Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Mesh distortion, locking and the use of metric trial functions for displacement type finite elements

  • Kumar, Surendra (CSIR Centre for Mathematical Modelling and Computer Simulation) ;
  • Prathap, G. (CSIR Centre for Mathematical Modelling and Computer Simulation)
  • Received : 2007.04.25
  • Accepted : 2008.03.21
  • Published : 2008.06.20
⟨ Previous Next ⟩

Abstract

The use of metric trial functions to represent the real stress field in what is called the unsymmetric finite element formulation is an effective way to improve predictions from distorted finite elements. This approach works surprisingly well because the use of parametric functions for the test functions satisfies the continuity conditions while the use of metric (Cartesian) shape functions for the trial functions attempts to 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, the issue of how to handle situations where there is locking along with mesh distortion has never been addressed. In this paper, we show that the use of a consistent definition of the constrained strain field in the metric space can ensure a lock-free solution even when there is mesh distortion. The three-noded Timoshenko beam element is used to illustrate the principles. Some significant conclusions are drawn regarding the optimal strategy for finite element modelling where distortion effects and field-consistency requirements have to be reconciled simultaneously.

Keywords

References

  1. Arnold, D.N., Boffi, D. and Falk, R.S. (2002), "Approximation of quadrilateral finite elements", Math. Comput., 71, 909-922 https://doi.org/10.1090/S0025-5718-02-01439-4
  2. Backlund, J. (1978), "On isoparametric elements", Int. J. Numer. Meth. Eng., 12, 731-732 https://doi.org/10.1002/nme.1620120418
  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. 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
  5. Prathap, G. and Naganarayana, B.P. (1992), "Field-consistency rules for a three-noded shear flexible beam element under non-uniform isoparametric mapping", Int. J. Numer. Meth. Eng., 33, 649-664 https://doi.org/10.1002/nme.1620330310
  6. Prathap, G. (1993), The Finite Element Method in Structural Mechanics, Kluwer Academic Press, Dordrecht
  7. Prathap, G., Senthilkumar, V. and Manju, S. (2006), Mesh distortion immunity of finite elements and the best-fit paradigm. Sadhana, 31, 505-514 https://doi.org/10.1007/BF02715909
  8. Prathap, G., Manju, S. and Senthilkumar, V. (2007), The unsymmetric finite element formulation and variational incorrectness. Struct. Eng. Mech., 26(1), 31-42 https://doi.org/10.12989/sem.2007.26.1.031
  9. 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
  10. 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(6), 767-788 https://doi.org/10.12989/sem.2004.17.6.767
  11. Rajendran, S. (2005), Personal communication
  12. 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

Cited by

  1. Use of unsymmetric finite element method in impact analysis of composite laminates vol.47, pp.4, 2011, https://doi.org/10.1016/j.finel.2010.12.016
  2. An improved parametric formulation for the variationally correct distortion immune three-noded bar element vol.38, pp.3, 2008, https://doi.org/10.12989/sem.2011.38.3.261
  3. Function space formulation of the 3-noded distorted Timoshenko metric beam element vol.69, pp.6, 2008, https://doi.org/10.12989/sem.2019.69.6.615