Acoustic properties of underwater acoustic metamaterials based on multi-physical field coupling model

In this paper, according to the internal structure of underwater acoustic metamaterial, a multi-physical field coupling model of underwater acoustic metamaterial is established by using finite element analysis. Based on the model, the influence of typical underwater acoustic metamaterial structure and material parameters on sound absorption performance is studied. The results show that increasing the height or radius of the local resonance mass unit in the metamaterial is beneficial to improve the low frequency sound absorption performance. With the increase of the distance between mass elements and the thickness of elastic material, the acoustic absorption performance increases first and then decreases. Increasing the elastic modulus of elastic material of the local resonance unit or matrix will lead to the decrease of low frequency sound absorption performance of the material and the improvement of high frequency sound absorption performance. Increasing the loss factor of elastic material of the local resonance unit or matrix material can broaden the sound absorption band. These rules can provide guidance for the design of underwater acoustic metamaterials.


Introduction
Acoustic meta-materials are artificial and structural materials that have emerged in recent years.They have special physical effects that are not found in natural materials, breaking the rule that traditional sound-absorbing materials consume sound energy proportional to the square of frequency, and providing new technical means for low-frequency noise control.With the acceleration of ocean development and utilization and the competition for sea power in various countries, acoustic meta-materials have also received more and more attention in the field of underwater acoustic materials, and have become one of the research hot spots in this field [1][2][3][4][5][6][7][8] .
Compared with traditional underwater acoustic materials, underwater acoustic meta-materials have the following characteristics: 1) Multi mechanism synergistic effect.In addition to the original acoustic mechanisms such as waveform conversion, cavity resonance, viscous loss, etc., due to the introduction of meta-material elements, local resonance and the enhancement of waveform conversion caused by resonance and the mechanism of interface bonding loss of multi-element materials should also be considered in the calculation.2) Complicated structure.Underwater acoustic meta-materials contain a large number of meta-material units and interfaces of different materials.The internal structure and layout of meta-material units are gradually becoming diversified and complex, resulting in the overall structure of acoustic meta-materials being quite complex.3) Diversified materials.Underwater acoustic meta-material not only includes matrix materials and cavities, but also includes one or more elastic materials and mass units in meta-material units.The number of materials involved has increased from traditional mono or binary materials to ternary or quaternary materials.The matching design between materials and the optimization of material parameters is difficult, and the period of model selection based on experiment optimization is long [9][10][11][12][13][14][15][16] .
Conducting acoustic village material design through finite element simulation is an important means to reduce research and development costs and achieve rapid product updates.Currently, research has been quite sufficient.For example, Li et al. [17] established a viscoelastic material acoustic model (H-N model) and studied the influence of acoustic parameters on the sound absorption performance of viscoelastic materials, and Sun et al. [18] established an underwater acoustic model of polyurethane/epoxy resin (PU/EP) elastomers, and used this model to explore the effects of elastic modulus, loss factor, and Poisson's ratio on their acoustic properties.However, due to the complexity of the structure of underwater acoustic meta-material, the traditional acoustic calculation model is no longer applicable to the prediction of the acoustic performance of underwater acoustic meta-material.How to establish an accurate prediction model of acoustic meta-material performance is a difficult point in the design of acoustic meta-material.At present, scholars such as Li et al. [19] have conducted relevant research on the problem of underwater radiation noise in the low-frequency domain of JC using a standard numerical calculation method based on the acoustic solid coupling dynamic equation.The study found that compared to the fluid solid coupling, the acoustic solid coupling mode has the advantages of low modeling difficulty and high model accuracy, making it the preferred algorithm for the numerical calculation of JC underwater acoustic radiation prediction.Leroy et al. [20] proposed a semi analytical model based on the equivalent medium theory to study the acoustic characteristics of periodically cavitating soft elastic media underwater.Meng et al. [21] used genetic algorithms and general nonlinear constraint algorithms to optimize the low-frequency underwater sound absorption design of a multi-layer acoustic meta-material plate.Zhao and Ren [22] simulated the damping acoustic material and the acoustic performance test environment-sound tube under normal pressure by using "Virtual lab-acoustics", and established a finite element model of acoustic vibration coupling, which can effectively guide the acoustic design and performance prediction of the damping acoustic material.
By analyzing the structure of underwater acoustic meta-material, a multi physical field coupling model was built, and based on this model, the acoustic performance of underwater acoustic meta-material was studied.The influence of different structural parameters and material parameters on the acoustic absorption performance of underwater acoustic meta-material was studied, providing guidance for the design of underwater acoustic meta-material.

