Methods for Learning Mechanics: Inverse Problems, Constitutive Modeling, and Design

Citation

Rajan, Aakila (2025) Methods for Learning Mechanics: Inverse Problems, Constitutive Modeling, and Design. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/caht-w798. https://resolver.caltech.edu/CaltechTHESIS:05262025-195518377

Abstract

Predicting the behavior of materials and structures under complex loading is a fundamental challenge in solid mechanics. Traditional techniques rely on idealized experiments, a priori information and are computationally intensive. This work introduces a unified framework for constitutive identification, multiscale modeling, and design optimization, grounded in physical laws and enhanced by machine learning.

We first focus on the inverse problem of identifying constitutive behavior from experimental data. We formulate this task as an optimization problem with the governing equations of the experiment as a constraint. This allows us to infer material parameters from full-field or contact-based measurements. This approach enforces physical laws and accommodates complex loading, noise, and limited data. We apply it to recover properties for a history-dependent material using both synthetic and experimental datasets. We further extend the framework using recurrent neural operators to learn constitutive responses directly from data, bypassing the need for an explicit model form.

In the second part of the thesis, we extend our focus to multiscale modeling and structural design. Neural operators are used to learn homogenized solutions of linear elliptic PDEs with discontinuous coefficients, eliminating the need to resolve fine-scale features during inference. For topology optimization, we develop reduced-order neural surrogates embedded within the design loop, achieving efficient yet accurate updates. Together, these contributions offer a cohesive, data-driven strategy for advancing modeling and design in solid mechanics.

Thesis Files