At present, researchers at home and abroad get the corresponding parameter distribution according to the coordinate transformation theory, and then design the acoustic cloak [24–30]. Figure 1 is a schematic of the coordinate transformation of the acoustic cloak. The virtual space X shown in Figure 1A is mapped to the physical space x shown in Figure 1B through the mapping function. In addition to ω ⁱⁿ, the mapping between virtual space and physical space is one-to-one. Outside the areas Ω^out and ω ^out, the mapping method shows identical transformation, and the density is ρ ₀, acoustic medium with bulk modulus of K ₀.

By designing the material parameters of the ω area, the sound wave can bypass the ω ⁱⁿ area, and at the same time, ensure that the sound pressure fields of the Ω^out and ω ^out areas are the same so as to realize the acoustic stealth function of the ω ⁱⁿ area.

Assuming that the outer boundaries of virtual space Ω and physical space ω are the same, rectangular coordinate systems XOY and xoy are established, respectively, and the mapping function is x = x (X). The stealth gradient of the virtual space is defined as F = ∂ x ∂ X = ( ∂ x ∂ X ∂ x ∂ Y ∂ y ∂ X ∂ y ∂ Y ) (1)

The mapping relationship can be deduced as follows: ∇ X = F T ∙ ∇ x ∇ X = J ∇ x ∙ ( J − 1 F ) (2) where J is the Jacobian determinant of F.

The fluid sound pressure control equation of the virtual space is v ˙ ( X ) = − ρ 0 − 1 ∇ X p ( X )   p ˙ ( X ) = − K 0 ∇ X ∙ v ( X ) (3) where v is the velocity vector and p is the fluid sound pressure.

By introducing the operator mapping in formula (2) into formula (3) and introducing the passive symmetric second order tensor S (satisfying S T = S ,   ∇ x ∙ S = 0 ), the following can be obtained: p ˙ ( X ) = − K 0 J ∇ x ∙ ( J − 1 F ∙ v ( X ) ) = − K 0 J S : ∇ x ∙ ( J − 1 S − 1 ∙ F ∙ v ( X ) ) , (4) J − 1 S − 1 ∙ F ∙ v ˙ ( X ) = − ρ 0 − 1 J − 1 S − 1 ∙ F ∙ ∇ X p ( X ) = − ρ 0 − 1 J − 1 S − 1 ∙ F ∙ F T ∙ ∇ x p ( X ) = − ( ρ 0 − 1 J − 1 S − 1 ∙ F ∙ F T ∙ S − 1 ) ∇ x ( p ( X ) S ) (5)

The elastic matrix of the underwater acoustic absorber material has only one non-zero eigenvalue, which is expressed as in the tensor form. In the x-coordinate system, according to Formula (4, 5), the material parameter distribution of the acoustic cloak in the physical space can be obtained as follows: ρ = ρ 0 − 1 J − 1 S − 1 ∙ F ∙ F T ∙ S − 1 K = K 0 J C = K S ⊗ S (6)

For acoustic cloaks with special shapes such as two-dimensional annular and three-dimensional spherical shells, the circular transformation area (0 < R < b) in the virtual space can be mapped to a minimal annular area (a<r < b) in the physical space through coordinate transformation, as shown in Figure 2.

Polar coordinate systems were established in both virtual and real spaces, considering the following mapping relationship between the virtual space ( R , φ ) and the real space ( r , θ ) . R = { r ( r ≥ b ) f ( r ) ( a < r < b ) δ ( r = a ) φ = θ (7)

In order to avoid the problem of parameter singularity, a minimal positive parameter δ ≪ 1 is usually selected to make the boundary conditions of radial mapping meet b = f ( b ) , δ = f ( a ) .

Under this mapping relationship, the transformation matrix F and its Jacobian determinant J can be expressed as follows: F = [ 1 f ( r ) 0 0 r f ( r ) ]   J = det ⁡   F = r f ′ ( r ) f ( r ) (8)

In the annular symmetric structure, the coordinate variable tensor between the virtual space and the physical space is symmetric, and the special stress tensor can be constructed in the form of S = J − 1 F . Then the material parameter distribution in the two-dimensional annular acoustic cloak can be derived by bringing it into Formula (6): ρ = ρ 0 f ′ ( r ) f ( r ) r   ,   K r = K 0 f ( r ) r f ′ ( r )   ,   K θ = K 0 r f ′ ( r ) f ( r ) (9) where K r is the radial stiffness of the material and K θ is the circumferential stiffness of the material.

