space dimensionality: 64), and Latent Bernoulli Autoencoder (LBAE, latent space dimensionality: 256) -- compared against conventional pixel parameterization. The designs were optimized with a BIPOP-CMA-ES optimizer. Note that the shown distributions do not add up to 1 due to the cut-off at 300%. Best, typical, and worst designs correspond to the best, median, and worst random initializations for each problem.
• Figure 4: An example of an out-of-distribution problem: classic L-shaped bracket problem, with clamped support along the top edge and a downward point load applied at the right edge.
• Figure 5: An example of a challenging out-of-distribution problem: a 'staircase' design with Z-shaped distributed load, with clamped support a the bottom edge.