Mathematical Analysis and Numerical Methods for Peridynamics and Nonlocal Models

计算机技术的进步正在推动复杂应用中更现实的数学模型的发展，包括非局部模型，它比传统的局部模型更现实，用于研究从物理学和生物学到材料和社会科学的各种现象。一个例子是周动力学模型，这是一种空间非局部力学理论，已成功地用于模拟材料缺陷和预测各种工程应用中的裂纹形成动力学。尽管新的非局部模型在各种应用中越来越受欢迎，但其背后的数学研究仍处于起步阶段，阻碍了计算工具的进一步发展。该项目的目标是开发高效可靠的数值方法来模拟非局部模型，并建立相关的数学分析，作为严格验证和验证过程的一部分。它旨在提高非局部建模的有效性和鲁棒性，同时保持建模精度。在教育方面，该项目将为学生提供数学和计算力学方面的培训。该项目旨在开发最先进的多尺度建模技术，以提高计算效率，同时保留预测动态裂缝的非局部模型的准确性，并建立新的理论方法来研究模型的分析特性。将讨论三种具体的多尺度建模方法，其中边界轨迹和界面条件的处理起着关键作用。首先是通过接口上非局部相互作用的异构定位实现非局部模型和局部模型的无缝耦合。利用一种新颖的非局部迹定理来保证耦合的适定性。目标是将接口视为基于解决方案开发的自由边界。第二种是受原子-连续体耦合方法启发的准非局部耦合方法。它的目的是在离散原子模型和连续非局部模型之间建立一座桥梁。三是设计合适的非局部边界条件，以减少无界域问题的计算量。本文将引入非局部诺伊曼边界条件的新概念，这将有助于阐明区域分解方法和非局部模型的耦合。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.