# Boundary Element Method Simulations

The Boundary Element Method (BEM) is a general method for solving problems that can be expressed as partial differential equations. (A good introduction can be found here.) The wave equation for sound is such a problem, and this means that we can for example find the sound field radiated by a structure, or the resulting sound field when sound from a source is scattered by an object. I have used this method a lot to simulate the sound field radiated by horns.

The basic principle used in BEM is that if you know the velocity potential (from which pressure and velocity can be derived) and normal velocity on the surface of a structure, you can calculate the field radiated by the structure anywhere. This has the advantage that only the structure surface has to be discretized (meshed), not the entire domain, which is required in other methods like FEM. This is an advantage when simulating sound that is radiated into an unbounded domain. The trick is, however, to find the velocity and velocity potential on the surface.

I will not go into details on how to do it here. For my own simulations, I have used BEM code that is available here. The code is in Fortran, which I have translated to Delphi to make a more userfriendly Windows program.  While Fortran is a little bit faster, this is to a large degree offset by the increased userfriendliness. BTW, this program will not be released to the public (yet, anyway), it is in a too unfinished state for most users, and there is no manual. Sorry.

While I started experimenting with BEM back in 2008, I have not published much of what I have done. Some simulation results are floating around on various forums, but that is about all. So I decided to put up a gallery with old and new BEM simulations, for the interest, and hopefully education, of my readers.

## General information

The gallery contains lots of plots and figures, starting with a three digit number (this is to make them appear in the order I want), the number is referenced to the descriptions below.

All simulations of horns have been done with a constant 1m/s throat velocity. Than means that no driver is involved, and that the responses are the responses of the horns alone. Most simulations are 150-8000Hz, with 100 log-spaced frequency points. Mesh bandwidth is usually the same as the upper frequency, and this means that there are six elements per wavelength (or a little bit more some places depending on how the discretization works out). Response curves are usually at a 3m distance, to be well into the far field of the horn.

Frequency response plots are given for the case of constant throat velocity. This causes ripples in the throat impedance to show up quite visibly as peaks in the response. These peaks do not change with angle. Response plots are shown for selected angles. Not all simulations have the same number of field points (points where the exterior pressure is calculated), so the resolution and angles are not always the same.

Traditional polar plots have only been given in a few cases. In these cases, the responses are normalized to the direction of max output. Traditional polars tend to get very messy unless you have just a few curves, and since you then have to select representative curves, it is easy to cheat and not display curves you are not satisfied with. Instead I give the polar map, which shows everything in one plot. This type of plot has been popularized by Earl Geddes, and has become fairly common. Three versions are shown: the response with constant velocity, with a source of constant resistive impedance of rho*c/St, called a matched source, and a plot normalized to the on-axis response. The matched source response is similar to driving the horn with a driver that is perfectly matched to the horn, and that does not have any upper or lower rolloff.

Throat impedance is also shown for all horns. I see that not all the plots go towards one, as they should, some end up a little lower, and some a little higher. I have not investigated this in detail, but it should not matter in judging the performance of the horns. It is, as far as I can tell, a numerical artefact, perhaps linked to the discretization of the boundary, which is done a little differently in the various cases.

Simulation methods are either AEBEM(A), ASEM(A) or BERIM(A). AEBEM is the standard exterior BEM method, and needs a closed boundary. ASEM is a BEM method that can simulate infinitely thin structures, like horn walls. BERIM is a BEM method that assumes the horn mouth to exit into an infinite baffle. Both were developed by Stephen Kirkup, and I have tested them out, and a paper on the BERIM method has been published in the Journal of Computational Acoustics. (A) stands for axisymmetric. All simulations are axisymmetric, unless otherwise noted.

The boundary and the field point distribution are given for most simulations. For BERIMA simulations, the mouth is "blocked", this is the surface that separates the interior of the horn from the exterior in front of the baffle. It does not mean that the mouth is closed.

