SUMMARY

Uncertainty exists in multiscale computational materials science tools, from quantum-level to macro-level. Similar to hierarchical multiscale modeling, uncertainty can also be coupled at different scales. Current methodologies are mainly focused on quantifying uncertainty as a single input-output mapping, but lack of a propagation scheme across scale, as well as using the available data to assess uncertainty on-the-fly. This proposal aims to contribute to uncertainty quantification in two scales: nano-scale and micro-scale. At nano-scale, we focus on molecular dynamics (MD) simulation and scale-bridging methodologies to propagate uncertainty in an upscaling manner. At micro-scale, a novel data-driven approach to quantify uncertainty is proposed to design and optimize a family of fractal metamaterials. The first research task focuses on a methodology to dynamically propagate uncertainty from MD simulation to a faster time-scale, using stochastic partial differential equations, as well as polynomial chaos expansion. The second research task focuses on the design and optimization of fractal metamaterials. Recent advances in 3D printing and additive manufacturing methods have enabled researchers to build parts at micro- and nano-scale, and thus open up many research opportunities in this area. The goal of second research task is to design and optimize the metamaterial architecture, to maximize the strength and minimize the mass of materials simultaneously. A novel method called distributed Gaussian process regression (GPR) is developed to incorporate discrete and continuous variables in design space, which includes both geometric model parameters and materials properties parameters. The distributed GPR is embedded within a modified Bayesian optimization framework to quantify uncertainty of the surrogate model. The advantages of the novel method include the ability to accommodate both discrete and continuous variables, dimensionality reduction, which reduces the dimensionality of the design space to the dimensionality of continuous variables by a decomposition scheme, and the reduction of computational cost that is naturally inherited by the decomposition scheme, which partially solves the scalability O(n^3) of the classical GPR. The third research task focuses on the generalization of the current technique toward 3D fractal metamaterials, and exploration in prototyping and testing the designed structures using additive manufacturing.