Skip to main content
SpringerLink
Log in

Modified and Trefftz unsymmetric finite element models

  • Published:
International Journal of Mechanics and Materials in Design Aims and scope Submit manuscript

Abstract

The unsymmetric finite element method employs compatible test functions but incompatible trial functions. The pertinent 8-node quadrilateral and 20-node hexahedron unsymmetric elements possess exceptional immunity to mesh distortion. It was noted later that they are not invariant and the proposed remedy is to formulate the element stiffness matrix in a local frame and then transform the matrix back to the global frame. In this paper, a more efficient approach will be proposed to secure the invariance. To our best knowledge, unsymmetric 4-node quadrilateral and 8-node hexahedron do not exist. They will be devised by using the Trefftz functions as the trial function. Numerical examples show that the two elements also possess exceptional immunity to mesh distortion with respect to other advanced elements of the same nodal configurations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  • Bachrach, W.E.: An efficient formulation of hexahedral elements with high accuracy for bending and incompressibility. Comput. Struct. 26(3), 453–467 (1987)

    Article  MATH  Google Scholar 

  • Bletzinger, K.U., Bischoff, M., Ramm, E.: A unified approach for shear-locking-free triangular and rectangular shell finite elements. Comput. Struct. 75(3), 321–334 (2000)

    Article  Google Scholar 

  • Cao, C.Y., Qin, Q.H., Yu, A.B.: Hybrid fundamental-solution-based FEM for piezoelectric materials. Comput. Mech. 50(4), 397–412 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Cardoso, R.P.R., Yoon, J.W., Mahardika, M., Choudhry, S., Sousa, R.J.A., Valente, R.A.F.: Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one-point quadrature solid-shell elements. Int. J. Numer. Methods Eng. 75(2), 156–187 (2008)

    Article  MATH  Google Scholar 

  • Cen, S., Fu, X.R., Zhou, M.J.: 8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shape. Comput. Method. Appl. Mech. Eng. 200(29–32), 2321–2336 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Cen, S., Zhou, G.H., Fu, X.R.: A shape-free 8-node plane element unsymmetric analytical trial function method. Int. J. Numer. Methods Eng. 91(2), 158–185 (2012)

    Article  MATH  Google Scholar 

  • Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J.: Concepts and Applications of Finite Element Analysis. Wiley, New York (2002)

    Google Scholar 

  • El-Abbasi, N., Meguid, S.A.: A new shell element accounting for through-thickness deformation. Comput. Methods Appl. Mech. Eng. 189(3), 841–862 (2000)

    Article  MATH  Google Scholar 

  • Felippa, C.A., Haugen, B., Militello, C.: From the individual element test to finite element templates: Evolution of the patch test. Int. J. Numer. Methods Eng. 38(2), 199–229 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Freitas, J.A., Moldovan, I.D.: Hybrid-Trefftz stress element for bounded and unbounded poroelastic media. Int. J. Numer. Methods Eng. 85(10), 1280–1305 (2011)

    MATH  Google Scholar 

  • Herrera, I.: Trefftz method: a general theory. Numer. Method Part. D. E. 16(6), 561–580 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Hughes, T.J.R.: Generalization of selective integration procedures to anisotropic and nonlinear media. Int. J. Numer. Methods Eng. 15(9), 1413–1418 (1980)

    Article  MATH  Google Scholar 

  • Kim, C.H., Sze, K.Y., Kim, Y.H.: Curved quadratic triangular degenerated- and solid-shell elements for geometric nonlinear analysis. Int. J. Numer. Methods Eng. 57(14), 2077–2097 (2003)

    Article  MATH  Google Scholar 

  • Liew, K.M., Rajendran, S., Wang, J.: A quadratic plane triangular element immune to quadratic mesh distortions under quadratic displacement fields. Comput. Method. Appl. M. 195(9–12), 1207–1223 (2006)

    Article  MATH  Google Scholar 

  • Liu, G.H., Sze, K.Y.: Axisymmetric quadrilateral elements for large deformation hyperelastic analysis. Int. J. Mech. Mater. Des. 6(3), 197–207 (2010)

    Article  MathSciNet  Google Scholar 

  • Macneal, R.H.: Derivation of element stiffness matrices by assumed strain distributions. Nucl. Eng. Des. 70(1), 3–12 (1982)