## 001: The AH425 project

During the fall of 2008, I collaborated with Lynn Olson and Martin Seddon (of Azurahorn) in the design of the AH425 LeCléac'h horns. My part in that project was mainly simulating the horn using Boundary Elements Method (BEM) SW, and to be a discussion partner. This horn was specifically designed to match the Altec 288 series of compression drivers. I did many simulations for this project, but I can't find all of the files. Here is anyway the simulation of the AH425, as it ended up. Cutoff is 425Hz, T is 0.707, and throat diameter is 35.5mm. ASEMA simulation.

## 002: AH425 in a baffle

Most horns get a smoother throat impedance when mounted in a baffle. Not so with LeCléac'h horns. At least not near cutoff. Higher up in frequency, the impedance is smoother than the plain AH425, though. BERIMA simulation.

## 003: 425Hz Tractrix

The tractrix horn is quite similar to the LeCléac'h horn. The mouth of a tractrix horn ends at 90 degrees, giving more reflections and diffraction than a LeCléac'h horn (that usually curves back a little), with the result that both the throat impedance and frequency response is less smooth. This is a BERIMA simulation, meaning that the horn is actually mounted in an infinite baffle, still it does not perform as well as one could hope for.

This horn was selected to have the same cutoff as AH425, 425Hz.

## 004: Conical horn

This is a plain vanilla conical horn. No baffle, no mouth radius. The mouth diameter is about 26cm, the wall angle +/-27 degrees. Length is 21.6cm. The response is ragged, the loading is poor below 1kHz, and directivity is not very good either, because of lots of mouth diffraction. ASEMA simulation.

## 005: Exponential horn

This is a 425Hz plain exponential horn of the same length as the conical horn, without baffle. The mouth is only  0.74 cutoff wavelengths (CIR = 0.74 in Hornresp), so the mouth termination is not very good. The results show it, this is a resonant horn with ragged response and directivity characteristics. ASEMA simulation.

## 006: OS Waveguide

This was an attempt to simulate an OS waveguide of approximately the same size as the AH425. It has no rollback or baffle, though it has a radiused mouth (10cm radius). Throat diameter is 35.6mm, mouth diameter of the OS part is 257.8mm, and at the edge, the diameter is 430mm. Despite having no baffle, it performs well in many respects. The directivity is fairly good, and far more constant than the horns above. ASEMA simulation.

## 007: OS Waveguide

An attempt to simulate the Summa 15in OS waveguide, including the internal flare of DE250. I had some communication with Earl Geddes about this, and due to some misunderstandings of where the 15" dimension was measured, etc, I don't think this is a complete replica of the Summa OSWG. The internal DE250 flare is probably too long too, giving rise to more resonances than what may actually be there. Here are the exact dimensions: the DE250 flare starts at a 15.1mm diameter, and flares conically out to 25.4mm/1in over 48.8mm, where it meets the OSWG. The OS is 142.2mm long, ending at 288.2mm (11.34in), where it meets a radius that flares out to 429.6mm (16.9 in). The radius is 70.7mm.

Still, the directivity performance is perhaps quite representative of this exact case, but since the box doesn't have any rounded corners, it is not representative of the Summa. AEBEMA simulation (exterior BEM, i.e. in a "box" as can be seen from the boundary). 300Hz-10kHz, 10kHz mesh bandwidth.

## 008: OS Waveguide sound field

Here are some plots of the nearfield of an OSWG in a baffle. They show the pressure amplitude contours in the front of the horn. These are some of the "pretty pictures" that probably do not tell you that much about the performance of the horn, but they are still interesting. The OS has a 1" throat, the OS part ends at a diameter of 38cm, after a length of 19cm, where it transitions to a 10cm radius. BERIMA simulation.

## 009: OS Waveguide in baffle

Another attempt to simulate the Summa waveguide. This one is mounted in a baffle, and is slightly larger. Perhaps still not a good representation, though. The DE250 flare starts at a 13.8mm diameter, and flares conically out to 25.4mm/1in over 5.51mm, where it meets the OSWG. The OS is 145mm long, ending at 293.7mm (11.56in), where it meets a radius that flares out to 435mm (17.1 in). The radius is 70.7mm. BERIMA simulation, 15kHz mesh bandwidth, simulated 300Hz-15kHz in 150 steps.

