PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Application of multi-objective optimization techniques to the designof sound diffusers
J. Redondo1, J. M. Herrero2; L. Godinho3, R. Patraquim4; T.J. Cox5
1234

Universitat Politècnica de València, Spain

ISISE, Department of Civil Engineering, University of Coimbra, Portugal
3

Acoustics Research Centre, University of Salford, United Kingdom

ABSTRACTMulti-objective optimization (MO) techniques have proven to be a very interesting methodology for solving
design problems in different areas of engineering. Diffuser design problems are another example. Whendesigning a diffuser, different features are to be optimized simultaneously, which are usually in opposition. It
is precisely the latter that encourages the use of MO techniques. The designer chooses the particular features
to be optimized independently, carries out the optimization, studies the resulting Pareto fronts, and finallychooses the optimal diffuser using his preferences a posteriori. This results in more informed and therefore
more conscientious decision making. This work shows how the MO techniques can be applied in the designof diffusers through a series of examples where different characteristics are taken into account: themaximization of the diffusion coefficient at a particular frequency range or the minimization of the depth ofthe diffuser. In the last case, we will explore the possibility of changing not only the depth of the wells but aswell their widths. The study of the Pareto fronts presented shows how the optimal diffuser to choose can besusceptible to be different depending on the characteristics most wanted by the designer.
Keywords: Sound diffusers, Room acoustics, Optimization

1. INTRODUCTION
Sound diffusers are widely used in the field of room acoustics in order to reduce the effect of low
frequency modes and to reduce echoes and focalizations and, as a result, improving the diffuseness ofthe sound field. The first sound diffusers were proposed by Schroeder in 1975 [1] and consist of a setof wells with different depths that come to modify the phase of sound. This is why they are known as
Schroeder or phase diffusers. When the variation of the depth of the wells is only in one dire ction, theresultant diffusers are called 1D Schroeder diffusers. Among the different types of Schroeder diffusers,
the most popular is the 7 wells QR (quadratic residue) diffuser. In this study, and without any loose ofgenerality, we will focus in this particular case. Figure 1 illustrates this sound diffuser.
It is possible to improve the acoustic performance of such devices using several optimization
techniques, but in all cases until recently, the optimization processes in the field of sound diffusers
have been carried out with a single-objective function. In 1995 Cox [2] suggested the use of iterative
methods as downhill simplex and quasi-Newton methods as tools to optimize these diffusers. In thiscase, the objective function used was the “diffusion parameter”, defined as the standard error of thesound pressure over the measurement positions, averaged over the desired working frequency range.
In order to avoid low performance for particular frequencies within that range, a penalty is introduced
adding the standard error to the frequency averaging. In recent years, volumetric diffusers based onarrays of cylindrical scatterers have been developed using evolutionary algorithms along to a
two-dimensional Fourier approximation [3]. In this case, the algorithms are used to minimize thestandard diffusion coefficient and the optimization is carried out by removing or varying the diameter
1

fredondo@fis.upv.esjuaherdu@isa.upv.es
3
lgodinho@dec.uc.pt
4
ricardo.patraquim@gmail.com
5
T.J.Cox@salford.ac.uk
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of a set of cylinders located in a periodic rectangular array, creating pseudorandom arrays with a highdiffusion performance.
More recently, unlike optimization processes used so far in which a single cost function isconsidered, two papers have demonstrated the potential of the use of multiobjective algorithms when
designing sound diffusers. In the first one [4], authors proposed the use of multiobjective algorithms
for the optimization of sound diffusers in a wide frequency range with a small variation of theperformance in that frequency range. In the second one [5] multiobjective algorithms were used to
optimize sound diffusers based on sonic crystals. In this case the aim was to optimize the performanceof the diffusers at low frequencies.
The following sections will briefly discuss the phases of the optimization process and present
different alternatives in the design objectives that will serve to illustrate the versatility of the use ofmulti-objective algorithms for the optimization of sound diffusers.
1.1 Quantification of the performance of sound diffusers
As will be shown below one of the crucial points in the optimization process is the prediction of theperformance of potential candidates. The quantification of the effect of a diffuser is given by two ISO
standards [6-7]. In this work we will follow the standard ISO 17497-2 2012 [7], based on themeasurement of the reflected sound pressure over a range of angles, between −90º and 90º in steps of
5º (37 measurements in total). For this purpose a microphone is sequentially positioned along asemi-circumference centered in the middle point of the test sample, which is compos ed by an array ofat least three diffusers (with seven wells each one). The original signal has to be windowed in order toseparate the reflected sound from the direct sound. The parameter measured using this technique isknown as the diffusion coefficient:2𝑛2𝑑′𝑗 = [(∑ 𝑝𝑖𝑗
)
𝑖=1

