Gradient reproducing kernel particle method

Abstract

This paper presents an innovative formulation of the RKPM (reproducing kernel particle method) pioneered by Liu. A major weakness of the conventional RKPM is in dealing with the derivative boundary conditions. The EFGM (element free Galerkin method) pioneered by Belytschko shares the same difficulty. The proposed RKPM referred to as GRKPM (gradient RKPM), incorporates the first gradients of the function in the reproducing equation. Therefore in three-dimensional space GRKPM consists of four independent types of shape functions. It is due to this feature that the corrected collocation method can be readily generalized and combined with GRKPM to enforce the EBCs (essential boundary conditions), involving both the field quantity and its first derivatives simultaneously. By considering several plate problems it is observed that GRKPM yields solutions of higher accuracy than those obtained using the conventional approach, while for a desired accuracy the number of particles needed in GRKPM is much less than in the traditional methodology.

On numerical aspects of different updating schedules for tracking fracture path in strain localization modeling

“…However, they are characterized by some numerical features that make them suitable for modeling strong discontinuities. Within this class, enriched Element-Free Galerkin formulation [Fleming et al, 1997], the Reproducing Kernel Particle Method [Liu et al, 1995;Klein et al, 2001;Boyce et al, 2014], the Gradient Reproducing Kernel Particle Method [Hashemian and Shodja, 2008], and smoothed-particle hydrodynamics [Batra and Zhang, 2004] are among the developed methods. A number of reviews regarding these methods can be found in [Belytschko et al, 1996;Liu et al, 1996Liu et al, , 1999Chen et al, 2011].…”

The nonlocal thermo‐hydro‐mechanical model for cracking behaviors of quasi‐brittle materials in the framework of advanced general particle dynamics

“…The commonly used meshless methods mainly include Smoothed Particle Hydrodynamic method (SPH), [30][31][32][33][34] moving least square approximation method (MLS), 35 and Reproducing Kernel Particle Method (RKPM). [36][37][38] RKPM has been applied to many large deformation problems, and it requires appropriate kernel support coverage of neighboring nodes to ensure kernel stable. A new reproducing kernel formulation with quasi-linear reproducing conditions, which overcomes the traditional large particle motion problem, was introduced by Yreux and Chen.…”

Application of RKP‐FSM in the buckling and free vibration analysis of thin plates with abrupt thickness changes and internal supports

“…Shodja and Hashemian , when dealing with beam‐column problems, noticed that the enforcement of derivative‐type essential boundary conditions in the context of the RKPM is troublesome. They proposed an incorporation of the first derivative of the function in the reproduction formula of the RKPM and named the method that evolved the Gradient RKPM . Li and Liu proposed a synchronized reproducing kernel interpolation method in which synchronized rates of convergence for the discrete functions and their derivatives can be turned.…”