Construction of a multi physical field coupling model
Using the finite element analysis software COMSOL, a prediction model for acoustic performance was constructed based on the acoustic performance testing conditions of underwater acoustic meta-materials, as shown in Figure 1.Underwater acoustic meta-material consists of a matrix and a local resonance unit.The local resonance unit consists of an elastic material and a mass unit.During the transmission of sound waves, there are multiple interfaces such as the water matrix material interface, the elastic material matrix material interface, the elastic material mass unit interface, and the mass unit air interface.Due to the complexity of underwater acoustic meta-material structures, it is impossible to accurately predict the transmission of sound waves in them using a single acoustic mechanism.Therefore, it is necessary to build a multi physical field coupling model containing multiple acoustic mechanisms, mainly including the following three acoustic mechanisms.

Pressure acoustics
For the propagation of sound waves in water and the air inside the local resonance unit, a pressure acoustic module is used for calculation.Its essence is to solve the Helmholtz equation in the frequency domain, and simulate the time-harmonic form of sound waves in the fluid domain through the Helmholtz equation.The expression is as follows: where p (N/m 2 ) is the time-harmonic sound pressure,    ,  is the density (kg/m 3 ) ,  is the angular frequency (rad/s), and c is the sound velocity (m/s).

Acoustic solid coupling
When sound waves are transmitted through water to the interface of underwater acoustic meta-materials, they will transform from pressure waves into elastic waves in the solid.Therefore, a series of interactions such as reflection and transmission of sound waves will occur at the interface.Simple pressure acoustic mechanisms cannot accurately simulate the transmission and transformation of sound waves.Therefore, the use of the sound solid coupling mechanism is considered to simulate the transmission of sound waves at the sound solid interface.The interface between water and underwater acoustic meta-materials is set as the sound solid coupling interface, and the transmission of elastic waves inside the solid is simulated using solid mechanics.
To simulate the interaction between sound waves and solid interfaces, it is necessary to apply the following loads at the boundary of the solid domain: where  refers to the unit normal vector outward from the solid domain.Considering the reaction of solids on sound waves and using the normal acceleration condition in the fluid domain, it can be obtained: where  is the unit normal vector outward from the fluid domain.

Acoustic thermal viscous acoustics
For underwater acoustic meta-material, due to the addition of soft materials, there are viscous boundary layer and thermal boundary layer near the hard soft hard wall, and the viscous loss gradient caused by shear and heat conduction is large, resulting in obvious heat loss and viscous loss effects.At the same time, the thermoacoustic effect is most evident at the resonance point, and the low-frequency resonance sound regulation mechanism of meta-materials is precisely based on unit localization resonance and energy localization.Therefore, considering the heat conduction effect and viscous loss effect in the control equation of meta-material acoustic calculation can obtain more accurate results.
Figure 2 shows a typical local resonance unit structure.The acoustic thermal viscous acoustic mechanism is used to calculate the acoustic propagation including thermal loss and viscous loss.By solving the linearized continuity equation, Navier Stokes equation and energy equation, the acoustic changes caused by pressure, velocity and temperature are obtained.It is suitable for accurate modeling of acoustic behavior in small structures, and has a good match with the acoustic calculation requirements under the current meta-material structure.When sound waves propagate within a narrow meta-material unit structure, thermal and viscous losses can lead to attenuation of sound waves.The losses occur in the sound thermal boundary layer and viscous boundary layer near the wall.In order to establish a model that accurately matches the experimental measurement results, it is necessary to consider and evaluate the impact of these losses on the thermal viscous acoustic system, especially for the large displacement and vibration characteristics of meta-materials at the wall interface, heat loss and viscous loss are indispensable.In addition, the thermoacoustic effect is most obvious at the resonance, which will enhance the thermoacoustic effect and reduce its frequency.In order to simulate these effects, the heat conduction effect and viscous loss must be added to the control equation.When sound waves propagate in the fluid around the wall, the solid surface will generate the so-called viscous boundary layer and thermal boundary layer.A non slip condition u=0 for the velocity field time and an isothermal condition T=0 for the temperature field can generate an acoustic boundary layer consisting of a viscous boundary layer and a thermal boundary layer.The thermal viscous acoustic cohesive boundary describes the continuity of meta-material in the normal stress and normal acceleration domains.An adiabatic boundary condition is preset in the total thermal field to match the physical assumptions of the pressure acoustic module.The coupling boundary conditions are set as follows: where  is the sound pressure variable of the thermoacoustic pressure field, and  is the sound pressure variable of the pressure acoustic field.
To sum up, by setting different physical fields in different areas and boundaries of the whole model, a multi physical field coupling model is established.The pressure acoustic physical field is used in the water area and air area, and the solid mechanics physical field is used in the solid area.The water area and underwater acoustic meta-material interface, internal cavity and mass unit interface are set as the acoustic solid coupling boundary, and the elastic material and matrix material interface in the local resonance unit.The interface between the elastic material and the mass unit is set as an acoustic thermal viscous boundary.
The material parameters required by the model are input, the incident surface uses plane wave radiation boundary conditions, and the sound pipe wall uses total reflection boundary conditions.The frequency domain solver is selected to define the solution frequency range for solving.The acoustic parameters of the model related boundary, such as sound pressure transmission coefficient, sound pressure reflection coefficient, etc., are extracted from the calculation results, and the sound absorption coefficient is further calculated.The specific process is shown in Figure 3.

