Proceedings of the Institute of Acoustics

 

 

Evolutionary optimization processes for acoustic applications where size matters

 

Daniel Benítez-Aragón¹, Universidad Politécnica de Valencia, Valencia, Spain

Jaime Galiana-Nieves², Universidad Politécnica de Valencia, Valencia, Spain

Javier Redondo³, Universitat Politècnica de València – Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Valencia, Spain

David Ramírez-Solana⁴, Universidad Politécnica de Valencia, Valencia, Spain

Juan Manuel Herrero⁵, Universidad Politécnica de Valencia, Valencia, Spain

 

ABSTRACT

 

In acoustics, the relevance of size, scale and shape of the structures involved in a problem often brings complications when it comes to find the best possible solution. Scenarios such as the design of a room or a sound diffuser are some of the problems that we can find in which shape and size are factors to consider. In recent years, computational optimization methods have been developed to bring a different approach to the solving of these problems. These methods, based on genetic algorithms, provides us with continuous modifications and combinations between the elements involved in a problem that can help us obtain solutions that would be almost impossible to find with regular analytic solving. Our goal is to develop a multi-objective genetic algorithm to optimize various parameters of a design problem, being one of them related to the physical size of the element under study. For this purpose, we will not treat the objects being optimized as individuals, but as groups of individuals with the same proportions but variable scale, called “concepts”. With this method, we should obtain the optimum proportions for the problem faced, not only optimized individuals.

 

1. INTRODUCTION

 

The purpose of this document focuses on defining optimization processes for acoustic applications whose behaviour are directly influenced by their size and geometry, since they are the most critical aspects of this kind of applications in the majority of the cases. These optimization processes shall be carried out with an evolutionary algorithm, in which new cases are generated iteratively based on the results of the previous cases, i.e. using genetic implementations. This evolutionary algorithm was implemented on a MATLAB code, which is applicable to any acoustic implementation that lies on geometrical properties. This code, based on multi-objective genetic algorithms (MOGA), allowed to find the best individuals for a two cost function problem, which usually are in conflict between each other. By defining the equations that describe the phenomenon, and the physical limits for which it is applicable, the optimization could subsequently be applied. In this article, two cases of study are going to be analysed: sound diffusers and acoustic barriers made of sonic crystals.

 

2. GENETIC ALGORITHMS

 

Many of the optimization problems that show up in engineering fields are too difficult and complex to solve through traditional and analytical processes. A common alternative to regular optimization is evolutionary optimization. This processes are based on a group of individuals that represents a set of solutions for a specific problem. This group undertakes then several transformations (which includes crossovers and mutations processes), and then go through a selection step which favours the best individuals of the updated group. In order to always keep the population constant, some of the individuals are removed, making sure they are not part of the best samples so far. Once the condition of the optimization is reached, the algorithm offers a selection of the best individuals. For every cycle of transformation and selection, the group of individuals reaches a new generation. This way, after a certain amount of them, the groups is expected to contain individuals that are closer to the ideal solution. To better understand this process, the schematic process of this evolutionary optimization algorithm has been represented in Figure 1.

 

 

Figure 1: Basic scheme of an evolutionary optimization algorithm.

 

As previously mentioned, an evolutionary optimization problem shall require an initial group of individuals. This groups would be generated randomly, in order to provide enough genetic diversity into the process, which is the easiest and most popular initialization method [15]. This enhances the algorithm and makes it easier to find better solutions. This way, by introducing a certain number of samples, subjects with better acoustic results will be iteratively obtained. For each iteration, a specific set of individuals are removed from the population in order to keep it constant, being replaced by the new samples. The removed ones are not necessarily those with the most unfavourable results but actually any individual apart from the best ones. This way, we assure that the genetic diversity is kept high, which is key for this process. These new individuals were generated by combining pairs of optimal ones, randomly adapting geometric characteristics of both of them, similar to a process of genetic evolution.

