Session: 13-22-02: CONCAM Distinguished Lectures on Computational Mechanics II

Paper Number: 173832

Constitutive Modeling in the Era of Ai. Part I: Constitutive Laws as Graphs and Trees 

The first part of this talk summarizes the various ways high-fidelity constitutive laws for a wide range of solids, such as soil, rock, alloys, and polymer composites, can be represented and how the choice of representations influences the accuracy, robustness, and data/computational efficiency for computer simulations of solids. Classical plasticity models typically rely on equations to express constitutive behaviors such as yield surfaces and elastic energy. However, deducing these equation-based models requires extensive trial-and-error and strong intuitions. While neural networks may enable the parameterization of material models, they require additional costs for supervised learning and may lead to slower inference. We introduce a generative artificial intelligence approach that does not require additional supervised learning to recover the yield surface. Instead, a generative latent diffusion model allows us to deduce yield points with maximum likelihood in the stress space, allowing the model to generalize from sparse or incomplete experimental data. To represent material models as meshes, we introduce a latent diffusion model where previous material models and experimental data are used to guide the reverse generation of models. This mesh-based material model is particularly efficient for non-smooth plasticity, where projection on segments can lead to significantly faster simulations. A new projection-based stress update algorithm designed specifically for the mesh-based models enables parallelized integration without the numerical instability or high computational cost of traditional return mapping schemes. Numerical benchmarks—ranging from classical plasticity models to data from discrete dislocation dynamics simulations of single-crystal copper—demonstrate that the proposed framework improves robustness, scalability, and inference accuracy. This work contributes a new class of data-driven plasticity models that are computationally efficient and well-suited for complex, path-dependent material behavior. Neural network models often require more CPT time per inference than hand-crafted constitutive laws. To address this problem, we represent material models as a tree. Rather than directly confronting NP-hard symbolic regression in the ambient strain space, HYDRA leverages a data-driven projection to map strain onto a hyperplane and a neural additive model to parameterize the hyperplane via univariate bases. This setting enables us to convert the univariate bases into symbolic forms via genetic programming with explicit control of the expressivity-speed trade-off. Additionally, the availability of analytical models provides the benefits of ensuring the enforcement of physical constraints (e.g., material frame indifference, material symmetry, growth condition) and enabling symbolic differentiation that may further reduce the memory requirement of high-performance solvers. This technique enables us to search for hyper-elasticity in high-dimensional space without sacrificing the expressivity of neural networks. We show that the proposed model may reproduce any polynomial of arbitrary orders and dimensions and thus achieve the universal approximation through the Stone-Weierstrass theorem. Through a series of 1D post-hoc symbolic regressions, we obtain symbolic material models that significantly reduce the inference time for hydrocodes. The pros and cons of these techniques for various practical applications will be discussed.

Presenting Author: Waiching Sun Columbia University

Presenting Author Biography: Dr. WaiChing "Steve" Sun is an associate professor of civil engineering and engineering mechanics at Columbia University. He received his PhD from Northwestern in 2011. From 2011 to 2013, He worked as a research engineer at Sandia National Laboratories. Sun’s research focuses on computational mechanics and scientific machine learning for material modeling. He received several awards, including the Walter Huber Prize and da Vinci Award from ASCE, the John Argyris Award from IACM, and the CAREER award from NSF, the Army, and the Air Force. Since April 1st, 2025, he has become an editor of the International Journal for Numerical Methods in Engineering.

Authors:

Waiching Sun Columbia University