2
𝑛
𝑛
2 2
2
2 2
− ∑(𝑝𝑖𝑗 ) ] / [(∑ 𝑝𝑖𝑗 ) − ∑(𝑝𝑖𝑗
) ]
𝑖=1
𝑖=1
𝑖=1
𝑛

(1)

where d’j is the diffusion coefficient for the j-th one-third octave band considered, pij is the reflectedsound pressure for the j-th one-third octave band considered at the i-th measurement position, and n is
the number of measurement positions (n = 37). This diffusion coefficient has to be averaged for
different incidence angles (in our case 3). To normalize this diffusion coefficient from zero to one, it iscompared with that of a flat surface. The purpose of normalization is to remove edge diffraction
scattering effects due to the limited size of the sample under analysis. The normalized diffusion
coefficient, dj , for the j-th one-third octave band considered, is defined as:
𝑑𝑗 = (𝑑′𝑗 − 𝑑𝑗,𝑟𝑒𝑓 )/(𝑑′𝑗 − 𝑑𝑗,𝑟𝑒𝑓 )(2)
where d j,ref is the diffusion coefficient of a flat panel for the j-th one-third octave band considered. As
a result, d j is equal to zero for all frequencies in the case of a flat surface.
All these coefficients require tedious measurements in an anechoic environment unless numericalmethods are available. In particular BEM is a well-established method that can be used for this purpose[8]. Since the classical geometry of a QR diffuser will be modelled, special care must be taken in order
to apply the BEM to solve the problem. Indeed, the presence of the walls separating the diffuser’s wells
originates very thin surfaces which typically lead the direct BEM formulation to degenerate and lead t ounstable equation systems. For this reason, a dual-BEM formulation is used here, in which the direct
BEM integral equation (see equation (3)) is complemented by the so-called hypersingular BEM
equation (see equation (4)). Details of this formulation can be found in [9], and thus only a general
overview is here given regarding the BEM.
The classical boundary integral equation can be derived from the Helmholtz equation in thefrequency domain by applying the reciprocity theorem, and in the case of rigid bou ndaries it can be
written as:
𝐶 𝑝(𝐱 0 , 𝜔) = − ∫ 𝐻(𝐱, 𝐱 0 , 𝜔, 𝐧) 𝑝(𝐱, 𝜔)𝑑Γ + 𝑝𝑖𝑛𝑐 (𝐱0 , 𝐱𝑆 , 𝜔)(3)

Γ

where  represents the Green’s function for the pressure defined before, and H is its first derivativewith respect to the normal direction to the boundary ; similarly, p and q are the pressure and its first
derivative in the normal direction to the boundary (n), at point x; p inc (x0 ,x s ,) represents the effect ofa possible acoustic source located at point x 0 . The factor C equals 1/2 if x, and 1 for points not in2298

the boundary but within the domain (x).
The hypersingular boundary integral equation can be derived by taking the first derivative of
equation (3) with respect to the surface normal, and thus the required additional integral equation canbe expressed as:
𝐴 𝑝(𝐱 0 , 𝜔) = − ∫ 𝐻′(𝐱, 𝐱 0 , 𝜔, 𝐧, 𝐧𝟐 ) 𝑝(𝐱, 𝜔)𝑑Γ + 𝑝′𝑖𝑛𝑐 (𝐱 0 , 𝐱𝑆 , 𝜔)

(4)

Γ

The Green's functions G’ and H’ can be seen as the derivatives of G and H with respect to thenormal to the boundary at the loaded point, n2 . In this equation, the factor A equals zero for piecewisestraight boundary elements.
This formulation is used to analyse each configuration of the diffuser, allowing to compute thesound pressure scattered by the diffuser at any point of the acoustic domain. It is thus called multiple
times from within the optimization algorithm, allowing the evaluation of the defined cost function foreach individual.

