Time step size in explicit dynamics

[henki]

Hi.

As far as I know, the time step size in explicit dynamics depends on the characteristic length "L" of an element and the wave speed "c" as follows: delta_t = L / c. This is also explained in the following link:

http://www.dynasupport.com/tutorial/ls-dyna-users-guide/time-step-size

I have done an explicit analysis of plate under blast loading using two different models: shell element model and beam element model. The beam model consists of only 2 beams which form a cross (this model is not very accurate, I know). With both models I used same element sizes (shell 8.5x8.5mm, beam 8.5mm). I performed the analyses using ABAQUS/Explicit.

Now here comes the problem: when using automatic time incrementation, the shell model uses time increments approximately of size 1.3E-6s, whereas the beam model uses time increments approximately of size 5E-8s. Therefore the beam model takes lot a lot longer to compute. If the time increment size depends only on the characteristic length (which is the same for both models) and on the wave speed, then how can there be so large difference between the two models? Is there something else that can also affect the time step size? Or is the wave speed calculated with some other method than mentioned in the provided link? I found some explanation of how dilatational wave speed is calculated using pressure increment and deviatoric stress increments from ABAQUS documentation, but I didn't quite understand it.

-henki

 

[IRstuff]

The time step is what the programmer thinks he can safely propagate the difference equations into the future and arrive at a convergent solution.

The use of L/c would seem to be an extremely gross value for a time step, since it ostensibly represents the time required for a wave to completely traverse the object's critical dimension. In general, lots of things happen while the wave propagates, so something substantially smaller than L/c would seem to be more consistent with what you should need.

While what you are trying to model might be the same, the approach used to converge the solutions might not.

TTFN

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[henki]

I agree, L / c is a gross value. I think the loading rate should also be considered when determining the time step, as the magnitude of the load can change substantially in small time. Also some other things could be considered. What I am trying to find out is that how do the explicit software determine the time step size.

The approximation delta_t = L / c yields the critical time step size which should not be exceeded. But as it seems that the chosen time step size is often smaller than the one given by L / c, then what are those criteria that affect the time step size? Where could I find information about this? As I mentioned, in ABAQUS documentation there was a small chapter on this subject, but I'm not sure if the particular chapter is the answer to this problem. And I haven't found anything useful from RADIOSS and LS-DYNA manuals. Or maybe I'm just stupid or blind .

I would be grateful if someone could point me to an article or book or anything that could have an answer to this problem .

-henki

 

[btrueblood]

Value of time step will vary with the type of solver, i.e. is the matrix solved by decomposition or by direct inversion. From (admittedly hazy) memory, linear beam FEA equations are directly invertible, but not so the shell elements...

It may also be that the programmers picked a value they knew would be very safe for the beam equations (say .05*c), and left it, thinking not many users would worry about execution time for "simple" beam models, or perhaps solver is fast due to different solution method. Alternatively, shell models being more generally complex and possibly very large, more effort is put into an algorithm for finding an optimal time step to keep solution stable whilst being as speedy as possible.

 

[henki]

btrueblood, what do you mean with this: "i.e. is the matrix solved by decomposition or by direct inversion"? I thought that in explicit method no matrix conversion and equation solving is needed. What matrix do you mean?

-henki

 

[IRstuff]

Regardless, the inner workings of an FEA time step selection is proprietary and subject to the constraints of their solver approach. While the fundamental differential equations might be the same, there's a variety of ways of skinning the solution cat.

TTFN

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[Drej]

The difference in the delta-t is that for the beam you have to account for the stiffness of the beam. Shells have a fairly straightforward delta-t criteria, set - as you point out - by a critical element dimension (usually the smallest cross-diagonal) and the wave speed (and intrinsically the density of the material).

