A Nonlocal Fractional Peridynamic Diffusion Model

Yuanyuan Wang,Siyuan Fan,Xiangnan Yu,Hongguang Sun,Yan Gu

doi:10.3390/fractalfract5030076

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.

Highlights

In recent decades, nonlocal models have attracted increasing attention for dealing with complex physical problems [1,2,3,4]
In our numerical analysis of PD and fractional peridynamic (FPD) diffusion models, the spatial integral term is discretized by a single-point Gauss quadrature scheme, and solved by using the peridynamic differential operator (PDDO) or fractional peridynamic differential operator (FPDDO) method
We developed a nonlocal FPD model together with a meshless FPDDO method to We developed a nonlocal FPD model together with a meshless FPDDO method to solve the anomalous diffusion problem

Summary

Introduction

Nonlocal models have attracted increasing attention for dealing with complex physical problems [1,2,3,4]. Numerical discretization of a traditional local model usually yields many degrees of freedom, which leads to excessive calculations and increases the computational complexity, despite providing low accuracy [8,9]. Nonlocal models have been proposed to facilitate numerical analysis [10,11,12]. Compared with the integer-order derivative model, the FDM has achieved great success in characterizing various anomalous phenomena in the physics and engineering fields [17,18,19] The fractional derivative model (FDM) has been widely used to describe the temporal history-memory and spatial nonlocality of physical behaviors [13,14,15,16].

Methods

Results

Discussion

Conclusion