## 010: LeCléac'h horn with 90 degree mouth

Here are the results of a series of simulations I did to see the effect of varying the rollback in a LeCléac'h horn. This one stops at 90 degrees (like a tractrix). I have used the AH425 parameters, and only varied the mouth termination. ASEMA simulation.

## 011: LeCléac'h horn with 180 degree mouth

The same as above, but ends at 180 degrees.

## 012: LeCléac'h horn with 270 degree mouth

The same as above, but ends at 270 degrees.

## 013: LeCléac'h horn with 360 degree mouth

The same as above, but ends at 360 degrees. Also under 013 are some plots comparing the throat impedance, zoomed in. It is clear that 90 degrees gives most reflections. 180 degrees helps a lot, and 270 degrees is as good as 360 degrees. The response curves for 270 degrees are unbelievably smooth, and they almost look fake! Still, this is what the simulation produces. 100 frequency points, no smoothing. Both the throat impedance and the response curves show that this horn has extremely little reflection, and the curves are also very smooth off-axis. This is clearly not a constant directivity device, though.

## 014: LeCléac'h horn with 360 degree mouth, sound field

Another series of pretty pictures, this time of the sound field near a LeCléac'h horn. Again, smooth and pretty, but perhaps not all that useful in judging the quality of the horn.

I also added a couple of sound field plots where the lines are isophase contours. These contours are lines between points that are in phase. Note that 1) this is not very frequency dependent, and 2) the lines correspond quite closely to the isophase wave front shapes used to calculate the LeCléac'h horn contour. In calculating the LeCléac'h contour, no assumption of wave front shape is made apart from that the wave fronts should be normal to the wall, and equidistant from each other. The wave front shapes that result from these assumptions, are very similar to what is shown in these plots.

## 015 and 016: Spherical wave horn

Also known as the Kugelwellen horn or Klangfilm horn. It is sometimes confused with the tractrix (which also assumes spherical wave fronts), but they are different. The tractrix assumes wave fronts of constant radius, that are normal to the walls. The wave fronts expand in an almost exponential way. The Spherical wave horn also assumes wavefronts of constant radius but this radius is twice the radius used in tractrix horns, and the wave fronts are not assumed to be normal to the walls. Wave front area expansion is explicitly exponential. See my Horn Theory articles for details on the two curves. I believe Avantgarde uses a profile similar to this, but that is just speculation.

Anyway, here are two simulations of Spherical wave horns, one with baffle (015) and one without (016). Notice the smoother response and throat impedance with the horn in a baffle.

