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

Technical Summary: Rank Reduction AutoEncoders for Mechanical Design

Problem Statement
Topology Optimization (TO) is a critical tool for synthesizing high-performance structures but remains computationally expensive, particularly when repeated analyses are required for design space exploration, uncertainty quantification, or inverse design. Traditional TO workflows involve solving high-dimensional Finite Element (FE) problems repeatedly, where optimized geometries are represented by dense density fields. While machine learning (ML) offers data-driven alternatives, standard approaches often suffer from limited interpretability, poor extrapolation, and difficulties in handling inverse problems where multiple geometries may yield identical performance metrics. Furthermore, standard autoencoders can introduce redundancy or overfitting when compressing high-dimensional TO data.

Methodology
The paper proposes a unified data-driven framework that combines Rank Reduction AutoEncoders (RRAEs) with neural latent-space mappings to model the relationship between optimized geometries and their mechanical responses (Quantities of Interest, or QoIs). The methodology is structured as follows:

1. Data Generation: A dataset is generated using the Solid Isotropic Material with Penalization (SIMP) method for a compliance minimization problem on a half MBB beam. The dataset includes optimized topologies (geometries) and corresponding mechanical responses (von Mises stress distributions) under varying volume constraints.
2. Dimensionality Reduction (RRAE): Instead of standard autoencoders, the framework employs RRAEs. This architecture encodes high-dimensional data (geometries or stress fields) into a latent space, which is then subjected to a truncated Singular Value Decomposition (SVD). This process retains only the most significant modes (latent coefficients), effectively identifying the dominant geometric and physical features while ensuring a low-rank approximation.
   • Separate RRAE models are trained for the geometry (using a CNN-RRAE for 2D density fields) and for the QoIs.
3. Latent Space Mapping: Multilayer Perceptrons (MLPs) are trained to map the low-dimensional latent coefficients of the geometry ($\alpha$) to those of the solution ($\beta$) and vice versa.
   • Direct Problem ($\text{geo} \to \text{sol}$): Predicting structural responses from a given geometry.
   • Inverse Problem ($\text{sol} \to \text{geo}$): Reconstructing geometries from prescribed performance targets.
4. QoI Variations: The study evaluates three distinct types of QoIs to determine their impact on model fidelity:
   • Scalar: Maximum von Mises stress (global metric).
   • 1D Field: von Mises stress values along the diagonal of the domain (main load path).
   • 2D Field: Full von Mises stress distribution across the domain.

Key Contributions
The paper makes three primary contributions:

1. Efficient Dimensionality Reduction: It introduces RRAEs as an adaptive tool for reducing the dimensionality of TO data, leveraging truncated SVD to maintain computational efficiency and robustness while capturing dominant modes.
2. Comparative Analysis of QoIs: It provides a systematic analysis of how the information content of the QoI (scalar vs. 1D vs. 2D fields) affects the accuracy and robustness of both forward and inverse surrogate models.
3. Generative Design Framework: It demonstrates a latent-space-based framework that facilitates the synthesis of new geometries and responses, enabling rapid exploration of optimized structural layouts without solving full FE problems.

Numerical Results
The framework was validated on a benchmark half MBB beam problem with 100 training samples and 20 test samples (varying volume fractions).

• Scalar QoI: This case proved the most challenging. The mapping from geometry to a single scalar value is inherently non-bijective. Consequently, the inverse problem suffered from overfitting (memorization) and produced geometries lacking physical meaning when extrapolating. The reconstruction loss for inverse problems was high ($\approx 20\%$).
• 1D Field QoI: Using the stress field along the diagonal significantly improved performance. The inverse problem reconstruction loss dropped to $\approx 7.6\%$ (test set), and the model successfully generated physically plausible geometries that matched the target stress curves.
• 2D Field QoI: This approach yielded the best overall performance. By utilizing the full stress distribution, the framework achieved high-fidelity predictions.
  • Direct Problem: The $R^2$ coefficient reached $0.999$ on the test set.
  • Inverse Problem: The reconstruction loss for geometries was $7.80\%$ (test set), with predicted volume fractions closely matching ground truth.
  • The results indicate that richer QoIs provide the necessary information to resolve the ill-posed nature of the inverse problem, leading to robust and accurate surrogate models.

Significance and Claims
The authors claim that this framework offers a computationally efficient surrogate for FE analysis in TO workflows. The study emphasizes that the richness of the QoI is the critical factor in the success of inverse design; compressing the solution too aggressively (e.g., to a scalar) renders the inverse problem ill-conditioned.

The methodology is presented as a foundation for Generative Design (GD). By operating in a low-rank latent space derived from SVD, the framework allows for the interpolation of new points to synthesize novel geometries and responses that remain physically plausible. The authors position RRAEs as a unifying tool for reduced-order modeling, inverse design, and data-driven exploration, noting that future work will focus on extending this to multi-parameter design spaces and incorporating physics-informed constraints.