# Simulation Overview - Loudspeakers Design via Shape and Topology Optimization

#### René - Acculution.com

##### Active Member
Technical Expert
-Introduction-

May 20 I will give a presentation on the COMSOL Acoustics Day on state-of-the-art loudspeaker simulation techniques, and I will give ASR readers a little sneak peek here. As always, feel free to ask away in the thread, and I will try and answer as best I can.

-Loudspeaker Simulations-
There are many different simulations that are relevant for exploring loudspeaker behavior. Below I have made a rough overview of what typically is being requested.

So we can deal with single-physics (acoustics, structural mechanics, electromagnetics) or multi-physics problems (any combination of the beforementioned). And the study types can vary from static, to steady-state where you basically run a frequency sweep, to a modal analysis where you find eigenfunctions and eigenvalues of your system (see e.g. my post on Room Gain), to transient analyses where you apply a time-varying input and look at the transient and the steady-state response together.

A static anaylysis could be relevant in conjunction with a structural mechanics to investigate the stiffness symmetry of spiders and surrounds as a function of displacement as shown below. The analysis takes into account the non-linear geometry, so that for small displacements the rolls 'unfold' with little strain, but at larger displacements you get more relative strain.

A typical steady-state problem would be a vibroacoustics (multi-physics) problem, where you model a complete driver and apply a harmonic voltage signal over a frequency range:

Assuming that all materical properties are correct, and the simulation in general is done correctly, you will get all information equivalent to Spinorama, plus much more. In the acoustics domain the only degree of freedom solved for is (complex valued) pressure, and that is all you need to calculate velocity, intensity, power, directivity index, and whatever else your heart pleases.

A modal analysis is relevant in several cases, for example to investigate the structural modes of a driver. Below a spider resonance is observed at a certain frequency, and so you can find all modes in the relevant frequency range and compare your findings to a steady-state frequency sweep to see if some modes lead to resonances.

Remember, modes exist independently of excitation, and not all modes are necessarily excited. So if you for example see a rocking mode in your modal analysis, you should remember how the excitation will work against such a mode.

Finally, transient responses can be relevant both to see the initial wave propagation, but also for investigating non-linear distortion coming primarily from the structural mechanics and electromagnetics. For the acoustics it can be relevant if you have high particle velocity/pressure, if you have small enclosures and drivers with large excursions, if you are looking at Doppler effects, and so on.

One technique that I have build into the software package COMSOL Multiphysics is Phase Decomposition, which allows me to dissect the cone and surround vibration to see which parts of the displacements add to, subtract from, or neither, the sound pressure in any observation point. A very underutilized technique in the loudspeaker industry.

Now, while the above analyses can be challenging enough in themselves, you can take things even further if start combining these analyses with formal, mathematical optimization. There are generally three groups of optimization, Parameter Optimization, Shape Optimization, and Topology Optimization (generally this is the ascending order of complexity).

With Parameter optimization you can control the geometry via parameters such as lengths and heights, but basic shapes are retained. With Shape optimization boundaries are described in a way that allows the basic outline/shape to be controlled, but topology remains. Finally, with Topology optimization the topology of the geometry is allowed to change so that domains and holes are controlled completely by the algorithm. The underlying mathematics of multi-physics optimization is quite involved, so I will instead show the potential via some examples.

First is an acoustics shape optimization case where a compression driver geometry is optimized. This particular combination of physics and optimization can also be used for waveguide design, but the compression driver design has some more challenging aspects to it. The blue lines on the initial geometry are allowed to change shape, and the resulting geometry has the curved phase plug channels seen on the right, with a nice resulting pressure response.

Next, a generic setup with vibroacoustics and shape optimization. This was to explore how to do this particular multiphysics problem with shape optimization and it is rare to see any work with full vibroacoustics modelling with optimization included. I ended up using the Rayleigh integral for the acoustics, which made life a lot easier. The cone was allowed to change its shape, and the result was less relevant, as this was more of a test case.

I have also done shape optimization combined with the magnetics system, where a boundary was allowed to change its shape. Again, the results are less relevant for test cases than for actual client cases that I would probably not be allowed to show anyway.

Moving into Topology optimization, here is a structural mechanics case, where I optimize the stiffness of a basket (for some given constraints). The holes that appear compared to the initial geometry completely grow out of the mathematics, which I find highly fascinating.

Next, an acoustic topology optimization where the complete multiphysics for the driver is included, but the optimization only takes place in the acoustics domain. With an objective to flatten the frequency response, a phase plug (in grey) has appeared in a domain in from of the driver assigned to be topology optimized. Again, this is a case I have never seen done before; including optimization in a full model of a realistic driver. It should of course be investigated how the off-axis response is affected, but imagine the time savings that are possible, compared to the traditional methods with clay modelling, and general trial and error with no guidance.