According to the density distribution formula, it can be concluded that the quality of the acoustic cloak was the same as that of the background medium. Evidently, if other objects to be invisible were placed in the internal cavity area, the invisibility cloak will not be suspended in the background medium.

The mapping methods between virtual space and physical space include equal density mapping, equal modulus mapping, and linear mapping. The first mapping requires the widest range of anisotropy values, the second mapping requires the widest range of density values, and the third mapping requires a moderate range of both values. Therefore, this study focuses on the performance of acoustic cloaks based on linear mapping.

The range of the acoustic cloak in the physical space is a circle with an internal radius of a and an external radius of b. After the acoustic cloak transformation, the range in the virtual space was a circle with an internal radius of δ and an external radius of b, and its circumferential angle belongs to an identical transformation, while the radial radius length of b- δ was linearly mapped from the radius length of b–a. Based on the definition of linear mapping, the linear mapping coordinate transformation function is derived as follows: f ( r ) = ξ r + η (10) where ξ = ( b − δ b − a ) ,   η = ( δ − a b − a ) b .

Equation (10) is substituted into equation (9) to obtain the distribution equation of cloak parameters using linear mapping, which can be written in the following form: ρ = ρ 0 ξ ( ξ r + η r ) , K r = K 0 ξ ( ξ r + η ξ r ) , K θ = K 0 ξ ( ξ r ξ r + η ) (11)

Finite element simulation software COMSOL was used to simulate the case of plane wave incident on an aluminum cylinder in an ideal infinite fluid medium, and then the total sound field and scattering sound field of the aluminum cylinder were analyzed after plane wave incidence. In this example, the three-dimensional cylindrical problem is simplified to a two-dimensional annular problem, which can effectively reduce the calculation load.

The simulated area is a square area with a side length of 5m, and there is an aluminum cylinder with a radius of 0.3 m in the center. The plane acoustic wave with a frequency of 2 kHz is incident from the left port of the area. In order to simulate the propagation characteristics of the plane acoustic wave in the ideal infinite fluid medium, in COMSOL physical field analysis, the four sides of the square area are selected when setting the boundary condition of “plane wave radiation,” which can make the plane wave directly pass through the boundary without any reflection so as to simulate the infinite fluid. In this example and the following sound field simulation covering the acoustic cloak, the fluid medium in the background is water ( ρ 0 = 1000 kg/m³, K 0 = 2.25 GPa).

Figure 3 shows the sound field distribution diagram of the plane sound wave without cloak covering with a frequency of 2 kHz when it hits the cylinder from the left. Figure 3A shows the scattering sound field distribution of a plane wave when it is interfered by a cylinder. Figure 3B shows the total sound pressure distribution of a plane sound wave when it is incident and scattered by a cylinder. It is the result of the superposition of the background sound pressure and the scattering sound pressure. The scattering effect at the front and back of the cylinder was serious, leading to bending and displacement of the wave front, and the intermittent scattering sound pressure distribution at the upper and lower sides made the wave front fractured.

Under the same conditions, the 10-layer acoustic cloak was covered on the obstacle. The sound field distribution in the flow area was studied, and its stealth ability was verified. The inner radius of the acoustic cloak is a = 0.3 m, and the outer radius is b = 1 m. After coordinate transformation, the target object with a radius of 0.3 m covered by the cloak is transformed into an equivalent scattering feature that is only equivalent to a circular microarea with a radius of δ = 0.06 m. The thickness of each layer of the cloak is 0.07 m, and the material parameters of each layer were obtained by substituting the radius of the center line of each layer into the linear mapping function and the calculation formula for the distribution of stealth material parameters.

The sound pressure distribution of the acoustic cloak covering 10 layers can be seen in Figure 4. The acoustic cloak with linear mapping has a certain acoustic stealth effect, and the aluminum cylinder is almost free of pressure. It can be seen from the scattering sound pressure diagram that there is still a relatively weak scattering effect in front of the cloak and behind the cloak. The scattering effect on both sides of the cloak was small. The compression distribution of the sound track inside the cloak was relatively uniform. The equal sound pressure lines at the inner boundary were not concentrated. The interaction with obstacles was very small, and its acoustic stealth performance was affected little by the inner boundary.

