A spectral method with volume penalization for a nonlinear peridynamic model

Abstract

The peridynamic equation consists in an integro‐differential equation of the second order in time which has been proposed for modeling fractures and damages in the context of nonlocal continuum mechanics. In this article, we study numerical methods for the one‐dimension nonlinear peridynamic problems. In particular we consider spectral Fourier techniques for the spatial domain while we will use the Störmer–Verlet method for the time discretization. In order to overcome the limitation of working on periodic domains due to the spectral techniques we will employ a volume penalization method. The performance of our approach is validated with the study of the convergence with respect to the spatial discretization and the volume penalization. Several tests have been performed to investigate the properties of the solutions.

A general and fast convolution-based method for peridynamics: Applications to elasticity and brittle fracture

“…Many linearized peridynamic models, featuring integral operators, can naturally be expressed via convolutions. Certain nonlinear models are also shown to have a convolutional structure [27,32]. We observe that in general, the following form of a possibly nonlinear PD integrand (e.g.…”

Qualitative Aspects in Nonlocal Dynamics

“…In the present paper, we exploit a qualitative analysis of a one-dimensional linear peridynamical model to highlight the behavior of the solutions of an initial value problem in dependence of the characteristics affecting the nonlocal properties of the equation. More precisely, we show through numerical investigations the way nonlocality rules a wide range of modes of wave propagation (see also [16,17]). Indeed, the dynamics predicted by the equation under study exhibits a behavior ranging from the hyperbolic-like to the dispersive-like propagation ( [18]), according to its highly nontrivial dispersive relation.…”

Qualitative aspects in nonlocal dynamics

“…In the present paper we exploit a qualitative analysis of a one-dimensional linear peridynamical model to highlight the behavior of the solutions of an initial value problem in dependence of the characteristics affecting the nonlocal properties of the equation. More precisely, we show through numerical investigations the way nonlocality rules a wide range of modes of wave propagation (see also [5,13]). Indeed, the dynamics predicted by the equation under study exhibits a behavior ranging from the hyperbolic-like to the dispersive-like propagation ( [21]), according to its highly nontrivial dispersive relation.…”