2. OPTIMIZATION OF SCHROEDER DIFFUSERS WITH MULTIOBJECTIVE
ALGORITHMS
In this section we will briefly explain the fundamental steps in an optimization process. The firststep is to parametrize the design, or in other words, to establish the genes of the candidates. The second
step is to stablish the metrics for the performance of each candidate. We will present here a few
examples of the so called cost functions than can be used in the particular case of sound diffusers.
2.1 Codification
The first step in using an evolutionary algorithm is to choose a genetic coding of the possible
candidates. In the case of a Schroeder diffuser the coding is esp ecially simple as there are no
constraints. Individuals will be coded using a series of genes proportional to the depth of each of thewell. Thus, in this case, each individual is given by 7 values (genes) that represent the set of depths of
the wells that conform the diffuser. A step of one centimeter has been chosen giving rise to 35 possible
values, given that, for convenience the depth has been limited to half a wavelength at 500 Hz, i.e. 34
cm. Thus, the value 1 represents a slot of maximum depth, 34 cm. Figure 1 illustrates the codificationof a 7 wells QR diffuser. For the sake of convenience, the width of the diffuser has been set to 1 meter.

Figure 1 – Schematic of the 7 wells diffuser to be optimized
2.2 Cost Functions
It can be said that the most important step of an optimization is to define the target. For this purpose
it is necessary to define cost functions, also called objective functions, that measure the "distance" tothe optimum. In multi-objective processes such as those discussed here, it is necessary to define atleast two objectives. For the sake of brevity in this work we will limit to the use of two cost functions.
However, we will define several objective functions that in the results section will be combi ned inpairs and compared to each other.

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The cost functions must be chosen by the designer taking into account the specific needs of theroom or rooms where the diffusers will be installed. In this fact lies the versatility of optimization in
general and multi-objective optimization in particular. We would like to highlight the flexibility of theapproach concerning the target and the frequency range of optimization, because the final choice in the
optimization process will depend on the particular problem to be solved by the diffusor, for instance
flutter echoes or room modes. In each case a cost function could result better than other one.
Without any loose of generality we will define all the cost functions as distances to the optimum
case, or in other words, the cost functions will take the value 0 in the best case. Also, where possible,
the value 1 will be assigned to the worst case.
The first cost function to be considered will be the average value of the diffusion coefficient in themean frequency range adapted to be 0 in the best case, i.e.:
The first three cost functions are defined to study the particular case where only the mean frequency
range is to be optimized. So, only the one third octave bands with central frequencies 400Hz, 500Hz,
630Hz, 800Hz, 1000Hz and 1250Hz (i.e, octave bands between 7 and 12) will be considered. We areassuming that the whole frequency range is formed by the low frequency range (bands from 1 to 6), themean frequency range (bands from 7 to 12) and the high frequency range (bands 13 to 18)
The first Cost Function that we suggest corresponds to the average of the normalized diffusion
coefficient in the mean frequency range, namely:
12

𝐽𝑚𝑒𝑎𝑛 = 1 − 𝑑𝑚𝑒𝑎𝑛 = 1 − ∑ 𝑑𝑗 /6

(5)

𝑗=7

As commented before in [2] authors suggest the use of a penalty in the cost function in order toavoid low performance. Two alternatives can be define in this sense.
The most evident is the use of the standard deviation of d in the optimization frequency range. In
our case:
12

𝐽𝑣𝑎𝑟 = √∑(𝑑𝑚𝑒𝑎𝑛 − 𝑑𝑗 )𝑑2 /5

(6)

