# Topology optimization of sound absorbing layer for the mid-frequency vibration of vibro-acoustic systems

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## Abstract

Due to the significant difference of dynamic properties between the fluid medium and the structure, when a vibro-acoustic system is subjected to a higher frequency excitation, it may typically exhibit mid-frequency behavior which involves different wavelength deformations and is very sensitive to the uncertainties of the system. This paper deals with optimized distribution of a sound absorbing layer for the mid-frequency vibration of vibro-acoustic systems by using hybrid boundary element analysis and statistical energy analysis. Based on the solid isotropic material with penalization approach, an artificial sound absorbing material model is suggested and the relative densities of the sound absorbing material are taken as design variables. The sound pressure level at a specified point in the acoustic cavity is to be minimized by distributing a given amount of sound absorbing material. An efficient direct differentiation scheme for the response sensitivity analysis is proposed. Then, the optimization problem is solved by using the method of moving asymptotes. A numerical example illustrates the validity and effectiveness of the present optimization model. Impact of the excitation frequency on optimized topology is also discussed.

## Keywords

Mid-frequency Vibro-acoustic system Sound absorbing layer Boundary element Statistical energy analysis Dynamic topology optimization## 1 Introduction

Vibro-acoustic systems are widely used in vehicles such as automobiles, trains, ships, and rocket launchers. These vehicles may be subjected to complex environmental excitations during their operation, resulting in strong structural vibration and harmful high noise levels. Early studies on vibration and noise control (see, e.g., Christensen et al. 1998a, b) mainly alter the shape and size of system components to control the generation of noise at the sound source. With the development of topology optimization techniques (Sigmund 2001; Bendsøe and Sigmund 2003; Allaire et al. 2004; Bendsøe 1989; Bendsøe and Kikuchi 1988; Wang et al. 2003; Zhou and Rozvany 1991) more and more researchers began to study the noise control problem of vibro-acoustic systems by using topology optimization techniques. Yoon et al. (2007) dealt with the problem of topology optimization of vibro-acoustic systems using a mixed finite element (FE) formulation (Zienkiewicz and Taylor 2000; Bathe 2008), in which the acoustic cavity is enclosed by a finite boundary. Kook et al. (2012) proposed a design method for acoustical topology optimization considering human hearing characteristics. Shu et al. (2014) studied the topology optimization of vibro-acoustic systems for minimizing sound pressure by using the level set method. To date, the studies on topology optimization of vibro-acoustic systems have been mainly focused on the optimized distribution of structural materials, while the topology optimization of damping or sound absorbing layers has been rarely involved.

However, the design of a large vehicle should not only consider vibration and noise reduction but also meet other design requirements including stiffness, strength, stability, aerodynamics, and hydrodynamics. Therefore, design methods for vibration and noise reduction by finding the optimized layout of structural material have some limitations. At present, a commonly used method is to place damping material on the surface of a structure and sound absorbing material on the sound propagation path to reduce the acoustic radiation and the reflection or transmission of sound waves, respectively. However, a large area of damping or sound absorbing material will cause a sharp increase in system weight, which will not only affect the system performance but also increase the manufacturing cost. In view of the above situation, topology optimization techniques are used to obtain the optimized layout of damping or sound absorbing layers in a vibro-acoustic system. Dühring et al. (2008) studied the optimized placement of damping panels on walls of acoustic cavities by using the solid isotropic material with penalization (SIMP) method (Sigmund 2001; Bendsøe and Sigmund 2003). The sound level can be significantly reduced by optimizing the distribution of the sound absorbing and reflecting material. Akl et al. (2009) developed a mathematical model to simulate fluid-structure interactions based on FE method. A good agreement was obtained between the results obtained from the mathematical model and those from the experiment. Zhang and Kang (2013) presented a topology optimization model to obtain the optimized layout of a damping layer for minimizing the acoustic radiation of damped thin-walled structures. In their paper, the dynamic coupling between the acoustic medium and the structure is neglected. Then, considering the velocity response of the structure which is calculated by FE method as an acoustic excitation, the sound pressure at a reference point is obtained by using the boundary element (BE) method (Ciskowski and Brebbia 1991; Wu 2000). They also proposed a sensitivity analysis scheme using the adjoint variable method. Zhao et al. (2017) studied the optimized design of sound absorbing material distribution within sound barrier structures based on the BE method and the optimality criteria method. A smoothed Heaviside-like function was developed to help the SIMP method to obtain a clear 0–1 distribution. The optimized distribution of the sound absorbing material is strongly frequency dependent according to the results obtained by authors, and the optimization in a frequency band was suggested. Du and Olhoff (2007, 2010) studied the topology optimization problem of vibrating bi-material elastic structures placed in an acoustic medium for minimizing the acoustic radiation and gave a corresponding sensitivity analysis scheme. Their papers assumed that the vibration frequency of a structure has a sufficiently high value, so that the radiation impedance at the structure boundary is approximately equal to the characteristic impedance of the acoustic medium (Lax and Feshbach 1947; Herrin et al. 2003). Thus, the sound pressure in the acoustic field can be easily obtained by using a high frequency boundary integral equation. Considering that resonance and wave-propagation problems are known to be highly sensitive towards parameter variations and the conventional robust topology optimization methods for structural problems are not suitable for the acoustic problem, Christiansen et al. (2015) suggested a new double filter approach and obtained highly robust designs for acoustic problem. Christiansen and Sigmund (2015) provide the experimental validation of an acoustic cavity designed using topology optimization with the goal of minimizing the sound pressure locally for monochromatic excitation.

