A Nonlocal Fractional Peridynamic Diffusion Model

Abstract

This paper proposes a nonlocal fractional peridynamic (FPD) model to characterize the nonlocality of physical processes or systems, based on analysis with the fractional derivative model (FDM) and the peridynamic (PD) model. The main idea is to use the fractional Euler–Lagrange formula to establish a peridynamic anomalous diffusion model, in which the classical exponential kernel function is replaced by using a power-law kernel function. Fractional Taylor series expansion was used to construct a fractional peridynamic differential operator method to complete the above model. To explore the properties of the FPD model, the FDM, the PD model and the FPD model are dissected via numerical analysis on a diffusion process in complex media. The FPD model provides a generalized model connecting a local model and a nonlocal model for physical systems. The fractional peridynamic differential operator (FPDDO) method provides a simple and efficient numerical method for solving fractional derivative equations.

Modelling of viscoelastic materials using non-ordinary state-based peridynamics

“…If the strain in the integral of ( 23) is substituted by (τ) = 0 (t)∕E 0 , we obtain where j is a normalised relaxation function given by j = E j ∕E 0 . To implement (22) in a numerical scheme such as in the framework of NOSBPD correspondence model will require a strategy for numerical integration of ( 23) or ( 24). This can be achieved using an algorithm [69] that discretises for example (24) in terms of a finite time interval.…”

“…In a research [19], it was demonstrated that PD can accurately predict size effect in geometrically equivalent structures across a whole range of sizes. PD has also been employed to study other nonlocal physical processes such as metal machining process [20], diffusion and other transport phenomena [21][22][23]. Peridynamic frameworks have also been proposed for homogenisation of heterogeneous materials [24][25][26][27][28][29][30][31] as well as multiscale [32][33][34][35][36][37][38] and Multiphysics modelling [39][40][41][42].…”

A well‐posed parameter identification for nonlocal diffusion problems

“…The constant coefficient fractional diffusion equation for $$$ 0<\alpha <1 $$$ can be written in the form $$$ {\partial}_tu\left(x,t\right)={\int}_{{\mathbb{R}}^d}\left(u\left(y,t\right)-u\left(x,t\right)\right)\phi \left(x-y\right) dy $$$, where $$$ \phi \left(x-y\right)=a\frac{\alpha }{\Gamma \left(1-\alpha \right)}{\left|u\right|}^{-\alpha -1} $$$ when $$$ x<y $$$ and $$$ \phi \left(x-y\right)=b\frac{\alpha }{\Gamma \left(1-\alpha \right)}{u}^{-\alpha -1} $$$ when $$$ x>y $$$. More details about the nonlocal diffusion and fractional diffusion can be seen in previous works 40–42 …”

“…More details about the nonlocal diffusion and fractional diffusion can be seen in previous works. [40][41][42]…”

Analysis of Longitudinal Forest Data on Individual-Tree and Whole-Stand Attributes Using a Stochastic Differential Equation Model

“…Considering the evolution of individual-tree and whole-stand growth paradigms over the past decades, we focus on the "diffusion process" theory of how trees in a stand can respond differently individually and collectively to both their internal and environmental factors, which is based on the Brownian motion model. Diffusion processes operate in many natural phenomena starting in the field of physics [6], but diffusion processes have been used in many biological, engineering, economic, and social phenomena [7][8][9]. In summary, a diffusion process is a stochastic continuous time process that satisfies the formalized stochastic differential equation.…”