I see that you have an axial resonance at 20509 Hz. I assume that this is the desired operating mode. What you should do is lengthen the sonotrode somewhat until the frequency is closer to 20000 Hz. If the sonotrode length must be fixed then you can remove material at a node to reduce the frequency. (I can provide details if needed.)
1. When the sonotrode is manufactured you should allow extra material for final tuning. This is common for all sonotrodes, particularly to allow for material variations, especially for titanium.
2. If you want to reduce your meshing and run time then you could reduce your mesh density without significantly affecting the results. I prefer using "absolute mesh size" rather than "percent of automatic". Using a 4.0mm mesh reduced the run time by more than half while only increased the axial resonance by 4 Hz.
The length of the sonotrode is not fixed. I told you that my generator works in a narrow range of frequency (between 19,4 and 20,4 kHz). Moreover I know that my piezoelectric transducer (alone) gives a frequency of 20,9 kHz. This means that the booster and the sonotrode must decrease the frequency of the equipment in order to set it in the range of the generator, isn't it?
The notes you wrote are very interesting, so I understand that in all cases I must allow extra material.
Yes, the sonotrode must be designed to compensate for any mis-tuning of the other components. However, the transducer may not be as mis-tuned as it appears. Acoustically, I consider that portion of the stud that protrudes into the transducer to be part of the transducer. (Similarly, I consider that portion of the stud that protrudes into the booster to be part of the booster.) With this assumption, the frequency of the transducer would be lower than bench-testing the transducer without a stud. You could check this by modeling the transducer without a stud and with a stud (with no stud protrusion beyond the transducer's face).
Yes, I would always recommend leaving extra material for tuning. If you are reasonably confident in your FEA material properties then you could adjust the pretune length in FEA to give a pretune frequency of ~19.5 kHz. This would leave 500 Hz for tuning in the machine shop. However, for the first sonotrode you might want to be more conservative and specify a pretune length that will give a pretune frequency of ~19.0 kHz. This would only require somewhat more material and tuning time. A tuned frequency specification of 20.0 kHz +/-100 Hz should be adequate.
I believe you, so I'm not going to model the transducer. I think that it would be very difficult to reproduce the performance of the piezoelectric rings.
To sum up, I have to design the booster and the sonotrode assuming that the frequency of the equipment is 20 kHz (although we know that my transducer alone gives a frequency of 20,9 kHz). Then, in order to do the final tuning, I have to leave extra material at the end of the sonotrode. Do you agree?