Type 16 shells and hourglassing
James B
l...@schwer.net
EQ.16: Fully integrated shell element (very fast)
NO hourglass stabilization is needed for fully integrated elements (shells or solids). Hourglass stabilization is only used for single point integration elements. --len
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James M. Kennedy
Dear James,
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Please see the following presentations:
Erhart, T., and Borrvall, T., "Drilling Rotation Constraint for Shell Elements in Implicit and Explicit Analyses", 9th European LS-DYNA Users Conference, Manchester, United Kingdom, May, 2013.
Erhart, T., "Drilling Rotation Constraint for Shell Elements in Implicit and Explicit Analyses", 2013 Developer Forum, Stuttgart-Filderstadt, Germany, September, 2013.
Haufe, A., Schweizerhof, K., and Du Bois, P., "Properties & Limits: Review of Shell Element Formulations", 2013 Developer Forum, Stuttgart-Filderstadt, Germany, September, 2013.
Kim, D.K., Ng, W.C.K., Hwang, O.J., Sohn, J.M., and Lee, E.B., “Recommended Finite Element Formulations for the Analysis of Offshore Blast Walls in an Explosion”, Latin American Journal of Solids and Structures, Vo. 15, No. 10, October, 2018.
https://www.scielo.br/scielo.php?script=sci_arttext&pid=S1679-78252018001000505
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https://ftp.lstc.com/anonymous/outgoing/jday/hourglass.pdf
https://www.dynasupport.com/howtos/element/hourglass
Accuracy:
From an accuracy standpoint, shell type 16 is preferred over the under-integrated formulations provided the following are true:
- initial element shape is reasonable
- element does not distort unreasonably during the simulation
- Used together with hourglass control type 8, the type 16 shell will give the correct solution for warped geometries.
Formulation 16 uses a Bathe-Dvorkin transverse shear treatment which eliminates w-mode hourglassing. Other modes of hourglassing are eliminated in the formulation 16 shell by virtue of the selective reduced (S/R) integration. The S/R integration here means that full integration (4 in-plane integration points) is used except for purposes of calculating transverse shear. To eliminate transverse shear locking, only 1 in-plane integration point is considered in calculating transverse shear.
Type 8 HG control applies only to shell formulation 16. This HG type activates warping stiffness in type 16 shells so that warping of the element does not degrade the solution. Type 16 shells will solve the so-called Twisted Beam problem correctly if HG type 8 is invoked.
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Please see section 11.6 Belytschko-Leviathan Projection in the LS-DYNA 2020 Theory Manual.
"LS-DYNA Theory Manual", LS-DYNA Dev/Revision 11261, Livermore Software Technology Corporation, Livermore, California, July, 2019.
http://ftp.lstc.com/anonymous/outgoing/jday/manuals/DRAFT_Theory.pdf
For warped configurations and since the geometry of the current shell element is flat, extremely flexible behavior can be expected for some modes of deformation. Following [Belytschko and Leviathan 1994], a 7-mode projection matrix 𝐏 (3 rigid body rotation modes and 4 nodal drill rotation modes) is constructed used for projecting out these zero energy modes. The explicit formula for the projection matrix is given by
𝐏 = 𝐈−𝐑(𝐑T𝐑)−1𝐑T, eq. (11.28)
where 𝐑 is a matrix where each column corresponds to the nodal velocity of a zero energy mode. This projection matrix operates on the nodal velocities prior to computing the strain rates, and also on the resulting internal force vector to maintain invariance of the internal power.
(older set of notes/older manual)
Since the Belytschko-Tsay element is based on a perfectly flat geometry, warpage is not considered. Although this generally poses no major difficulties and provides for an efficient element, incorrect results in the twisted beam problem, See Figure 7.2, are obtained where the nodal points of the elements used in the discretization are not coplanar. The Hughes-Liu shell element considers non-planar geometry and gives good results on the twisted beam, but is relatively expensive. The effect of neglecting warpage in typical a application cannot be predicted beforehand and may lead to less than accurate results, but the latter is only speculation and is difficult to verify in practice. Obviously, it would be better to use shells that consider warpage if the added costs are reasonable and if this unknown effect is eliminated. In this section we briefly describe the simple and computationally inexpensive modifications necessary in the Belytschko-Tsay shell to include the warping stiffness. The improved transverse shear treatment is also described which is necessary for the element to pass the Kirchhoff patch test. Readers are directed to the references [Belytschko, Wong, and Chang 1989, 1992] for an in depth theoretical background.
