Stabilized conforming nodal integration: exactness and variational justification

Abstract

In most Galerkin mesh-free methods, background integration cells partitioning the problem domain are required to evaluate the weak form. It is therefore worthwhile to consider these methods using the notions of domain decomposition with the integration cells being the subdomains. Presuming that the analytical solution is admissible in the trial solution, domain and boundary integration exactness, which depend on the orders of the employed trial solution and the required solution exactness, are identified for the strict satisfaction of traction reciprocity and natural boundary condition in the weak form. Unfortunately, trial solutions constructed by many mesh-free approximants contain non-polynomial terms which cannot be exactly integrated by Gaussian quadratures. Recently, stabilized conforming (SC) nodal integration for Galerkin mesh-free methods was proposed and illustrated to be linearly exact. This paper will discuss how linear exactness is ensured and how spurious oscillation encountered by direct nodal integration is suppressed in SC nodal integration from a domain decomposition point of view. Moreover, it will be shown that SC nodal integration can be formulated by the Hellinger-Reissner Principle and thus justified in the classical variational sense. Applications of the method to straight beam, plate and curved beam problems are presented.

Adaptive element-free Galerkin method applied to the limit analysis of plates

“…The smoothing technique proposed in [28] is one of the most efficient nodal integration methods available, and has been applied successfully to various analysis problems [9,10,[33][34][35]. In this technique nodal values are determined by spatially averaging field values using the divergence theorem.…”

Limit analysis of plates and slabs using a meshless equilibrium formulation

“…The strain smoothing method was then modified to allow stabilization in nodal integration schemes, leading to the so-called stabilized conforming nodal integration (SCNI) scheme [11]. The SCNI scheme was then successfully applied to both elastic analysis [12,13] and plastic analysis [9] problems. It has been shown that the SCNI scheme results in an efficient and truly mesh-free method, and also to numerically stable solutions.…”

“…The main idea of the scheme is that nodal values are determined by spatially averaging field values using the divergence theorem. The scheme has been applied successfully to various analysis problems [9,[12][13][14]. It is shown that, when the SCNI scheme is applied, the solutions obtained are accurate and stable, and the computational cost is much lower than when using Gauss integration.…”

A cell‐based smoothed finite element method for kinematic limit analysis

“…It is known as the stabilized conforming nodal integration (SCNI) scheme. The SCNI scheme has been applied successfully to various problems, for instance, elastic analysis [12][13][14], plastic limit analysis [15], error estimation [16] and a stabilized mesh-free equilibrium model for limit analysis [17]. It is shown that, when the SCNI scheme is applied, the solutions obtained are accurate and stable, and locking problems can also be prevented.…”