Rank Reduction AutoEncoders for Mechanical Design: Advancing Novel and Efficient Data-Driven Topology Optimization

TL;DR

This work tackles the computational burden of topology optimization by introducing a data-driven framework that leverages Rank Reduction Autoencoders (RRAEs) to compress high-dimensional geometry and response data into low-dimensional latent spaces. Separate RRAEs for geometry and various quantities of interest (QoIs) are connected via neural latent-space mappings, enabling fast direct (geometry-to-response) and inverse (response-to-geometry) analyses. A comprehensive study on a half MBB beam with SIMP-generated data shows that scalar QoIs degrade inverse predictability, while 1d and especially 2d von Mises QoIs yield high accuracy in both forward and inverse tasks, with 2d providing the best performance. The approach also lays groundwork for Generative Design by enabling latent-space exploration to synthesize new topologies and responses, potentially accelerating design optimization and uncertainty-aware workflows. The framework offers a principled, data-driven surrogate for TO that can be extended to multi-parameter spaces and physics-informed latent models to enhance robustness and applicability.

Abstract

This work presents a data-driven framework for fast forward and inverse analysis in topology optimization (TO) by combining Rank Reduction Autoencoders (RRAEs) with neural latent-space mappings. The methodology targets the efficient approximation of the relationship between optimized geometries and their corresponding mechanical responses or Quantity of Interest (QoI), with a particular focus on compliance-minimized linear elastic structures. High-dimensional TO results are first compressed using RRAEs, which encode the data into a low-rank approximation via Singular Value Decomposition (SVD), obtained in this sense the most important features that approximate the data. Separate RRAE models are trained for geometry and for different types of QoIs, including scalar metrics, one-dimensional stress fields, and full two-dimensional von Mises stress distributions. The resulting low-dimensional latent coefficients of the latent space are then related through multilayer perceptrons to address both direct problems -- predicting structural responses from geometry -- and inverse problems -- recovering geometries from prescribed performance targets. The proposed approach is demonstrated on a benchmark TO problem based on a half MBB beam, using datasets generated via density-based Solid Isotropic Material with Penalization (SIMP) optimization. Numerical results show that the framework enables accurate and computationally efficient surrogate models, with increasing robustness and fidelity as richer QoIs are considered. The methodology also provides a foundation for generative mechanical design by enabling the synthesis of new geometries and responses through latent-space exploration.

Figures (19)

• Figure 1: CNN RRAE architecture applied to a topology optimization example, consisting of an encoder $\mathcal{E}$, performing the SVD and decode $\mathcal{D}$. $L\gg T$ is the high-rank latent space dimension, $k_{\max}$ is the low-rank latent size, i.e., number of modes retained from the truncated SVD, and $n_s$ is the number of samples in the data.
• Figure 1: $\mathcal{NN}_{s}^d$ MLP model. Prediction of the scalar QoI $\sigma_{\max}^{vM}$.
• Figure 2: Pipeline to address the direct geo2sol$_{\star}$ or inverse sol2geo$_{\star}$ problems, for (a) scalar and (b) 2d field QoIs. There are three models in each case that are trained separately: geometry RRAE $(\bm U\bm A)$, solution RRAE $(\bm V \bm B)$, and the mappings between both latent coefficients, either $\mathcal{NN}^d_{\star}:\bm\alpha\rightarrow \bm\beta$ or $\mathcal{NN}^i_{\star}: \bm\beta \rightarrow \bm\alpha$.
• Figure 2: $\mathcal{NN}_{s}^i$ MLP model. Prediction of the geometries latent coefficients $\alpha_0$ and $\alpha_1$.
• Figure 3: Domain $\Omega$ and boundary conditions of the half MBB beam for the TO problem.