Mathematical Analysis and Numerical Methods for Peridynamics and Nonlocal Models

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Improvements in computer technology are fueling the development of more realistic mathematical models for complex applications, including nonlocal models that are more realistic than conventional local models for studying various phenomena from physics and biology to materials and social sciences. An example is the peridynamics model, a spatially nonlocal mechanics theory, which has been successfully used to model material defects and predict dynamics of crack formation in various engineering applications. Although new nonlocal models have been gaining popularity in various applications, the study of mathematics behind them is still at the nascent stage, impeding the further development of computational tools. The goal of this project is to develop efficient and reliable numerical methods for simulating nonlocal models and establish related mathematical analysis as part of the rigorous validation and verification process. It aims to improve the effectiveness and robustness of nonlocal modeling while retaining modeling accuracy. On the educational side, this project will provide training to students in both mathematics and computational mechanics. This project aims to develop state-of-the-art multiscale modeling techniques to improve computational efficiency while retaining the accuracy of nonlocal models for predicting dynamic fractures, with new theoretical methodologies built to study the analytic properties of the models. Three specific methods of multiscale modeling will be addressed, in which the treatment of boundary traces and interfacial conditions plays pivotal roles. The first is the seamless coupling of nonlocal and local models via heterogeneous localization of nonlocal interactions at the interface. A novel nonlocal trace theorem is used to ensure the well-posedness of the coupling. The goal is to treat the interface as a free boundary based on the development of the solution. The second is a quasi-nonlocal coupling method inspired by the atomistic-to-continuum coupling method. It is aimed at building a bridge between the discrete atomistic model and continuous nonlocal model. The third is to design appropriate nonlocal boundary conditions to reduce the computational cost for problems on unbounded domains. A new notion of nonlocal Neumann boundary condition will be introduced, which will shed light on domain decomposition methods and the coupling of nonlocal models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.