Acoustic performance research
Based on the above multi physical field model, the influence of underwater acoustic meta-material structure parameters and material parameters on its sound absorption performance is studied.When studying the impact of one parameter, keep the other parameters unchanged.
A typical underwater acoustic meta-material is selected as the research object.Its structure is shown in Figure 4.It is mainly composed of the matrix and the internal local resonance unit, which is composed of elastic materials and mass units.The material thickness is 80 mm, the diameter is 120 mm, and the pressure is atmospheric.

Research on the Influence Law of Structural Parameters
The influence of structural parameters such as mass unit radius r, mass unit spacing d, mass unit height h, and elastic material thickness e on sound absorption performance in local resonance units was studied.The calculation results are as follows: From Figure 5, it can be seen that as the radius of the mass unit increases, the first sound absorption peak gradually shifts towards low frequencies, which is beneficial for improving low-frequency acoustic performance, which is consistent with the local resonance sound absorption mechanism.
From Figure 6, it can be seen that increasing the spacing of quality units appropriately can improve low-frequency performance, but excessive spacing of quality units is not conducive to broadband sound absorption performance in the full frequency band.When the spacing is 0, low-frequency performance decreases significantly, indicating that the spacing of quality units should be designed reasonably according to performance requirements.
The results in Figure 7 show that as the height of the mass unit increases, the resonance frequency of the system decreases, and the absorption peak shifts towards low frequencies, which is consistent with the local resonance absorption mechanism.
The results in Figure 8 show that as the thickness of the elastic material increases, the sound absorption peak shifts towards low frequencies.However, further increasing the thickness is equivalent to reducing the space and weight of the mass unit, and increasing the fixed frequency of the resonance unit vibration leads to a decrease in low-frequency sound absorption performance.Therefore, under the premise of determining the maximum outer diameter, the thickness of the elastic material should be reasonably designed.