Based on deterministic methods, such as the FE method, several of the papers mentioned above studied the topology optimization of vibro-acoustic systems. As the frequency increases, the deformation wavelength of the system components will decrease significantly. A fine mesh is required to capture the detailed deformation, typically six to eight elements per wavelength (Simmons 1991; Steel and Craik 1994), which leads to a large number of degrees of freedom (DOF). The computational cost of element-based techniques typically increases due to decreasing wavelengths and multiple reanalyses in the optimization process (Cotoni et al. 2007). Moreover, as the frequency increases, the response of a system will be more and more sensitive to the uncertainties which are inevitably generated during the manufacturing process. Systems with the same nominal geometric and material parameters may produce different responses. At this point, it makes no sense to analyze only one system, and an estimate of average behavior of an ensemble of similar systems with the same nominal properties might be preferred (Ladeveze et al. 2012). As a common statistical method, statistical energy analysis (SEA) (Lyon and DeJong 1995) can give an average prediction for the statistical behavior of systems with little computational cost. However, the assumptions introduced in SEA can only be satisfied when the system is subjected to sufficiently high frequencies (Lyon and DeJong 1995; Langley 1989a). In addition, based on SEA, the properties of a system may be highly generalized as some parameters are independent of the material topology layout, which makes topology optimization impossible.