Belytschko, T., and Leviathan, I., "Projection Schemes for One-Point Quadrature Shell Elements", Computer Methods in Applied Mechanics and Engineering, Vol. 115, Issues 3-4, pp. 277-286, May, 1994.
A new projection scheme to improve the performance of one-point quadrature shell elements is presented. The projection is performed element wise and enforces invariance of the internal power to rigid body motion and drill rotation. The projection is especially suitable for 6 degree-of-freedom per node explicit programs, and can be easily implemented as an integral part of the subroutine that calculates the internal forces on the element level. Problems that could not be solved accurately with one-point quadrature elements, such as the twisted beam problem, show excellent results even for very coarse meshes when the projection is used.
https://www.sciencedirect.com/science/article/abs/pii/0045782594900639
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An example that can be modified to look at different element formulation and various hourglass types.
https://www.dynaexamples.com/introduction/examples-manual/control/shell-2
Description:
A beam twisted 90 degrees about its length is constrained on on e edge and has a point load prescribed normal to the opposite end of the beam.
Model:
The beam measures 12.00 * 1.10 * 0.32 cubic inches. A concentrated load is applied to one node on the end in the x-direction and the other node on the end in the z-direction.
Input:
This model uses the Hughes-Liu five through the thickness integration points (*CONTROL_SHELL, *SECTI ON_SHELL). The element has the shell normal update calculation performed at each nodal fiber every cycle (*CONTROL_SHELL). Note: This is another example that will not work correctly with the B-T shell formulation (unless warping stiffness is added).
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Sincerely,
James M. Kennedy
KBS2 Inc.
December 31, 2020
From: ls-d...@googlegroups.com [mailto:ls-d...@googlegroups.com] On Behalf Of James B
Sent: Thursday, December 31, 2020 9:52 AM
To: LS-DYNA2 <ls-d...@googlegroups.com>
Subject: [LS-DYNA2] Type 16 shells and hourglassing
Hi,
--
James M. Kennedy
P.S.
Suggest reading the following *CONTROL_SHELL entry.
*CONTROL_SHELL
Purpose: Provide controls for computing shell response.
PROJ - Projection method for the warping stiffness in the Belytschko-Tsay shell (the BWC option above) and the Belytschko-Wong-Chiang elements (see Remark 1 below). This parameter applies to explicit calculations since the full projection method is always used if the solution is implicit and this input parameter is ignored.
EQ.0: drill projection
EQ.1: full projection
Notes
1. Drill versus Full Projections for Warping Stiffness. The drill projection is used in the addition of warping stiffness to the Belytschko-Tsay and the Belytschko-Wong-Chiang shell elements. This projection generally works well and is very efficient, but to quote Belytschko and Leviathan:
"The shortcoming of the drill projection is that even elements that are invariant to rigid body rotation will strain under rigid body rotation if the drill projection is applied. On one hand, the excessive flexibility rendered by the 1-point quadrature shell element is corrected by the drill projection, but on the other hand the element becomes too stiff due to loss of the rigid
body rotation invariance under the same drill projection."
They later went on to add in the conclusions: "The projection of only the drill rotations is very efficient and hardly increases the computation time, so it is recommended for most cases. However, it should be noted that the drill projection can result in a loss of invariance to rigid body motion when the elements are highly warped. For moderately warped configurations the drill projection appears quite accurate."
In crashworthiness and impact analysis, elements that have little or no warpage in the reference configuration can become highly warped in the deformed configuration and may affect rigid body rotations if the drill projection is used, that is, DO NOT USE THE DRILL PROJECTION. Of course, it is difficult to define what is meant by "moderately warped." The full projection circumvents these problems but at a significant cost. The cost increase of the drill projection versus no projection as reported by Belytschko and Leviathan is 12 percent and by timings in LS-DYNA, 7 percent, but for the full projection they report a 110 percent increase and in LS-DYNA an increase closer to 50 percent is observed.
In Version 940 of LS-DYNA, the drill projection was used exclusively, but in one problem the lack of invariance was observed; consequently, the drill projection was replaced in the Belytschko-Leviathan shell with the full projection and the full projection is now optional for the warping stiffness in the Belytschko-Tsay and Belytschko-Wong-Chiang elements. Starting with version 950 the Belytschko-Leviathan shell, which now uses the full projection, is somewhat slower than in previous versions. In general, in light of these problems,
the drill projection cannot be recommended. For implicit calculations, the full projection method is used in the development of the stiffness matrix.
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