On approximation theory of nonlocal differential operators

Abstract

Recently, several types of nonlocal discrete differential operators have emerged either from meshfree particle methods or from nonlocal continuum mechanics, such as peridynamics. In this article, we discuss the mathematical formulation as well as construction of the nonlocal discrete differential operators. Based on a least‐square minimization procedure and the associated Moore–Penrose inverse, we have found a general form of the shape tensor and a unified expression for the first type nonlocal differential operators. We then conduct a convergence study, which provides the interpolation error estimate for the first type discrete nonlocal different operators. We have shown that as the radius of the horizon approaches to zero, the first type nonlocal differential operators will converge to the local differential operators. Moreover, we have demonstrated the computational performance of the first type nonlocal differential operators in several numerical examples.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

“…We compared the accuracy of each numerical method by defining the following error shown in Eq. (23).…”

“…The peridynamic differential operator (PDDO) method, a nonlocal differential operator [21][22][23], has been developed in recent years based on the concept of the peridynamics (PD) theory [24,25]. It connects local partial derivatives and nonlocal integrals through Taylor series expansions and the property of orthogonal functions.…”

Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

“…For instance, Dorduncu devised a nonlocal stress analysis model for functionally graded sandwich panels using PDDO [22], Gao et al developed a nonlocal model for fluid flow and heat transfer coupling using PDDO [23], and Li et al introduced a nonlocal model for steady-state thermoelastic analysis of functionally graded materials with PDDO [24]. Additionally, Li et al compared the PDDO with the other nonlocal differential operators and proposed some improvements for PDDO [25][26][27].…”

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Two-Dimensional Population Balance Equations in Crystallization

“…In this study, we attempt to extend the EE-PDDO method to the simulation of 2D PBEs. The PDDO method is an effective numerical method for solving differential equations that has been developed in recent years by Madenci et al [23][24][25] based on the Peridynamics (PD) theory [26][27][28]. The PDDO is a nonlocal differential operator, and the main advantage of the PDDO method is that it can deal with the discontinuities or singularities without additional processing.…”