Due to the significant difference of dynamic properties between the fluid medium and the structure, when a vibro-acoustic system is subjected to a higher frequency excitation, it may typically exhibit mid-frequency behavior in which some subsystems are large compared with a wavelength, while others are small compared with a wavelength (Shorter and Langley 2005b). At present, neither the FE method nor SEA can describe the motion of vibro-acoustic systems well. To address this situation, three types of improved methods have been proposed for the mid-frequency vibration of complex systems. The first type aims to extend the effective frequency range of traditional deterministic methods to the mid-frequency domain (see, e.g., Langley 1989b; Van Vinckenroy and De Wilde 1995; Harari and Avraham 1997; Pluymers et al. 2007; Hinke et al. 2009; Ma et al. 2015b). The second type aims to relax the assumptions in SEA to extend its application to the mid-frequency domain (see, e.g., Keane and Price 1987; Langley 1992; Maxit and Guyader 2003; Mace 2005). The third type combines the deterministic and statistical methods to develop a hybrid model for the mid-frequency vibration of complex systems (see, e.g., Zhao and Vlahopoulos 2000; Shorter and Langley 2005a, b; Ji et al. 2006; Vergote et al. 2011; Zhu et al. 2014; Ma et al. 2015a; Gao et al. 2018). As the most popular hybrid approach, the hybrid FE-SEA method proposed by Shorter and Langley (2005b) divides a complex system into a number of deterministic and statistical subsystems according to the deformation wavelength. The so-called deterministic subsystem which is subjected to long wavelength deformation can be modeled by using the FE method, while the so-called statistical subsystem which is subjected to short wavelength deformation can be modeled by using SEA. The dynamic coupling between the two types of subsystems is described as the reflection and transmission of the vibration wave. Based on the diffuse field reciprocity principle (Shorter and Langley 2005a), a non-iterative relationship between the deterministic and statistical subsystems can be established. Due to the combination of the FE method and SEA, the hybrid FE-SEA method can give an average prediction for the mid-frequency vibration and deal with actual engineering systems. Considering the BE method to describe the motion of an acoustic cavity, Gao et al. (2018) proposed the hybrid BE-SEA method for the mid-frequency vibration of vibro-acoustic systems. Due to the nature of the BE method, the hybrid BE-SEA method not only satisfies the Sommerfeld radiation condition at infinity for exterior acoustic problem but is also more efficient in the modeling stage. Since hybrid approaches are more appropriate than the traditional method for the mid-frequency of complex systems, Muthalif and Langley (2012) studied the active control of mid-frequency vibration by using the hybrid FE-SEA method as an analysis tool. The optimized skyhook damping value and its location were calculated by using the MATLAB GADS toolbox with combined pattern search and genetic algorithms. By using the hybrid FE-wave based (WB) method, Goo et al. (2017) proposed an efficient topology optimization method for bounded acoustic problems. Their method employs the FE method and WB method to respectively model the design and non-design domains to increase computational efficiency and can thus be applied to higher frequency applications that conventional method takes considerable computation time to manage.

The present work studies optimized distribution of a sound absorbing layer for the mid-frequency vibration of vibro-acoustic systems by using the hybrid BE-SEA method. In the topology optimization model, an artificial sound absorbing material model is established by employing the SIMP approach. The design objective is the sound pressure level at a specified point in the acoustic cavity, and the design variables are the relative densities of the sound absorbing material. The corresponding sensitivity analysis scheme is derived by direct differentiation. The basic principles of the hybrid BE-SEA method are outlined in Sect. 2. The topology optimization problem formulation and the corresponding sensitivity analysis scheme are developed in Sect. 3. In Sect. 4, a numerical example is presented to illustrate the efficiency of the hybrid BE-SEA method and the validity of the proposed topology optimization model. The impact of the excitation frequency on optimized topology is also discussed. Finally, conclusions are given in Sect. 5.

## 2 Basic principles of hybrid BE-SEA method

The hybrid BE-SEA method was proposed by Gao et al. (2018) for the mid-frequency vibration of vibro-acoustic systems based on the concept of the hybrid FE-SEA method. Due to the use of the BE method, the hybrid BE-SEA method provided an appropriate model with modeling advantages when the deterministic part of the model is a relatively large acoustic domain.

*Ω*, as shown in Fig. 1, of which the boundary surface

*Γ*

_{a}contains a velocity surface

*Γ*

_{v}, an impedance (sound absorbing) surface

*Γ*

_{Z}and an elastic thin-walled structural surface

*Γ*

_{s}. The velocity and impedance boundary conditions are expressed by a generalized equation which can be written as (Wu 2000)

**p**and

**v**

_{n}respectively represent the sound pressures and normal velocities at nodal points on the boundary of the acoustic cavity.

**C**

_{α}and

**C**

_{β}are constraint coefficient diagonal matrices corresponding to sound pressure and normal velocity, respectively.

**C**

_{γ}is a constraint coefficient vector. The hybrid BE-SEA method (Gao et al. 2018) may be employed for the mid-frequency vibration of the system. The acoustic cavity modeled by the BE method is treated as the deterministic subsystem, while the thin-walled structure modeled by SEA is treated as the statistical subsystem. According to the hybrid BE-SEA method, the response of the statistical thin-walled structure is viewed as the superposition of the direct and reverberant fields (see Shorter and Langley 2005a). Considering the velocity and impedance boundary conditions and the coupling interaction between the acoustic cavity and the direct field of the thin-walled structure, the governing equation of the system can be written as (Gao et al. 2018)

**,**\( \overset{\sim }{\mathbf{H}}=\mathbf{H}+\mathbf{G}{\mathbf{C}}_{\beta }{\mathbf{C}}_{\alpha } \)

**, H,**and

**G**are the influence matrices of sound pressure and normal velocity, respectively.

**D**

_{dir}and

**u**are the dynamic stiffness matrix and displacement vector of the direct field of the thin-walled structure, respectively. \( {\mathbf{f}}_{\mathrm{rev}}^{\mathrm{s}} \) is the vector of the blocked reverberant forces.

