Linear elements stiffer in bending than quadratic ones (not the other way around as it should be)

[Rocketeer3k]

Dear all,

I have a peculiar case in which the exact opposite happens from what is expected. I am modelling crack propagation using cohesive elements. In the beginning I am just modelling a mixed-mode bending test in plane strain to calibrate my model parameters. The model is bending dominated, much like a cantilever with tip-load. Trying different mesh sizes and elements for the bulk material I found the following:

1. Quadratic plane strain elements (CPE8) deliver the most accurate solution. (Closest to analytic solution)
2. Linear elements (CPE4) are softer than quadratic elements
3. With increasing mesh density the solution with linear elements approaches the quadratic solution, thus the model becomes stiffer.

This is exactly contrary to what I know, that linear elements are softer in bending than quadratic ones and that the response stiffens with mesh refinement.

Can anyone explain this?

 

[IceBreakerSours]

There is nothing wrong in what you've found; your prior knowledge, it turns out, is inaccurate.

When a mathematical model (in this case, an FE model) is determined to be "stiff", it means there are not enough degrees of freedom available for the variation in the field variable (such as stress). So, when you increase mesh density or increase the order of the elements, you provide the model with more degrees of freedom making it "less stiff". This "stiffness" is different from the typical definition of stiffness being equal to delta_load/delta_displacement in the linear region of the experimental data from, say, a uniaxial tensile test.

http://www.eng-tips.com/faqs.cfm?fid=376

http://www.eng-tips.com/faqs.cfm?fid=1083

 

[Rocketeer3k]

Dear IceBreaker,

Its the other way around: More elements --> Stiffer

 

[DanStro]

If you are disagreeing with IceBreakerSours point then I'm sorry but I have to agree with IceBreakerSours. Below is a quote from an article in MachineDesign (

http://machinedesign.com/article/a-closer-look-at-model-meshing-1104).

I don't have any of my texts to copy a page that describes, more or less, the same thing.

If you are just reiterating what you are seeing in your model then I guess just ignore this response.

QUOTE: "Consider a cantilever beam meshed with three different densities to see how the maximum deflection changes. To make this comparison fair, even the first mesh (the rough one) must be able to model bending stiffness.

The beam deflects further with finer meshing. This proves that the finite-element model becomes "softer" with a rise in element density. The effect arises because artificial constraints become less imposing with mesh refinement. ..."

Dan

Han primo incensus

 

[IceBreakerSours]

How do you know the "model" is "stiffer"? What variables (Stress/Strain, Load/Displacement .. at a specific location, for example) are you looking at to make this determination?

http://www.eng-tips.com/faqs.cfm?fid=376

http://www.eng-tips.com/faqs.cfm?fid=1083

 

[Rocketeer3k]

Dear DanStro,

I do not disagree with IceBreaker, neither do I disagree with you. What both of you wrote is also what I have experienced so far.

But you both have misread my post, or I have not clearly expressed myself. My model behaves the other way around from what you describe:

- Common knowledge: Finer mesh --> softer response

- My model: Finer mesh --> stiffer response

I do not understand why. That is my question.

 

[Rocketeer3k]

@IceBreaker,

I am looking at plots of applied force vs tip displacement. The more elements I use through the thickness of the specimen, the more load is measured for the same applied displacement, thus the response is stiffer (steeper plot).

Thank you btw.

 

[Rocketeer3k]

I have attached a plot to show what I mean. t stands for number of elements through the thickness, all else being equal.

http://files.engineering.com/getfil...3-39e8-4f3c-b6d7-da79bf9a4b72&file=sample.png

 

[Rocketeer3k]

This image shows what I am modelling:

 

[IceBreakerSours]

a) Firstly, check the .msg and .dat files when t=1. My guess is that there may be some warning messages. Even if they are not, output at t=1 is incorrect because one linear element can only allow constant strain through it. In a problem such as yours, there is a variation in strain going through the thickness. Bottomline: t=1 is not even a correct solution. This is a classic "stiff" problem but the solution (from the curve) is, if I use your terminology, compliant/soft.

b) Secondly, as I said before, this may be just a matter of terminology used by mathematicians (FEA developers) vs. engineers. Stiffness of the structure and a mathematical problem/model being "stiff" are not the same thing. You are converging to the analytical solution as you increase mesh density, which is what is expected. So, if one is observing your load-displacement curves, then yes, the structure is becoming stiffer as you increase mesh density but that is not the same thing as the problem being "stiff" or not.

http://www.eng-tips.com/faqs.cfm?fid=376

http://www.eng-tips.com/faqs.cfm?fid=1083

 

[Rocketeer3k]

dear IceBreaker, thx for your answers

then I am confused as to the terminologies of stiff in FEA.

From the document that DanStro posted I quote: "The beam deflects further with finer meshing"

In my weird model the opposite holds true: My model deflects less with finer meshing. Definition of stiffness be as it may, but that fact remains that my model behaves the contrary to common believe.

 

[IceBreakerSours]

Do you have the nonlinear geometry flag on or off? I am also not sure about how you are post-processing the output.

If you wish, uploading the INP will help us get a closer look.

http://www.eng-tips.com/faqs.cfm?fid=376

http://www.eng-tips.com/faqs.cfm?fid=1083