A spectral method with volume penalization for a nonlinear peridynamic model

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 Stormer-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.