Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion models

TL;DR

This work tackles the inverse design of nonlinear metamaterials by leveraging a video denoising diffusion model trained on full-field deformation data from FE simulations of 2D periodic cellular structures. The method learns to map target nonlinear stress–strain responses to plausible microstructural designs while simultaneously predicting the full deformation path and internal stress distributions, including buckling and contact, in a single framework. It demonstrates accurate forward predictions (full-field σ22 and the resulting stress–strain curves) and exposes the probabilistic design space, enabling multiple viable designs for a given target. The approach offers a path to rapid, physically interpretable material design for soft robotics, biomedical implants, and impact mitigation, with potential extensions to other physics and design representations.

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.

Figures (3)

• Figure 1: Metamaterial generation process.(a) A 2D cellular UC is generated by sampling from a 2D Gaussian random field, applying a varying threshold to extract a binary field, and mirroring the resulting pattern when connectivity to the boundaries is ensured. (b) To obtain the stress-strain response, we place the UC between two rigid plates with periodic boundary conditions in the horizontal direction and apply a compressive strain of up to $20\%$. The corresponding stress and displacement fields within the UC are computed by FE simulations, and the overall effective stress-strain response $\sigma_{\text{eff.}}$ is extracted from the nodal reaction forces, though they can be equally obtained from the full-field data. A representative selection of responses of the generated designs is plotted in gray.
• Figure 2: Denoising diffusion model architecture. The denoising diffusion model is based on the 3D U-Net video architectureHo2022 which iteratively adds information to a Gaussian prior. To include a temporal dimension, each spatial convolution and attention layer is followed by temporal attention computed over the eleven strain steps. We condition the model by transforming the stress-strain response to a token embedding, which is added via cross-attention into both spatial and temporal attention layers.
• Figure 3: Metamaterial synthesis for four stress-strain responses not represented in the training dataset.(a-d) The model is conditioned on four technically relevant, challenging target responses. Validation of the predicted effective stress response $\sigma_{\text{eff.}}$ ('Fwd. eval.'; NRMSE with respect to the target response in brackets) of the generated designs is achieved by FE simulations ('FE eval.'), agreeing with the predicted response and significantly outperforming the best match in the training dataset ('Best match'). We additionally compare the predicted full-field $\sigma_{22}$-distribution (indicated in MPa in the Eulerian frame) with the FE ground truth and provide the corresponding relative $L_2$-errors. To highlight the range of responses in the training dataset, we plot a representative selection in gray in (a). $^*$The relative $L_2$-error is numerically inflated due to the small magnitude of the stress field and is hence not truly indicative (but included for completeness).