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An improved parametric formulation for the variationally correct distortion immune three-noded bar element

  • Mukherjee, Somenath (Central Mechanical Engineering Research Institute, CSIR) ;
  • Manju, S. (National Aerospace Laboratories, CSIR)
  • Received : 2010.03.31
  • Accepted : 2010.12.17
  • Published : 2011.05.10

Abstract

A new method of formulation of a class of elements that are immune to mesh distortion effects is proposed here. The simple three-noded bar element with an offset of the internal node from the element center is employed here to demonstrate the method and the principles on which it is founded upon. Using the function space approach, the modified formulation is shown here to be superior to the conventional isoparametric version of the element since it satisfies the completeness requirement as the metric formulation, and yet it is in agreement with the best-fit paradigm in both the metric and the parametric domains. Furthermore, the element error is limited to only those that are permissible by the classical projection theorem of strains and stresses. Unlike its conventional counterpart, the modified element is thus not prone to any errors from mesh distortion. The element formulation is symmetric and thus satisfies the requirement of the conservative nature of problems associated with all self-adjoint differential operators. The present paper indicates that a proper mapping set for distortion immune elements constitutes geometric and displacement interpolations through parametric and metric shape functions respectively, with the metric components in the displacement/strain replaced by the equivalent geometric interpolation in parametric co-ordinates.

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References

  1. Backlund, J. (1978), "On isoparametric elements", Int. J. Numer. Meth. Eng., 12, 731-732. https://doi.org/10.1002/nme.1620120418
  2. Felippa, C.A. (2006), "Supernatural Quad4; a template formulation", Comput. Meth. Appl. Mech. Eng., 195, 5316-5342. https://doi.org/10.1016/j.cma.2005.12.007
  3. Gifford, L.N. (1979), "More on distorted isoparametric elements", Int. J. Numer. Meth. Eng., 14, 290-291. https://doi.org/10.1002/nme.1620140212
  4. Kumar, S. and Prathap, G. (2008), "Mesh distortion, Locking and the use of metric trial functions for the displacement type finite elements", Struct. Eng. Mech., 29(3), 289-300. https://doi.org/10.12989/sem.2008.29.3.289
  5. Mukherjee, S. and Jafarali, P. (2010), "Prathap's best-fit paradigm and optimal strain recovery points in indeterminate tapered bar analysis using linear element", Int. J. Numer. Meth. Biomedical Eng., 26, 1246-1262.
  6. Norrie, D.H. and De Vries, G. (1978), An Introduction to Finite Element Analysis, Academic Press, New York.
  7. Prathap, G. (2007), "Stay Cartesian or go natural? A comment on the Article "Supernatural Quad4; a template formulation" by C. A. Felippa", Comput. Meth. Appl. Mech. Eng., 196, 1847-1848. https://doi.org/10.1016/j.cma.2006.09.018
  8. Prathap, G., Manju, S. and Senthilkumar, V. (2007), "The unsymmetric finite element formulation and variational correctness", Struct. Eng. Mech., 26, 31-42. https://doi.org/10.12989/sem.2007.26.1.031
  9. Prathap, G. and Mukherjee, S. (2003), "The Engineer grapples with Theorem 1.1 and Lemma 6.3 of Strang and Fix", Current Science, 85(7), 989-994.
  10. 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.
  11. 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
  12. Simmons, G.F. (1963), Introduction to Topology and Modern Analysis, Mc Graw Hill Inc.
  13. Strang, G. and Fix, G.J. (1973), An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, NJ.
  14. 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