At present, the method of layered discrete and parameter equivalent was adopted in the preparation of the invisibility cloak, and the material parameters of each layer will directly affect the invisibility effect of the cloak. In order to quantitatively describe the invisibility of the cloak, the dimensionless parameter scattering cross-section σ is defined to describe the invisibility effect of the invisibility material. σ = E s E b , (12) where E s is the acoustic energy incident on the surface of the stealth material and E_h is the scattered acoustic energy.

For a layered design of the acoustic cloak, the number of layers is an important factor to be considered. Under the condition that the thickness of the cloak is constant and the frequency is 2 kHz, the stealth performance of the cloak with different layers was studied. The sound field simulation results are shown in Figure 5.

Figure 5 shows the sound field diagram of uniform layering when the number of layers of the acoustic cloak is 10, 20, and 40 at the frequency of 2 kHz. It can be seen from the scattering sound pressure diagram that with the increase of the number of layers N, the backscattered wave was significantly weakened, and the disturbance of the scatterer to the sound wave was getting smaller and smaller. When the number of layers of the acoustic cloak is 40, the stealth effect is better.

It can be seen from Table 1 that when the frequency is constant, the cloak thickness is the same, and the layers are evenly layered. With the increase in the number of layers N, the scattering cross-sectional value becomes smaller, and the stealth effect becomes better. However, due to the limited thickness of the cloak, the preparation difficulty will also increase significantly. Moreover, too few layers make the cloak’s stealth performance difficult to meet the use requirements. Therefore, the reasonable number of layers selected is of great significance to the practical application of the acoustic cloak.

This study selects a cloak with a frequency of 2 kHz and five layers of acoustic cloak to optimize the layer thickness distribution with the goal of optimal stealth performance. The function extreme value optimization of the neural network genetic algorithm is mainly divided into two steps: BP neural network training fitting and genetic algorithm extreme value optimization. The algorithm flowchart is shown in Figure 6. A suitable BP neural network was constructed according to the characteristics of the optimization function. A total of 1,000 groups of radius values of each layer and corresponding scattering cross-sectional values calculated by COMSOL were imported to the BP neural network, of which 800 groups were selected as the training samples of the BP neural network, and the remaining 200 groups were used as the prediction input of the trained BP neural network. The prediction results of the trained BP neural network are used as individual fitness values during the optimization of the extreme value of the genetic algorithm, and the global optimal values and corresponding input values of the function are found through selection, crossover, and mutation operations.

The optimization model in this study has four input parameters and one output parameter, so the BP neural network structure is 4–11–1. A total of 1,000 groups of input and output data of the function are taken, of which 800 groups of data are randomly selected to train the network, 200 groups of data are selected to test the network performance, and then the trained network is used to predict the output.

The real number encoding method was used to encode individuals in the genetic algorithm. Since the number of input parameters of the optimization function is four, the proposed individual length is 4. The fitness value of the individual is set as the prediction value of the BP neural network. When the fitness value is smaller, the individual is better. The selection operator, crossover operator, and mutation operator are consistent, with a crossover probability of 0.4 and a mutation probability of 0.2.

Figure 7 shows the convergence curve of the optimization algorithm. In the early stage of calculation, the convergence is fast, and the optimal value is basically reached in about 20 generations. The radius distribution obtained by optimization calculation is 0.3, 0.4009, 0.5315, 0.6820, 0.8477, and 1 m. Table 2 shows the layer thickness parameters of the cloak after optimization and before optimization. Figure 8 is the schematic diagram of the optimized sound field. Compared with the material distribution and sound field schematic diagram of the cloak before optimization (Figure 9), it can be seen that the cloak scattering effect decreases when compared with the average layer thickness. By comparing the acoustic field before and after optimization, it can be seen that the impact of the optimized incident plane wave is smaller and the scattering effect is weaker. The scattering cross area of the optimized layer thickness distribution is σ = 3.3 × 10⁻³ and that of the uniform layer thickness is σ = 1.23 × 10⁻². After structural optimization, the scattering cross area of the cloak is reduced by 73%.

Based on the optimization simulation results of COMSOL, the density, shear modulus, Poisson’s ratio, and bulk modulus parameters of each layer of the stealth material are obtained as shown in Table 3 so as to prepare a polyurethane underwater sound absorber.