𝑗=7

The second option is to include the minimum value of the diffusion coefficient in the optimization
frequency range, namely:
𝐽𝑚𝑖𝑛 = 1 − 𝑑𝑚𝑖𝑛
(7)Thus, J mean should be used in combination with either J var or J min to avoid the appearance of very
irregular frequency dependencies along the frequency domain being optimized.
It is well known that the minimum frequency with significant diffusion depends on the dimensionsof the diffuser, fundamentally on the maximum depth of the diffuser. As a result, when attempting tosolve the problems associated with the formation of room modes that usually are more problematic at
low frequencies, the use of excessively deep diffusers is involved. In order to reduce the minimum
working frequency without excessively increasing the depth, we propose two additional costfunctions:
𝐽𝑓𝑚𝑖𝑛 = 𝑓𝑚𝑖𝑛(8)
where f min is the lowest frequency with a value of the diffusion coefficient larger than 0.35. The valueof 0'35 is arbitrary and has been chosen as it is approximately half the maximum value observed in aquadratic residue diffuser. Additionally another cost function is introduced to take into account thedepth of the diffuser.
𝐽𝑑𝑒𝑝𝑡ℎ = 𝑚𝑎𝑥𝑑𝑒𝑝𝑡ℎ(9)The problem with the definition of Jfmin is that the diffusion coefficient can oscillate once thethreshold is exceeded, falling back below that value (see figure 2). In the following, the highest cut -offvalue between the line defined by the diffusion coefficient and the threshold, i.e. the poi nt to the rightof figure 2, will be used.

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Figure 2 – Normalized diffusion coefficient versus frequency of a candidate with a very uneven behavior.
Red dots correspond to three crossings with the threshold 0.35.
2.3 Optimization Process
The optimization process itself begins by generating an initial population, in principle randomly. In
this population, cost functions are evaluated and the initial set of optimal individuals is detected. This
group is known as the Pareto front. The optimal individuals are those that are dominant over the rest of
the population, that is, there is no other individual whose values of any of the two Cost Functions (J)considered in each case are lower (Figure 3).

Figure 2 – Representation of the dominance criterion. Individuals in red are dominant, since noother individual is better for both criteria simultaneously.
Then begins the process of optimization by crossing individuals. Two individuals are randomly
chosen from the initial population and their genes are randomly combined, just as in nature children
randomly possess part of the genetic information of their two parents. Also, with a certain probability,
in general very low, mutations are established from individuals also chosen at random. Subsequently,
the values of the cost functions of the new individuals thus obtained are calculated. To maintain thesize of the population throughout the process, the appearance of new individuals implies theelimination of individuals previously present in the population from among those who are not part of
Pareto's front. The whole process is repeated a finite number of times, or in other words, a certain
number of generations.

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In the present work an initial population of 2000 individuals has been considered and the system
has been allowed to evolve a number of generations of the same order. When it is observed that
Pareto's front tends to stabilize which is a first indicator that the process has been saturated and no
better solutions can be found than those already found.
In this work we have run three different optimization processes. The first two ones correspond to
the typical optimization in a relatively wide frequency range. The latter consists of minimizing the
thickness of the diffuser without shifting the working range of the diffuser to high frequencies.
Namely,a. J mean and J var
b. J mean and J minc. J fmin and Jdepth
For convenience in this paper we will use a well stablish multi objective evolutionary techniquecalled evMOGA. Further details can be found at [10].
2.4 Selection of optimals
In order to choose the best individual we suggest the addition of a supplementary criterion:
robustness. This parameter measures the degree in which the values of the cost functions are affectedby small changes in depths of the wells that appears in the process of manufacturing a sound diffuser.
It is very inadequate to choose a sound diffusers that is too sensitive to these small changes. For the
shake of brevity we will not include in this paper any additional plot with robustness analysis. Further
details can be found at [5].

