Erhart, T., "Review of Solid Element Formulations in LS-DYNA: Properties,
Limits, Advantages, Disadvantages", 2011 Developers' Forum, Stuttgart,
Germany, October, 2011.
http://www.dynamore.de/en/training/conferences/past/forum11/entwicklerforum-2011/erhart.pdf
also in
http://ftp.lstc.com/anonymous/support/PRESENTATIONS/solid_formulations.pdf
(Looks at both hex and tet formulations in modeling of metal and rubber parts.)
The following paper (undated but apparently rather old) includes a look at some shells and solids:
"Comparison of different element types in structural analysis",
K. Elsäßer, B. Keding, H. Müllerschön, C. Pedrazzi,
http://ftp.lstc.com/anonymous/outgoing/support/FAQ_docs/compare_element_formulations.pdf
__________________________________________________________
RE: Unexpected stresses in S/R solid formulations
Question:
On 09/02/2014 09:06 PM, jd wrote:
> Hi, Thomas.
>
> Was wondering why the elastic x-stress, y-stress, and z-stress in 8 integration point solids are not even close to what I was expecting.
> What I expected for this end-loaded cantilever where the end load (z-force) = 20:
>
> x-stress = +- Mc/I where c = sqrt(3)/3 * half the depth of the section = 0.866 and I= 6.75
> y-stress = 0
> z-stress = z-force/area = 2.22 (all integration points).
>
> Stresses are reported in eloutdet.
>
> The end displacement is reasonably close to the theoretical PL^3/3EI.
Input deck: http://ftp.lstc.com/anonymous/outgoing/support/FAQ_kw/elform_neg1.k
------------------------
Answer:
the quick answer is probably that the pressure is constrained to be constant within the element, so basically two integration points in the same element MUST have the same pressure. That means that if one integration point has (as you write)
sigxx=+Mc/l
sigyy=0
sigzz=f/a
the pressure in that integration point would be (-Mc/l-f/a)/3. And if in another integration point we would have
sigxx=-Mc/l
sigyy=0
sigzz=f/a
the pressure in there would be (+Mc/l-f/a)/3. This is impossible in an S/R element (type -1,-2,2 solids), equilibrium is obtained imposing the additional constraint that p=constant in all integration points... which of course makes it not so straightforward to analyze analytically, and is something I haven't done.
The stresses may seem counterintuitive, but I do believe that equilibrium holds... at least two elements through the thickness would be necessary to get a somewhat decent stress profile... that way the pressure can vary through the thickness... but I doubt it would be perfect without having a few more than just two...
tb
___________________________________________________________
ELFORM -2 was discovered by Satish to be invarient and Thomas Borrvall implemented
a tentative fix in 971 Dev r75926 (~ 8/23/12). After further testing, the fix
will be included in R6.
Note this comes AFTER the release of R6.1.0.
*** Update for 971 R4.2.1 ***
Solid ELFORM
.EQ.-2: 8 pt selective reduced integration for elements with poor element aspect ratio, accurate formulation
.EQ.-1: 8 pt selective reduced integration for elements with poor element aspect ratio, fast formulation
Elements -1 and -2 are variants of the selective reduced integration element type 2 that are
intended to be used for elements with initially poor aspect ratio, i.e., elements for which
one side is significantly smaller or larger than the other two.
These formulations alleviate the transverse shear locking tendencies that otherwise is
an unfortunate side effect for this fully integrated brick, and are in this sense more accurate.
The element theory is based on heuristics but in a vague sense type -2 may be regarded an
accurate formulation that suffers from high computational expense (~2x or more cost of type 2)
whereas type -1 (~1.2x cost of type 2) is
computationally more efficient but may be too soft for certain deformation modes, owing to
some coefficients used in calculating the strain-displacement matrix being adjusted to
make the matrix more sparse.
http://ftp.lstc.com/anonymous/outgoing/support/FAQ_docs/7EC_ThomasBorrvall.pdf is conference paper describing
these new solid element formulations.
The same paper is found at
http://www.dynamore.de/documents/papers/euro2009/G-I-02.pdf
See also 02_elements_contact.pdf
downloaded from http://www.dynamore.de/2010_new_methodologies
The developer had this to say... "In deriving type -2 it turned out to be computationally inefficient so I made some simplifications in the algorithm to render it considerably cheaper, this ended up as type -1. A consequence of changing the kinematics is that a certain mode became less stiff when compared to the others. But this mode is still part of the element kinematics and is not to be seen as a stabilization term, from a theoretical perspective the element may be prone to deformations that look like an hourglass mode but it really isn't."
*** End update ***
For hex elements, I prefer ELFORM 1 over ELFORM 2 and most definitely over ELFORM 3.
The reasons are:
- speed
- absence of shear locking behavior for less than ideal element aspect ratios
- robustness (less likely to encounter negative volumes)
That said, there are hourglassing issues to be considered for ELFORM 1.
See http://ftp.lstc.com/anonymous/outgoing/support/FAQ/hourglass_condensed
-------------------------------------------------------------------
For accounts of element formulations studies:
1. Erhart, T., "Review of Solid Element Formulations in LS-DYNA: Properties,
Limits, Advantages, Disadvantages", 2011 Developers' Forum, Stuttgart,
Germany, October, 2011.
http://www.dynamore.de/de/download/papers/forum11/entwicklerforum-2011/erhart.pdf
2. Schwer, L.E., Key, S.W., Pucik, T.A., and Bindeman, L.P.,
"An Assessment of the LS-DYNA Hourglass Formulations via
the 3D Patch Test," 5th European LS-DYNA User's Conference,
Birmingham, United Kingdom, May, 2005.
http://www.dynalook.com/european-conf-2005/Code_Developments/Schwer.pdf
3. http://ftp.lstc.com/anonymous/outgoing/support/FAQ_docs/compare_element_formulations.pdf
4. http://ftp.lstc.com/anonymous/outgoing/support/FAQ_docs/dubois-foam-tets.pdf
-------------------------------------------------------------------
Other references:
1. elform=2 formulation:
Nagtegaal, J.C., Parks, D.M., and Rice, J.R., "On Numerically
Accurate Finite Element Solutions in the Fully Plastic Range",
Computer Methods in Applied Mechanics and Engineering, Vol. 4,
Issue 2, pp. 153-177, 1990.
2. LS-DYNA Theory Manual (www.lstc.com/download/manuals).
3. ./solid_elform3 (text document available on request; contains UNofficial
notes concerning solid formulation 3.
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V29-482B533-1S&_user=10&_coverDate=09%2F30%2F1974&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8560087107f879d5571e13e690154bd8
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19740020296_1974020296.pdf
4. ./solid_elforms_2_-1_-2_18 (text document available on request; contains UNofficial
notes concerning solid forms 2,-1,-2,18.