Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation

Abstract

State-based peridynamics is a non-local reformulation of solid mechanics that replaces the force density of the divergence of stress with an integral of the action of force states on bonds local to a given position, which precludes differentiation with the aim to model strong discontinuities effortlessly. A popular implementation is a meshfree formulation where the integral is discretized by quadrature points, which results in a series of unknowns at the points under the strong-form collocation framework. In this work, the meshfree discretization of state-based peridynamics under the correspondence principle is examined and compared to traditional meshfree methods based on the classical local formulation of solid mechanics. It is first shown that the way in which the peridynamic formulation approximates differentiation can be unified with the implicit gradient approximation, and this is termed the reproducing kernel peridynamic approximation. This allows the construction of non-local deformation gradients with arbitrary-order accuracy, as well as non-local approximations of higher-order derivatives. A high-order accurate non-local divergence of stress is then proposed to replace the force density in the original state-based peridynamics, in order to obtain global arbitrary-order accuracy in the numerical solution. These two operators used in conjunction with one another is termed the reproducing kernel peridynamic method. The strong-form collocation version of the method is tested against benchmark solutions to examine and verify the highorder accuracy and convergence properties of the method. The method is shown to exhibit superconvergent behavior in the nodal collocation setting.

A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods

“…In the correspondence formulation, a kinematic variable (e.g., the deformation gradient F) is computed using integration, rather than differentiation, and then used to evaluate its energy conjugate (e.g., the first Piola-Kirchhoff stress) through local material models. It was shown in [4,15] that there is a close connection between the PD correspondence model and meshfree discretizations of local theories, such as the reproducing kernel (RK) particle method [10,22]. [15] showed the equivalence between the PD differential operator [24] and the RK implicit gradient operator [11], which both can be employed to obtain more accurate meshfree gradients, such as a deformation gradient of higher-order accuracy.…”

“…It was shown in [4,15] that there is a close connection between the PD correspondence model and meshfree discretizations of local theories, such as the reproducing kernel (RK) particle method [10,22]. [15] showed the equivalence between the PD differential operator [24] and the RK implicit gradient operator [11], which both can be employed to obtain more accurate meshfree gradients, such as a deformation gradient of higher-order accuracy. Reproducing kernel enhanced approaches have been proposed to improve meshfree integration in peridynamic models [15,19,28].…”

“…[15] showed the equivalence between the PD differential operator [24] and the RK implicit gradient operator [11], which both can be employed to obtain more accurate meshfree gradients, such as a deformation gradient of higher-order accuracy. Reproducing kernel enhanced approaches have been proposed to improve meshfree integration in peridynamic models [15,19,28].…”

“…In Part I of this two-part paper, we focus on a unified meshfree approach that considers non-local integration as a regularization of classical continuum mechanics to obtain more accurate local derivatives. The two methods of RK-PD [15] and generalized moving least squares (GMLS) [35] serve our goal here. The RK and GMLS approximations were shown to have a close connection under a uniform-grid discretization [18].…”

Mixed peridynamic formulations for compressible and incompressible finite deformations

“…A stresspoint method to overcome the rank deficiency was presented in Luo and Sundararaghavan [25]. Hillman et al [15] unified the peridynamic deformation gradient with the implicit gradient approximation, studied the convergence of the differential operations and proposed a reproducing kernel peridynamic method.…”