A stabilized conforming nodal integration for Galerkin mesh-free methods

2000/01/01 by Jiun-Shyan Chen, Jiun‐Shyan Chen, Cheng-Tang Wu +3 · 1,328 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Boundary value problem #Discretization #Electromagnetic Simulation and Numerical Methods #Finite element method #Galerkin method #Gaussian quadrature #Mathematical analysis #Mathematics #Numerical integration #Numerical methods in engineering #Nyström method #Smoothing

paper · doi:10.1002/1097-0207(20010120)50:2<435::aid-nme32>3.0.co;2-a

published in International Journal for Numerical Methods in Engineering 50(2), 435-466 (Wiley)

openalex publication_date 2000/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/27

Abstract

Domain integration by Gauss quadrature in the Galerkin mesh-free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh-free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh-free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh-free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh-free method as presented in several numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.

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