A final example is actual a heat conduction topology optimization case. As heat affects the material parameters of the structural mechanics and magnetics domains (and also the acoustics to some degree), it is desireable to lead heat away from the driver. So I thought this could be an interesting challenge. While I had never done any heat conduction cases, I spend a day reading relevant papers and setting up the simulation, and set the computer to work over the night. And I got this pretty heat sink.

-Closing remarks-
The above techniques are not all being utilized in the loudspeaker industry yet, but with the interest I am experiencing from several of them it will just be a matter of time, before we see more designs that are aided by formal optimization. Also, there are more simulations that I have not touched upon in the above, but I am working on composites with anistropic layer, metamaterials, and additional optimization cases, that will benefit the loudspeaker industry, so stay tuned.

- About me -
René Christensen, Denmark, BSEE, MSc (Physics), PhD (Microacoustics), FEM and BEM simulations specialist in/for loudspeaker, hearing aid, and consultancy companies. Own company Acculution, blog at acculution.com/blog

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#### Jim Matthews

##### Major Contributor
Forum Donor
Thank you for posting this. The graphics help explain the concepts to enthusiasts like myself who can barely remember where I left the car keys.

#### Grotti

##### Active Member
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Had to read it twice to get an idea of the whole picture. Very interesting since the approach starts at the source of it all: the misbehavior of drivers in almost all regards.

But is some of that analysis not done for years now? As far as I know, resonances of the drivers surface and surrounding as well as of enclosures can be made by laser inferiometric (hope this is the right translation) for instance.

But of course having all the possibilities in a simulation before building a certain driver seems to be desirable and promising.

I certainly will watch this thread

#### MrPeabody

##### Addicted to Fun and Learning
Wow, René, there is an awful lot there. I like the driver basket partly because it is so different from conventional approaches. With some of the large woofers we see occasionally in subwoofers, the basket design appears to be stamped from sheet steel, formed into shape and a number of coin-sized holes punched out all over. There is no way that anyone can tell just from looking what advantages or disadvantages this kind of basket would have, compared to baskets made by casting and machining. Nevertheless, when someone starts a new thread presenting a new subwoofer using a basket made by stamping steel rather than by casting aluminum, there will inevitably be someone quick to assert that the driver is cheaply made because it is made by stamping steel rather than by casting aluminum. Not sure why I'm mentioning this, you just reminded me of this debate that arises from time to time. There are so many variables with this kind of question that it seems folly to expect anything like a general answer; only answers that are based on a number of assumptions are possible. Nevertheless, one cannot help but wonder whether a basket made of stamped steel will be comparable in terms of stiffness and avoidance of standing waves within the structure, and if so, how much more massive the stamped steel basket would need to be. I'm just wondering out loud. I suspect that the number of variables is such that it doesn't really make a lot of sense to even pose the question.

#### pierre

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But is some of that analysis not done for years now? As far as I know, resonances of the drivers surface and surrounding as well as of enclosures can be made by laser inferiometric (hope this is the right translation) for instance.

This is different. Usually for a given driver or speaker, the software or the experiment will give you a result.
here the question is more: what would be the optimal shape such that this properties are optimized for?
Usually you have an idea (my horn will be like this) and compute a freq response. Then you iterate manually. In the above, the software give you the best possible shape in 1 run. This is of course more cpu intensive and you need to compute more gradients etc

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#### René - Acculution.com

##### Active Member
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Had to read it twice to get an idea of the whole picture. Very interesting since the approach starts at the source of it all: the misbehavior of drivers in almost all regards.

But is some of that analysis not done for years now? As far as I know, resonances of the drivers surface and surrounding as well as of enclosures can be made by laser inferiometric (hope this is the right translation) for instance.

But of course having all the possibilities in a simulation before building a certain driver seems to be desirable and promising.

I certainly will watch this thread
Some of the simulations have been done for years by some companies, while others are still not doing them. Those who are doing them sometimes have issues with doing it correctly though. Shape optimization is slowly creeping into the top companies, but topology optimization is not.

Laser vibrometry is a measurement technique which can give a lot of insight into the driver behavior, but simulations let you see much more. On the other hand, simulations depend on measurements for some of the material input and are more complex to set up. Measurements and simulations should go hand in hand. With a simulation you trust, you can do virtual prototyping without having to wait for new builds, expensive tooling, and so on.

