Novel Decompositions and Fast Numerical Methods for Peridynamics

周波理论是一种新兴的建模工具，用于工程应用中分析材料的不连续性，包括动态断裂，自发裂纹形成和破碎。尽管有越来越多的实验证据支持周波模型，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使用。在计算方面，标准的数值方法可以是昂贵的，由于需要处理较长范围的相互作用，而在理论方面，需要在理解的来源和演变的不连续性，并在确定的本地限制行为和连接到非线性弹性动力学的进展。此外，虽然在实际工程应用中，边界条件的精确应用是必不可少的，非局部边界条件仍然知之甚少。该项目通过开发新的数学和计算策略来解决这些挑战，这些策略可以推进周波建模并增加其对工程应用的吸引力。该项目为本科生和研究生提供应用数学和计算数学方面的培训机会。本计画介绍以傅立叶频谱分析为基础，统一系统地研究线性与非线性周期模式及其应用。该方法是建立在分析和计算方法的基础上的线性周波，然后解除非线性周波。研究人员将研究周波算子的傅里叶乘子，并开发出有效的算法来计算它们。这种傅立叶乘数分析将用于开发规律性结果，周期性设置中的谱求解器，以及边界值条件的傅立叶延拓方法。介绍的谱方法是有效的，非常适合研究周期性的，因为它们解耦的非局部性参数和网格大小，在有限差分或有限元方法，其中的非局部性尺度与网格大小。从应用的角度来看，傅里叶分析方法将被用于研究非局部边界条件，调查的来源和演变的不连续性，以及更好地理解非局部方程与其局部经典方程之间的联系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准。

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.