I have for the past couple of years tried to push the development of Microacoustics Optimization (especially Topology Optimization) and get more focus on its research and engineering development. Even though it is still very much an unexplored topic, there has been work done with Microacoustics and another optimization type; Shape Optimization. This blog post will describe some of this development.

This week Peter Risby Andersen got his PhD degree (http://www.act.elektro.dtu.dk/news/2018 link), and part of his work was to combine microacoustics (i.e. acoustics with thermoviscous losses included), via an accurate Kirchhoff decomposition formulation of the Navier-Stokes equations implemented in a Boundary Element Method framework, with shape optimization. With this capability he has shown examples where the acoustics losses are indeed accurately included while doing the optimization, and also how standard acoustical shape optimization fails for the same examples. Very nice work indeed, congrats, and good luck to Peter with his post-doc :-)

Any optimization with the thermoviscous losses implemented accurately for any geometry will inherently be time-consuming, and so adding optimization on top of such a method will of course be computationally heavy. However, an alternative to a general thermoviscous implementation is the so-called Boundary Layer Impedance (Bli) model. Since the thermovisous losses take place in a boundary layer, and thus become less and less important as we move into the bulk (the domain away from the boundaries), it is natural to try and implement the entire effect into a *boundary condition*. It has been shown in previous publications [1, 2, 3, 4] how an impedance boundary condition can be formulated when assuming that the boundary layer is narrow compared to all local geometry variations (which is problematic for small geometries/low frequencies), but also corners or other non-smooth geometry variations can cause issues (which is probably worse at high frequencies). On the other hand, if you are careful with fulfilling these requirements, the geometry can be arbitrarily shaped, which is an advantage compare to several other methods.

I have never really been too fond of this method; there will cases where you already know in advance that it will not work, e.g. for small geometries found many places in hearing aids and similar applications, or you are in a grey area, where it might work, but you are not sure. The latter will especially be a problem, if you don't have much experience with thermoviscous losses and how model them. So you have to only use it in a 'Goldilock'-zone, where your geometry is not so big that microacoustics is overkill, and not so small that the method breaks down. On the other hand, it is a fairly simple, and hence fast, method, which is of course a big advantage when doing engineering tasks.

Recently, a couple of new papers have been published [5, 6] describing and using the method again, so I got some renewed interest in it. The boundary impedance is related to a specific, normalized acoustic boundary admittance described as

The specifics of this admittance expression can be found in the listed papers; for now just note the first part of the sum is related to viscous losses, and the second to thermal losses. Both are scalar values, but as can be seen the viscous part involves solving a so-called tangential Laplacian (solving this directly was suggested already in [4], possibly even before), which is not readily available in most simulation software. Luckily, COMSOL Multiphysics comes to the rescue once again, and so with the assumptions mentioned earlier in mind, we are ready to try out the method. I have done some testing on the method before, around 9 years ago, but it was limited to certain simple geometries, so it is nice to finally have a complete implementation for general geometries. Anyway, we go straight to a case with two tubes with equal dimensions; small enough that thermoviscous losses do matter; and large enough that the Bli method should work.

One tube has the Bli applied, whereas the other is calculated using COMSOL's narrow region acoustics. For a certain input velocity at one end of the tubes, the sound pressure levels at the other end match perfectly:

Now to the fun part; with the Bli we should be able to do very efficient shape optimization, since the method includes losses in a boundary condition, instead of in a formulation where the entire bulk is considered. As long as we don't misuse the method, we should be good to go, and so we can for example come up with an objective like "*please dampen the sound pressure level by 3 dB at the resonance frequency without changing the tube length at all, nor the volume too much*", albeit in a more rigorous mathematical form, of course. We can already gather that what needs to happen is that the shape has to change in such a way that the inner surface area is increased in order to increase the thermoviscous losses. And that is exactly what happens. First the two responses:

where the deformation has caused a 3 dB damped response at 1 kHz, and with the two tubes where one is now shape optimized:

I have never seen this done before, but we seem to have proof of concept that this is valid approach to take when it comes to shape optimization with thermoviscous losses included, under the given assumptions. One thing missing is to actual run a simulation using a more general microacoustics method with the newly obtained geometry to make absolutely sure that we have not violated any assumptions in our optimization process, but this is more time-consuming and will be omitted for now. As a quick check, though, you could opt for running the analysis just at the resonance frequency.

My colleague Junghwan Kook suggested that I try the Bli method for a metamaterial case that he was already looking at, and the dimensions were perfect for the method to be applied. His aim was to have a design with a certain band gap in the acoustic response, and adding losses did in fact change the band gap quite a bit. Hopefully, he and I will publish more on this next year. Acoustic metamaterial design seems to often be placed in the Goldilock zone, when it comes to physical dimensions, and I imagine that the method will be adapted in this field of work in the coming years.

So in conclusion, the Boundary Layer Impedance method is an interesting alternative to the number of other methods. You have to be careful when you use it, as to not violate any of its assumptions, but aside from that, you should get accurate results very quickly. The method lends itself well to many metamaterial cases, and if you have such, or similar, cases, give it a try.

[1]: "Acoustics: An Introduction to Its Physical Principles and Applications", A.D. Pierce, Acoustical Society of America, 1989

[2]: "Hybrid numerical and analytical solutions for acoustic boundary problems in thermo-viscous fluids", Bossart et al., Journal of Sound and Vibration 263(1): 69-84, May 2003

[3]: "Viscothermal acoustics using finite elements: Analysis tools for engineers" W.R. Kampinga, PhD thesis, Twente University, June 2010

[4]: "Viscothermal wave propagation", M.J.J. Nijhof, PhD thesis, Twente University, December 2010

[5]: "Acoustic boundary layers as boundary conditions", M. Berggren et al., Umeå University, January 2018

[6]: "Boundary Layer Impedance model to analyse the visco-thermal acousto-elastic interactions in centrifugal compressors", J. Jith and S. Sarkar, Journal of Fluids and Structures 81, August 2018