    Article  Google Scholar 

  • Macneal, R.H., Harder, R.L.: A proposed standard set of problems to test finite element accuracy. Finite Elem. Anal. Des. 1(1), 3–20 (1985)

    Article  Google Scholar 

  • Ooi, E.T., Rajendran, S., Yeo, J.H.: A 20-node hexahedron element with enhanced distortion tolerance. Int. J. Numer. Methods Eng. 60(15), 2501–2530 (2004)

    Article  MATH  Google Scholar 

  • Ooi, E.T., Rajendran, S., Yeo, J.H.: Remedies to rotational frame dependence and interpolation failure of US-QUAD8 element. Commun. Appl. Numer. Methods Eng. 24(11), 1203–1217 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Pian, T.H.H., Sumihara, K.: Rational approach for assumed stress elements. Int. J. Numer. Methods Eng. 20(9), 1685–1695 (1984)

    Article  MATH  Google Scholar 

  • Pian, T.H.H., Tong, P.: Relations between incompatible displacement model and hybrid stress model. Int. J. Numer. Methods Eng. 22(1), 173–181 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  • Qin, Q.H.: Variational formulations for TFEM of piezoelectricity. Int. J. Solids Struct. 40(23), 6335–6346 (2003)

    Article  MATH  Google Scholar 

  • Rajendran, S., Liew, K.M.: A novel unsymmetric 8-node plane element immune to mesh distortion under a quadratic displacement field. Int. J. Numer. Methods Eng. 58(11), 1713–1748 (2003)

    Article  MATH  Google Scholar 

  • Simo, J.C., Rifai, M.S.: A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 29(8), 1595–1638 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Sze, K.Y.: Efficient formulation of robust hybrid elements using orthogonal stress/strain interpolants and admissible matrix formulation. Int. J. Numer. Methods Eng. 35(1), 1–20 (1992)

    Article  MathSciNet  Google Scholar 

  • Sze, K.Y.: On immunizing five-beta hybrid-stress element models from `trapezoidal locking’ in practical analyses. Int. J. Numer. Methods Eng. 47(4), 907–920 (2000)

    Article  MATH  Google Scholar 

  • Sze, K.Y., Chow, C.L., Chen, W.J.: On invariance of isoparametric hybrid or mixed elements. Commun. Appl. Numer. Methods 8(6), 385–406 (1992)

    Article  MATH  Google Scholar 

  • Sze, K.Y., Liu, G.H., Fan, H.: Four- and eight-node hybrid-Trefftz quadrilateral finite element models. Comput. Method. Appl. M. 199(9–12), 598–614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Sze, K.Y., Zheng, S.J., Lo, S.H.: A stabilized eighteen-node solid element for hyperelastic analysis of shells. Finite Elem. Anal. Des. 40(3), 319–340 (2004)

    Article  Google Scholar 

  • Taylor, R.L., Beresford, P.J., Wilson, E.L.: A non-conforming element for stress analysis. Int. J. Numer. Methods Eng. 10(6), 1211–1219 (1976)

    Article  MATH  Google Scholar 

  • Taylor, R.L., Simo, J.C., Zienkiewicz, O.C., Chan, A.C.H.: The patch test—a condition for assessing FEM convergence. Int. J. Numer. Methods Eng. 22(1), 39–62 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  • Yuan, K.Y., Huang, Y.S., Pian, T.H.H.: New strategy for assumed stresses for 4-node hybrid stress membrane element. Int. J. Numer. Methods Eng. 36(10), 1747–1763 (1993)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong

    Q. Xie, K. Y. Sze & Y. X. Zhou

  2. Faculty of Civil Engineering & Mechanics, Jiangsu University, Zhenjiang, 212013, People’s Republic of China

    Y. X. Zhou

Authors
  1. Q. Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Y. Sze
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Y. X. Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Y. Sze.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xie, Q., Sze, K.Y. & Zhou, Y.X. Modified and Trefftz unsymmetric finite element models. Int J Mech Mater Des 12, 53–70 (2016). https://doi.org/10.1007/s10999-014-9289-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10999-014-9289-3

Keywords

Search

Navigation