Analytical Aspects of Nonlocal Models in Applications

应用中非局部模型的分析方面

• 批准号: 2206252
• 负责人: Tadele Mengesha
• 金额: $ 27.68万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206252&HistoricalAwards=false
• 关键词: Analytical; Aspects; Nonlocal; Models; Applications

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.

力学和其他应用领域中的非局部模型是相对较新的模型，但已被证明对于模拟某些具有挑战性的现象非常有效，例如固体力学中的断裂。了解材料的行为、失效和变形时的强度，对于正确使用材料和设计新材料至关重要，这可能会对制造、材料工程和相关技术产生影响。这个项目的目的是开发基本的数学技术，为所谓的动力学模型以及连续介质力学的其他非局部模型提供良好的分析基础。这些发现有望适用于具有类似结构的其他模型，并在工程科学中应用。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仿真工作将更加定量和可靠。该项目将为博士生提供研究培训机会。该项目的中心是对非局部模型在应用中的严格研究，主要是在力学方面。所考虑的模型的特点是在单一的数学框架内通过使用积分方程而不是微分方程式来有效地描述连续和不连续的场。这些模型已经成功地应用于描述跳跃随机过程、超扩散和次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。这个项目致力于探索一些分析问题，同时为未来非局部模型的研究奠定必要的数学基础。需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，解作为作用力和材料系数的函数的正则性理论的发展，以及由非局部模型控制的系统的最优设计和最优控制问题。所考虑的数学方法包括摄动法、变分法和非线性泛函分析。这项研究旨在使基于非局部和动态的建模和仿真在实际应用中更加数学上的一致性、量化和预测性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

A note on estimates of level sets and their role in demonstrating regularity of solutions to nonlocal double phase equations

关于水平集估计及其在证明非局部双相方程解的规律性中的作用的注释

• 发表时间: 2020
• 作者: J. Scott;T. Mengesha
• 通讯作者: T. Mengesha