001 AH425 Boundary
001 AH425 polar map, constant throat velocity
001 AH425 polar map, matched impedance source
001 AH425 polar map, normalized
001 AH425 frequency response, constant throat velocity
001 AH425 throat impedance
002 AH425 Boundary (in baffle)
002 AH425 polar plot (in baffle)
002 AH425 (in baffle) polar map, constant throat velocity
002 AH425 (in baffle) polar map, matched impedance source
002 AH425 (in baffle) polar map, normalized
002 AH425 (in baffle) frequency response, constant throat velocity
002 AH425 (in baffle) throat impedance
003 425Hz Tractrix Boundary
003 425Hz Tractrix polar map, matched impedance source
003 425Hz Tractrix polar map, normalized
003 425Hz Tractrix polar map, constant throat velocity
003 425Hz Tractrix frequency response, constant throat velocity
003 425Hz Tractrix throat impedance
004 Conical horn Boundary
004 Conical horn polar map, matched impedance source
004 Conical horn polar map, normalized
004 Conical horn polar map, constant throat velocity
004 Conical horn frequency response, constant throat velocity
004 Conical horn throat impedance
005 Exponential horn Boundary
005 Exponential horn polar map, matched impedance source
005 Exponential horn polar map, normalized
005 Exponential horn polar map, constant throat velocity
005 Exponential horn frequency response, constant throat velocity
005 Exponential horn throat impedance
006 OSWG Boundary
006 OSWG polar map, matched impedance source
006 OSWG polar map, normalized
006 OSWG polar map, constant throat velocity
006 OSWG frequency response, constant throat velocity
006 OSWG throat impedance
007 OSWG+DE250 Boundary
007 OSWG+DE250 in box, polar plot
007 OSWG+DE250 in box, polar map, matched impedance source
007 OSWG+DE250 in box, polar map, normalized
007 OSWG+DE250 in box, polar map, constant throat velocity
007 OSWG+DE250 in box, frequency response, constant throat velocity
007 OSWG+DE250 in box, throat impedance
008 OSWG 1kHz sound field
008 OSWG 2.2kHz sound field
008 OSWG 4kHz sound field
008 OSWG 6kHz sound field
009 OSWG+DE250 Boundary
009 OSWG+DE250 in baffle, polar map, constant throat velocity
009 OSWG+DE250 in baffle, polar map, matched impedance source
009 OSWG+DE250 in baffle, polar map, normalized
009 OSWG+DE250 in baffle, frequency response, constant throat velocity
009 OSWG+DE250 in baffle, throat impedance
010 AH425 90 deg mouth Boundary
010 AH425 90 deg mouth polar map, constant throat velocity
010 AH425 90 deg mouth polar map, matched impedance source
010 AH425 90 deg mouth polar map, normalized
010 AH425 90 deg mouth frequency response, constant throat velocity
010 AH425 90 deg mouth, throat impedance
011 AH425 180 deg mouth Boundary
011 AH425 180 deg mouth polar map, constant throat velocity
011 AH425 180 deg mouth polar map, matched impedance source
011 AH425 180 deg mouth polar map, normalized
011 AH425 180 deg mouth frequency response, constant throat velocity
011 AH425 180 deg mouth, throat impedance
012 AH425 270 deg mouth Boundary
012 AH425 270 deg mouth polar map, constant throat velocity
012 AH425 270 deg mouth polar map, matched impedance source
012 AH425 270 deg mouth polar map, normalized
012 AH425 270 deg mouth frequency response, constant throat velocity
012 AH425 270 deg mouth, throat impedance
013 AH425 360 deg mouth Boundary
013 AH425 360 deg mouth polar map, constant throat velocity
013 AH425 360 deg mouth polar map, matched impedance source
013 AH425 360 deg mouth polar map, normalized
013 AH425 360 deg mouth frequency response, constant throat velocity
013 AH425 360 deg mouth, throat impedance
AH425 360 deg (pale) vs AH425 180 deg, throat impedance
AH425 360 deg (pale) vs AH425 270 deg, throat impedance
AH425 360 deg (pale) vs AH425 90 deg, throat impedance
014 AH425 360 deg mouth sound field 1150Hz
014 AH425 360 deg mouth sound field 1510Hz
014 AH425 360 deg mouth sound field 2620Hz
014 AH425 360 deg mouth sound field 3500Hz
014 AH425 360 deg mouth sound field 500Hz
014 AH425 360 deg mouth sound field 6000Hz
014 AH425 360 deg mouth sound field 660Hz
014 AH425 360 deg mouth sound field 870Hz
014 AH425 360 deg mouth isophase contours 2620Hz
014 AH425 360 deg mouth isophase contours 6000Hz
425Hz Spherical horn in baffle, polar map, constant throat velocity
425Hz Spherical horn in baffle, polar map, matched impedance source
425Hz Spherical horn in baffle, polar map, normalized
425Hz Spherical horn in baffle, throat impedance
425Hz Spherical horn in baffle, Boundary
425Hz Spherical horn in baffle, frequency response, constant throat velocity
Throat impedance comparison: with baffle (pale) and without baffle.
425Hz Spherical horn, free field, polar map, constant throat velocity
425Hz Spherical horn, free field, polar map, matched impedance source
425Hz Spherical horn, free field, polar map, normalized
425Hz Spherical horn, free field, throat impedance
425Hz Spherical horn, free field, Boundary
425Hz Spherical horn, free field frequency response, constant throat velocity