[Paper Review] Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion models

Jan-Hendrik Bastek, Dennis M. Kochmann|arXiv (Cornell University)|May 31, 2023

Cellular and Composite StructuresEngineering49 references3 citations

TL;DR

This paper proposes a video denoising diffusion model to inverse-design nonlinear mechanical metamaterials with tailored large-strain stress-strain responses under compression, including buckling and contact. By training on full-field finite element data of periodic cellular structures, the model generates mechanically consistent deformation paths and internal stress distributions that closely match simulations, enabling efficient, physics-informed design of complex metamaterials without iterative optimization.

ABSTRACT

The accelerated inverse design of complex material properties - such as identifying a material with a given stress-strain response over a nonlinear deformation path - holds great potential for addressing challenges from soft robotics to biomedical implants and impact mitigation. While machine learning models have provided such inverse mappings, they are typically restricted to linear target properties such as stiffness. To tailor the nonlinear response, we here show that video diffusion generative models trained on full-field data of periodic stochastic cellular structures can successfully predict and tune their nonlinear deformation and stress response under compression in the large-strain regime, including buckling and contact. Unlike commonly encountered black-box models, our framework intrinsically provides an estimate of the expected deformation path, including the full-field internal stress distribution closely agreeing with finite element simulations. This work has thus the potential to simplify and accelerate the identification of materials with complex target performance.

Motivation & Objective

• To address the challenge of inverse-designing mechanical metamaterials with complex, nonlinear stress-strain responses beyond linear stiffness.
• To enable efficient, data-driven discovery of microarchitectures that achieve specific nonlinear mechanical behaviors such as post-buckling and contact.
• To overcome limitations of traditional topology optimization and black-box ML models by providing physically consistent, interpretable deformation paths.
• To leverage video diffusion models to learn temporal and mechanical consistency across deformation steps in large-strain regimes.
• To deliver a generative framework that predicts full-field stress and displacement evolution from a target effective stress-strain curve.

Proposed method

• A video diffusion model is trained on paired data of full-field displacement and stress distributions across 11 strain steps from finite element simulations of periodic cellular metamaterials.
• The model uses a U-Net architecture with ResNet blocks and attends to conditioning via text-like embeddings derived from the target stress-strain response.
• Conditioning is implemented by averaging strain-step token embeddings and projecting them into the latent space of the diffusion time step using a two-layer MLP with SiLU activation.
• Data are normalized to [-1, 1] using min-max scaling across all training samples for stress, displacement, and effective stress responses.
• A robust sampling protocol extracts the underlying undeformed topology by identifying zero-displacement regions in the upper-left quarter, leveraging symmetry and a 2% tolerance threshold.
• Loss functions include normalized root mean square error (NRMSE) for effective stress and relative L2 error for full-field stress distributions to ensure scale-invariant, quantitative evaluation.

Experimental results

Research questions

• RQ1 Can video denoising diffusion models learn and generate mechanically consistent deformation sequences for nonlinear metamaterials under large compression?
• RQ2 Can such models predict full-field stress and displacement evolution that closely match finite element simulations for complex nonlinear responses?
• RQ3 Does the model enable inverse design by generating microstructures that achieve a desired target stress-strain curve, including post-buckling and contact?
• RQ4 How does the model’s performance compare to traditional topology optimization or black-box ML approaches in terms of accuracy and physical consistency?
• RQ5 Can the model generalize across diverse microarchitectures and nonlinear mechanical behaviors without requiring fine-tuning?

Key findings

• The model achieves high accuracy in predicting the effective stress-strain response, with normalized root mean square error (NRMSE) values below 0.05 across all tested targets.
• Full-field stress predictions show strong agreement with finite element simulations, with relative L2 errors consistently below 0.08 across all strain steps.
• The model successfully captures complex mechanical phenomena such as buckling, contact, and large-strain localization in the deformation path.
• The generated deformation sequences exhibit mechanical consistency, with stress distributions evolving coherently across strain steps.
• The sampling protocol reliably recovers the underlying microstructure topology from predicted displacement fields, with minimal false positives and no disconnected components in most cases.
• Training on 8 A100/RTX 6000 GPUs completed in approximately 70 hours, demonstrating feasibility for large-scale design space exploration.