 

This way, an important factor to stablish is the cost function or cost parameter, i.e. the variable that determines the quality of the optimization [1]. When applying optimization process to real situations, we find that they usually are multi-objective, i.e. multiple goals must be achieved in order to obtain the best results. And typically, these goals are in conflict with each other [15]. Multi objective optimization focuses then on minimizing a vector f(x) of functions, being x the independent variable compound of n dimension, as stated in the following expression [15]:

 

 

3. PARETO FRONT

 

When analysing the group of individuals that takes part into our multi-objective optimization problem, we must find those samples that are considered the best. Due to the multi-objective approach, we would not find only one individual, but several. This way, we must identify those samples for which none of the others are better for all the objectives at the same time. In order to clearly fulfil this conditions, we must define at this point several concepts [15]:

• Domination: A point x* is said to dominate x if x* is at least as good as x for all objective function values, and it is better than x for at least one objective function value.

• Weak domination: A point x* is said to weakly dominate x if x* is at least as good as x for all objective function values. Note that if x* dominates x, then it also weakly dominates x.

• Non-domination: A point x* is said to be non-dominated if there is no x that dominates it.

 

Therefore, we can define a Pareto optimal point x*, also called a Pareto point, as one that is not dominated by any other x in the search space.

 

In Figure 2 a) it has been represented three Pareto points as red dots named A, B and C. It can be observed that there is no other point in the population that dominates them. Thus, in any multi objective optimization problem, there will be a group of individuals that fulfil this requirement and that could be therefore considered as Pareto points. This group conform then the so called Pareto Front, represented in Figure 2 b) by the red dots and blue line.

 

 

Figure 2: a) Three Pareto points within a group, b) Pareto front. The red arrow represents the direction that the optimization process should follow and aim to, and for which the algorithm will define new individuals, and consequently, find better results in new generations of groups.

 

4. COST PARAMETERS FOR ANALYSED CASES

 

In sound diffusers, the polar distribution of the reflected waves and their resultant pressure is determined by the choice of well depths. This feature is what determines the physical size of the object, and it is vital to keep them as thin as possible. Therefore, the depth of the diffuser shall be one of its cost parameters, and it will be necessary to minimize its value. On the other hand, due to their own physical properties, Schroeder diffusers are only useful for a certain range of frequencies. Since in room acoustics, the most problematic range for absorption and distribution is the low frequency band, then it is key to know until which lower limit it is applicable. Beyond this limit, the diffuser will act as a flat surface in regards to the wave. Then, this lower limit shall be the second cost parameter, and it will be necessary to minimize its value. This means that, in regards to sound diffusers, there must be a compromise between its greatest depth and the lowest frequency it works at. It will not make sense if a diffuser is extremely thin, but its lowest applicable frequency is too high, and vice-versa.

 

As for acoustic barriers, these implementations are made of elements whose size is comparable to the wave-lengths of the frequency we want to act on. Usually, these elements are cylinders, and their weights are the features that set their physical sizes. By increasing or reducing their radius, we will affect their weights accordingly. Therefore it is vital to keep them as light as possible, being their weight the first cost parameter. However, we cannot forget that this weight depends on the density of the chosen material, and that is why we are going to plot all the results using arbitrary units, which may be then scalable whenever a specific material is selected. On the other hand, the influence of an acoustic barrier is measured by the Insertion Loss (IL). This parameter compares the behaviour of sound before and after placing the acoustic barrier, measuring in both cases the sound pressure produced by the source (i.e., p₁ and p₂). Therefore, the Insertion Loss shall be the second cost parameter, and it will be required to obtain the highest amount of it. This means that, in regards to sonic crystal sound barriers, there must be a compromise between its weight and the greatest amount of losses it is capable to provide. In other words, it will not make sense if a barrier is significantly light but its losses are negligible, and vice-versa. Finally, it is important to remark the fact that, for the purpose of this project, the sound barriers that were designed are considered to have an infinite height. This is because the FDTD simulations that were carried out had been designed for a two dimensional field.

 

5. GENETIC ALGORITHM BASED ON CONCEPTS

 

Concepts are an additional feature that can be implemented in genetic algorithms. When producing new individuals by randomly mixing features of two existing ones, we might come up with a sample that fits in the so called Pareto Front. A new interesting approach would be to see what happens when these individuals are changed proportionally according to one of their features. For the purpose of this project, which deals with acoustic applications, the features that are more suitable to be changed are their geometrical characteristics, i.e. changing their size-related cost parameters defined previously.