**A**is a coupling coefficient matrix which converts the sound pressure vector of the acoustic cavity into the nodal force vector of the direct field of the thin-walled structure.

**T**is the transformation matrix resulting from the non-conforming grids appearing at the fluid-structure coupling face,

*ω*is the angular frequency, and \( \mathrm{i}=\sqrt{-1} \). For the sake of simplicity, (2) will be written as

If there is sufficient uncertainty in the statistical subsystem, the statistics of the blocked reverberant force tend to zero (see Shorter and Langley 2005a). Rewriting (3) in cross-spectral form and averaging over an ensemble of statistical thin-walled structures gives

**S**

_{qq}represents the cross-spectrum matrix of the deterministic DOF. #

^{H}is the Hermitian transpose of #, and 〈#〉 is the ensemble average of #. The subscripts

*p*and

*u*stand for the DOF of the acoustic cavity and the direct field of the thin-walled structure. Also

^{−H}represents the Hermitian transpose of the inverse matrix. Considering the diffuse field reciprocity principle (see Shorter and Langley 2005a), (6) can be rewritten in terms of the reverberant field energy

*E*and the modal density

*n*

_{m}of the thin-walled structure as

The cross-spectrum matrix of the deterministic DOF **S**_{qq} can be obtained by using (4)–(9). The only unknown quantity *E*, at this time, can be calculated by employing the power balance equation of the reverberant field of the thin-walled structure, which is given by (see Gao et al. 2018)

*h*

_{diss}respectively represent the total energy leaving the reverberant field of the thin-walled structure into the acoustic cavity and the total energy dissipating by the damping of the thin-walled structure per unit modal energy density in the reverberant field of the thin-walled structure. \( {P}_{\mathrm{in}}={P}_{\mathrm{in}}^{\mathrm{dir}}+{P}_{\mathrm{in}}^{\mathrm{ext}} \) is the total power input to the statistical thin-walled structure, where \( {P}_{\mathrm{in}}^{\mathrm{dir}} \) is the power arising from the force applied to the acoustic cavity and \( {P}_{\mathrm{in}}^{\mathrm{ext}} \) is the power caused by other sources applied directly to the statistical thin-walled structure. The above parameters can be expressed as

*η*is the damping loss factor of the thin-walled structure, and

**C**

_{sa}is a coupling coefficient matrix which connects the shape functions of the grids of the acoustic cavity and the direct field of the thin-walled structure (see Gao et al. 2018). #

^{∗}stands for the complex conjugate of #.

By substituting (11)–(13) into (10), the ensemble average energy of the reverberant field can be obtained. The energy of the direct field of the thin-walled structure can be neglected (see Shorter and Langley 2005b). Hence, by using (4)–(9), **S**_{qq} can be calculated. Selecting some points inside the acoustic cavity and calculating the corresponding coefficient matrices **g** of sound pressure and **h** of normal velocity, the cross-spectrum matrix of sound pressure at these points can be expressed as (see Gao et al. 2018)

Now inserting (15)–(21) into (14), \( {\mathbf{S}}_{pp}^{\mathrm{in}} \) can be obtained, and then the sound pressure level at the points inside the acoustic cavity can be calculated.

