Novel Decompositions and Fast Numerical Methods for Peridynamics

周期动力学理论是一种新兴的工程应用建模工具，用于分析材料的不连续性，包括动态断裂、自发裂纹的形成和破碎。尽管越来越多的实验证据支持周期动力学建模，但仍然存在一些数学和计算挑战，可能会阻碍其在应用中的潜在广泛应用。在计算方面，由于需要处理更长范围的相互作用，标准的数值方法可能昂贵得令人望而却步，而在理论方面，在理解不连续的来源和演变以及在识别局部极限行为和与非线性弹性动力学的联系方面需要取得进展。此外，尽管对于实际工程应用来说，边界条件的精确应用是必不可少的，但对非局部边界条件的理解仍然很少。该项目通过开发新的数学和计算策略来解决这些挑战，这些策略促进了动态建模并增加了其对工程应用的吸引力。该项目为本科生和研究生提供了应用数学和计算数学的培训机会。本项目介绍了一种基于傅里叶谱分析的研究线性和非线性动力学模型及其应用的统一而系统的方法。该方法建立在线性周期动力学的分析和计算方法的基础上，然后将其提升到非线性周期动力学。研究人员将研究周期动力学算子的傅里叶乘子，并开发有效的算法来计算它们。这种傅里叶乘子分析将用于开发正则性结果、周期设置中的谱求解器以及用于边值条件的傅立叶连续方法。与非局域性参数随网格大小成比例的有限差分或有限元方法相比，谱方法是有效的，非常适合于研究周期动力学，因为它们将非局域性参数与网格尺寸解耦。从应用的角度来看，傅立叶分析方法将在周期动力学的背景下用于研究非局部边界条件，调查不连续的来源和演化，并更好地了解非局部方程与其本地经典方程之间的联系。该奖项反映了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。

Peridynamic theory is an emerging modeling tool used in engineering applications to analyze material discontinuities including dynamic fracture, spontaneous crack formation, and fragmentation. Despite a growing body of experimental evidence in favor of peridynamic modeling, there remain several mathematical and computational challenges that may hinder its potential widespread use in applications. On the computational side, standard numerical methods can be prohibitively expensive due to the need to handle longer range interactions, while on the theoretical side, progress is needed in understanding the sources and evolution of discontinuities and in identifying the local limiting behavior and the connection to nonlinear elastodynamics. Furthermore, while for practical engineering usage, precise application of boundary conditions is essential, nonlocal boundary conditions are still poorly understood. This project addresses these challenges by developing new mathematical and computational strategies which advance peridynamic modeling and increase its appeal for engineering applications. The project provides training opportunities in applied and computational mathematics for undergraduate and graduate students. This project introduces a unified and systematic approach based on Fourier spectral analysis for studying linear and nonlinear peridynamic models and their applications. The approach is built on a foundation of analytical and computational methods for linear peridynamics, which are then lifted to nonlinear peridynamics. The investigators will study the Fourier multipliers of peridynamic operators and develop efficient algorithms to compute them. This Fourier multipliers analysis will be used to develop regularity results, spectral solvers in periodic setups, and a Fourier Continuation method for boundary value conditions. The spectral methods introduced are efficient and well-suited to study peridynamics as they decouple the nonlocality parameter and the grid size, in contrast to finite difference or finite element methods in which the nonlocality scales with grid size. From the applications point of view, the Fourier analysis approach will be utilized in the context of peridynamics to study nonlocal boundary conditions, to investigate the sources and evolution of discontinuities, and to better understand the connection between nonlocal equations and their local classical counterparts.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.