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In Reply to: OK so what was posted by EGeddes on January 27, 2006 at 18:53:41:
The description was the file name - a 1kHz tractrix horn with a 1" entrance. If you browsed that directory, there was a shot of the model with a similar file name. The input wave was flat with equal amplitude across it, and yes, I realize this is not likely to be the case in real life with a real compression driver providing the input to the horn. I've also played with curved wavefronts (normal to the horn wall at the entrance) still with equal amplitude across the wavefront and not seen much difference. From what you mentioned the other day, the real problem might be that the magnitude varies across the wavefront...? Any thoughts on getting input data for this, a good approximation to use, or is it really just a case of having to simulate the entire system?Btw, I'm a little fuzzy on the difference between FEA and BEM (I do know what the acronyms stand for). In BEM, are you only specifying conditions at boundaries, or do you still have a mesh with nodes, etc.?
John
Follow Ups:
With waveguide theory the wave equations in waveguides and the sound radiation can all be done analytically, which has a major advantage over FEA or BEM. With both methode to go to very high wavenumbers requires a very large number of elements and matrix sizes. Even if this is doable, the results get unstable due to round-off errors in the matrix crunching.With the analytic approach you can go much higher in wavenumber than with any numerical approach.
So the tradeoff is an accurate solution of an approximation geometry or an inaccurate solution of an exact geometry. I prefer the former. And the analytic approach is reversable, and global (in the sense that once solved all frequencies are known) and the numerical approaches are not.
These have all led me over the last couple of decades away from numerical methods to the analytic approach. It was not until I found an analytic approach that I really came to understand waves in conduits (horns, waveguides). You don't get that insight from numerical approachs.
I would bet that one could never understand horns and waveguides until they understood the analytical solutions.
Earl Geddes
First let me ask you a question which will lead into the BEM description.How do you terminate the FEA model? Obviously you don't take the mesh to infinity. So where does it end and what are the boundary conditions there?
Depending on how you terminated the FEA model there can be a lot of errors. You can look up a complete description of this problem, and its solution in a paper I did in the AES in about 1985.
BEM meshs only the exterior boundary of the problem. Using Green's theorem one can transcribe the wave equation in three dimensions into an integral at the bounding surface of that space, i.e. two dimensions. Now for radiation problems this is ideal since the integral of the boundary at infinity is zero - has to be. So this leaves an integral only over the actual radiating surfaces. Rayleighs equation is a special case of BEM where the radiating surface is flat and finite.
So for an axi-symmetric problem, like yours, one would only have to mesh a line representing the horn contour, but then some form of enclosure would have to be assumed and meshed. The order of the BEM is always one dimension lower than that of the same FEA problem so it should be faster, but, unlike FEA which has a banded matrix structure, BEM is full. So it has a smaller but full matrix to FEA's larger but banded matrix. In essence the run times are about the same.
There is one catch. The BEM equations go singular at any internal resonance and so the equations must be suplimented by an additional equation for each resonance. There are some very nice BEM programs available (let me know if you are interested) and for a time the Navy gave away their BEM source code - in FORTRAN. I still have a copy, just as I have a source code copy of an old FEA program.
Earl Geddes
Thanks for the explanations.My termination in that model is an arc at some radius from the origin (a half circle for an axisymmetric problem for example) with the Sommerfeld boundary condition applied. So that's an approximation of infinity that becomes better as the radius increases, iirc.
I'll take a look at your article when I get more time. If it's from 1985, I do have it.
I guess where as you say you prefer an accurate solution with approximate geometry, I gravitate towards the exact geometry because I like trying a lot of different geometries. While I'm sure you could come up with analytic solutions for all of them, I'm much faster at drawing things in CAD and then working with them in an FEA-type program. Either way you eventually still have to build it, measure it, and listen to it to make sure it does what you simulate (unless you've spent years validating all your models, which I suppose you probably have).
I'm always interested in some new software to play with. Email me through the link here if you want.
John
"So that's an approximation of infinity that becomes better as the radius increases, iirc."This is correct, but as you go further and further out the matrices get bigger and bigger and less and less accurate so your not really getting anywhere. In our paper we derive a true infinite element which is accurate at even the smallest possible radius's that encompass the geometry, thus allowing small matrices for higher wavenumber solutions.
"I gravitate towards the exact geometry because I like trying a lot of different geometries."
To me this is eactly the problem, you keep trying things hoping to find something good. By doing the problem analytically I was able to go directly to the best geometry, which, as you point out, was built, tested, listened to, and low and behold, wonder of wonders, it works! And I can now prove how and why nothing else can work better. You can never do that with "cut and try".
I have always found that people in audio ask the wrong questions. They are always asking "What will happen if I do X?" A better, but more difficult approach is to ask the question "What do I want to do and how do I do it?" The second question will lead to real results far more often and far faster than the first will.
FEA does the first question well but analytical approachs are usually required to answer the second one. Thats why I moved away from FEA and BEM. I did my PhD in FEA problems in Acoustics way back in 1979. I used it for about 5 years after that, always asking the "what if" questions, but I never seemed to find anything important. Then I started asking the other question and I found that FEA was not well suited to that one.
So if your having fun, then great, but if you want to move the needle forward, I'd look to the analytical methods. They are far more difficult, but far more effective.
Earl Geddes
Hi Earl and Johnsome may not know what they want other than something new to try (which might be good overall for audio business?) rather than set about an approach to system goals and results obtainable for X smount of money, type of amplifier and bulk-constaints.
keep trying to push needle forwards (and remember some old-stuff which was deemed by some to have problems many years back may stick around for awhile longer - not sure now much = nostalgia and how much = actual preferences-?)
now much improvement might be possible 5-10 years from now? - what more can be done on the loudspeaker's end?
Freddy
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