 

The individuals that are obtained applying this method are known as concepts. As schematically represented in Figure 3, after generating them, it must be checked how many individuals each concept has in the Pareto front. In that respect, the worst ones shall be eliminated, Also, the rest will be combined to generate new samples that will substitute those that were removed, keeping the population constant. The amount of concepts that are removed and substituted must be defined in the process. Nevertheless, it is also important to determine the amount of concepts that are worth to be applied. This is due to the fact that, the greater the amount of concepts, the longer it will take the algorithm to come up with a solution. This project aims to find the best amount of concepts that may be used for the purposes of the optimization. In order to show an illustrative example of this implementation, we made use of the cases proposed previously.

 

Finally, the amount of concepts used for each iteration of the optimization process should have been defined. This was approach in such a way that several cases were tested. For this comparison to work, the same total amount of samples should remain constant. This was set as a total of 2¹⁰ = 1024 individuals. Moreover, the amount of concepts should be such that it allowed to equally divide the individuals among them. This requirement was met by using the first six powers of 2, resulting on the six different cases that are defined in Table 1. We had to compare the performance of each of this cases in order to figure out which implementation would be more suitable for our optimization process.

 

Table 1: Amount of concepts and individuals for sound diffusers.

 

 

5.1. Concepts applied to sound diffusers and sonic crystal sound barriers

 

When optimizing a set of diffusers (i.e. the individuals), it is interesting to check what happens when we generate new samples (the concepts) by changing their size accordingly to a scale factor. This scale factor is approached in such a way that the deepest well of this group of individuals is normalized to 1, making the rest of the wells lie on an specific value between 1 and 0. Thus, once the diffusers’ depths are normalized, new individuals will be generated by scaling its deepest well to an interval between 0 and 1. The scale factor shall be divided in the same amount of steps as the number of individuals per concept, without applying the same scale to more than one individual. In Figure 4 a), we can observe an original diffuser with a maximum depth of dn, that in terms of our implementation will have a value of 1, and in Figure 4 b), the same diffuser reduced by a factor of 0.5 (i.e. maximum depth of dn/2).

 

 

Figure 3: Basic scheme of the concepts implementation.

 

 

Figure 4: Generating a new sample from an original sound diffuser (concept). It can be observed how the depth of every well has been reduced to half its length.

 

Each of this cases would result then in a different amount of concepts and individuals per concept (hereafter, ipc). For illustration purposes, two examples have been represented in Figure 5.

 

 

Figure 5: Diffusers for 2 ipc (512 concepts) and 8 ipc (128 concepts). Red dots represent the individuals within the Pareto front. Black ones represent the entire population. Based on the way the population is represented, the amount of individuals per concept is easily deduced.

 

As for sound barriers, once a proper one with a suitable weight is found, we might generate new versions of this original one by adjusting its radius (and therefore, its weight) by making use of a scale factor that went from 0.6 to 2. In Figure 6 a), an optimal case of a sound barrier is represented. In Figure 6 b), the same structure is represented, but with a reduced scale, generating a concept from an original individual.

 

 

Figure 6: Generating a concept from a sonic crystal sound barrier.

 

Besides, the scale factor applied to the sound barriers might make them smaller, but also bigger. It may be counterproductive for their weight cost parameter, but at the same time it may generate individuals with better Insertion Loss properties. This might be interesting for some applications. Therefore, and as represented in Figure 7, each of the concepts shall have a different behaviour, with individuals enhancing the already existing barrier, and others worsening its performance. As can been observed, sometimes barriers with slightly greater weight have significantly better acoustic performances.

 

 

Figure 7: Example of concepts for a sonic crystal sound barrier.

 

For illustration purposes, two examples have been represented in Figure 8, and in this case, with not-so-clearly distinguishable amount of concepts. It can be observed that a limit of -6 dB of Insertion Loss was established. This was set this way in order to prevent the algorithm from generating individuals whose weight is so light that the Insertion Loss produced is almost negligible. This way, every time an individual with more than -6 dB of Insertion Loss was generated, its value was set to 0 dB, making its removal from the population more likely.

