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A stabilized conforming nodal integration for Galerkin mesh‐free methods
Author(s) -
Chen JiunShyan,
Wu ChengTang,
Yoon Sangpil,
You Yang
Publication year - 2000
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/1097-0207(20010120)50:2<435::aid-nme32>3.0.co;2-a
Subject(s) - numerical integration , discretization , smoothing , galerkin method , mathematics , gaussian quadrature , finite element method , mathematical analysis , nyström method , boundary value problem , statistics , physics , thermodynamics
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.