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Bulletin of the Polish Academy of Sciences Technical Sciences

The Journal of Polish Academy of Sciences

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Volume 61, Issue 1 (Mar 2013)


Application of grid convergence index in FE computation

L. Kwaśniewski
  • Corresponding author
  • Faculty of Civil Engineering, Warsaw University of Technology, 16 Armii Ludowej Ave., 00-637 Warszawa, Poland
  • Email:
Published Online: 2013-05-01 | DOI: https://doi.org/10.2478/bpasts-2013-0010


This paper presents an application of the grid convergence index (GCI) concept based on the Richardson extrapolation to a selected simple problem of a cantilever beam loaded with vertical forces at the tip end. The GCI method, popular in computational fluid dynamics, has been recently recommended for finite element (FE) applications in solid and structural mechanics. Based on the results obtained usually for three meshes, the GCI method enables one to determine, in an objective manner, the order of convergence to estimate the asymptotic solution and the bounds for discretization error. The example shows that the characteristics of the convergence depend on the selection of the quantity of interest, which can be local or a global functional such as the deflection considered here. The results differ for different FE formulations, and the difference is bigger when the nonlinearities (e.g., due to plastic response) are taken into account

Keywords : discretization error; grid convergence index; finite element; mesh refinement; verification

  • [1] J.T. Oden, T. Belytschko, J. Fish, T.J. Hughes, C. Johnson, D. Keyes, A. Laub, L. Petzold, D. Srolovitz, and S. Yip, “Revolutionizing engineering science through simulation”, in NationalScience Foundation Blue Ribbon Panel Report 65, National Science Foundation, Arlington, 2006.

  • [2] L.F. Konikow and J.D. Bredehoeft, “Ground-water models cannot be validated”, Advances in Water Resources 15 (1), 75-83 (1992).

  • [3] L. Kwaśniewski, “On practical problems with verification and validation of computational models”, Archives of Civil Eng. LV (3), 323-346 (2009).

  • [4] I. Babuska and J.T. Oden, “Verification and validation in computational engineering and science: basic concepts”, ComputerMethods in Applied Mechanics and Eng. 193 (36-38), 4057-4066 (2004).

  • [5] U.S. Department of Defense, Verification, Validation, andAccreditation (VV&A) Recommended Practices Guide, Defense Modeling and Simulation Office, Office of the Director of Defense Research and Engineering, Virginia, 1996, www.dmso.mil/docslib.

  • [6] AIAA, Guide for the Verification and Validation of ComputationalFluid Dynamics Simulations, American Institute of Aeronautics and Astronautics, AIAA-G-077-1998, Reston, 1998.

  • [7] ASME, Guide for Verification and Validation in ComputationalSolid Mechanics, American Society of Mechanical Engineers, Virginia, 2006.

  • [8] J. Zhu and O. Zienkiewicz, “A posteriori error estimation and three-dimensional automatic mesh generation”, in Finite Elementsin Analysis and Design, vol. 25, no. 1-2, pp. 167-184, Elsevier, London, 1997.

  • [9] L.E. Schwer, “Is your mesh refined enough? Estimating discretization error using GCI”, 7th LS-DYNA Anwenderforum I-I-45-54, CD-ROM (2008).

  • [10] P.J. Roache, “Verification and validation in computational science and engineering”, Computing in Science Eng. 1, 8-9 (1998).

  • [11] L.F. Richardson, “The approximate arithmetical solution by finite differences of physical problems including differential equations. with an application to the stresses in a masonry dam”, Philosophical Trans. Royal Society of London A 210 (459-470), 307-357 (1911).

  • [12] NASA NPARC Alliance Verification and Validation, ExaminingSpatial (Grid) Convergence, http://www.grc.nasa.gov/WWW/wind/valid/tutorial/spatconv.html.

  • [13] J.O. Hallquist, LS-DYNA Keyword User’s Manual-Version971, Livermore Software Technology Corporation, Livermore, 2007.

  • [14] ABAQUS Users Manual V. 6.10-1, Dassault Systemes Simulia Corp., Providence, 2010.

  • [15] J. Gere and S. Timoshenko, Mechanics of Materials, Brooks, Cole Publishing, New York, 2000.

About the article

Published Online: 2013-05-01

Published in Print: 2013-03-01

Citation Information: Bulletin of the Polish Academy of Sciences: Technical Sciences, ISSN (Print) 0239-7528, DOI: https://doi.org/10.2478/bpasts-2013-0010. Export Citation

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