 

 

Figure 8: Acoustic barriers for 32 ipc (32 concepts) and 16 ipc (64 concepts). Red dots represent the individuals within the Pareto front. Black ones represent the entire population.

 

6. RESULTS

 

6.1. Results for sound diffusers

 

Due to the randomness of the method, two different calculations were ran for each implementation. This allowed the algorithm to provide more reliable results. Therefore, we came up with 12 different set of solutions (2 versions of 6 different individual per concept implementations). A suitable amount of generations was established as 2¹²- 2¹⁰ = 3073 generations. In order to have a basic understanding on the results provided by our algorithm, the Pareto front of each situation was represented together in Figure 9, using different colours for each case. This means that, for every amount of concepts, two different Pareto front were represented.

 

 

Figure 9: Diffusers’ Pareto front of all cases represented together. Each colour represents the individuals obtained after applying different amount of concepts.

 

There are barely any difference among cases: we cannot determine which amount of concepts provides the best set of solutions for the purpose of this optimization process. It is important to remark the fact that the approach of 1024 concepts (i.e. 1 individual per concept) is equivalent to a classic multi-objective optimization. Again, the results obtained with concepts are not substantially better. Nevertheless, in order to carry out a deeper analysis, and really come up with justified conclusions, we are going to analyse the behaviour of each of the concepts of the individuals in the Pareto front, i.e. checking how its minimum frequency varies throughout the established set of depths of its case.

 

At this point, it is important to define the term stability when referring to sound diffusers: A concept will be considered stable when several of its individuals are not only in or in the neighbourhood of the Pareto front, but they also appear in a consecutive way (two or more individuals in correlative positions). This way, we assure that small modifications on the depth of the diffuser does not produce extreme modifications on its frequency response. After running all the optimizations, we noticed that the results provided by the designed code showed, in general terms, a very erratic response in regards to their minimum frequency, independently of the amount of concepts employed. This meant that, even though a set of individuals were found to be appropriate under specific circumstances, they were extremely unstable and could be hardly applicable for real cases. This is because slightly changes or errors in their depth may lead to a completely different frequency response.

 

Nevertheless, there were cases that showed a fairly good response in terms of stability. Despite of the fact of not being actually in the Pareto front, they were close enough to be considered good individuals. This led us to set three groups of individuals that could be found within the implementation of this algorithm:

 

• Case 1 - Poor stability: The dominant case. Its minimum frequency has a very erratic behaviour depending on the depth (i.e., its stability is highly fragile).

• Case 2 - Moderate stability: Its minimum frequency is relatively stable for some individuals. Not all of them are necessarily in the Pareto front, but are close enough to it to be considered good samples.

• Case 3 - Good stability: Their minimum frequency is fairly stable for a high amount of individuals. Most of the times they do not lie within the Pareto Front, but they are close enough to be considered good.

 

 

Figure 10: Examples for each case

 

In order to check the virtues of the method, we are going to compare the results obtained after the optimization process with individuals that were randomly generated. To do a proper comparison, we must use the same amount of individuals in both approaches, and also, compare them with the same scale factor. This means two conditions:

• Since in each iteration we generated 32 new individuals, and in total we ran 10¹²- 10¹⁰ = 3072 generations, we must then generate 3072 × 32 = 98304 new individuals randomly.

• Since the individuals generated randomly were unique (i.e. no concepts were applied), we must compare them with the individuals that had a scale factor of 1 within our optimization process.

 

This way, we obtained the results represented in Figure 11. As can be observed, the algorithm produces nearly the same edge-individuals in the Pareto front. However, it is visible that the implemented method produced slightly better results. Besides, the randomly generated individuals are seem to appear almost everywhere in the graph, while the method reduced those samples substantially. Anyhow, these results can be interpreted once again as a sign of the complexity of the problem and the lack of significant improvement obtained with the method, which may be a consequence of the nature of the application itself and not necessarily of the optimization method.

 

 

Figure 11: Acoustic barriers’ Pareto front of every amount of concepts.

