#045: Reduced Order Models and Microacoustics

Oftentimes you do not need, or want, a very complex model to describe a system. While I enjoy working with multiphysics simulations in COMSOL Multiphysics, I am also very fond of reduced order models like lumped models and transmission line modelts hat the engineers can use to get quick and quite accurate result, and make their decisions from, instead of having to wait half and whole days for simulation job to complete.

In some cases, however, some unique knowledge may be needed to find a good simplication. If we for example look at a circular and a triangular tube, or hole, with acoustic propagation in them, they will have very similar behavior in most cases.

What really counts is the area of the cross-section, and not its shape. So a common transmission line is enough to cover all cross-sectional shapes (within some reason).

S is the area, and l is length, and you are basically done. However, when the tubes or holes are very small, certain boundary effects influence the overall acoustics behavior with the tubes/holes, and the standard acoustics approaches start to break down. So engineers can go a couple of ways: Ignore these effects, and use the model anyway, knowing that the model is not quite accurate. Try and find a more accurate solution for each geometry they encounter. Or find a middle ground; establish a more accurate solution for 'prototypical' case only and view other geometries as perturbations of this case.

For tubes and holes, the microacoustics variables describing is most well-established for the circular cross-sections, and this cross-section then serves as the 'parent' to all similar geometries. The variables describing the behavior for this tube/hole could look something like:

These expression will serve to inject the boundary layer effects into the analytical models to a very high accuracy from an acoustics perspective, ignoring the geometrical varations in the bulk and boundary layers seen below, and focusing on the averaged response.

Now, you could take the middle ground, and focus on the circular tube, and use the expression for other shapes just taking into account the overall area. This will be better than not including the effects at all. One the other we can clearly see that the boundary layer distribution cannot be exactly the same for any shape. So do accept the compromise or not? For the lumped models one can deduce a classical result of:

We typically focus only on the viscous and inertial effect, so two lumped components are needed, with an underlying assumption of a wide boundary layer. The engineer can then enter the values into SPice or other software, get accurate for the circular hole situation, or get okay results for other cross-section.

But if you are really a stickler for accuracy, hunting the last fraction of a dB, and at the same need an extremely quick model, you need to opt for another approach, where you directly target the geometry that you are using. So if you for example have a plate with triangular holes in it, you would need to establish the variables below.

This can then be used in a transmission line model, and you would now get even more accurate results than using the circular cross-section expression. For the lumped you can show (as I did years back and published much later (https://www.ingentaconnect.com/content/dav/aaua/2019/00000105/00000006/art00040) that the relevant lumped parameters for this case reduce to:

You can then investigate analytically which change occur if you use triangular holes in your headphone, or whatever application, and achieve that extra insight and accuracy. And different methods like Taylor expansion, pertubation theory, or inital calculations in two-dimensional simulations, can push forward more accurate variable inputs for different cross-sections as needed. You will see overall levels, peaks and resonance frequencies align a little better, and perhaps that is what pushed ahead your particular application.

The lumped models and transmission line models are ideel if the engineer has something like Spice or MATLAB availble and acoustics phenonema are to be investigated. But what if you have access to a simulation software package... Is it worth still doing the reduced models? I would argue yes. The models will provide insight and intuition that you not extract by only working with advanced simulations. The models are much quicker to run, so changing parameters and re-running the analysis will only add seconds to your day. Also, you can combine the two methods, so that you lump whatever you can, and couple that model to the more complex models, getting the best of both worlds, and saving any time possible. If you are starting out with this, it may be a good idea to start with COMSOL's Narrow Region Acoustics approach, and take it from there.

If you are dealing with these issues in general regarding when to use which model for your physics/products, I am availble as a consultant, to just reach out.

(23/9 2022)