## 3 Topology optimization problem formulation

### 3.1 Topology optimization model

**ρ**is the vector of the relative density design variables describing layout of the sound absorbing material.

*m*

_{Z}represents the total number of boundary elements in the design domain, and each element has one design variable. \( {\overset{\sim }{S}}_{pp}^{\mathrm{in}}\left({\mathbf{r}}_{\mathrm{in}}\right) \) represents the objective function and is a diagonal element of \( {\mathbf{S}}_{pp}^{\mathrm{in}} \) representing the power spectral density of sound pressure (PSDSP) at a specified point

**r**

_{in}inside the acoustic cavity. It is important to point out that the objective function can also be written as the sum of the PSDSPs at more points (i.e., \( f=\sum \left\langle {p}_i^{\ast }{p}_i\right\rangle, i=1,2\dots n \)) to obtain an overall sound pressure reduction in the acoustic cavity (Du and Olhoff 2010).

*f*

_{V}represents the volume fraction and \( {V}_k^0 \) is the volume of sound absorbing material in the

*k*th boundary element when

*ρ*

_{k}=1.

*ρ*

_{min}is the lower bound of the relative density variables, which is set to be 10

^{−6}to avoid possible numerical singularity.

Based on the framework of the SIMP approach, **C**_{α}, **C**_{β}, and **C**_{γ} can be respectively written as

*k*th element of the sound absorbing layer. The penalty factor

*N*>1 is set to be

*N*=3 in this study.

**E**

_{v}and

**E**

_{Z}are location vectors corresponding to velocity and impedance boundary conditions, respectively. \( {\mathbf{v}}_0^{(k)} \) is the velocity vector of the

*k*th element of the velocity surface, and

*m*

_{v}represents the total number of boundary elements on the velocity surface. From (24)–(25), it is clear that

**C**

_{β}and

**C**

_{γ}remain unchanged during the topology optimization process.

### 3.2 Sensitivity analysis

For solving the optimization model of (22) with a gradient-based mathematical programming algorithm, it is necessary to perform sensitivity analysis of the objective and constraint functions with respect to the design variables. The sensitivity equation for the PSDSP is derived by direct differentiation, as follows.

Differentiating the objective function in (22) with respect to the *k*th design variable gives \( \frac{\partial {\overset{\sim }{S}}_{pp}^{\mathrm{in}}}{\partial {\rho}_k} \), a diagonal element of \( \frac{\partial {\mathbf{S}}_{pp}^{\mathrm{in}}}{\partial {\rho}_k} \) which can be expressed as

By using (15)–(21), \( \frac{\partial \boldsymbol{\Delta}}{\partial {\rho}_k} \), \( \frac{\partial \boldsymbol{\Xi}}{\partial {\rho}_k} \), and \( \frac{\partial \boldsymbol{\Pi}}{\partial {\rho}_k} \) in (26) can be, respectively, written as

*k*th design variable, \( \frac{\partial {\mathbf{C}}_{\alpha }}{\partial {\rho}_k} \) can be written as

*k*th design variable and written as

*k*th design variable respectively appear in (37) and (38). By using (2), (3), and (33), \( \frac{\partial {\mathbf{D}}_{\mathrm{tot}}}{\partial {\rho}_k} \) can be expressed as

*k*th design variable, yields

- (i)
The plate energy sensitivity with respect to design variables \( \frac{\partial E}{\partial {\rho}_k} \) is obtained by inserting (42) and (43) into (41).

- (ii)
Substituting (41) into (37), and using (35) and (36), \( \frac{\partial {\mathbf{S}}_{qq}}{\partial {\rho}_k} \) is then calculated.

- (iii)
Inserting (27)–(29) into (26), and using (33)–(35), one can obtain the sensitivity of the cross-spectrum matrix of sound pressure at inner points with respect to design variables \( \frac{\partial {\mathbf{S}}_{pp}^{\mathrm{in}}}{\partial {\rho}_k} \).

- (iv)
The objective function sensitivity with respect to design variables is calculated by using (26).

The sensitivity of the constraint function in the optimization model (22) with respect to the *k*th design variable *ρ*_{k} equals \( {V}_k^0 \).

## 4 Numerical example

*c*

_{0}=340 m/s and

*ρ*

_{a}=1.225 kg/m

^{3}, respectively. Two thin plates with the same dimensions of 0.7 m × 1.0 m × 1 mm are connected respectively to the front and rear surfaces of the acoustic cavity (the gray areas in Fig. 3). The edges of the two plates are all simply supported, and the in-plane deformation of the two plates is ignored. The two plates are made of aluminum, of which the mass density, Young’s modulus, Poisson’s ratio, and damping loss factor are 2700 kg/m

^{3}, 7.1 × 10

^{10}Pa, 0.33, and 0.01, respectively.

