UC San Diego

Physics-based numerical simulation remains challenging as the complexity of today’s high-fidelity models has dramatically increased. Model order reduction (MOR) and data-driven modeling, based on the emerging techniques of data learning and physical modeling, present a promising way to tackle the computational bottleneck related to the computational intensity and model complexity.

Nevertheless, MOR has proven to be significantly more difficult for parameterized mechanics systems that exhibit a wide variety of parameter-dependent nonlinear behaviors or that involve localized essential features. The first objective of this work is to develop robust, physics-preserving MOR methods. As constructing a low-dimensional MOR model can be considered as the hybrid data-physics approach, one can optimize it through a learning process using both data and physical models. As such, we first propose a MOR method based on decomposed reduced-order projections that well preserve the essential near-tip characteristic for fracture mechanics. Moreover, we develop an enhanced reduced-order basis to construct a low-dimensional subspace, deriving from a generalized manifold learning framework that allows the employment of local information in the data structure during the learning phase. This approach can yield a robust reduced-order model against noise and outliers and is well suited for parameterized nonlinear physical systems. Finally, a nonlinear MOR for a meshfree Galerkin formulation with the stabilized conforming nodal integration (SCNI) scheme is developed to yield a pure node based MOR that is particularly effective for hyper-reduction techniques. A numerical example of two-phase hyperelastic solid with perturbed loading conditions is used to validate the effectiveness of the proposed reduction method.

The second goal of the dissertation is to develop a robust data-driven computational framework, which provides an alternative to conventional scientific computing for complex materials. This framework aims at performing physical simulation by directly interacting with material data via machine learning procedures instead of employing phenomenological constitutive models, and especially addressing the robustness issue associated with noisy and scarce data. To this end, we propose to search data solutions from a locally reconstructed convex hull associated with the k-nearest neighbor points, which leads to robustness to noisy data and ensures convergence stability. The accuracy and robustness of the proposed data-driven approach are demonstrated in the modeling of linear and nonlinear elasticity problems. In addition, we present a preliminary result of data-driven modeling of biological tissue using material data collected from laboratory testing on heart valve tissue, showing the potential of data-driven simulation by integrating physical modeling and machine learning techniques.