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Proceedings of the Institute of Acoustics
Vol. 45. Pt. 2. 2023
ZONE TO ZONE REFLECTOR OPTIMISATION USING A
GENETIC ALGORITHM
J O’Keefe O’Keefe Acoustics
1 INTRODUCTION
A number of studies have recently been published by the author using Genetic Algorithms (GA) to
optimise acoustic reflectors1,2,3,4. The challenge in those early studies being not so much the acoustic
performance of the finished reflector design but the geometry of how one constructs the reflector
inside the computer in the first place. The geometric methods have been streamlined along the way
in an effort to increase computational speed and thus allow a GA to do what it does best: evolve
solutions from the largest populations as possible, over as many generations as possible. The
present study, it might be argued, takes the optical assumptions of geometric acoustics to the limit.
The goal not being to calculate any of the standard acoustic parameters but, rather, to simply ask the
question: can everyone in a given zone of receivers (i.e. in a seating area) see an entire zone of
sources (e.g. an orchestra on stage) if they were looking at the reflector as if it were an (optical) mirror.
2 REFLECTOR CONSTRUCTION
Traditionally, the perturbation of a geometry like an acoustic reflector has been done inside a simple
six sided rectilinear box, often referred to as a Bounding Box. The methods previously developed by
the author, and described in references [1] and [2], allow for controlled perturbation of a reflector
inside any arbitrarily chosen volume. These volumes are created from Non-uniform Rational B-Spline
(Nurb) curves. Surfaces or volumes created by Nurb-curves like this are referred to as Boundary
Representations or Breps. The new perturbation control volumes, first documented in ref. [1] are
referred to as Bounding Breps or B-Breps. This study will present yet another refinement of the
geometric methods used to build a B-Brep and perturb a reflector design inside it.
*
In the example to be demonstrated here, we will be optimising a side wall balcony facia. We start
with an elliptically shaped cylinder as our B-Brep. Please see Figure 1. A “spinal curve” is drawn
through the centre of the B-Brep and a number of points are established on the curve. In the example
shown here there are only four points (Figure 1b) but any appropriate number may be chosen. At
each one of the points, a “spinal plane” is created perpendicular to the spinal curve. Then the
intersections between these planes and the original B-Brep volume are calculated. This gives us four
Bounding Nurb Curves (BN Curves), one for each plane (Figure 1c). Each spinal plane has its own
U and V axes. Four points are established on the U axes of each of the four planes. These are the
points that will be used to build the reflector surface. The points are perturbed along the V axes of
their respective spinal planes, their maximum and minimum perturbations being limited by their
associated BN Curve (Figure 1d).
In the previous geometric methods1, 2, 3, the control points were free to move in the U, V and W
directions. (The W direction being along the spinal curve). This proved to be a complicated process
and one that was very expensive in terms of computer time. In references [4] and [5], the author
found that increasing the speed of the Genetic Algorithm (GA), and thus increasing the number of
optimising generations, was more important than the accuracy of calculations within each generation.
For this reason, in the simplified geometric method described here, the perturbation of the control
points is only performed along the V axes of each plane. This, along with the method of using pre-
*
Copyright: ©2023 John O’Keefe. This is an open-access article distributed under the terms of the Creative
Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original author and source are credited
Proceedings of the Institute of Acoustics
Vol. 45. Pt. 2. 2023
determined reflection points, introduced in ref. [4], has greatly increased the speed of GA process
and, eventually, the quality of the solutions.
(a) Initial B-Brep and spinal curve.
(b) Divide the spine and create perpendicular
planes.
(c) Intersection of planes with the BBrep
creates the BN curves.
(d) Control points (blue X’s) after the first
perturbation along the V-axis. Dashed
lines indicate the interpolated construction
curves.
(e) Build the reflector surface from the
construction curves.
(f) Create parameterized reflections points on
the reflector surface.
Figure 1. Construction of a reflector surface
Proceedings of the Institute of Acoustics
Vol. 45. Pt. 2. 2023
Having created and populated the spinal planes, we now have a grid of 16 control points, four points
on each of the four planes. Nurb curves are interpolated from these points, roughly parallel to the
spinal curve. These are referred to as the construction curves. The curves span between spinal
planes, as shown in Figures 1d and 1e. Adjacent pairs of the construction curves are then used to
create surface “strips”. Then, finally, these strips are joined together to create the reflector surface,
as shown in Figure 1e.
3 CALCULATION PROCEDURE
3.1 Overview
The inspiration for the calculation procedure came during a concert, while sitting on the end balcony.
The question being: if the facia of the side wall balcony on the opposite side of the room was a mirror,
could a listener on the end balcony see all of the orchestra on stage, inside the mirror. And, perhaps
more to the point, could all the listeners on the end balcony see the orchestra inside the mirror.
