2025 Theses Doctoral

Learning Non-Euclidean Representations for Solid Mechanics

Xiao, Mian

In the recent advancement of applying machine learning (ML) methods to solid mechanics modeling, ML models have made a great leap over conventional hand-crafted mechanical models in terms of model capacity and generalizability. However, these models could still fail to predict locally, especially in engineering extreme event cases, when material behaviors are significantly complicated by large deformations and high strain rates. Hence, geometric learning, a branch of ML, demonstrates a greater potential due to its ability to incorporate the non-Euclidean representations of the data obtained from mechanical solutions.

This dissertation proposes a series of geometric-learning-assisted computational mechanics models covering both microscale constitutive modeling and macroscale structural modeling. The objective is to extract the non-Euclidean representations of mechanics problems’ solutions for building smart, powerful, and efficient solid mechanics models, which better suit the demanding model capacity required by extreme event engineering modeling.Firstly, the author demonstrates how to learn non-Euclidean representations for both microscale constitutive models and macroscale structural behaviors. In the case of microscale constitutive modeling, the yield surface is represented as a manifold reconstructed from multiple overlapping coordinate charts, each with a specific local parametrization.

This approach enhances model generalizability by enabling adaptive resolution without overly complicating the parametrization. Additionally, a stress integration algorithm that projects trial stress directly onto local patches mitigates issues related to non-smoothness and convexity. In the case of macroscale structural modeling, a graph-manifold iterative algorithm is developed to predict the configurations of geometrically exact shells under external loading. Finite element solutions are embedded into a low-dimensional latent space through a graph isomorphism encoder, allowing the deformed configuration to be predicted without additional simulations. By enforcing balance laws on a reconstructed manifold, this method balances computational efficiency with solution fidelity, especially in critical regions of the shell structure.

The author then proposes an integrated continuum-scale finite element solver with constitutive laws represented in a non-Euclidean manner (specifically, mesh triangulations). The geometry of the yield surfaces is inferred as synthetic data points from priors trained on material databases and augmented with heuristics from limited high-fidelity experimental or numerical data, using a generative AI approach. These points form an edge-weighted graph that discretizes the yield surface, enabling a novel stress integration algorithm that remains stable even under unstable plastic flow. More importantly, this framework facilitates a global stress update scheme, significantly accelerating computations compared to classical return mapping algorithms. The effectiveness of this approach is demonstrated by inferring yield surface evolution from discrete dislocation dynamics (DDD) simulations, recovering a complete plasticity model that captures both the yield criterion and its evolution under isotropic strain hardening.

Finally, this work incorporates a material point method (MPM) to model specific extreme events at the continuum scale, including burial explosion and thermo-contact-fragmentation under high loading rate. This MPM customization considers the mechanics of material anisotropy, capturing of shock waves, thermal-sensitive contact, and fracturing in our computational framework. This framework is also flexible with the constitutive models, so that the capability of the customized MPM can be further improved by replacing the phenomenological material models with models obtained from geometric learning. The resulting numerical solution can capture the rapid evolution of material configuration under high loading rates and complex behaviors related to anisotropic shock propagation and dynamic fragmentation.