Coupled problems are notoriously more difficult to handle than single-physics problem. Sometimes the software can do the coupling automatically, but other times you have to do it manually, which of course is an error-prone process. The meshing procedure may differ for the two or more physics, the desired element order may be different as seen from the different physics, computational times are typically higher, and so on. In certain situations, however, you can handle one or more physics in a simpler fashion. We will have a look at such a situation for the usual Acculution application; the loudspeaker driver.

A loudspeaker driver is an electromagnetic-mechanic-acoustic sequentially coupled device. The electromagnetic part is complex, in the sense of non-linear effects, but for a harmonic analysis the EM part is sufficiently included as so-called lumped parameters, with the physics being reduced to a few discrete component. The acoustics and the structural mechanics are sometimes also modelled as lumped parameters, where the total moving mass of all parts is lumped into one component, and ditto with the elastic and lossy parts. If the driver is mounted in a cabinet the internal volume can also make up a component. The lumped approach is outlined below.

There are different approaches to take when it comes to lumped parameter modelling. Here, I have mapped all components into the electrical domain. The input voltage source connects to a resistor and an inductor related to the electromagnetics. From here, five components are parallel, since they share a velocity; the only velocity in the system, since this is a single DOF system (if you add a port, this is no longer true, just as break-up modes are not 1DOFs). Three components are related to the moving mass of the driver (in vacuo), the inherent stiffness, and the mechanical resistance, respectively. The final two components are related to the acoustic forces from having to compress the air in the cabinet in the rear, and the radiation problem in front of the driver.

If you work with loudspeaker design, you should be able to construct the above circuit with realistic values, if you have the datasheet for the driver. But if the volume is intricate, and/or if you have multiple drivers that couple to each other, the lumped strategy may not be accurate enough. For the external volume similar arguments can be made; the outer surfaces of the cabinet will affect (as I will investigate in an upcoming post, I just want to make a full tweeter CAD, which will be another post) the output pressure in a given point, and multiple drivers will also couple to each other externally.

Now, in these more complicated cases, simulations engineers usually step in, and (try to) set up a fully-coupled model, that takes all effects into account in a continuous domain approach, though discretized via the finite element method (FEM) typically. This is not trivial, and it can be overkill when it comes to the insight you were after in the first place. With a standard FEM approach, you would mesh the air in the cabinet and outside, mesh the structural parts, and try to fit the many material parameters to get a behavior of your model, which is similar to what you measure. This can take days to get right, and once you have it, it may not be very suited for e.g. parameter studies, simply because of computation time being large, especially if you are working with a 3D problem. One way to simplify things is to go with a pure acoustics approach, where the air is still meshed, but the structural mechanics is only included via an analytical approach. This is done by setting up the proper coupling between the air and the moving surface(s) on the driver. As usual, I turn to COMSOL Multiphysics, where coupling of different physics is probably as easy as it is possible, although certainly not trivial, especially if you have not done it before. Knowing the analytical equations related to the circuit above, you can two-way couple these equations and the acoustic interior/exterior. Below, I show the sound pressure level and the impedance, for a fully-coupled vibroacoustics finite element analysis and the FEM-analytical-FEM. Again, for the former, the structural mechanics domains are meshed, and a certain density and stiffness is input for each domain, whereas for the FEM-Ana-FEM analysis, the structural mechanical parts are included as equivalent lumped components via equations.

The calculation time for this 2Daxi problem goes down one order of magnitude; very advantageous if you want to do parameter studies, if you have limited computer power in the first place, if you want to do topology optimization in the air domain, and so on. Internal pressures and the coupling between multiple drivers could be estimated very accurately, and with much less effort than a fully-coupled approach.

Since the driver is my own creation with no physical counterpart, I had to estimate the mechanical parameters. This also illustrates nicely another advantage of this method: You can try out designs that have not yet been made (cough cough or competitor designs cough cough...), as long as you know the Thiele-Small parameters. Btw., I have had several companies reach out to me regarding a simple model for the possible disadvantages of letting multiple woofers share a volume, and the approach I describe should be suited for this.

If you have a 3D problem, the same technique can be applied. Here, I have changed things up a bit by having the outer air region be modelled via the new Boundary Element Method (BEM) functionality in COMSOL Multiphysics, whereas both the structural mechanics parts AND the rear air are modelled analytically. I also tried a BEM-Ana-FEM coupling, where the rear air is modelled with FEM. This worked fine also, and this is the approach you should take if the internal setup of your speaker is not simple.

This is the beauty of starting out simple; once you have got the fundamentals in place, you can build more elaborate models that give you even more insight. More on that below... You cannot yet do 2Daxi BEM, that is why I chose to go with a 3D model. If you are fortunate enough to have a geometry as the one above, you can of course take advantage of the symmetry. Just be careful when setting up the coupling in that case.

Rant warning!

As always, Acculution.com is about finding the smartest compromise between complex methods and good engineering practice to get the optimal physics insight in the least amount of time. I have seen plenty an engineer struggle with complex simulations, because they didn’t know how to do simpler, but much more telling and efficient, analytical work, and I have seen engineers struggling with complex software without having the necessary mathematical knowledge. When this happens, things that could have been done in hours (as the above simulations), takes days or weeks, and things that could have taken days, take weeks or months. This of course costs a company a ton money, and it slows down innovation, which is so sought after these days. It is okay when starting out to struggle; I certainly did. Just make sure you don't constantly go towards the same solution approach all the time, since it will not always be optimal. Sometimes it even results in a sort of “negative innovation”; I have often heard the sentence “Oh yeah, we tried to do simulations in our company, but they did not work, so we abandoned it”. Really? But others can do it. And that is one thing I know helps and motivates; to see others having succeeded with a certain tasks* that you yourself are trying to do. So in the Acculution blog posts, I try to show *what* can be done (but not necessarily *how*) to motivate both students and engineers to both get more proficient in mathematics and physics, but also to realize, when the simpler method is actually the better method. There are certainly more complicated applications than loudspeakers, and we should all strive towards getting a little better each day, to be able to take on these problems.

**Try and look up “4 minute mile” and see how feats that were once thought impossible, can in a short time be much more common, once it has been proven that they can be done after all.*