Novel Decompositions and Fast Numerical Methods for Peridynamics

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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.