3. RESULTS AND DISCUSSION
3.1 Optimization of the diffusion coefficient in a wide frequency range
The results of the first two optimizations, J mean &J var and J mean &J min, are illustrated in figures 3 and4. Actually figure 4 is just a zoomed version of figure 3. For convenience a single objective
optimization has been carried out with the cost function J mean .
The first thing to note is that all optimizations produce diffusers with better performance than theconventional QR diffuser. When comparing the two multi-objective optimizations it is detected thatpart of the individuals of each optimization are dominant for the other. Each optimization seems not tofind individuals with extreme values of the second cost function, i.e., low standard deviation for thesecond optimization or high d min for the first. It would therefore be possible to combine the results ofboth and start a new optimization.

Figure 3. Optimizations results. Left, Jmean vs J var . Right, J mean vs J min. Black dots: initial
population; blue points: Pareto front of Jmean &J var ; red points: Pareto front of J mean &J min ; green square,conventional QR diffuser; green diamond, mono-objective optimization result. Dotted rectangle, zoom
for figure 4.
On the other hand, single objective optimization provides a high average value of the diffusion
coefficient (low J mean ) surpassing the results of multiobjective optimizations for this parameter.
Moreover, in the case of the second optimization, the value of d min is also exceeded. This seems to2302

indicate that the optimization J mean &J min is not adequate in this case. This is not the case for J mean &J var
where the optimal individual of the single objective optimization would be no more than one
additional point on Pareto's front far from those points with more constant diffusion coefficients
throughout the optimization frequency range.

Figure 4. Zoomed version of figure 3
Traditionally, the use of several criteria in a mono-objective optimization process was achieved byaveraging all the criteria to be considered. For instance in [2] two different cost functions w ere use,
one to estimate the diffusion of a given surface averaged over the optimization frequency range, andanother to avoid large variations of the diffusion in that range. This is equivalent to add the two Cost
Functions so that J=J mean + Jvar . In doing so, we are assigning the same relevance to both criteria. A
more general definition will add a weighting coefficient to give different relevance to each criteria,
i.e.: J=·J mean + (1-) J var .
In a conventional single-objective optimization using a weighted combination of Cost Functions,the relative relevance of each of the objectives () has to be fixed before running the process. On thecontrary, in a multi-objective optimization, the relative relevance is stablished after the optimization
process, during the decision making, when one of the individuals of the Pareto front is selected.
3.2 Minimization of the thickness of the diffuser
The optimization results for this case are summarized in Figure 5. It can be seen that optimized
diffusers can extend the low frequency working limit about one octave if compared with conventional
QR diffusers.

Figure 5. Jdepth vs Jfmin for QR sound diffuser (width 1m) for different total depths (black continuousline). Families of optimized diffusers: black circles, blue diamonds and red squares.
It is very normal that when doing an optimization the optimal individuals are very similar to eachother. The case presented here is no exception. Figure 6 illustrates the 3 families of diffusers that make2303

up the Pareto front. Three clearly differentiated families are observed, the first appears for high depths
and is formed by a group of two wells with very shallow depths and another group of 3 wells with very
high depths separated from each other by wells with intermediate depths. In the second family there is
only one intermediate deep well. Finally in the last family, which covers the lower depth area, there arebasically only two different depths.

.
Figure 6. Appearance of the families of optimized diffusers. Left: Family 1, the deepest diffusers.
Middle: Family 2. Medium depth. Right: Family 3. Less deep.

4. CONCLUSIONSIn the present work we have demonstrated that multi-objective evolutionary algorithms areespecially indicated in the case of sound diffusers design. First of all because of the simplicity of thegenetic coding. And secondly because its use is flexible and adapts to the specific needs of the designer.
The multiobjective approach gives the designer the possibility to consider several properties at thesame time, without any previous knowledge of the relevance of each criteria. This has been
demonstrated by using two pairs of objectives very different from each other and in all cases highperformance diffusers can be obtained if compared with conventional ones.

ACKNOWLEDGEMENTS
This work was supported by the Ministerio de Economía y Competitividad (Spain) [grant number
RTI2018-096904-B-I00]

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