Research on the Influence Law of Material Parameters
The underwater acoustic meta-material consists of a matrix material and a local resonance unit.The effects of the material parameters (elastic modulus and loss factor) of the matrix material and the elastic material in the local resonance unit on the acoustic performance are studied respectively.The elastic modulus of the elastic material and the matrix material are E1 and E2, and the loss factors are S1 and S2, respectively.
Figure 9 shows the curve of the acoustic absorption coefficient of underwater meta-material corresponding to the elastic modulus of elastic material of 0.05 MPa, 0.1 MPa and 0.3 MPa respectively.From the figure, it can be seen that as the elastic modulus of the elastic material increases, the first absorption peak gradually splits into two, and the first and second absorption peaks synchronously move towards high frequencies.This is because the stiffness of the elastic material itself and the local resonance meta-material increases with the increase of the elastic modulus of the elastic material, hence the high order modal frequency and the low order bending vibration modal frequency move to the high frequency.Figure 10 shows the effect of changes in the loss factor of elastic materials on the sound absorption coefficient.From the calculation results, it can be seen that as the loss factor increases, the positions of the first and second absorption peaks remain unchanged, but the first absorption peak gradually flattens and merges from multiple peaks into a broadband absorption bee.The peak value of the second absorption peak decreases and gradually exhibits broadband absorption characteristics.Since the loss factor does not affect the resonant mode frequency of meta-material, the absorption frequency band can be widened by increasing the material loss factor without changing the absorption frequency point.Figure 11 shows the influence of the elastic modulus of the base material on the acoustic absorption coefficient of the underwater meta-material.According to the calculation results, as the elastic modulus of the matrix material decreases, the first and second absorption peaks gradually shift towards low frequencies and two absorption peaks gradually decompose into multiple absorption peaks.As the elastic modulus of the base material decreases, the overall stiffness of the underwater acoustic meta-material decreases, and the resonance frequency moves to the low frequency.At the same time, by reducing the property modulus of the matrix material, more local modes can appear in the underwater acoustic meta-material, which makes the mode denser in a certain frequency band, resulting in more peaks of sound absorption.As shown in Figure 12, the influence of matrix material loss factor on the sound absorption performance of meta-material.From the calculation results, it can be seen that as the loss factor of the matrix material increases, the positions of the first and second sound absorption peaks do not change, but the first sound absorption peak decreases and the second sound absorption peak increases.With the increase of the loss factor of the matrix material, the first and second absorption bandwidths are increasing, and the matrix material loss is significantly stronger than the elastic material loss factor due to its impact on the bandwidth, which is mainly because the matrix material accounts for a larger proportion in meta-material.This is also consistent with the trend of the influence of elastic material loss factor on sound absorption performance.The material loss factor does not affect the resonant modal frequency of meta-material, and increasing the material loss factor can broaden the sound absorption band.

Conclusion
Firstly, increasing the height or radius of the local resonance mass unit in meta-material is conducive to improving the low-frequency sound absorption performance.Increasing the spacing of quality units appropriately can improve low-frequency sound absorption performance, but excessive spacing of quality units is not conducive to broadband sound absorption performance in the full frequency band.This indicates that the spacing of quality units should be designed reasonably according to performance requirements.As the thickness of the elastic material increases, the sound absorption peak shifts towards low frequencies.However, excessive thickness of the elastic material can lead to a decrease in low-frequency sound absorption performance.Therefore, a reasonable design should be made for the thickness of the elastic material under the premise of determining the maximum outer diameter.
Secondly, with the increase of elastic material modulus in the local resonance unit, the first sound absorption bees of meta-material gradually split into two, and the first and second sound absorption peaks moved synchronously to the high frequency, that is, the low frequency sound absorption performance decreased, and the high frequency sound absorption performance improved.With the increase of elastic loss factor, the sound absorption peak position of meta-material almost does not change, but gradually presents broadband sound absorption characteristics, indicating that increasing the elastic loss factor can broaden the sound absorption band.
Lastly, with the decrease of the elastic modulus of the matrix material, the overall stiffness of the acoustic meta-material decreases, and the resonance frequency moves to the low frequency, which is conducive to the low-frequency sound absorption performance.At the same time, the material presents more local modes, and more sound absorption peaks appear.With the increase of the loss factor of the matrix material, the location of the sound absorption peak of the meta-material almost does not change, but it shows an obvious broadband sound absorption characteristic, indicating that increasing the loss factor of the matrix material can significantly broaden the sound absorption band.

Figure 2 .
Figure 2. Schematic diagram of local resonance unit.

Figure 3 .
Figure 3. Calculation process of acoustic solid thermal viscous multi physical field coupling model.

Figure 4 .
Figure 4.A typical underwater acoustic meta-material model.

7 Figure 5 .
Figure 5.Effect of different mass unit radii r on sound absorption coefficient.

Figure 6 .
Figure 6.Effect of different quality unit spacing d on sound absorption coefficient.

Figure 7 . 8 Figure 8 .
Figure 7. Effect of different mass unit heights h on sound absorption coefficient.

Figure 9 .
Figure 9.Effect of elastic modulus of elastic materials on sound absorption coefficient.

Figure 10 .
Figure 10.Effect of elastic material loss factor on sound absorption coefficient.

Figure 11 .
Figure 11.Effect of elastic modulus of matrix material on sound absorption coefficient.

Figure 12 .
Figure 12.Effect of loss factor on sound absorption coefficient.