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Dispersion Diagrams & Floquet-Periodicity - Theoretical Question
Posted Jun 13, 2022, 6:08 a.m. EDT Structural & Acoustics, General, Acoustics & Vibrations Version 6.0 2 Replies
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Dear all,
the following questions is a beginner's one, but yet one that I am puzzled about. I am trying to compute the band diagram/ dispersion curves of a periodic structure. As I am not familiar with Bloch-Floquet periodicity, I devised a validation case (see figure attached):
Two simple rectangular unit cells are computed, which only differ concerning their length in the y-direction where I am imposing Bloch-Floquet periodicity (
). My intial assumption was that both cases should lead to an identical band structure as both unit cells lead to the same infinite structure when translated periodically.
However, this is not exactly the case. When comparing both diagrams representing the first irreducible Brillouin zone (IBZ, see Fig. a1 and b), the band structures differ. When comparing both diagrams for identical wave numbers (Fig. a2 and b), the slopes for the compression and shear waves are the same as expected, but the branches originating at the corners of the IBZ are different.
Can anyone give a word of advice where my error in reasoning is? Or recommend literature concerning the choice of unit cells? Perhaps I have a fundamental missconception of waves in periodic media...
Many thanks for your help and ideas!
Attachments:
Hello Lukas
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Hi,
I may not be able to explain Floquet Bloch theory in simple terms here, but in case you are still looking for an answer to the question you posted, I will try to point out a few things which might help you think about it. First of all, you could have considered a unit cell of length L/2 and obtained another dispersion curve which would then have an x axis with more range than Fig. a1. I think it is safe for me to say that there is no 'correct' unit cell for the infinitely long beam. In a way, the solution obtained for a 'small' unit cell already has the solutions of the 'large' unit cell. It is useful to plot two of these results on the same axis and realize that one of them is simply the other solution folded or unfolded about the end of the x-axis (zone folding is something you might want to read up on eventually). Physically, if k varies from 0 to pi/L then wavelength varies from 2L to infinite, whereas if k varies from 0 to pi/2L then wavelength varies from 4L to infinite. So, the smaller unit cell solution already contains all possible wavelengths resulting from the larger unit cell solution (and has more solutions - from 2L to 4L).
Best, Sai
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Dear Sai,
I almost missed checking on this thread, but your answer is most appreciated! Thank you very much for this detailed and concise answer! I had indeed found that I had not captured Floquet Bloch theory when posting this question...especially the fact that the folding of the dispersion branches is associated with the choice of the unit cell, and that this folding may be unraveled when the zone is extended beyond this first Brillouin zone.
From a mechanical engineer's point of view, I found the analytical solutions for infinite mass-spring chains with 1D-translational periodicity very helpful in understanding that the folding is more a question of how the dispersion relation is represented, but that the information itself is not changed. If I am correct, considering a "super cell" with an extended period 
will result in the corresponding N-times folding of the dispersion branch in the reduced Brillouin zone 
- I guess this is essentially what you explained by relating the wavenumber of the considered wavelengths in the small and large unit cell (which is a great hint!). Further, I would say that the choice of the unit cell is not unique such that any multiples of the smallest periodic unit are permissible.
Since you seem to have a strong background in this field: Can you recommend any "standard" literature on wave propagation in periodic structures & Floquet-periodicity?
Thank you very much again for your help! Kindest regards Lukas