In conventional reflector design, one starts at a point source location then shapes or orientates the
reflector to direct sound towards a receiver point or a receiver zone. In this procedure, we reverse
the process, starting at a receiver point, casting rays towards the reflector (in this case the side wall
balcony facia) and then determine how many of the reflected rays intercept the source zone (i.e. the
orchestra on stage). This is repeated for a number of receiver points. Then, starting with a flat
reflector, the Genetic Algorithm (GA) is used to create and perturb a number of surfaces to, eventually,
find a single reflector that will best satisfy all the receiver points (i.e. the seating locations).
3.2 Calculating Reflections
In ref. [4], the author adopted an atypical approach for calculating the reflection of rays off a surface.
The method proved much faster than the traditional ray bundle/intersection method and, in so doing,
allowed the GA to work to its strengths. Namely, employing high population counts, optimised over
significantly more generations.
The reflection scheme is based on pre-determined reflection points. Each 3-dimensional reflector
surface is parameterised into its standard (s, t) parameters. Each parameter pair (si, ti) represents a
point on the reflector. An example of parameterised, pre-determined reflection points is shown in
Figure 1f.
The reflection for the ith reflection point is calculated as follows:
1. Evaluate the perturbed surface at the parameter pair (si, ti). This will generate the ith reflection
point.
2. Draw a line from the receiver point to the reflection point. This is the incident path.
(Remember that the listener at the receiver point is “looking” for the orchestra on stage. What
would normally be the acoustic reflection path is now the optical incident path.)
3. Calculate the normal to the surface at the ith reflection point.
4. Create a plane with an origin at the ith reflection point and the normal vector that was
determined in Step 3.
5. Create a virtual (mirror image) source point using the receiver point and the plane created in
Step 4.
6. Calculate the vector between the virtual image source and the ith reflection point. Use this to
generate an infinite ray with its origin at the ith reflection point.
7. Search for an intersection with the source zone surface. If successful, this will be the reflected
path.
The advantages of this approach are two-fold. First and foremost, computer intensive intersection
calculations are no longer required between a ray bundle and the reflector surface. The second
Proceedings of the Institute of Acoustics
Vol. 45. Pt. 2. 2023
advantage to this approach is that, no matter how the reflecting surface is perturbed, the reflection
points will always be in the same relative positions with respect to the surface’s centre and its edges.
This allows for a simple and systematic evaluation of diffraction effects using the methods developed
by Rindel6. An advantage that has been put to use in ref. [5], studying the diffraction effects on flat
surface stage reflector arrays.
4 FITNESS FUNCTIONS
Most of the Genetic Algorithm (GA) optimisations in this study were performed with a three objective
routine, a modification of the Non-dominated Sorting Genetic Algorithm (NSGA-II)7. An early version
of the work employed a two-objective optimisation version of NSGA-II but closer analysis of the
results suggested the need for a third objective in the optimisation procedure. The fitness functions
for the three objectives are presented below. Briefly summarised, the goals are to: (i) maximise the
source zone area seen by the listeners on the balcony; (ii) minimise the amount of energy scattered
by the reflector away from the listeners; and (iii) ensure that the listeners’ view of the source zone is
uniform (i.e. avoiding “hot spots” or “dead zones” on stage).
4.1 Area Fitness
As is often the case in GA optimisation, the goal will be to minimise the fitness functions. So, although
we want to maximise the visible area, the actual function works towards a minimum. The area fitness
function becomes:
𝐴𝑟𝑒𝑎%𝐹𝑖𝑡𝑛𝑒𝑠𝑠% = !
"#!"#$%&
∑-
1 − $'(&()*"+(
$&,-.#"/0,1"
0
#!"#$%&
%&'
(1)
where:
NRecPts = the number of receiver points in the receiver zone
Svisible,i = the surface area of the source zone that is visible from the ith receiver point.
Ssource zone = the total surface area of the source zone.
4.2 Scattering Fitness
In an effort to maximize the useful reflection area of the optimized reflector, the second goal will be
to minimize the scattered energy, employing the fitness function:
𝑆𝑐𝑎𝑡𝑡𝑒𝑟𝑖𝑛𝑔%𝐹𝑖𝑡𝑛𝑒𝑠𝑠% =% !
#!"#$%&
∑
!
#!"2$%&
∑
𝑃𝑜𝑖𝑛𝑡𝑠()%*+,
#!"2$%&
-&'
#!"#$%&
%&'
(2)
where:
NRecPts = the number of receiver points in the receiver zone
NRefPts = the number of pre-determined reflection points on the reflector
Pointsfailed = the number of reflection points that failed to cast reflections to the
source zone from the ith receiver point
4.3 Spreading Fitness
The procedure, as described above, casts rays from a receiver point, to the reflector and, hopefully,
to the source zone. The goal of the spreading fitness function is to ensure that the intersections of
the reflected rays with the source zone are distributed as uniformly as possible. A similar fitness
function was developed in ref. [1], which we have modified and renamed here:
Proceedings of the Institute of Acoustics
Vol. 45. Pt. 2. 2023
𝑆𝑝𝑟𝑒𝑎𝑑𝑖𝑛𝑔%𝐹𝑖𝑡𝑛𝑒𝑠𝑠% =% !
