Need help in modeling indentation stress relaxation of cartilage

[vgupta0603]

Dear Dr. Ateshian,

Thanks for your reply.

I added the log option to my optimization file. Now, I can see that the optimization terminated due to the following error (iteration 3): -

************************************************** ***********************
* ERROR *
* *
* Max nr of reformations reached. *
************************************************** ***********************


------- failed to converge at time : 13.2418

Max. nr of retries reached.


Any suggestions to fix this?

Thanks and Regards
Vineet

[vgupta0603]

Dear Dr. Ateshian

I tried increasing the number of reformations and retries in my model file but still the optimization terminates with the above (please see previous message) error message. Please let me know what should I try to overcome the problem.


Thanks and Regards
Vineet

[ateshian]

Hi Vineet,

I suspect most of the trouble you are having is with the sharp edge of the flat indenter. Can you try rounding off that edge a little? I realize this may not match exactly with you experiment, but it might improve convergence.

Also, I notice that you are trying to fit for Poisson's ratio of the ground matrix. I suspect you will not find a strong dependence on this material parameter. Instead you should try to optimize on the fiber modulus ksi (and don't use a value as high as 10 MPa for ksi, that's too stiff). I also recommend that you use beta=2 for the fibers.

Hopefully these changes will improve the curve-fitting analysis.

Best,

Gerard

[vgupta0603]

Dear Dr. Ateshain,

Thanks for your reply and my apologies for the delay in getting back to you.

As per your suggestions, I changed the value for ksi (0.1) and beta (2). Also, now I am not trying to fit the Poisson's ratio (assumed it to be 0). The problem converges but the experimental and FEBio results don't match very well (please see attached figure). I think by rounding off the sharp edges of the indenter the issue may be resolved (as you suggested) but the problem is I am not sure how to round off the edges using Preview. Can you please give me a quick overview on how to do that?

Thanks and Regards
Vineet

[ateshian]

Hi Vineet,

Which material parameters are you fitting in your optimization analysis? I recommend that you fit for ksi, Young's modulus and the permeability.

The suggestion to round the indenter edge is to avoid premature termination of an analysis due to negative jacobians. If you are now able to optimize without running into this problem, there is no need to round the edge. If you do want to round the edge this cannot be done within PreView (please recall that PreView is not a full-fledged modeling and meshing software). You need to use some CAD package to generate the geometry, mesh it, export it to a standard file format (e.g., Abaqus) and import it into PreView.

Best,

Gerard

[vgupta0603]

Dear Dr. Ateshian,

I am fitting those parameters only (ksi, Young's modulus and permeability). I have attached the files in case you want to have a look.

Thanks and Regards
Vineet

[ateshian]

Hi Vineet,

Can you tell me which optimal values you found for ksi, E and k, and how they compare to the bounds that you placed on these parameters in your optimization? Is it possible that the optimization was proceeding well but was ultimately constrained by the bounds?

Best,

Gerard

[vgupta0603]

Dear Dr. Ateshian,

I got the following optimal values: -

k = 0.01
E = 0.1
ksi = 0

And my bounds were: -
k (0.001, 0.01)
E (0.1, 5)
ksi (0, 1)

Please let me know if changing anything will improve the results.

Thanks and Regards
Vineet

[ateshian]

Hi Vineet,

So it does seems that your optimal values hit the bounds during the optimization, which is why the fit did not improve. Your optimal k value hit the upper bound on k, whereas E hit the lower bound. You should therefore expand these bounds (increase the upper bound on k and decrease the lower bound on E). The lower bound of ksi=0 cannot be changed since ksi cannot become negative.

Best,

Gerard

[vgupta0603]

Dear Dr. Ateshian,

Thanks for your suggestion. It really did improve the results. The curve fit is much better than what I got last time. Do you think the fit can be improved further?
I am getting the following optimal values: -
k = 0.028; Bounds (0.001, 0.04)
E = 0.042; Bounds (0.01, 5)
ksi = 0.0045; Bounds (0, 1)

Thanks and Regards
Vineet

[ateshian]

Hi Vineet,

I am glad to see that you got these improvements. I do think that the solution should be able to converge better. Normally, since the optimization uses a least-squares scheme, I would expect that the converged FEBio solution should intersect (interlace) with the experimental response. The graph you have provided shows that the FEBio solution lies entirely on the same side (more negative) than the experimental response. Did the optimization abort prematurely? In principle it is okay to restart the optimization starting with the final solution of the previous run.

Best,

Gerard

[vgupta0603]

Dear Dr. Ateshian,

The optimization did not terminate prematurely and I reran it with the solutions from the previous run and the results I got are exactly same. Moreover, the FEBio curve also didn't intersect the experimental curve.

Anything else you would recommend that I should try?

Thanks and Regards
Vineet

[ateshian]

Hi Vineet,

Another possibility is that the magnitude of the objective function in the parameter optimization is a small number (compared to unity) and that might be throwing off the optimization algorithm. Try to change the units of force in your model to increase the magnitude of the objective function. For example, if your current units are N for force and mm for length (implying MPa for stress and moduli), switch force to mN but keep the length scale to mm. Now your force magnitude should scale up by 1000. Keep in mind that moduli will be in mN/mm^2 or kPa, and permeability will be in mm^4/mN.s. So you'll have to scale up your experimental data curve in the optimization file and also adjust the initial guesses and ranges of all the material properties to the proper units.

Let me know if that helps.

Best,

Gerard

[vgupta0603]

Dear Dr. Ateshian,

As per your suggestion, I changed the units of force to mN and kept the length in mm in the model file. Moreover, I made the corresponding changes in the experimental data files as well. Still the fit doesn't look that good (I think the previous fit was better). But the FEBio curve seems to be intersecting the experimental file.

I have attached the files for you to review. Please let me know what you think I should try further.

Thanks and Regards
Vineet

[ateshian]

Hi Vineet,

The fact that the unit change caused this difference in the fit suggests that the optimization algorithm is sensitive to the scale of the objective function, so it is good that you changed the units.

What were the optimal parameter values that you found for this fit?

Looking at your optimization file, I cannot make full sense of your choices for each of the parameters. For example, in the case of permeability, you specify

<param name="Cartilage.permeability.perm">8e-005, 0.0000001, 0.0001, 0.00000005</param>

This means that your initial guess is 0.8e-4, the minimum value you allow is 1e-7, the maximum is 1e-4 and the scale factor is 5e-8. For best results, the scale factor should be comparable to your initial guess, whereas in your analysis they differ by several orders of magnitude. It would also be helpful to keep the minimum and maximum values within two orders of magnitude of each other at most.

If you could provide the optimal parameters you found in your fit, I'll be able to suggest more specific guidelines for restarting your optimization.

Best,

Gerard