This letter develops a simple approach of duct mode identification and reconstruction based on genetic algorithms, which can extend the azimuthal mode order range compared to the conventional method based on the (spatial) discrete Fourier transform. The underlying principle is reconstructing the dominant mode from the modal identification forward model through optimization by exploiting the sparsity of the mode amplitude vector. The performance is experimentally demonstrated for detections of one and two azimuthal modes under noisy conditions with nondominant modes. Overall, the proposed genetic-algorithm-based framework for solving acoustic inverse problems is beneficial to duct acoustic testing, particularly design evaluations of fan blades and acoustic liners for aeroengines.

## 1. Introduction

Acoustic mode identification, also known as mode detection and mode decomposition, is a widely used testing method to reveal the generation, propagation, radiation, and attenuation of turbomachinery noise, which, thus, becomes an effective way of evaluating designs of fan blades and acoustic liners for aeroengines.^{1}

The acoustic mode can be sorted as azimuthal or radial orders (*m*,*n*). Because acoustic energy is mainly concentrated in the azimuthal direction, azimuthal mode detection is usually preferred in practical tests. The implementation is rather simple, which generally adopts a ring of uniformly spaced acoustic sensors, in-duct or out-of-duct, and then infers modal information from the measurement through spatial Fourier decomposition.^{2} This method has to obey the Shannon-Nyquist sampling theorem in the spatial sampling domain, i.e., the azimuthal mode order under examination is at most half the number of the spatial sampling points. Given a fixed number of sensors (limited by testing capabilities or analyzing existing data from past tests), extending the detectable mode range is seemingly a wild wish in the view of traditional sampling theories but has already come true thanks to certain techniques. Early work^{3} proposed a method based on the conjugate gradient method, which was able to detect azimuthal modes $ | m | \u2264 79$ with only 100 acoustic sensors by suppressing the mode sidelobes. Progress in compressive sensing has also offered various solutions. Readers of interest can refer to a series of investigations^{4–8} on this topic. By exploiting the sparsity of the mode amplitude vector, the compressive-sensing-based methods solve an underdetermined system of equations through optimization. However, this kind of method usually involves very complex mathematical theories^{9} and, therefore, sets a high threshold for beginners. By contrast, this work proposes a similar but much simpler method based on the well-known genetic algorithm (GA), which borrows only a few easily understood concepts from nature evolution and can be readily implemented on some simply accessed platforms/toolboxes. Based on this idea, our recent work^{10} has successfully developed a beamforming method. As a follow-up study, this paper aims to present an extended application to the acoustic mode identification problem.

## 2. Methodology

### 2.1 The conventional method

*P*is the complex sound pressure,

*m*and

*n*are the azimuthal and radial mode order, respectively,

*A*$ m n$ is the amplitude of the (

*m*,

*n*)th order mode,

*k*$ z$ is the axial wavenumber, and $ \zeta m n$( $\theta $,

*r*) is the mode shape function described by

*J*$ m$ is the

*m*th order Bessel function of the first kind,

*k*$ m n$ is the

*n*th radial wavenumber and can be obtained from the hard-wall boundary condition.

*F*$ m n$ is the modal normalization constant introduced to satisfy the orthogonality condition over the duct cross-section area. Practical acoustic testing frequently considers the azimuthal mode alone at a certain cross section ( $ z = constant$) and measures acoustic data at the duct wall (

*r*=

*R*). Then, Eq. (1) can be simplified into

*A*$ m$ denotes the amplitude of the

*m*th azimuthal mode. This expression can be written as the following matrix form:

*N*wall-flush mounted sensors uniformly spaced along the circumferential direction. Let

*P*$ j$ and $ \theta j$ denote the measured (complex) sound pressure and azimuthal angular location of the

*j*th sensor in the array ( $ j = 1 , 2 , \u2026 , N$). According to the spatial Fourier decomposition, the mode amplitude,

*A*$ m$, can be easily calculated by

### 2.2 The proposed method

A major part of the turbomachinery noise originates from rotor-stator interactions. The order of the produced azimuthal mode can be described by the Tylor-Sofrin selection law.^{11} That is, there are only a few modes propagating in the duct, and the vector of the mode amplitude, ** A**, is sparse. By exploiting the sparsity, an optimization method can be established based on GAs, the target of which is to search for a good solution,

**, to the modal identification model of Eq. (4). Before describing the process of the proposed method, some essential concepts of a GA are introduced as follows.**

*A*#### 2.2.1 Chromosome representation

Specifically, this concept refers to encoding and decoding. Assuming a mode amplitude vector, ** A**, containing

*s*dominant modes (

*s*sparsity), the encoding is to transform

**into a new matrix, $ A \u2032 \u2208 C 2 \xd7 s$. The first row of $ A \u2032$ stores the index of the dominant mode in**

