# Shell Formulations

The following discussion is within the context of explicit analysis.

In terms of speed and robustness I would rank shell formulations as follows:
1. `type 2`
2. `type 2` with `BWC` warping stiffness and full projection (see `BWC` and `PROJ` in `*CONTROL_SHELL`)
3. `type 10`
4. `type 16` (Type 16 shells require approximately 2.5 times more CPU than type 2 shells.)
5. `type 7`
6. `type 6` [1]

### Robustness:

'Robustness' is meant here as an ability to remain stable under adverse conditions such as poor element shapes and large deformation/distortion. Choices 2 and 3 above are, as you might expect, close to a tie in terms of performance and speed. The last 3 formulations listed above are fully-integrated (4 in-plane integration points) and thus do not suffer from hourglass modes [2]. Generally speaking, the underintegrated elements tend to be a little too soft. By using stiffness-based hourglass control (`HG type 4`) and a reduced hourglass coefficient (say, .03 to .05), the behavior is stiffened slightly and so this hourglass combination is generally recommended for most applications of the underintegrated shells. For very high velocity/rate problems, viscosity-based hourglass control is recommended.

### Accuracy:

From an accuracy standpoint, shell `type 16` is preferred over the underintegrated 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.

[1] `Formulation 6` with `IRNXX` set to -2 in `*CONTROL_SHELL`, while expensive, has been observed to give accurate springback response subsequent to a transient simulation involving large rotations, e.g., spinning blade. Also, this formulation is able to represent a tapered (nonuniform) thickness in the element.

[2] `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 itegration (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. For certain composite materials, laminate shell theory can be invoked by setting `LAMSHT`=1 in `*CONTROL_SHELL`. This option removes the usual assumption of uniform shear strain through the thickness of the shell -- this is important for sandwich composites with soft cores.

## Miscellaneous:

`NFAIL1` and `NFAIL4` in `*CONTROL_SHELL` can be invoked to automatically delete highly distorted shells (negative jacobians) before they lead to an overall instability.

When `ESORT`=1, all triangular shells which are not assigned a triangular element formulation by the user, e.g., `ELFORM 3`, will automatically take on the C0 formulation (`ELFORM 4`).
Triangular shells assigned `ELFORM 16` will automatically become `ELFORM 4` regardless of the value of `ESORT`.

It is generally recommended that invarient node numbering be invoked by setting `INN`=2 or 4 in `*CONTROL_ACCURACY`.  This is especially important when the material is orthotropic.

### REFERENCES LS-DYNA Theory Manual 2006:

• Section 7 in the 2006 Edition of the Theory Manual addresses formulation 2.
• Shell formulation 16 is discussed in Section 9 of the Theory Manual.
• Section 10 addresses formulation 1 with Section 10.6 extending the discussion to formulations 6 and 7.
• Laminate shell theory (`LAMSHT` in `*CONTROL_SHELL`) is discussed in Section 11 of the Theory Manual.