Numerical Methods for the Nonlocal Wave Equation of the Peridynamics

TL;DR

This paper reviews and develops numerical methods for solving the linear and nonlinear nonlocal wave equations in peridynamics, introducing higher-order and spectral techniques validated through numerical tests.

Contribution

It introduces new higher-order spatial numerical methods and applies spectral discretization to the linear peridynamics model, extending to nonlinear cases.

Findings

Higher-order numerical methods improve accuracy

Spectral methods effectively discretize linear problems

Numerical tests validate the proposed techniques

Abstract

In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering because seems to provide an effective approach to modeling mechanical systems avoiding spatial discontinuous derivatives and body singularities. In particular, we will consider the linear model of peridynamics in a one-dimensional spatial domain. Here we will review some numerical techniques to solve this equation and propose some new computational methods of higher order in space; moreover we will see how to apply the methods studied for the linear model to the nonlinear one. Also a spectral method for the spatial discretization of the linear problem will be discussed. Several numerical tests will be given in order to validate our results.