*A***, and the second row is the (complex) amplitudes of the mode. With this operation, the algorithm can search for the optimal solution more efficiently. Decoding is the inverse transformation, i.e., from $ A \u2032$ to**

*A***, according to the modal index and amplitude stored in $ A \u2032$. The operation is demonstrated in Fig. 1. In the classic terminology of GAs,**

*A***and $ A \u2032$ refer to the phenotype and genotype of the solution, respectively.**

*A*#### 2.2.2 Genetic operators

Genetic operators include crossover, mutation, and selection. As demonstrated in Fig. 1, crossover produces new individuals ( $ A \u2032$) by exchanging some columns of two randomly selected individuals. Mutation occurs at a small probability, which is empirically set in the algorithm. With this operation, a single individual is arbitrarily chosen, and the corresponding values are randomly modified. The crossover and mutation operations produce additional individuals. To maintain the size of the population, those individuals leading to a large fitness value are eliminated.

#### 2.2.3 Fitness function

**. The former term is determinate while the latter varies with $\alpha $ and $\beta $. These two parameters were previously carefully evaluated,**

*A*^{10}and $ \alpha = constant$ and $ \beta = 0$ yielded results with the best accuracy. For the present problem, which is largely different, selections of $\alpha $ and $\beta $ also deserve a study. The relevant analysis will be presented in Sec. 3.

#### 2.2.4 Termination criterion

Two stopping criteria are used in this work: (i) the fitness value from Eq. (6) reaches the predefined threshold of 5%, and (ii) the iteration steps reach the predefined upper limit of 1000. The algorithm ends when either condition is satisfied.

The whole process of the GA mode identification is presented in Fig. 1. First, an initial population of 1000 individuals is created. The population size here is empirically determined. Note that a large number of individuals leads to computational burdens while a small scale of populations has difficulties in quickly converging to optimal solutions, both of which influence efficiency. For each iteration of the population, the fitness is evaluated based on Eq. (6). If the stopping criterion is not met, crossover, mutation, and selection are operated to produce the next generation, $ A \u2032$. Finally, an optimal solution can be found and then decoded for further analysis.

## 3. Experimental validation

A duct acoustic rig designed with a controllable source system is used for validations, as shown in Fig. 1. The testing facility is newly built in an anechoic chamber at the Northwestern Polytechnical University. Flow conditions have not yet been realized. The experimental system mainly includes a cylindrical duct with a radius of $ R = 300 $ mm and a total length of3.75*R*, which is terminated with sound-absorbing materials. To produce acoustic modes, an azimuthal mode synthesizer with a ring of 16 speakers (BMS 5531ND type compression drivers, BMS Speakers GmbH, Hannover, Germany) is deployed. The signals feeding to these speakers are generated by an NI PCI-6723 card (National Instruments, Austin, TX) and then amplified by two eight-channel QSC CX168 type power amplifiers (QSC Pro Audio, Costa Mesa, CA). The produced modes are limited to $ | m | < 8$, owing to the Shannon sampling theorem, and most of them show a sufficiently high signal-to-noise ratio (SNR). A circumferential sensor array consisting of 24 GRAS 40PH microphones (GRAS Sound & Vibration, Holte, Denmark) is used to acquire in-duct acoustic data and enable modal identification. The conventional method based on discrete Fourier transform (DFT) adopts the whole array to identify mode spectra, although fewer sensors can be used if the mode order, *m*, is not high. The GA processes data only from randomly selective channels therein. The accuracy of the proposed method will be analyzed by making comparisons to the conventional method of Eq. (5).

First of all, the conventional method and proposed method are compared for a single-mode case and a dual-mode case under different frequencies. Representative results are shown in Fig. 2. The GA is performed with only six sensors at the angular positions of $ \theta = [ 1 , 5 , 9 , 13 , 17 , 21 ] ( 2 \pi / 24 )$. Here, the fitness function is tentatively determined with $ \alpha = 0.1$ and $ \beta = 0$. Figure 2(a) presents the mode spectrum of *m* = 6 at $ f = 4$ kHz. According to the conventional method, the synthesized mode, *m* = 6, has a well-dominant amplitude of $ 123.8$ dB, whereas the other modes are much weaker with an amplitude below $ 110$ dB and, thus, not shown. The proposed method accurately identifies the dominant mode with an amplitude deviation less than $ 0.5$ dB. Similar observations can be made in Fig. 2(b), where the deviations between the two methods are also less than $ 0.5$ dB. Such a small disparity in the mode amplitude only makes slight visual differences in the reconstructed modal fields at the measurement cross section, as shown in Figs. 2(c) and 2(d).

