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I am using a constant damping factor for the Automatic Stabilization and it works well in converging my analysis.
As I increase the damping factor (the default value is 0.0002) it even works better.
I wonder what is the damping factor doing and how much I can increase or decrease it from the default value.
The damping is ramped down to 0 at the end of the solution step but could cause inaccurate results is some cases. Typically I check the energies of the solution to see if the energies associated with the damping is deemed significant compared to the total energy for my case. You can also do s sensitivity to see if changing the value has a significant effect on the results of interest. I hope this helps.