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#### René - Acculution.com

##### Active Member
Technical Expert
Wow, René, there is an awful lot there. I like the driver basket partly because it is so different from conventional approaches. With some of the large woofers we see occasionally in subwoofers, the basket design appears to be stamped from sheet steel, formed into shape and a number of coin-sized holes punched out all over. There is no way that anyone can tell just from looking what advantages or disadvantages this kind of basket would have, compared to baskets made by casting and machining. Nevertheless, when someone starts a new thread presenting a new subwoofer using a basket made by stamping steel rather than by casting aluminum, there will inevitably be someone quick to assert that the driver is cheaply made because it is made by stamping steel rather than by casting aluminum. Not sure why I'm mentioning this, you just reminded me of this debate that arises from time to time. There are so many variables with this kind of question that it seems folly to expect anything like a general answer; only answers that are based on a number of assumptions are possible. Nevertheless, one cannot help but wonder whether a basket made of stamped steel will be comparable in terms of stiffness and avoidance of standing waves within the structure, and if so, how much more massive the stamped steel basket would need to be. I'm just wondering out loud. I suspect that the number of variables is such that it doesn't really make a lot of sense to even pose the question.
Manufacturability is very important, and of course so is cost. If current methods are good enough, then that is surely fine. But with optimization you can at least explore the solution space more freely, and adapt the findings to current manufacturing methods. I thought it was a good example to merge academia with industry, which is what I am all about.

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#### René - Acculution.com

##### Active Member
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I have a presentation on shape and topology optimization on my channel that goes into more detail:

#### NTK

##### Major Contributor
Forum Donor
Wow, René, there is an awful lot there. I like the driver basket partly because it is so different from conventional approaches. With some of the large woofers we see occasionally in subwoofers, the basket design appears to be stamped from sheet steel, formed into shape and a number of coin-sized holes punched out all over. There is no way that anyone can tell just from looking what advantages or disadvantages this kind of basket would have, compared to baskets made by casting and machining. Nevertheless, when someone starts a new thread presenting a new subwoofer using a basket made by stamping steel rather than by casting aluminum, there will inevitably be someone quick to assert that the driver is cheaply made because it is made by stamping steel rather than by casting aluminum. Not sure why I'm mentioning this, you just reminded me of this debate that arises from time to time. There are so many variables with this kind of question that it seems folly to expect anything like a general answer; only answers that are based on a number of assumptions are possible. Nevertheless, one cannot help but wonder whether a basket made of stamped steel will be comparable in terms of stiffness and avoidance of standing waves within the structure, and if so, how much more massive the stamped steel basket would need to be. I'm just wondering out loud. I suspect that the number of variables is such that it doesn't really make a lot of sense to even pose the question.
Manufacturability is very important, and of course so is cost. If current methods are good enough, then that is surely fine. But with optimization you can least explore the solution space more freely, and adapt the findings to current manufacturing methods. I thought it was a good example to merge academia with industry, which is what I am all about.
The structural design of the basket is, interestingly, not only for acoustic performance. There is a COSMOL Conference (2017) presentation by Sonos that discussed improving the strength of the basket to minimize structural failures during handling and transportation. Imagine having a 20 lb magnet hanging off the basket, and the speaker is bouncing around in a truck running over potholes, or getting dropped onto the pavement at the loading dock.
https://www.comsol.com/blogs/keynote-video-using-simulation-to-develop-reliable-audio-transducers/

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#### René - Acculution.com

##### Active Member
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The structural design of the basket is, interestingly, not only for acoustic performance. There is a COSMOL Conference (2017) presentation by Sonos that discussed improving the strength of the basket to minimize structural failures during handling and transportation. Imagine having a 20 lb magnet hanging off the basket, and the speaker is bouncing around in a truck running over potholes, or getting dropped onto the pavement at the loading dock.
https://www.comsol.com/blogs/keynote-video-using-simulation-to-develop-reliable-audio-transducers/
Very good point. A lot of these practical points are just as, if not more, important than the aspects discussed here. In the hearing aid industry there are tons of issues with leakages, nanocoating getting into the microphones, and other problems that delay productions, and the impact is much bigger than squeezing 0.1 dB performance in the acoustics. And microphones can be great on paper and even after production, and then you fly them somewhere, and the all break because there is no pressure equalization hole in them, so again, you need to look broad within these aspects.

#### alanca3

##### Member
This is so fascinating, I look forward to your future posts.

I've been searching the web for COMSOL tutorials regarding loudspeaker design. So far there is not much out there (aside from a few research papers).

#### alex-z

##### Addicted to Fun and Learning
This is excellent work, I hope driver OEM's all adopt driver CAD so that the quality goes up across the board. Highly optimized woofers like the Purifi models exist but the price leaves them less accessible.

