Shell Elements Overview

This element is very efficient if the spurious singular modes, called “hourglass modes”, which result from the reduced integration are stabilized.

The stabilization approach consists of providing additional stiffness so that the spurious singular modes are suppressed. Also, it offers the possibility of avoiding some locking problems. One of the first solutions was to generalize the formulation of Kosloff and Frazier ¹ for brick element to shell element. It can be shown that the element produces accurate flexural response (thus, free from the membrane shear locking) and is equivalent to the incompatible model element of Wilson et al. ² without the static condensation procedure. Taylor ³ extended this work to shell elements. Hughes and Liu ⁴ employed a similar approach and extended it to nonlinear problems.

Belytschko and Tsay ⁵ developed a stabilized flat element based on the $\gamma$ projections developed by Flanagan and Belytschko ⁶. Its essential feature is that hourglass control is orthogonal to any linear field, thus preserving consistency. The stabilized stiffness is approached by a diagonal matrix and scaled by the perturbation parameters $h_i$ which are introduced as a regulator of the stiffness for nonlinear problems. The parameters $h_i$ are generally chosen to be as small as possible, so this approach is often called, perturbation stabilization.

Belytschko ⁷ extended this perturbation stabilization to the 4-node shell element which has become widely used in explicit programs.

Belytschko ⁸ improved the poor behavior exhibited in the warped configuration by adding a coupling curvature-translation term, and a particular nodal projection for the transverse shear calculation analogous to that developed by Hughes and Tezduyar ⁶, and MacNeal ¹⁰. This element passes the Kirchhoff patch test and the Twisted Beam test, but it cannot be extended to a general 6 DOF element due to the particular projection.

Belytschko and Bachrach ¹¹ used a new method called physical stabilization to overcome the first drawback of the quadrilateral plane element. This method consists of explicitly evaluating the stabilized stiffness with the help of 'assumed strains', so that no arbitrary parameters need to be prescribed. Engelmann and Whirley ¹² have applied it to the 4-node shell element. An alternative way to evaluate the stabilized stiffness explicitly is given by Liu et al. ¹³ based on Hughes and Liu's 4-node selected reduced integration scheme element ⁴, in which the strain field is expressed explicitly in terms of natural coordinates by a Taylor-series expansion. A remarkable improvement in the one-point quadrature shell element with physical stabilization has been performed by Belytschko and Leviathan ¹⁴. The element performs superbly for both flat and warped elements especially in linear cases, even in comparison with a similar element under a full integration scheme, and is only 20% slower than the Belytschko and Tsay element. More recently, based on Belytschko and Leviathan's element, Zhu and Zacharia ¹⁵ implemented the drilling rotation DOF in their one-point quadrature shell element; the drilling rotation is independently interpolated by the Allman function ¹⁶ based on Hughes and Brezzi's ¹⁷ mixed variational formulation.

The physical stabilization with assumed strain method seems to offer a rational way of developing a cost effective shell element with a reduced integration scheme. The use of the assumed strains based on the mixed variational principles, is powerful, not only in avoiding the locking problems (volumetric locking, membrane shear locking, as in Belytschko and Bindeman ¹⁸; transverse shear locking, as in Dvorkin and Bathe ¹⁹), but also in providing a new way to compute stiffness. However, as highlighted by Stolarski et al. ²⁰, assumed strain elements generally do not have rigorous foundations; there is almost no constraint for the independent assumed strains interpolation. Therefore, a sound theoretical understanding and numerous tests are needed in order to prove the legitimacy of the assumed strain elements.

The greatest uncertainty of the one-point quadrature shell elements with physical stabilization is with respect to the nonlinear problems. All of these elements with physical stabilization mentioned above rely on the assumptions that the spin and the material properties are constant within the element. The first assumption is necessary to ensure the objectivity principle in geometrical nonlinear problems. The second was adapted in order to extend the explicit evaluation of stabilized stiffness for elastic problems to the physical nonlinear problems. It is found that the second assumption leads to a theoretical contradiction in the case of an elastoplastic problem (a classic physical nonlinear problem), and results in poor behavior in case of certain crash computations.

Zeng and Combescure ²¹ have proposed an improved 4-node shell element named QPPS with one-point quadrature based on the physical stabilization which is valid for the whole range of its applications (see Shell Formulations). The formulation is based largely on that of Belytschko and Leviathan.

Based on the QPPS element, Zeng and Winkelmuller have developed a new improved element named QEPH which is integrated in Radioss 44 version (see 3-Node Shell Elements).