Handbook of Peridynamic Modeling

Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

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Numerical methods for nonlocal and fractional models

In this paper we study variational problems on a bounded domain for a nonlocal elastic energy of peridynamic-type which result in nonlinear systems of nonlocal equations. The well-posedness of variational problems is established via a careful study of the associated energy spaces. In the event of vanishing nonlocality we establish the convergence of the nonlocal energy to a corresponding local energy using the method of -convergence. Building upon existing techniques, we prove an Lp-compactness result (on bounded domains) based on near-boundary estimates that is used to study the variational limit of minimization problems subject to various volumetric constraints. For energy functionals in suitable forms, we find the corresponding limiting energy explicitly. As a special case, the classical Navier-Lame potential energy is realized as a limit of linearized peridynamic energy offering a rigorous connection between the nonlocal peridynamic model to classical mechanics for small uniform strain.

https://iopscience.iop.org/article/10.1088/0951-7715/28/11/3999/pdf

On the variational limit of a class of nonlocal functionals related to peridynamics

We found that the mathematically correct specification of a fractional differential equation on a bounded domain requires specification of appropriate boundary conditions, or their fractional analogue. In this paper we discuss the application of nonlocal diffusion theory to specify well-posed fractional diffusion equations on bounded domains.

/pdf/fractional-diffusion-on-bounded-domains-4d2l59r7na.pdf

Fractional diffusion on bounded domains

In this paper, a reaction–diffusion system is proposed to model the spatial spreading of West Nile virus in vector mosquitoes and host birds in North America. Transmission dynamics are based on a simplified model involving mosquitoes and birds, and the free boundary is introduced to model and explore the expanding front of the infected region. The spatial-temporal risk index $$R_0^F(t)$$
 , which involves regional characteristic and time, is defined for the simplified reaction–diffusion model with the free boundary to compare with other related threshold values, including the usual basic reproduction number $$R_0$$
 . Sufficient conditions for the virus to vanish or to spread are given. Our results suggest that the virus will be in a scenario of vanishing if $$R_0\le 1$$
 , and will spread to the whole region if $$R_{0}^F(t_0)\ge 1$$
 for some $$t_0\ge 0$$
 , while if $$R^F_0(0)<1<R_0$$
 , the spreading or vanishing of the virus depends on the initial number of infected individuals, the area of the infected region, the diffusion rate and other factors. Moreover, some remarks on the basic reproduction numbers and the spreading speeds are presented and compared.

Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary.

We introduce the Cauchy and time-dependent volume-constrained problems associated with a
 linear nonlocal convection-diffusion equation. These problems are shown to be well-posed and correspond to
 conventional convection-diffusion equations as the region of nonlocality vanishes.
 The problems also share a number of features such as the maximum principle,
 conservation and dispersion relations, all of which are consistent with their corresponding local counterparts. Moreover, these problems are the master equations for a class of finite activity Levy-type processes with nonsymmetric Levy measure. Monte Carlo simulations and finite difference schemes are applied to these nonlocal problems, to show the effects of time, kernel, nonlocality and different volume-constraints.

Zhan Huang

Papers

Nonlocal convection-diffusionvolume-constrained problems and jump processes