Structural Engineering and Mechanics

Techno-Press (테크노프레스)
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The unsymmetric finite element formulation and variational incorrectness

  • Prathap, G. (CSIR Centre for Mathematical Modelling and Computer Simulation) ;
  • Manju, S. (National Aerospace Laboratories) ;
  • Senthilkumar, V. (CSIR Centre for Mathematical Modelling and Computer Simulation)
  • Received : 2005.07.14
  • Accepted : 2006.10.12
  • Published : 2007.05.10
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Abstract

The unsymmetric finite element formulation has been proposed recently to improve predictions from distorted finite elements. Studies have also shown that this special formulation using parametric functions for the test functions and metric functions for the trial functions works surprisingly well because the former satisfy the continuity conditions while the latter ensure that the stress representation during finite element computation can retrieve in a best-fit manner, the actual variation of stress in the metric space. However, a question that remained was whether the unsymmetric formulation was variationally correct. Here we determine that it is not, using the simplest possible element to amplify the principles.

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References

  1. Amold, D.N., Boffi, D. and Falk, R.S. (2002), 'Approximation by quadrilateral finite elements', Math. Comp., 71, 909-922 https://doi.org/10.1090/S0025-5718-02-01439-4
  2. Babuska, I. and Stroubolis, T. (2001) The Finite Element method and its Reliability, Clarendon Press, Oxford
  3. Backlund, J. (1978), 'On isoparametric elements', Int. J. Numer. Meth. Eng., 12, 731-732 https://doi.org/10.1002/nme.1620120418
  4. Gifford, L.N. (1979), 'More on distorted isoparametric elements', Int. J. Numer. Meth. Eng., 14, 290-291 https://doi.org/10.1002/nme.1620140212
  5. Jafarali, P. (2005), Personal Communication
  6. Ooi, E.T., Rajendran, S. and Yeo, J.H. (2004), 'A 20-node hexahedron element with enhanced distortion tolerance', Int. J. Numer. Meth. Eng., 60, 2501-2530 https://doi.org/10.1002/nme.1056
  7. Prathap, G. (1993), The Finite Element Method in Structural Mechanics, Kluwer Academic Press, Dordrecht
  8. Prathap, G. and Mukherjee, S. (2003), The Engineer Grapples with Theorem 1.1 and Lemma 6.3 of Strang and Fix, Current Sci., Vol. 85, 7, 989-994
  9. Prathap, G. and Mukherjee, S. (2004), Management-by-stress Model of Finite Element Computation. Research Report CM 0405, CSIR Centre for Mathematical Modelling and Computer Simulation, Bangalore, November 2004
  10. Prathap, G., Senthilkumar, V. and Manju, S. (2005), Mesh Distortion Immunity of Finite Elements and the Bestfit Paradigm. Research Report CM 0501, CSIR Centre for Mathematical Modelling and Computer Simulation, Bangalore, February 2005
  11. Rajendran, S. and Liew, K.M. (2003), 'A novel unsymmetric 8-node plane element immune to mesh distortion under a quadratic field', Int. J. Numer. Meth. Eng., 58, 1718-1748
  12. Rajendran, S. and Subramanian, S. (2004), 'Mesh distortion sensitivity of 8-node plane elasticity elements based on parametric, metric, parametric-metric, and metric-parametric formulations', Struct. Eng. Mech., 17, 767-788 https://doi.org/10.12989/sem.2004.17.6.767
  13. Rajendran, S. (2005), Personal Communication
  14. Strang, G. and Fix, G.J. (1973) An Analysis of the Finite Element Method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, NJ
  15. Stricklin, J.A., Ho, W.S., Richardson, E.Q. and Haisler, W.E. (1977), 'On isoparametric vs. linear strain triangular elements', Int. J. Numer. Meth. Eng., 11, 1041-1043 https://doi.org/10.1002/nme.1620110610

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