#!"#$%&
∑
!
#3.(4
∑
1 −%𝑒5466
74(&%
#3.(4
-&'
#!"#$%&
%
(3)
where:
NRecPts = the number of receiver points in the receiver zone
Ngrid = the number of evenly distributed grid points on the source zone surface
dNN = distance from the jth grid point to its nearest neighbour (NN) intersection point
tdist = the convergence coefficient associated with distance
5 RESULTS
The geometry for the example demonstration
discussed here is shown in Figure 2. The
audience layout is meant to be representative of
a typical 20 m wide shoebox shaped concert hall
with a single balcony. The source zone is similar
to that of an orchestra on stage. The receiver
zone is restricted to only one half of the balcony.
The idea being that the (blue) side wall balcony
facia is best used to create lateral reflections on
the opposite side of the room. The receiver zone
is sparsely populated with only 9 receiver points,
primarily for computational speed.
As mentioned above, the original intention of this study was to employ a standard two-objective
optimisation. Figure 3 shows some of the results from one of the early 2 objective optimisations and
demonstrates the need for a 3 objective optimisation search. In this case, the optimisation was driven
by Equations 1 and 2, i.e. for Area and Scattering Fitness but not Spreading Fitness. The Pareto
Optimisation graph in Figure 3a may be interpreted as follows.
The big blue and yellow dot is the reference point. Diffusion or scattering studies often use a sphere
or cylinder as a point of reference. The blue and yellow dot indicates the fitness function value for a
semi-cylindrical facia reflector the same size and in the same location as our optimised reflectors.
The semi-cylinder has an Area/Scattering Fitness value of (0.33, 0.93). This suggests that with a
semi-cylindrical facia reflector, listeners on the opposite (end) balcony will not be able to see 33% of
the orchestra on stage if they tried to find it as a mirror reflection off the side wall balcony facia. The
scattering fitness value of 0.93 suggests that the semi-cylindrical reflector scatters 93% of the visibility
rays away from the intended receiver points on the balcony. Or, put another way, only 7% of the
reflector surface is directing sound towards the nine listeners in receiver zone.
The solid and open circles in Figure 3a indicate how the solutions have been sorted into the first and
second Pareto Fronts by the Non-dominated Sorting Genetic Algorithm (NSGA-II)7. The solid circles
are for Pareto Front 1, the open circles for Pareto Front 2. The goal of the optimisation was to
minimise the Area and Scattering Fitness Functions and all of the solutions on both Pareto fronts are,
indeed, closer to the graph’s origin than the reference point for the semi-cylinder.
Figure 3b shows the results from one of the better solutions in Figure 3a. The one indicated by the
arrow. It has an Area/Scattering Fitness of (0.20, 0.71). The view of the stage, in Figure 3b, is for
the end seat in the 2nd row, the one that has been circled in yellow. The grey spheres on the right
side of Figure 3b indicate where the visibility rays from the balcony front have intersected the source
zone. The white surface indicates the part of the source zone that is visible from the balcony seat,
as calculated by the intersection points. Their irregular distribution leads to an erroneous over
estimation of the visible source zone area. Hence the need for a 3 objective optimisation search.
Figure 2 Perspective view of the example room
geometry.
Proceedings of the Institute of Acoustics
Vol. 45. Pt. 2. 2023
Figure 3a. Pareto graph for a 2 objective
optimisation search. The blue and yellow dot
indicates the performance of the reference
semi-cylindrical reflector. Solid circles are for
the 1st Pareto Front, open circles for the 2nd.
Figure 3b. Image of the view of the stage from
the seat circled on the balcony, using the
reflector given by the solution indicated by the
arrow in Figure 3a. The yellow dots on the blue
reflector indicate the reflection points. The grey
spheres indicate intersections with source zone.
A 3 objective optimisation was therefore performed on a population of 315 over 50 generations. The
resulting Pareto Graphs are shown in Figures 4a and 4b. Quite impressively, all of the solutions have
been sorted onto the first Pareto Front and appear to have reached their best possible fitness values.
This would not have been possible without such a high population count or the high number of
optimising generations. And neither of those would have been possible without the computer time
saving methods described above, notably the concept of pre-determined reflection points.
Figure 5 presents a more in depth analysis of one of the better solutions taken from this optimisation.
