A nonlocal operator method for solving partial differential equations

Abstract

We propose a nonlocal operator method for solving partial differential equations (PDEs). The nonlocal operator is derived from the Taylor series expansion of the unknown field, and can be regarded as the integral form "equivalent" to the differential form in the sense of nonlocal interaction. The variation of a nonlocal operator is similar to the derivative of shape function in meshless and finite element methods, thus circumvents difficulty in the calculation of shape function and its derivatives. The nonlocal operator method is consistent with the variational principle and the weighted residual method, based on which the residual and the tangent stiffness matrix can be obtained with ease. The nonlocal operator method is equipped with an hourglass energy functional to satisfy the linear consistency of the field. Higher order nonlocal operators and higher order hourglass energy functional are generalized. The functional based on the nonlocal operator converts the construction of residual and stiffness matrix into a series of matrix multiplications on the nonlocal operators. The nonlocal strong forms of different functionals can be obtained easily via support and dual-support, two basic concepts introduced in the paper. Several numerical examples are presented to validate the method.

A peridynamic Reissner‐Mindlin shell theory

“…Moreover, in the present PD Reissner‐Mindlin shell theory and formulation, we only need the linear polynomial basis to construct shape tensor, and because the Reissner‐Mindlin shell theory only requires C 0 continuity. However, if one wishes to extend the current approach to the Kirchhoff plate and shell theories, one may need to apply higher order PD operators as discussed in Ren et al 22 and Yan et al 63 …”

“…An energy functional operator is introduced to control the zero-energy modes in an implicit or explicit simulation. 22 Several control approaches of zero-energy modes have been proposed in the dynamic analysis of the non-ordinary state-based PD, for example, References 14,51…”

“…The is defined as the internal force state of the material point i due to the interaction with the particle X j . An energy functional operator is introduced to control the zero‐energy modes in an implicit or explicit simulation 22 . Several control approaches of zero‐energy modes have been proposed in the dynamic analysis of the non‐ordinary state‐based PD, for example, References 14,51 The occurrence of zero energy mode of NOSB‐PD is similar to the FEM's hourglass mode which can be restrained by implementing the stress points, for example Reference 52.…”

“…19 Hu et al reported a state-based PD modeling of ductile fracture for solid plates. 20 In fact, there have been some shell formulations based on meshfree particle methods developed over the years, for example, the Nonlocal Operator Method proposed by Rabczuk et al 21 and Ren et al, 22 which can solve higher-order partial differential equations in structural mechanics such as the Kirchhoff plate problem. Peng et al 23,24 developed a Mindlin-Reissner shell theory based on the Galerkin weak formulation of the reproducing kernel particle method, and they used it to calculate elastoplastic deformation of ship structures.…”

Coupling XFEM and Peridynamics for brittle fracture simulation: part II—adaptive relocation strategy

“…The dual-horizon methodology presented in [14] is such an approach that also avoids the appearance of spurious reflections. Furthermore, the nonlocal operator method, that can be considered a generalization of the dual-horizon PD model, has been proposed in [15,16]. Our study is limited to methodologies that combine the FE method with PD as it is envisaged to take advantage already established FE solvers and potentially port PD models to commercially available FE packages.…”