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 onedimensional 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.
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.
We reproduce the capacity drop phenomenon at a road merge by implementing a non-local point constraint at the junction in a first order traffic model. We call capacity drop the situation in which the outflow through the junction is lower than the receiving capacity of the outgoing road, as too many vehicles trying to access the junction from the incoming roads hinder each other. In this paper, we first construct an enhanced version of the locally constrained model introduced by Haut et al. (Proceedings 16th IFAC World Congress. Prague, Czech Republic 229 (2005) TuM01TP/3), then we propose its counterpart featuring a non-local constraint and finally we compare numerically the two models by constructing an adapted finite volumes scheme.
In this paper we validate the implementation of the numerical scheme proposed in [3]. The validation is made by comparison with an explicit solution here obtained, and the solutions of Riemann problems for several networks. We then perform some simulations in order to qualitatively validate the model under consideration.Such results represent also a first step for the validation of the finite volumes scheme introduced in [9].
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