With the beam, the delta-t calculation is different and includes "stiffness" - beams in explicit analysis are notorious for delta-t problems, which is why in some respect they're hardly used (or when they are you have to be very careful with the impact on the delta-t). If your beam is considered extremely stiff (stubby, with high second moment of area, and/or high modulus) then the delta-t will be impacted. You'll need to bear this in mind when you mesh your structure using these elements. Might be useful for you to check the documentation for delta-t calcs for a beam.


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[Drej]

As a postscript, the web link you provide above talks about time step size for beams (about half way down the page). As you'll notice, the calculation for a beam critical delta-t is based on I and A: the second moment of area and the cross section, respectively, as well as the wave speed. My guess, as I say, is that your delta-t is being impacted by the stiffness (I, A).


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[GregLocock]

When I model a tire turning during a parking manouevre the time step often drops to as little as 1e-6 to 1e-5. The vibrations seen in practice are of the order of hundreds of Hz. Therefore I need ~ hundreds of time steps per cycle of vibration. Admittedly that is a stick slip friction problem.

Cheers

Greg Locock


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[henki]

Drej, I calculated the time step for beam element using the equation that takes into account "I" and "A". The dimensions of the beam cross-section are:
-width = 176.8mm
-height = 7.1mm
Now I haven't found anything of time step with beam elements from ABAQUS documentation, but I will now assume that it uses the same formulation as LS-DYNA beam element.

I first only calculated the time step using the second moment of area obtained by (width*height^3)/12 and this gave time step size of approximately 1.3E-6s. This is the "I" that determines the stiffness against the loading. BUT when I calculated the "I" in transverse direction, the time step size decreased to 8E-8s. This seems to determine the time step size, even though this latter stiffness has no effect on the results. I ran the beam analysis again using a fixed time increment of 1E-6s and the results were same as those obtained with 5E-8s. So this answers my question. Thank you!

When using the automatic time incrementation, the time step size in my model decreases from 5E-8s to 2.5E-8 (halves) during the analysis. Now I suppose this is simply caused by the changes in the beam cross-section, which in turn changes the stiffness. Or could there be something else affecting this phenomenon?

-henki

 

[IRstuff]

Stepsize is also driven by solution convergence, i.e., does one get the same answer within the convergence tolerance. If not, the stepsize is decreased until it does.

If you loosen the convergence tolerance values in your program, you should see a reduction in number of iterations and a commensurate decrease in runtime.

TTFN

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[henki]

IRstuff, okay, now you got me confused. Are you sure you are not talking about implicit procedure? I was under the impression that in explicit analysis no iterations are performed, as the chosen timestep size is so small that the solution is assumed to converge by itself. Explicit method is supposed to be as efficient as possible, therefore it would seem odd that additional iterations would be performed.

Disclaimer: I have just done my master's degree and haven't got much experience and knowledge yet, so it may be that I don't know what I'm talking about . If I am completely wrong, could you please point me to a book or another reference where I could find more info on this subject. I have Belytschko's book "Nonlinear finite elements of continua and structures" but I haven't read the chapter of stability in explicit methods with much thought yet, just browsed through it. Maybe I should do it the first thing tomorrow.

-henki

 

[Drej]

The delta-t in explicit is (initially) conditionally stable, as per the requirement of the explicit algorithm - this conditionally stable delta-t is obtained from the procedure discussed above. With implicit, on the other hand, the delta-t is unconditionally stable (but that is another story).

In both implicit and explicit the delta-t can cut-back if it is required to for convergence - that is, in explicit, the delta-t can cut-back from the initially conditionally stable delta-t to attempt to reach convergence - based on the response of the model. The cut-back can occur owing to non-linear phenomena such as NLGEOM, plasticity or highly non-linear contact (slip/stick, for example). To be able to do this, explicit has to calculate a solution based on the stable delta-t and attempt to iterate to convergence - if this is not possible then it has to cut back (usually an initial attempt of some percentage of the stable/current delta-t), and it carries out this process many, many more times compared to the implicit procedure.

At least, this is my understanding.


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[henki]

Okay, now I understand. Thank you all for your answers!

-henki