Mathematical Theory of Peridynamics and Nonlocal Models

Mengesha1312809首席研究员和他的同事研究非局部模型的数学性质，特别是连续介质力学的动力学模型。该项目专注于建立涉及非局部边界(体积约束)值条件的动态和平衡问题的线性和非线性模型的适定性；将新出现的不连续性分类并发展关于非局部模型解的正则性理论作为给定数据的函数；以及研究能够在动力学方程和其他非局部模型中形成和传播奇点的条件。研究人员使用了从微扰法到变分法的各种分析工具。该项目解决了传统本地方法所不存在的技术挑战，导致将经典数学概念和技术扩展到非本地环境。该项目为研究非局部算符和方程提供了一个数学框架，这对研究许多常见的非局部现象可能是有用的。它有助于为非局部模型建立必要的理论基础。研究人员将项目发现整合到课堂教学和其他教育活动中。研究人员的长期目标是与机械工程师、物理学家、生物学家和材料科学家合作，使非局部模型成为研究各种科学和工程应用中复杂系统的有用工具，如反常扩散、非局部热传导、相变和生物聚集。

Mengesha1312809 The principal investigator and his colleague study the mathematical properties of nonlocal models, particularly the peridynamic model of continuum mechanics. The project focuses on establishing the well-posedness of both linear and nonlinear models for dynamic and equilibrium problems involving nonlocal boundary (volume-constrained) value conditions; classifying emerging discontinuities and developing a regularity theory for solutions of nonlocal models as functions of given data; and studying conditions that enable nucleation and propagation of singularities in peridynamic equations and other nonlocal models. The investigators use analytical tools ranging from perturbation methods to calculus of variations. The project addresses technical challenges that are absent from the traditional local approach, leading to extensions of classical mathematical concepts and techniques to the nonlocal setting. The project offers a mathematical framework for studying nonlocal operators and equations that may be useful to the study of many common nonlocal phenomena. It helps establish the necessary theoretical footing for nonlocal models. The project findings are integrated by the investigators into classroom teaching and other educational endeavors. The investigators' long-term goal is to work with mechanical engineers, physicists, biologists and materials scientists to make nonlocal models useful tools for studying complex systems in various scientific and engineering applications such as anomalous diffusion, nonlocal heat conduction, phase transition, and biological aggregation.