The solution has an Area/Scatter/Spreading Fitness value of (0.08, 0.75, 0.82). It is being compared
to the reference point semi-cylindrical reflector on a seat by seat basis. The open bars are for the
semi-cylinder, the solid bars for the optimised reflector. In every seat, the optimised reflector provides
Figure 4a. Area/Scatter fitness Pareto
analysis.
Figure 4b Area/Spreading fitness Pareto
analysis.
Proceedings of the Institute of Acoustics
Vol. 45. Pt. 2. 2023
a significantly better view of the source zone than the benchmark semi-cylinder. In the range of 90
to 95%.
Figure 6 shows the visible area of the stage from the same seat as depicted in Figure 3b, this time
using the reflector analysed in Figure 5. Including the 3rd objective (Spreading Fitness) has produced
a much more uniform view of the stage and, hence, a more accurate estimation of the visible surface
area.
6 CUT-ON FREQUENCIES
As mentioned at the outset, the optimisations described here have centred around the optical visibility
of the source zone. The acoustical implications should, of course, be taken into account. With that
in mind, a brief study of the reflecting panels’ cut-on frequencies was undertaken. Rindel has outlined
a method for approximating the diffraction attenuation of rectangular reflectors similar to the ones
considered here, i.e. reflectors with a large ratio between length and height8. Two cut-on frequencies
are calculated, one for the short dimension (fg1) and one for the long dimension (fg2). Above these
frequencies, the reflector will cast uniform reflections, at lower frequencies diffraction effects will
attenuate the reflections at a rate of 3 dB per doubling of frequency between fg1 and fg2 and 6 dB per
doubling below fg2 .
A cursory study of the cut-on frequencies was
carried out as follows. A single source-receiver
combination was chosen, with the source and
receiver points at the centroids of the source and
receiver zones respectively. Reflections were
cast off of the original flat panel at 1089 pre-
determined reflection points. For the long
dimension (11.4 m), the fg2 cut-on frequencies are
between 15 and 18 Hz and, hence, are not a
concern. For the short dimension (1.3 m) the cut-
on frequencies range from 1545 to 7554 Hz. A
contour map of these frequencies is shown in
Figure 7. As indicated by the yellow dots in
Figure 3b and Figure 6, most of the successful
reflections are cast from points on the reflector
Figure 5. Percentage of visible areas from the
9 balcony seats. Solid bars for the optimized
reflector, open bars for the reference semi-
cylindrical reflector.
Figure 6. View of the stage from the same
seat as depicted in Figure 3b. Including the
3rd optimization objective provides a more
uniform view of the stage.
Figure 7 Contour map of fg1 cut-on
frequencies superimposed on the balcony
facia.
Proceedings of the Institute of Acoustics
Vol. 45. Pt. 2. 2023
that are closest to the stage. The contour map in Figure 7 suggests that the cut-on frequencies in
this area are in the range of 2000 to 2500 Hz. An octave below these frequencies (i.e. in the 1000
Hz octave band) reflection levels will be attenuated by 3 dB. In the 500 Hz octave band they will be
attenuated by 6 dB, and so on into the lower frequencies.
7 CONCLUSIONS
A new procedure has been presented to optimise a reflector casting reflections from one zone to
another zone. If one accepts the inherent limitations of geometric acoustics, i.e. the optical
equivalency, the new procedure does, indeed, improve how well a group of listeners might “see”
another group of sound sources as a mirror reflection inside the reflector. Using a 3 objective Genetic
Algorithm (NSGA-II), the receiver zone’s view of the source zone has been significantly optimised. A
standard semi-cylindrical reflector provided a view of 67% of the source zone. An optimised reflector
of the same size and in the same location provided a view of 92% of the source zone.
8 REFERENCES
1. O’Keefe, J. Geometric Algorithms for Machine Based Optimisation of Acoustic Reflectors,
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Reflectors, Proc. of Euronoise, 2021
3. O'Keefe, J., A Two Stage Embedded Genetic Algorithm to Optimise Ceiling Reflections. Proc.
of Euroregio BNAM 2022 Joint Acoustics Conference; 9th-11th May 2022; Aalborg,
Denmark.
4. O’Keefe, J., Genetic Algorithm Fitness Functions for acoustic reflectors in performing arts
venues, Proc. 24th ICA (2022), Seoul, South Korea.
5. Rindel JH. Attenuation of Sound Reflections due to Diffraction, Proc. of NAM Conference 20-
22 August 1986.
6. O’Keefe, J. A Genetic Algorithm to Optimise Stage Reflectors for Self and Other Reflections,
Proc. 10th Forum Acusticum, Torino (2023).
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algorithm: NSGA-II IEEE Transactions on Evolutionary Computation. 6 (2), 2002, p.182
8. Rindel, J.H., Acoustic Design of Reflectors, Proc. of IOA, Vol. 14 (2), 1992.
