#042: Microacoustics in Any FEM Software Package

(20 August, 2022)

Microacoustics, or thermoviscous acoustics, is a topic that I have engaged in for many years now. At milimeter/submilimeter scale, thermal and viscous effects lead to losses that are not included in standard acoustics. These days, many finite element packages have implemented one or more variations of microacoustics, but some still have no such implementation. So, you either need to simply ignore the losses in the simulations, and accept that measurement and simulation results will not quite match, or find a way to implement the losses yourself. Below is a strategy for implementing microacoustics in basically any FEM software package that lets you modify two material variables, such as the typical sound speed and density.

When doing acoustic analyses it is sometimes necessary to include viscothermal losses, especially for small and/or narrow geometries. The viscothermal losses occur in a boundary layer extending into the bulk from the relevant boundaries. The depth of this boundary layer is frequency dependent, so that for low frequencies the depth is large (characterized as a “narrow tube”), and for high frequencies the boundary layer depth is small (“wide tube”). In general, the viscothermal losses have to be included by solving equations involving both the acoustic pressure, the temperature variation and the velocity components. One of the only (if not the only) commercial software available which has the full implementation valid for basically any geometry is COMSOL Multiphysics. However, for certain distinct geometries, the losses can be included via a complex and frequency dependent density and ditto bulk modulus. All cases that fall under the so-called Low Reduced Frequency Model (LRFM) can be tackled in this manner, which includes tube with circular, rectangular and triangular cross-sections as well as certain layer geometries. In COMSOL Multiphysics this implementation is called Narrow Region Acoustics, whereas in other software packages such as ANSYS it will likely have another name.

Some of the tube cross-sections in question are shown below.

For these tubes and slits, analytical expressions can be found for the inertia/viscous and the compression/thermal effects. These four effects are contained within two complex variables, such as a complex density and a complex bulk modulus, or any other pair out of many. In general, these two variables are written as

for the complex density, and

with some details left to the reader. For a circular tube with radius a, the function F can be found as

and so for a particular tube, one can calculate the complex variables needed for an accurate result with losses included.

When working in the hearing aid industry I sometimes worked with the software package ABAQUS, which barely has standard acoustics, let alone microacoustics. And yet, I was including microacoustics whenever it was needed for tubes and slits. I used the strategy oulined here: The tabular values of the complex denisty and the complex bulk modulus were calculated in MATLAB or similar. The part or cell to be designated a complex valued material was assigned a material with no properties and a name, which could be easily recognized in the file. The tabular values were entered via Model->Edit Keywords->(Model-1 or similar)

Under the line above the complex variables were entered, and I ended up with a file like this.

With this strategy, I would be doing calculations on triangular tubes, slits, and all kinds of combinations of tubes and slits.

For your particular simulation software, the microacoustics may already be implemented, but if it is not, it is highly likely that you can implement at least some aspects of thermoviscous losses, and probably in a simpler way than for ABAQUS. Try it out, and let me know how it went.