An Effective Way to Control Numerical Instability of a Nonordinary State‐Based Peridynamic Elastic Model

Abstract

The constitutive modeling and numerical implementation of a nonordinary state-based peridynamic (NOSB-PD) model corresponding to the classical elastic model are presented. Besides, the numerical instability problem of the NOSB-PD model is analyzed, and a penalty method involving the hourglass force is proposed to control the instabilities. Further, two benchmark problems, the static elastic deformation of a simple supported beam and the elastic wave propagation in a two-dimensional rod, are discussed with the present method. It proves that the penalty instability control method is effective in suppressing the displacement oscillations and improving the accuracy of calculated stress fields with a proper hourglass force coefficient, and the NOSB-PD approach with instability control can analyze the problems of structure deformation and elastic wave propagation well.

Refined bond-based peridynamics for thermal diffusion

“…The In addition, it is noteworthy that the present simulations adopt the uniform orthogonal particle discretization, and the thermal conduction simulations based on non-uniform particle distributions will be further investigated. Although the particle distributions influence the accuracy of PD solid models (Gu et al, 2017a(Gu et al, , 2017bNikravesh and Gerstle, 2018), according to the authors' experience, the PD differential operator can reproduce the derivatives even with non-uniform discretization, so that the non-uniform particle distributions probably only weaken the accuracy slightly.…”

“…As the horizon size approaches to zero, the interaction becomes local, and the PD theory reduces to the classical local theory (Silling and Lehoucq, 2008;Tian and Du, 2014;. Since its inception by Silling et al (2007), the PD theory can be divided into the bond-based peridynamic (BB PD), the ordinary state-based peridynamic (OSB PD) (Le et al, 2014;Madenci and Oterkus, 2016) and the non-ordinary state-based peridynamic (NOSB PD) (Warren et al, 2009;Foster et al, 2010;Gu et al, 2017bGu et al, , 2018.…”

A time‐discontinuous peridynamic method for transient problems involving crack propagation

“…38 Gu et al then proposed a penalty instability control method involving hourglass force to control instability to increase the accuracy of the stress field in elastic wave propagation problems. 39 Silling also presented a method for computing the attenuation coefficient explicitly to study the effect of spatial nonlocality on the decay of waves in a dissipative material. 40 As for damage problems, Silling and Askari proposed the critical stretch for the brittle materials, which is related to the critical energy release rate of the materials.…”

“…Silling analyzed the propagation of large amplitude nonlinear waves in a PD solid, which can propagate as solitary waves that move without dispersion at speeds much greater than the linear wave speed 38 . Gu et al then proposed a penalty instability control method involving hourglass force to control instability to increase the accuracy of the stress field in elastic wave propagation problems 39 . Silling also presented a method for computing the attenuation coefficient explicitly to study the effect of spatial nonlocality on the decay of waves in a dissipative material 40 .…”