Reduced Order Multiscale Modeling of Nonlinear Heterogeneous Materials

Predicting the multiscale nonlinear behavior of heterogeneous materials is critical to many engineering fields, but requires computationally intensive techniques such as Computational Homogenization (CH). Reduced Order Model (ROM) surrogates have been developed to address the demands of multiscale modeling, but thus far most are limited to single scale or linear behavior. To this end, this dissertation develops the physical and numerical framework for a novel multiscale ROM that bypasses the computational requirements of scale and nonlinearity inherent to heterogeneous material simulation. It is formulated in a first-order CH framework, with reduction carried out at the finest scale as it is the source of most of the workload. For irreversible processes, this requires reduction of both nonlinear Partial Differential Equations (PDEs) responsible for the geometric nonlinearity, and nonlinear Ordinary Differential Equations (ODEs) responsible for material evolution. PDE reduction is carried out using a Manifold-based Reduced Order Model (MNROM), where new states can be predicted cheaply with respect to a manifold of prior simulations. Here we extend MNROM application to incompressible materials and develop an a-priori one step cutoff for enrichment. ODE reduction is carried out using the Adaptive Discrete Empirical Interpolation Method (ADEIM) with adaptive sampling, where we model only at sampled evaluation points to adapt bases and project for information elsewhere. Here we extend ADEIM application to nonscalar quantities for complex tensorial problems. Both PDE and ODE schemes are joined together using operator splitting to tackle coupled problems. We demonstrate the coupled ROM - investigating both PDE and ODE reduction components individually and together - by examining elastoviscoplastic behavior in a particulate composite with a nearly incompressible matrix. We verify it for a complex 2D microstructure over a large range of strains, with large plastic deformation.