Optimizing the Cross-Sections of Lattice Structures Using Variational Autoencoders

Akshay Kumar, Saketh Sridhara, Krishnan Suresh
Department of Mechanical Engineering, University of Wisconsin–Madison

This repository accompanies the paper:

Optimizing the Cross-Sections of Lattice Structures Using Variational Autoencoders

and provides a complete, end-to-end workflow for the design, optimization, and reconstruction of lattice structures with free-form, manufacturable beam cross-sections.

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Overview

This work presents a data-driven framework for optimizing lattice beam cross-sections under linear elastic loading. Beam cross-sections are parameterized using B-spline curves and optimized in a learned latent space obtained via a variational autoencoder (VAE).

Direct optimization in the B-spline space is challenging due to high dimensionality and geometric validity constraints. To address this, a VAE is trained on datasets of both valid and invalid cross-section shapes, enabling it to implicitly learn valid geometries. The resulting latent space is smooth, low-dimensional, and well-suited for gradient-based optimization.

Optimized latent variables are decoded back into B-spline representations and used to generate watertight STL files for additive manufacturing, enabling a fully digital design-to-production workflow.

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Workflow

The framework consists of two primary stages:

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1. Data Generation and VAE Training

Use trainVAE.py to generate cross-section datasets and train the variational autoencoder.

This script allows you to:

• Define B-spline geometry parameters:
  • Number of control points (n_cp)
  • Spline degree (k)
• Generate datasets containing both valid and invalid cross-section shapes
• Specify VAE architecture and training parameters
• Train a VAE that maps B-spline control points and cross-section properties to a compact latent space

Output:
A trained VAE model whose decoder produces valid, manufacturable B-spline cross-sections.

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2. Lattice Optimization Using a Trained VAE

Latent-space optimization is performed using optBeam.ipynb.

This notebook allows you to:

• Load a previously trained VAE model
• Define structural optimization problems:
  • Single beams
  • Three beams
  • Lattice structures
  • More
• Specify boundary conditions, loading, and volume constraints
• Perform gradient-based optimization directly in the latent space
• Decode optimized latent variables into B-spline cross-sections
• Automatically generate watertight STL files suitable for 3D printing

Example problems are provided and can be easily extended to custom geometries and loading conditions.

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Key Features

• Free-form beam cross-section representation using B-splines
• Data-driven enforcement of geometric validity via a VAE
• Low-dimensional latent space for efficient optimization
• Differentiable decoder enabling automatic differentiation
• Support for solid and hollow cross-sections
• Automated generation of watertight STL files for additive manufacturing

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Installation

1. Create a Conda Environment

conda create -n LatticeCSOpt python=3.10
conda activate LatticeCSOpt

2. Install Dependencies

pip install -r requirements.txt

See requirements.txt for the complete list of dependencies.

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Citation

If you use this code in your research, please cite the associated paper. (Citation information will be added upon publication.)

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License

This project currently has no license. All rights reserved.

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Acknowledgments

This work was supported by the U.S. Office of Naval Research under the PANTHER award N00014-21-1-2916, monitored by Dr. Timothy Bentley.