Robust Topology Optimization Using Multi-Fidelity Variational Autoencoders

TL;DR

This work tackles the high computational cost of robust topology optimization under loading uncertainty by introducing a three-stage, data-driven framework. It uses variational autoencoders to map high-dimensional topologies into a low-dimensional latent space and a dense surrogate to predict robust compliance, enabling gradient-based optimization in latent space. A multi-fidelity extension further cuts data-generation and training costs by coupling low- and high-resolution models. Across L-bracket scenarios with single and multiple loading uncertainties, the approach yields designs that improve upon training samples while substantially reducing computational effort, highlighting practical potential for rapid, robust structural optimization under uncertainty.

Abstract

Robust topology optimization (RTO), as a class of topology optimization problems, identifies a design with the best average performance while reducing the response sensitivity to input uncertainties, e.g. load uncertainty. Solving RTO is computationally challenging as it requires repetitive finite element solutions for different candidate designs and different samples of random inputs. To address this challenge, a neural network method is proposed that offers computational efficiency because (1) it builds and explores a low dimensional search space which is parameterized using deterministically optimal designs corresponding to different realizations of random inputs, and (2) the probabilistic performance measure for each design candidate is predicted by a neural network surrogate. This method bypasses the numerous finite element response evaluations that are needed in the standard RTO approaches and with minimal training can produce optimal designs with better performance measures compared to those observed in the training set. Moreover, a multi-fidelity framework is incorporated to the proposed approach to further improve the computational efficiency. Numerical application of the method is shown on the robust design of L-bracket structure with single point load as well as multiple point loads.

Figures (18)

• Figure 1: An overview of the neural network structure of a VAE with a two dimensional latent space. It consists of an encoder network and decoder network. The input of the encoder network is the image of a topology and the output of the decoder network is the corresponding reconstructed topology. The encoder maps the high dimensional input data to a low dimensional vector, $[z_1, z_2]$, which becomes the input to the decoder network.
• Figure 2: The complete neural network architecture for topology optimization. The top image shows the training of the VAE network and the compliance neural network surrogate. The bottom image shows how gradient descent is applied for finding the optimal design for the compliance minimization problem.
• Figure 3: An overview of the VAE architecture with multi-fidelity approach. A low-fidelity VAE network is trained using a sample of low-resolution designs, $\theta_L$. This is used as a pre-trained network to which additional fully connected layers are added to the encoder and decoder parts to form the high-fidelity VAE. The additional layers are added to match the dimensions of high-resolution designs, $\theta_H$, where $|\theta_H|>>|\theta_L|$.
• Figure 4: An example of the structure used in our study. The top part of the L-shaped structure has a fixed boundary. The load is applied at a single point with uncertainty in the loading angle.
• Figure 5: Total testing loss for various training sample sizes, $n_\text{train}$. Total loss is the sum of reconstruction error and KL Divergence. The results are shown for $|\bm z|=2$.