Analytical Aspects of Nonlocal Models in Applications

力学和其他应用领域的非局部模型是相对较新的，但已被证明在模拟某些具有挑战性的现象（如固体力学中的断裂）方面非常有效。了解材料在变形时的行为、失效和强度，对于材料的正确使用和新材料的设计至关重要，对制造业、材料工程和相关技术都有潜在的影响。该项目旨在发展基本的数学技术，为所谓的周期动力学模型以及连续介质力学的其他非局部模型提供可靠的分析基础。研究结果有望应用于具有类似结构的其他模型，并在工程科学中得到应用。这些活动将有助于提高这些模型的有效性，并确保未来基于这些非局部理论的建模和模拟工作将更加定量和可靠。本项目将为博士生提供科研培训机会。该项目集中在应用的非局部模型的严格研究，主要是在力学。所考虑的模型的特点是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，超扩散和亚扩散，以及固体中裂纹的自发形成和扩展，仅举几个例子。然而，它们也给科学界带来了新的数学挑战。本项目致力于探索一些分析性问题，同时为今后非局部模型的研究奠定必要的数学基础。要解决的问题包括涉及各向异性和异质性的线性化非局部模型的适定性，具有振荡系数的线性和非线性非局部模型的渐近分析，作为施加力和材料系数函数的解的正则理论的发展，以及由非局部模型控制的系统中的最佳设计和最优控制问题。考虑的数学方法包括微扰方法、变分法和非线性泛函分析。该研究旨在使非局部和基于周围动力学的建模和仿真在实际应用中更具数学一致性、定量和预测性。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。

Nonlocal models in mechanics and other application areas are of relatively recent vintage but have proved to be very effective for modeling certain challenging phenomena, such as fracture in solid mechanics. Understanding how materials behave, their failure as well as their strength when deformed, is crucial for their proper usage and for the design of new materials, with potential impact in manufacturing, materials engineering, and related technologies. This project aims to develop fundamental mathematical techniques that will deliver a sound analytical footing for the so-called peridynamic model, as well as other nonlocal models of continuum mechanics. The findings are expected to be applicable to other models having similar structure, with applications in engineering sciences. The activities will contribute to improved effectiveness of these models and ensure that future modeling and simulation efforts based on these nonlocal theories will be more quantitative and reliable. The project will provide research training opportunities for doctoral students. The project centers on a rigorous study of nonlocal models in applications, primarily in mechanics. The models under consideration are characterized by their effective description of continuous as well as discontinuous fields within a single mathematical framework by using integral equations instead of differential equations. These models have been successfully applied to better describe jump stochastic processes, super- and sub- diffusion, and spontaneous formation and propagation of cracks in solids, to name a few applications. However, they have also presented the scientific community with new mathematical challenges. This project is devoted to exploring some analytical issues while at the same time laying the necessary mathematical foundation for future studies of nonlocal models. Issues to be addressed include well-posedness of linearized nonlocal models that involve anisotropy and heterogeneity, asymptotic analysis of linear and nonlinear nonlocal models with oscillatory coefficients, development of a regularity theory of solutions as a function of applied force and material coefficients, and optimal design and optimal control problems in systems governed by nonlocal models. The mathematical approaches considered include perturbation methods, calculus of variations, and nonlinear functional analysis. The research aims to make nonlocal and peridynamics-based modeling and simulation more mathematically consistent, quantitative, and predictive in practical applications.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.