Abstract
One point quadrature shell elements are being widely used in the numerical simulation of shell structures, including sheet forming, because essentially of their computational efficiency. Nowadays, the purpose of using one point quadrature shell elements is not only related to computational efficiency but also because these elements have shown to be simultaneously robust and accurate in the simulation of complex sheet metal forming processes. The main objective of this work is to study the convergence behavior of different one-point quadrature shell elements and their ability to pass the membrane and bending patch tests. For comparison purposes, two new elements include a new formulation for the membrane strain field in order to further improve the membrane behavior of the element developed in previous work of (in Cardoso et al. Comput Meth Appl Mech Eng 191:5177, 2002). The original convective membrane strains of Cardoso et al. (Comput Meth Appl Mech Eng 191:5177, 2002) (in the stabilization matrices only) are thus replaced by new membrane strains, constructed directly at the co-rotational coordinate system (located at the element’s center). It is thus proved that with this new membrane formulation the elements pass now all the patch tests but, for warped (or curved) element geometries, their accuracy is not as good as the original element of (Cardoso et al. in Comput Meth Appl Mech Eng 191:5177, 2002) based on the convective coordinate system. In the numerical results presented in this paper, comprehensive comparison and discussion of these formulations are made for well known linear benchmark examples.
Similar content being viewed by others
References
ABAQUS (2002) Theory manual, Version 6.3. Hibbitt, Karlsson & Sorensen, Inc. Rhode-Island USA
Andelfinger U, Ramm E (1993) EAS-Elements for two-dimensional, three-dimensional, plate and shells and their equivalence to HR-elements. Int J Numer Meth Eng 36: 1413–1449
Batoz JL, Dhatt G (1992) Modélisation des structures par Éléments Finis, Vol 3. Coques. Hermés, Paris
Bazeley GP, Cheung YK, Irons BM, Zienkiewicz OC (1965) Triangular elements in bending-conforming and non-conforming solutions. In: Conference of matrix methods in structural mechanics, Air Force Inst. Tech., Wright-Patterson AF base
Belytschko T, Bachrach WE (1986) Efficient implementation of quadrilaterals with high coarse-mesh accuracy. Comput Meth Appl Mech Eng 54:279
Belytschko T, Lin JI, Tsay CS (1984) Explicit algorithms for the nonlinear dynamics of shells. Comput Meth Appl Mech Eng 42:225
Belytschko T, Leviathan I (1994) Physical stabilization of the 4-node shell element with one point quadrature. Comput Meth Appl Mech Eng 113:321–350
Belytschko T, Leviathan I (1994) Projection schemes for one-point quadrature shell elements. Comput Meth Appl Mech Eng 115:277
Cardoso RPR, Yoon JW, Grácio JJA, Barlat F, César de Sá JMA (2002) Development of a one point quadrature shell element for nonlinear applications with contact and anisotropy. Comput Meth Appl Mech Eng 191:5177
Cardoso RPR, Yoon JW, Valente RAF (2006) A new approach to reduce membrane and transverse shear locking for one point quadrature shell elements: linear formulation. Int J Numer Meth Eng 66:214–249
Chen XM, Cen S, Long YQ, Yao ZH (2004) Membrane elements insensitive to distortion using the quadrilateral area coordinate method. Comput Struct 82:35–54
Dvorkin E, Bathe KJ (1984) A continuum mechanics based four-node shell element for general nonlinear analysis. Eng Comput 1:77
Hughes TJR (1987) The finite element method: linear static and dynamic analysis. Prentice-Hall, Englewood Cliffs
Ibrahimbegović A, Taylor RL, Wilson EL (1990) A robust quadrilateral membrane finite element with drilling degrees of freedom. Int J Numer Meth Eng 30:445–457
Kasper EP, Taylor RL (2000) A mixed-enhanced strain method. Part I: Geometrically linear problems. Comput Struct 75:237–250
Kemp BL, Cho C, Lee SW (1998) A four-node solid shell element formulation with assumed strain. Int J Numer Meth Eng 43:909–924
Knight NF (1997) The Raasch challenge for shell elements. AIAA J 35:375–381
MacNeal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1:1–20
MacNeal RH (1994) Finite elements: their design and performance. Marcel Dekker New York
Parisch H (1991) An investigation of a finite rotation four node assumed strain shell element. Int J Numer Meth Eng 31:127–150
Simo JC, Fox DD, Rifai MS (1989) On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Comput Meth Appl Mech Eng 72: 267–304
Simo JC, Fox DD, Rifai MS (1989) On a stress resultant geometrically exact shell model. Part II: The linear theory—computational aspects. Comput Meth Appl Mech Eng 73: 53–92
Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Meth Eng 29:1595–1638
Slavković R, Miroslav Z, Kojić M (1994) Enhanced 8-node three-dimensional solid and 4-node shell elements with incompatible generalized displacements. Commun Numer Meth Eng 10:699–709
Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. Int J Numer Meth Eng 10:1211–1219
Wilson EL, Taylor RL, Doherty WP, Ghabussi T (1973) Incompatible displacement models. In: Fenven ST et al (eds) Numerical and Computer Methods in Structural Mechanics Academic, New York, pp 43–57
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cardoso, R.P.R., Yoon, JW. One point quadrature shell elements: a study on convergence and patch tests. Comput Mech 40, 871–883 (2007). https://doi.org/10.1007/s00466-006-0148-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-006-0148-4