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#### René - Acculution.com

##### Active Member
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This is so fascinating, I look forward to your future posts.

I've been searching the web for COMSOL tutorials regarding loudspeaker design. So far there is not much out there (aside from a few research papers).
Thanks a lot. COMSOL has some good application cases on how to model loudspeakers, but the design side of things is up to the manufactures.

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#### René - Acculution.com

##### Active Member
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This is excellent work, I hope driver OEM's all adopt driver CAD so that the quality goes up across the board. Highly optimized woofers like the Purifi models exist but the price leaves them less accessible.
Thank you. The CAD is just the first step, they need to have the simulation software and proper skillsets to set up the model. If they were to spend the amount of time that Purifi puts into their drivers, prices would go up accordingly. Going with a consultant could be a good approach for OEM companies.

#### KSTR

##### Major Contributor
@René - Acculution.com, I just went to all the avaliable material on your blog/website and I'm deeply impressed. Acoustic demultiplexer, cloaking, mode conversion, ..., wow! One can only imagine the implications of this.

And a hidden gem is the pole/zero<--->transfer function post. While every engineer knows that a mag/phase Bode plot, a Nyquist plot and a real/imag plot are all the same thing, a transfer function, in different projections, the identity of pole/zero and transfer function is much less clear.
To quote you
"Now what I really want to convey with this blog post, is that the plots above are actually one and the same, just seen from different perspectives! The first time I saw this visualized, was in an excellent DSP book, and it blew my mind. I couldn't believe that this had never dawned on me, as I had actually finished my MSc at the time".
Bingo. I've been dealing a lot with poles/zeros and TF's for 30+ years now... and today, the lightbulb moment, finally.

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#### René - Acculution.com

##### Active Member
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@René - Acculution.com, I just went to all the avaliable material on your blog/website and I'm deeply impressed. Acoustic demultiplexer, cloaking, mode conversion, ..., wow! One can only imagine the implications of this.

And a hidden gem is the pole/zero<--->transfer function post. While every engineer knows that a mag/phase Bode plot, a Nyquist plot and a real/imag plot are all the same thing, a transfer function, in different projections, the identity of pole/zero and transfer function is much less clear.
To quote you
"Now what I really want to convey with this blog post, is that the plots above are actually one and the same, just seen from different perspectives! The first time I saw this visualized, was in an excellent DSP book, and it blew my mind. I couldn't believe that this had never dawned on me, as I had actually finished my MSc at the time".
Bingo. I've been dealing a lot with poles/zeros and TF's for 30+ years now... and today, the lightbulb moment, finally.
Thanks a lot, I have been working on that blog for several years now, and it was meant to be a first a step towards being more known, and down the line going self-employed based on the shown fields of interest. And last month it came to fruition

I am happy to hear that, that is the experience that makes it worth writing. I have later seen it done in other books, and perhaps it is well-known to some (in my experience engineers are often not even up to speed with complex numbers, so maybe not...), but it makes a lot of sense once you see it.

I am contemplating making a post that takes into consideration poles/zero placement, LTI systems, causality, stability, region of convergence, modes, transient/steady-state reponse,..., and shows how it all fits together. If only engineers were more inclined to really embrace these learnings, companies could save soooo much time, but if I can inspire just a few people then that is also something.

#### Joseph Crowe

##### Member
Does the motor simulation predict flux modulation in the voice coil gap?

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#### René - Acculution.com

##### Active Member
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Does the motor simulation predict flux modulation in the voice coil gap?
Yes if you run a transient study taking the time dependency into account.

#### NTK

##### Major Contributor
Forum Donor
Does the motor simulation predict flux modulation in the voice coil gap?
COMSOL is very flexible in this regard. You can add whatever level of complexity to the model you wish. You will need have license(s) for the analysis option(s), e.g. acoustics, AC/DC magnetics, structural mechanics, heat transfer, etc. The complexity of the model is generally proportional to the power of the number physical quantities/equations you want to solve. Very often highly coupled non-linear problems are numerically intractable and COMSOL won't be able to give you a solution (or worse, incorrect solutions).

Here is a speaker driver example problem from COMSOL. This example requires the optional acoustics and AC/DC modules (on top of the base COMSOL program).
https://www.comsol.com/model/download/791771/models.aco.loudspeaker_driver.pdf

Here are a couple examples on the types of analyses Samsung (parent company of Harman) did with COMSOL.
https://www.researchgate.net/public..._Simulations_of_Loudspeaker_Transducer_Motors
https://www.researchgate.net/public...s_for_Acoustic_Elements_in_Loudspeaker_Design

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