Mathematical Analysis on Peridynamic Models

该奖项支持首席研究员正在进行的关于“非局部模型”的密切分析检查的研究项目，非局部模型是一种相对较新的数学模型，已被证明对某些具有挑战性的现象建模非常有效，例如固体力学中的断裂。例如，了解材料在变形时的行为、失效和强度，对材料的正确使用以及对制造、材料工程和相关技术有潜在影响的新材料的设计至关重要。为此目的，过去提出了各种成功程度不同的模型。本研究项目旨在发展基本的数学技术，为最近提出的连续介质力学的“周动力学模型”提供适当的分析基础。这些发现也将适用于其他类似结构的非局部模型，并在社会和生物科学中得到应用。这些活动不仅将有助于建模开发和实验验证的成功和有效性，而且将确保未来基于这些非局部理论的建模和仿真工作将更加定量和可靠。该研究项目将为研究生的培养提供机会和支持。首席研究员将把研究结果整合到课堂教学和其他教育工作中。本项目涉及一般非局部模型的理论和技术的发展，特别是周动力模型。这些模型的特点是通过使用积分方程代替微分方程，在单一数学框架内有效地描述连续和不连续场。这些模型已经成功地应用于更好地描述跳跃随机过程、异常扩散以及固体中裂纹的自发形成和扩展，仅举几例。然而，这些模型也给科学界带来了新的数学挑战。本研究在对一些分析问题进行探索的同时，也为今后非局部和周期动力学模型的分析奠定了必要的数学基础。要解决的问题包括研究活动，以提高对实际感兴趣的线性化周动力模型分析的知识；非局部方程解随作用力、初始数据和系数的正则性以及对基于周期动力学的非线性行为的理解。这些方法涉及到将经典数学概念和技术扩展到非局部环境的各种工具，包括微扰方法、变分法和非线性泛函分析。此外，将制定的基本数学基础设施可能会影响有效可靠的有限元方法和其他数值方案的发展，以通过周动力学来解决复杂的工程问题。该研究将使基于非局部和周动力学的建模和仿真在数学上更加一致，并有助于在实际应用中使建模和仿真更具定量化和预测性。

This award supports the ongoing research program of the Principal Investigator on the close analytical examination of "nonlocal models," a type of mathematical model of relatively recent vintage that has proved to be very effective for modeling certain challenging phenomena, such as fracture in solid mechanics. For example, understanding how materials behave, their failure as well as their strength when deformed, is crucial for their proper usage and also for the design of new materials with potential impact on manufacturing, materials engineering, and related technologies. For this purpose, models with varied levels of success have been proposed in the past. This research project aims to develop basic mathematical techniques that will deliver proper analytical footing for the recently proposed "peridynamic model" of continuum mechanics. The findings will also be applicable to other nonlocal models of similar structure, with applications in social and biological sciences. The activities not only will contribute to the success and effectiveness of attempts on modeling development and experimental validation but also will ensure that future modeling and simulation efforts based on these nonlocal theories will be more quantitative and reliable. The research project will provide opportunities and support for the training of graduate students. The Principal Investigator will integrate the findings of the project into classroom teaching and other educational endeavors.This project concerns the development of theory and techniques for nonlocal models in general and for the peridynamic model in particular. These models are characterized by their effective description of continuous as well as discontinuous fields within a single mathematical framework by using integral equations in lieu of differential equations. The models have been successfully applied to better describe jump stochastic processes, anomalous diffusion, and spontaneous formation and propagation of cracks in solids, to name a few applications. However, the models have also presented the scientific community with new mathematical challenges. This research is devoted to exploring some analytical issues while at the same time laying the necessary mathematical foundation for future analyses on nonlocal and peridynamic models. Issues to be addressed include research activities that advance knowledge on the analysis of linearized peridynamic models of practical interest; regularity properties of solutions of nonlocal equations as a function of applied force, initial data, and coefficients; and understanding of peridynamic-based nonlinear behavior. The approaches involve various tools that lead to extensions of classical mathematical concepts and techniques to the nonlocal setting, including perturbation methods, calculus of variations, and nonlinear functional analysis. Furthermore, the basic mathematical infrastructures that will be worked out are likely to impact the development of effective and reliable finite element methods and other numerical schemes to solve complex engineering problems via peridynamics. The research will make nonlocal and peridynamics-based modeling and simulation more mathematically consistent, and it will contribute to making such modeling and simulation more quantitative and predictive in practical applications.