The results from different combinations of $ \alpha = 0.01 , 0.1 , 1$ and $ \beta = 0 , 0.5 , 1 , 2$ are analyzed. Given a fixed group of $\alpha $ and $\beta $, statistical analysis based on the Monte Carlo method is adopted. Specifically, provided with single-mode test data, a certain number (*N*) of the sensors are randomly selected from the whole array to perform the GA. This process is repeated 200 times, and then the corresponding averaged error [Eq. (7)] is calculated. Figure 3 presents the results of *E* with varying sensor numbers for different fitness functions. For the case with $ \alpha = 0.1$, the error falls with the increase in the sensor number, which is overall below 10% for all the cases. The reason for this behaviour can be attributed to the coherence of the modal matrix, which is similar to the compressive sensing. With the increase in *N*, the coherence is statistically decreased. Moreover, different choices of $\beta $ result in an insignificant disparity in the accuracy. However, the observation becomes different for the other two cases in Figs. 3(a) and 3(c). For the smaller $\alpha $ of 0.01, $ \beta = 2$ gives a very large error approaching 50%. This is because $ \beta = 2$ cannot warrant a solvable sparse solution of ** A**. Similar results for $ \beta = 2$ also appear in Fig. 3(c). However, for this case with a relatively large $\alpha $ of 1, the error from $ \beta = 1$ becomes abnormal, which might be attributed to a large residual error of the corresponding fitness function. In addition, the case with $ \beta = 0$ yields a lower error than $ \beta = 0.5$.

From the analysis here and together with our previous study,^{10} some consistent conclusions can be drawn regarding the choices of $\alpha $ and $\beta $. Specifically, $ 0 \u2264 \beta \u2264 1$ can be the first choice to constitute a convex regularization term of the fitness function. In the previous beamforming problem and current mode identification application, $ \beta = 0$ always yields the most accurate and reliable results. In addition, a small $\alpha $ (i.e., below 0.1) is suggested. The following results to be discussed are obtained under $ \alpha = 0.1$ and $ \beta = 0$.

Finally, the effect of the SNR on the reconstruction error is investigated. The noise interference can be originated from various factors in practical tests^{6} and collectively reflected as “noise modes” in the mode spectrum. Stronger noise modes basically result in a less sparse mode amplitude, $ A m$. Hence, SNR is directly linked to the sparseness and, thus, influences the performance of our method. To measure the SNR, a parameter named dynamic range ( $\Delta $) is used, which is defined as the difference in the amplitude between the target mode and strongest noise modes.^{6} For instance, the strongest noise mode in Fig. 2(a), although not explicitly shown, is at *m* = 7 with $ 105.5$ dB, resulting in $ \Delta = 123.8 \u2212 105.5 = 18.3$ dB. The reconstruction error for different cases with representative $\Delta $ is presented in Fig. 4. As indicated in the legend, these data are obtained at different frequencies, and $\Delta $ varies between $ 14.7$ dB and $ 23.6$ dB. It is observed that the results for $ \Delta \u2265 20$ dB are almost identical. Given a fixed sensor number, *E* is overall decreased with the increase in $\Delta $, which means improved accuracy under the weakened noise interference.

The analysis here indicates the importance of the sparseness to the proposed method. For extreme cases in which the SNR is low and the mode spectrum is not sufficiently sparse, one can adopt the second order model of modal identification developed in our previous work^{12} [see Eq. (13) therein] to enhance the sparsity. Then, it will be very straightforward to apply this GA to reconstruct the mode power spectrum, $ A m 2$.^{2} In addition, an extension of the present method for radial mode identification is also possible once the forward model is prepared. Relevant investigations for these ideas are still ongoing.

## 4. Summary

This paper develops a novel acoustic mode identification method based on the well-known GA, which recovers the mode spectrum by making use of the sparsity. This method is very simple without esoteric theories but can effectively extend the range of mode detection compared to the conventional method.

The performance of the proposed method is experimentally demonstrated. The results indicate that the mode amplitude can be accurately reconstructed. The error is decreased with the increase in the sampling points and SNR. This work shows an extended application of our proposed method.^{10} With these efforts, the potential of the GA for solving certain acoustic inverse problems has been well demonstrated, which can be particularly beneficial to acoustic testing areas.

## Acknowledgments

The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 12202364 and 12072277), the Fundamental Research Funds for the Central Universities (Grant Nos. G2022WD01008 and G2022KY0608), the Basic Research Program of Taicang (Grant No. TC2022JC05), the Laboratory of Aerodynamic Noise Control of China Aerodynamics Research and Development Center (Grant No. ANCL20220304), and the Key Laboratory of Aeroacoustics of AVIC Aerodynamics Research Institute (Grant No. XFX20220204).

## Author Declarations

### Conflict of Interest

The authors have no conflicts of interest to disclose.

## Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

*A Mathematical Introduction to Compressive Sensing*