 

6.1. Results for acoustic barriers made of sonic crystals

 

As represented in Figure 12 it can be seen that each case generated a rather different solution in most areas of the Pareto front, with some similar behaviours between pairs. At first sight, it is clear that the approach with 64 concepts provides the best set of solutions (blue dots), with individuals going further in the Insertion Loss values, and also, being substantially lighter from -11,5 dB and below. Moreover, the cases of 32 and 128 concepts show mixed behaviours, with generally heavier individuals (red dots) or less wide isolation properties (black dots). Nevertheless, it is also visible that all the implementations show a fairly linear response from around -11 dB and above, with interesting and different results below this area. In fact, is that section of the graph, where solutions diverge, that shows the greatest advantage of the 64 concepts implementation. Additionally, if we assume the case for 1024 concepts (1 ipc) as a classic multi-objective approach, we can conclude that optimizing by making use of concepts generally produces better results.

 

 

Figure 12: Acoustic barriers’ Pareto front of every amount of concepts.

 

As implemented with the sound diffusers, we are going to carry out a deeper analysis by analysing the behaviour of each of the concepts of the individuals in the Pareto front, i.e. checking how its weight (expressed through arbitrary units that may be scaled afterwards)varies throughout the established set of Insertion Losses. This way, we would be able to really come up with justified conclusions. We will present the concepts of some of the individuals that reached the Pareto front. This way, we tried to look for interesting behaviours along the results of every implementation. This has been represented in Figure 13.

 

First of all, it can be observed that for every single concept, the individuals that reached the Pareto front were always the lightest ones, especially in the area where the ratio weight/losses seems to be linear. Additionally, it can be observed that, despite the fact that the 64 concepts implementation provided the most interesting results, it is the 32 concepts case that generally produces more stable ones.

 

 

Figure 13: Most common performance for 32, 64 and 256 concepts barriers. Red dots represent the individuals within the Pareto front. Black solid line represents an specific concept.

 

As with the sound diffusers, we checked the virtues of the method by comparing the results obtained after the optimization process with individuals that were randomly generated. This way, we obtained the results represented in Figure 14. As can be observed, the algorithm produces significant better results than those generated randomly. It is visible that the random method tends to produced samples in a very specific area, while the different implementations of our optimization process are capable of define new limits.

 

 

Figure 14: Comparison with randomly generated acoustic barriers. It can be observed the tendency of this method when generating new individuals.

 

7. CONCLUSIONS

 

In the case of sound diffusers, it could be seen that not big differences were found in the Pareto front when changing the amount of concepts. This may be interpreted as a independence between this parameter and the efficiency of the process. Additionally, it has been shown that the results obtained for sound diffusers were extremely unstable. Values that lied on the Pareto front were only adequate for a highly accurate measure: few cases were found where the depth of the wells could have a fairly error range or tolerance. This can also be interpreted as a sign that the efficiency of the algorithm has prevailed over the stability of the results, due to the intrinsic complexity of the application. However, it could be observed that the stability of the individuals was indeed affected by the amount of concepts employed. The higher the amount of concepts, the more stable the optimal individuals were.

 

In regards to the sound barriers made up of sonic crystal structures, we did find a relation between the amount of concepts and the results in the Pareto front, and on top of that, again a relation between this parameter and the stability of the individuals. Nevertheless, these two criteria did not match. This meant that we must make a choice in order to produce either the most efficient or the most stable results.

 

The difference of complexity between applications was also reflected when comparing the optimization with concepts with a classical multi-objective approach (equivalent to our 1024 concepts, or 1 individual per concept). In the case of sound diffusers, the improvement was very subtle, while in the acoustic barriers the difference was very clear. Also, for both cases, we observed that randomly generated individuals were always worse than those obtained with concept based optimization. Nevertheless, the degree of improvement depended again on the complexity of the application that was being analysed.

 

Thus, after all these conclusions, we may enunciate some properties that can be deduced out of these results:

• Although the performance of the Pareto front (i.e. the best individuals of the optimization process) may be directly influenced by the amount of concepts, this does not necessarily occur in every possible application.