The design domain of a sound absorbing layer is connected to the right surface of the acoustic cavity (the hexagon filled area in Fig. 3). A unit velocity excitation is applied over a square region (see the diagonal area in Fig. 3) of 0.04 m^{2} on the left surface of the acoustic cavity located at the point (0, 0.5, 0.25). The other areas are considered to be acoustically rigid.

### 4.1 Response analysis for the mid-frequency vibration of the vibro-acoustic system

In order to show the efficiency of the hybrid BE-SEA method, the responses of the vibro-acoustic system calculated by employing the hybrid BE-SEA method are compared with those calculated by using Monte Carlo simulation.

The impedance of the sound absorbing material is set to be a large real number 10^{40}, and all element-relative densities of the sound absorbing material in the design domain are set to be 1. The frequency range considered here is from 1 to 400 Hz with a frequency step of 1 Hz. Since the acoustic cavity and each thin plate respectively have 7 and 79 modes below 400 Hz, which illustrates the modal density of the acoustic cavity and each plate are significantly different, the systems will exhibit typical mid-frequency vibration behavior consisting of a deterministic acoustical behavior and a statistical structural behavior, within the frequency range of interest. In the hybrid BE-SEA model, the acoustic cavity is modeled by using the BE method, while the two plates are modeled by using SEA. A pure FE model is employed in the Monte Carlo simulation, and a regular fine FE mesh requires to be established to capture detailed deformation. Considering the influence of the uncertainties of the system, an ensemble consisting of 500 samples is generated by randomly choosing 200 points within each plate and adding 0.1% of the mass of one plate at each point.

Details of two analysis models

Analysis model | Element type | Element size (m) | Number of elements per wavelength at 400 Hz | Number of DOFs | Calculation time (h) | |
---|---|---|---|---|---|---|

Hybrid BE-SEA | Cavity | 4-node quadrilateral | 0.050 | 17 | 1242 | 7.9 |

Direct field | 4-node quadrilateral | 0.025 | 6 | 1189 × 2 | ||

Monte Carlo simulation | Cavity | 8-node hexahedral | 0.025 | 34 | 24,969 | 61.4 |

Plate | 4-node quadrilateral | 0.020 | 6 | 5508 × 2 |

As the frequency increases, the system response becomes very sensitive to the uncertainties of the system, and the resulting curves of 500 samples become dispersed. The hybrid BE-SEA method predicts well the average trend of the pure FE method calculations with perturbed plate mass at higher frequencies. Furthermore, according to the principles of the hybrid BE-SEA method, there is sufficient uncertainty in a statistical subsystem. Hence, it should be pointed out that the average of the Monte Carlo results, which cannot involve all uncertainties (sufficient uncertainty), may have discrepancies with the results obtained by hybrid BE-SEA method even at some higher frequencies (Cotoni et al. 2007; Shorter and Langley 2005b).

In this paper, the discrepancies at lower frequencies may be neglected since the attention is only on the mid-frequency vibration of vibro-acoustic systems.

### 4.2 Sensitivity analysis and topology optimization of sound absorbing layer for mid-frequency vibration of the vibro-acoustic system

Setting the excitation frequency to be 415 Hz, the partition of the system can be performed by wavelength analysis for the acoustic cavity and the two plates. Here, the acoustic cavity is modeled using the BE method, while the two plates are modeled using SEA. Element sizes chosen for the parts modeled using element-based techniques are the same as those in Table 1. The design domain is discretized by 200 (20 × 10) uniform-sized square elements. Hence, there are 200 design variables *ρ*_{k}(*k*=1, 2, …, 200). The method of moving asymptotes (MMA) (Svanberg 1987, 2002; Johnson 2008, 2014) is employed to update the design variables. The optimization process is stopped when the relative difference of the PSDSP between two adjacent iteration steps is less than 10^{−6}.

