Abaqus/Explicit energy growth issue with S4RS elements

[henki]

I am trying to compare different shell element formulations in Abaqus/Explicit and I stumbled upon an energy growth issue. Here are the specifics of my test case:

-2m x 2m rectangular plate with 5mm thickness
-Element size 0.1m x 0.1m (total of 400 elements)
-All edges fixed (U1 = U2 = U3 = UR1 = UR2 = UR3 = 0)
-Elastic material (steel properties)
-Short duration pressure pulse
-Default hourglass formulations
-Section integration before analysis AND another case where section integrated using 3 Gauss points
-Time step calculated using global stable increment estimator AND another case using a very small time step (time scaling factor 0.15)

I know this is quite coarse mesh and simple test case, but I am using the student version of Abaqus which allows only a maximum of 1000 elements. Nevertheless, I believe it is good enough for comparison purposes.

I ran the test case using S4R (finite strain), S4RS (small strain) and S4RW (small strain, warping considered) elements. The displacement history of plate middle point is quite similar for each model during the first few oscillations (max. displacement approximately 80mm, max. discrepancy between models <2%), but strain contour plots show a slightly larger variance. This of course can be attributed to the different element formulations. What I can't understand is the energy balance. For finite strain elements (S4R) the total energy remains low and negative, as it is supposed to. But for small strain elements (S4RS and S4RW) the total energy is large and positive and continues to increase boundlessly, which is a clear sign of instability. And because of this, the displacement amplitude also continues to increase.

Any idea what might be causing this instability? I always thought that using the small strain elements might give slightly inaccurate results when used in relatively large deformation applications, but I never realized that they could result in fully unstable results.

 

[henki]

Okay, specifying second order accuracy in "element controls" section for S4RS elements solved the energy balance issue, although it had minimal effect on displacement results (<0.1%). I still don't understand why disabling second order accuracy leads to energy creation from nowhere instead of just slightly inaccurate results.

 

[Drej]

You will need to assess all of the components that make up the total energy to identify first which energy is adding to the total energy balance. Any one of these components may be indicating that you have a problem but you need to find which one or ones (for example, is the mesh showing signs of hourglassing? a plot the hourglass energy with help).


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[henki]

The energies present in this case are:

-Kinetic energy
-Elastic strain energy
-Work done by external forces
-Hourglass energy

Hourglass energy is negligible and the mesh shows no signs of hourglassing. Kinetic energy and strain energy oscillate and their sum continues to increase. Therefore, the total energy continues to increase (as the work done by external forces remains constant after the pressure has gone to zero). The thing that wonders me, is that this happens even when I'm using small enough pressure load to excite only 20mm mid-point displacement, which in my opinion is quite small value compared to plate dimensions. Even using extremely low time increments does not remove this behavior even though the displacement increments during these time increments remain very small. Now, the second order accuracy setting removes this behavior, but also increases the cost of the analysis when using larger models. Otherwise I would suspect something wrong in my test case, but the fact that using second order accuracy OR using S4R elements removes the energy issue, is making me distrust the S4RS element (which apparently is the default shell element formulation in other software such as LS-DYNA and Radioss).

The reason I am inspecting this case is that I am trying to check my own FE-code and my implementation of Belytschko-Tsay shell element (which is named S4RS in Abaqus/Explicit). I am just wondering if this element is not appropriate for this kind of test case or if the energy issue is always present when using this element formulation.

 

[Drej]

Do you have damping in your model?


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[henki]

No damping. But I'm pretty sure that is not the problem in this case, since the S4R model and S4RS with second order accuracy give reasonable results. Besides, I think that damping would only have the same effect as plasticity: it would just dissipate the already erroneous energy, making it harder to spot the energy issue.

 

[henki]

I googled this for a while from another perspective and it seems the problem is caused by the use of Green-Naghdi stress rate, which is not work-conjugate with any finite strain measure and, therefore, causes energy error depending on the problem at hand. I suspect that the first order accurate strain measure of S4RS element used with Green-Naghdi stress rate creates this energy issue.