Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems

Abstract

Galerkin meshfree methods can suffer from instability and suboptimal convergence if the issue of quadrature is not properly addressed. The instability due to quadrature is further magnified in high strain rate events when nodal integration is used. In this paper, several stable and convergent nodal integration methods are presented and applied to transient and large deformation impact problems, and an eigenvalue analysis of the methods is also provided. Optimal convergence is attained using variationally consistent integration, and stability is achieved by employing strain smoothing and strain energy stabilization. The proposed integration methods show superior performance over standard nodal integration in the wave propagation and Taylor bar impact problems tested.

An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics

“…However, directly integrating at nodes results in stability issues as well as sub‐optimal convergence , and requires special techniques in order to overcome these difficulties. Several nodal integration methods have been developed that are convergent and stable , but in the end sacrifice efficiency because of the approaches taken to ensure accuracy and stability. Another issue with these methods is the tuning of stabilization parameters for optimal accuracy and convergence.…”

“…However, directly integrating at nodes results in stability issues [13][14][15] as well as sub-optimal convergence [13,15,17], and requires special techniques in order to overcome these difficulties. Several nodal integration methods have been developed that are convergent and stable [13,25,28,29], but in the end sacrifice efficiency because of the approaches taken to ensure accuracy and stability. Another issue with these methods is the tuning of stabilization parameters for optimal accuracy and convergence.The instability in direct nodal integration is due to the fact that for discretizations with spacing h, oscillating modes of wavelength 2h are admitted in the solution with little or no energy due to gradients being sampled only at the nodes [13][14][15].…”

Meshfree Methods: Progress Made after 20 Years

“…Nodal integrations with corrective VCI procedures and stabilization can be formulated such that they do not detract from the meshfree character of the method (Chen et al 2013;Hillman et al 2014;Puso et al 2008;Rüter et al 2013;Wu et al , 2015.…”

“…• VC-MSNNI: variationally consistent MSNNI (Hillman et al 2014); • VC-NSNI: variationally consistent NSNI (Hillman and Chen 2016); • WLS: weighted least squares;…”

Impact Resistance of Concrete Structures

“…Commonly, element-deletion techniques [25,26] were applied to allow for large deformations and perforations. A strong competitor to finite elements is mesh-free methods [27][28][29][30][31][32][33][34][35][36][37] that allow for a more robust and accurate prediction of dynamic fracture and fragmentation; see [38] for a recent overview of mesh-free methods and their applications. Reference [39], for instance, was able to predict the impact resistance of the experiments carried out by [40] with mesh-free methods quite accurately, while the FE simulations done in [40] led always to an underestimation of the impact resistance.…”