*Z*

_{0}= 975 + 8798i kg/(m

^{2}s) (see Siemens Product Lifecycle Management Software Inc. 2014), and all element-relative densities of the sound absorbing material in the design domain are set to be 0.6. The relative errors of the sensitivities of the objective function with respect to the design variables \( \partial {\overset{\sim }{S}}_{pp}^{\mathrm{in}}/\partial {\rho}_k \)(

*k*=1, 2, …, 200), calculated by using the present method and the finite difference method (FDM) with 10

^{−4}perturbation, are given in Fig. 5.

As can be seen from Fig. 5, the comparison shows good agreement. The FDM requires one solution of the linear system of equations for the original value plus one solution (or two if using a central perturbation method) for each design variable. For the present method, one solution of the linear system of equations for the original value is required, and then, according to Sect. 3.2, the derivative (sensitivity) can be calculated directly by a few matrix product operations, without other solutions of the linear system of equations for each design variable. Therefore, compared with FDM, the present method requires less computation time.

*Z*

_{0}= 4

*ρ*

_{a}

*c*

_{0}. All initial design variables are set to be 0.4, and the upper limit of the volume fraction of the sound absorbing material is given as

*f*

_{V}=0.5. The optimization procedure converged after 21 iterations, and the iteration histories of the objective function and volume fraction are shown in Fig. 6. As can be seen, the PSDSP decreases significantly from 196,065.61 Pa

^{2}/Hz in the initial design to 52,063.696 Pa

^{2}/Hz in the final optimized design, and the volume fraction of the sound absorbing material reaches the upper limit. The sound absorbing layer layout and the contour of the PSDSP of the design domain for the initial and the optimized design are shown in Fig. 7.

As can been seen, the sound absorbing material is concentrated in the places where there is a high PSDSP in the inital design, which indicates that incident sound waves reflect strongly at these areas

It is seen from Fig. 7c, d that the PSDSP of the overall design domain has significantly reduced.

### 4.3 Influence of excitation frequencies on optimized designs

In the following, the influence of excitation frequencies on optimized designs is considered. The point (0.60, 0.50, 0.25) is adopted as the reference point. The impedance of the sound absorbing material is set as *Z*_{0} = 4*ρ*_{a}*c*_{0}. All initial design variables are set to be 0.4, and the upper limit of the volume fraction of the sound absorbing material is given as *f*_{V}=0.5. The optimization process is performed by selecting the excitation frequencies as 348 Hz, 381 Hz, 425 Hz, 466 Hz, 491 Hz, 300 Hz, 400 Hz, and 500 Hz. It is seen from Fig. 8 that the first five selected frequencies correspond to the three peaks and two valleys of the curve of the PSDSP, and the last three selected frequencies correspond to the beginning, middle, and end points of the frequency range of interest.

Consider now the topology optimization over the whole frequency band. Selecting three sampling frequencies (348 Hz, 425 Hz, and 491 Hz) in the frequency band of interest, an envelope of the objection function, which is constructed by using a composite Kreisselmeier–Steinhauser objective function (Kreisselmeier and Steinhauser 1979; Wrenn 1989) with the aggregation parameter *η* = 1000, is taken as the objective function to be minimized.

## 5 Conclusions

This paper performs the sensitivity analysis and topology optimization of a sound absorbing layer for minimizing the PSDSP at a specified point in the acoustic cavity when a vibro-acoustic system exhibits mid-frequency behavior. In the topology optimization model, an artificial sound absorbing material model is employed using the SIMP approach and the relative densities of the sound absorbing material are taken as design variables. The PSDSP of the acoustic cavity are calculated by using a hybrid BE-SEA method. In this context, the sensitivity analysis scheme of the PSDSP at a given reference point is developed by using the direct differentiation method. The optimized designs obtained under different excitation frequencies and using an envelope function as the objective function are also compared. The optimization process gives essentially the same optimized designs over a relatively wide frequency range. Moreover, due to the strong reflection of sound waves, the central area in the design domain which faces the region at which the velocity is applied is always covered with sound absorbing material.

## Notes

### Funding information

The authors are grateful for support under grants from the National Science Foundation of China (11672060) and the Cardiff University Advanced Chinese Engineering Centre.

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