Analytical and computational foundations of peridynamics: regularity theory and the resolution of discontinuities

Abstract

This dissertation develops analytical and computational foundations for the study of peridynamic and nonlocal models, with a particular focus on solution regularity, nonlocal-to-local limits, and the analysis and numerical treatment of discontinuities. We first investigate the regularity of solutions to the nonlocal wave equation in the framework of periodic distributions. The spatial operator is given by a nonlocal Laplace operator with a compactly supported kernel, encompassing both integrable and singular interactions. Using a unified Fourier multiplier approach, we characterize the spatial and temporal regularity of solutions in terms of the initial data and forcing terms. This framework extends naturally to nonlocal pseudo-differential operators and applies in arbitrary spatial dimensions. Furthermore, we establish convergence of solutions to those of the classical wave equation under two limiting regimes: vanishing nonlocality and increasing kernel singularity toward a critical, dimension-dependent threshold. Building on this scalar framework, we then study the peridynamic equilibrium and evolution equations for linear state-based models in multi-dimensional settings. Working in the space of periodic vector-distributions, we analyze both integrable and singular kernels through the asymptotic behavior of the Fourier multipliers associated with the peridynamic operator. This analysis yields explicit regularity results for solutions in terms of the data, kernel singularity, and spatial dimension. In addition, we establish nonlocal-to-local convergence results for both the operator and the corresponding solutions, again under regimes of vanishing horizon and increasing kernel singularity, thereby connecting peridynamic models to classical linear elasticity. Finally, we investigate the formation and propagation of discontinuities in the one-dimensional nonlocal Poisson equation with discontinuous source terms. We show that the regularity of the solution is governed by the interplay between the smoothness of the kernel and that of the source term. In particular, singularities of the kernel at the origin induce blow-up in derivatives of the solution, while discontinuities of the kernel or its derivatives at the horizon generate cascades of derivative discontinuities at shifted locations. Motivated by this analysis, we develop a semi-analytic spectral method for accurately resolving discontinuities. The method employs explicit smoothing transformations to remove the discontinuities, enabling high-order spectral approximation and reconstruction of the original solution. Numerical experiments demonstrate that this approach effectively mitigates Gibbs oscillations and significantly improves both accuracy and convergence. Together, these results provide a unified analytical and computational framework for understanding regularity, singular behavior, and efficient numerical approximation in peridynamic and nonlocal models.