# Shell Elements Overview

The historical shell element in Radioss is a simple bilinear Mindlin plate element coupled with a reduced integration scheme using one integration point. It is applicable in a reliable manner to both thin and moderately thick shells.

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.

- The parameters ${h}_{i}$ are user-inputs and are generally problem-dependent.
- Poor behavior with irregular geometries (in-plane, out-of-plane). The stabilized stiffness (or stabilized forces) is often evaluated based on a regular flat geometry, so they generally do not pass either the Patch-test or the Twisted beam test.

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).

^{1}Kosloff D. and Frazier G., “Treatment of hourglass pattern in low order finite element code”, International Journal for Numerical and Analytical Methods in Geomechanics, 1978.

^{2}Wilson L.T., “Incompatible displacement models”, page 43. Academi Press, New York, 1973.

^{3}Taylor R.L., “Finite element for general shell analysis”, 1979.

^{4}Hughes T.J.R. and Liu W.K., “Nonlinear finite element analysis of shells: Part I: Three-dimensional shells”, Computer Methods in Applied Mechanics and Engineering, 26:331-362, 1981.

^{5}Belytschko T. and Tsay C.S., “A stabilization procedure for the quadrilateral plate element with one-point quadrature”, Computer Methods in Applied Mechanics and Engineering, 55:259-300, 1986.

^{6}Flanagan D. and Belytschko T., “A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control”, Int. Journal Num. Methods in Engineering, 17 679-706, 1981.

^{7}Belytschko T. and Leviathan I., “Physical stabilization of the 4-node shell element with one-point quadrature”, Computer Methods in Applied Mechanics and Engineering, 113:321-350, 1992.

^{8}Belytschko T., Wong B.L. and Chiang H.Y. “Advances in one-point quadrature shell elements”, Computer Methods in Applied Mechanics and Engineering, 96:93-107, 1989.

^{9}Hughes T.J.R. and Tezduyar T.E., “Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element”, J. of Applied Mechanics, 48:587-596, 1981.

^{10}MacNeal R.H., “Derivation of element stiffness matrices by assumed strain distributions”, Nuclear Engrg. Des., 70:3-12, 1982.

^{11}Belytschko T. and Bachrach W.E., “Efficient implementation of quadrilaterals with high coarse-mesh accuracy”, Computer Methods in Applied Mechanics and Engineering, 54:279-301, 1986.

^{12}Engelmann B.E. and Whirley R.G., “A new elastoplastic shell element formulation for DYNA3D”, Report ugrl-jc-104826, Lawrence Livermore National Laboratory, 1990.

^{13}Liu W.K., Law E.S., Lam D. and Belytschko T., “Resultant-stress degenerated-shell element”, Int. Journal Num. Methods in Engineering, 19:405-419, 1983.

^{14}Belytschko T. and Leviathan I., “Projection schemes for one-point quadrature shell elements”, Computer Methods in Applied Mechanics and Engineering, 115:277-286, 1993.

^{15}Zhu Y. and Zacharia T., “A new one-point quadrature, quadrilateral shell element with drilling degree of freedom”, Computer Methods in Applied Mechanics and Engineering, 136:165-203, 1996.

^{16}Allman D.J., “A quadrilateral finite element including vertex rotations for plane elasticity problems”, Int. Journal Num. Methods in Engineering, 26:717-739, 1988.

^{17}Hughes T.J.R. and Brezzi F., “On drilling degrees of freedom”, Computer Methods in Applied Mechanics and Engineering, 72:105-121, 1989.

^{18}Belytschko T. and Bindeman L., “Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems”, Computer Methods in Applied Mechanics and Engineering, 88:311-340, 1991.

^{19}Dvorkin E. and Bathe K.J. “A continuum mechanics four-node shell element for 35 general nonlinear analysis”, Engrg Comput, 1:77-88, 1984.

^{20}Stolarski H., Belytschko T. and Lee S.H., “A review of shell finite elements and corotational theories”, Computational Mechanics Advances, 2:125-212, 1995.

^{21}Zeng Q. and Combescure A., “A New One-point Quadrature, General Nonlinear Quadrilateral Shell Element with Physical Stabilization”, Int. Journal Num. Methods in Engineering 42, 1307-1338, 1998.