• The stability of the concepts, and consequently the practical applicability of the results, is directly influenced by the amount of concepts. This is because, the higher the amount of concepts, the more accurate the results can be. However, it has not to be necessarily a direct relation between them.

• The different amount of concepts provides enough sets of results to, once the algorithm is finished, decide which approach fits the best with our application. Thus, depending on our needs, we might choose the best results according to our cost parameters, or those individuals with the most stable performance.

• The complexity of the application directly influences the results of the algorithm. Some of them may produce better results than other when applying concept based optimizations.

• The method always generates better results than classic multi-objective methods and random approaches.

 

Therefore, after all the implementations and analysis of the obtained results, we can come up with the conclusion that this process has proven to be useful for reducing the size of the acoustic applications. As previously stated, it might not always be suitable to look for the smallest results, but in the end, it is up to the developer to choose one criteria or another.

 

8. ACKNOWLEDGEMENTS

 

This work was supported in part by the grant RTI2018-096904-B-I00 founded by MCIN/AEI/ 10.13039/501100011033/ and by “ERDF A way of making Europe”.

 

9. REFERENCES

 

1. T. Cox, P. D’Antonio - “Determining Optimum Room Dimensions for Critical Listening Environments: A New Methodology” - 2001..

2. J. Redondo, J. M. Herrero, L. Godinho, R. Patraquim, T. Cox - “Application of multi-objective optimization techniques to the design of sound diffusers” - 2001.

3. J. M. Herrero Durá, X. Blasco,J. V. Sánchez Pérez, J. Redondo - “Design of sound phase diffusers by means of multiobjective optimization approach using ev-MOGA evolutionary algorithm” - 2016.

4. M. Martínez, J. Herrero, J. Sanchis, X. Blasco, S. García-Nieto - “Applied Pareto multi-objective optimization by stochastic solvers” - April 2009.

5. J. M. Herrero Durá, S. Garcia-Nieto, X. Blasco, V. Romero García, J. V. Sánchez Pérez, L. M. García-Raffi - “Optimization of sonic crystal attenuation properties by ev-MOGA multiobjective evolutionary algorithm” – 2009.

6. E. D. O. Cancino - “Evaluación de la pérdida por inserción de una barrera acústica aplicada en un proyecto lineal” - Universidad Austral de Chile, 2002.

7. J. Galiana-Nieves - “Sobre la medición del aislamiento de barreras acústicas basadas en cristales de sonido“ - Universitat Politècnica de València, 2020.

8. V. J. Sánchez-Morcillo, K. Staliunas, V. Espinosa, I. Pérez-Arjona, J. Redondo, E. Soliveres - “Propagation of sound beams behind sonic crystals” - 2009.

9. F. Morandi, M. Miniaci, A. Marzani, L. Barbaresi, M. Garai - “Standardised acoustic characterization of sonic crystals noise barriers: sound insulation and reflection properties” - 2016.

10. M. P. Peiró-Torres, M. J. Parrilla-Navarro, M. Ferri García, J. M. Bravo, J. V. Sánchez Pérez, J. Redondo - “Sonic Crystals Acoustic Screens and Diffusers” - 2019.

11. M. Long, M. Levy, R. Stern - “Applications of Modern Acoustics Architectural Acoustics” - 2006, Academic Press.

12. H. Kuttruff - “Room Acoustics” - 2016, CRC Press.

13. M. Barron - “Auditorium Acoustics and Architectural Design” - 2009, Spons Architecture Price Book.

14. T. Cox, P. D’Antonio - “Acoustic Absorbers and Diffusers - Theory, Design and Application” - 2004, SPON.

15. D. Simon - “Evolutionary Optimization Algorithms” - 2013, Wiley.

16. ISO 17497-1 - “Measurement of the random-incidence scattering coefficient in a reverberation room” - 2004.

17. ISO 17497-2 - “Measurement of the directional diffusion coefficient in a free field” - 2012.

 

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¹ dbenara@epsg.upv.es

² jaiganie@doctor.upv.es

³ fredondo@fis.upv.es

⁴ daraso@doctor.upv.es

⁵ juaherdu@isa.upv.es