Three-dimensional three-layer type isotropic five-mode metamaterial, obtaining method thereof and application of metamaterial to low-frequency regulation and control of underwater sound waves

Abstract

The invention discloses a three-dimensional three-layer type isotropic five-mode metamaterial, and simultaneously provides an acquisition method of the metamaterial and application of the metamaterial to low-frequency regulation and control of underwater sound waves. A unit cell structure of the metamaterial is a face-centered cubic lattice formed by intersecting 16 symmetrical three-dimensional five-mode symmetrical double cones; each symmetrical three-dimensional five-mode symmetrical double-cone is of a three-layer nested double-cone structure, each three-layer nested double-cone structure is formed by compositing double cones made of three different materials, and the three-layer nested double-cone structure sequentially comprises a first hard material layer, a soft material layer and a second hard material layer from the outer layer to the inner layer. The hard-soft-hard composite structure provided by the invention accords with the spring-mass theorem, the influence of structural parameters on the acoustic characteristics of the five-mode metamaterial can be effectively analyzed in the simulation experiment process, and the five-mode metamaterial has excellent pressure resistance under water pressure.

Description

Technical Field

[0001] This invention belongs to the technical field of topological properties of acoustic metamaterials, specifically relating to a three-dimensional, three-layer, isotropic five-mode metamaterial, its acquisition method, and its application in low-frequency modulation of underwater acoustic waves. Background Technology

[0002] Underwater acoustics, especially the manipulation of low-frequency sound waves, has extremely important applications in marine exploration, underwater communications, and national defense. However, it is also an area where technological bottlenecks have long existed. Traditional methods mainly rely on viscoelastic materials, which attenuate sound wave energy through dissipation and scattering within the material. However, these materials have significant limitations: they are usually thick and heavy, and while their performance is acceptable in the mid-to-high frequency range, their performance drops sharply once the target frequency drops to the low-frequency range of several hundred hertz.

[0003] To overcome this physical-scale contradiction, acoustic metamaterials have seen tremendous development opportunities. Among them, five-mode metamaterials are considered a highly promising solution. Inspired by the simulation of fluid dynamics, their core lies in macroscopically simulating fluid behavior through meticulously designed rigid three-dimensional microstructures. This fluid-like property allows researchers to precisely control the equivalent sound velocity and density at the subwavelength scale by adjusting the geometry of the microstructures, such as the thickness, length, and arrangement of the rods, thus achieving arbitrary manipulation of the sound wavefront. This provides a theoretical possibility for solving the low-frequency control problem. However, while five-mode metamaterials have demonstrated remarkable acoustic manipulation capabilities in air, their practical application underwater, especially in deep-sea environments, still faces a significant gap between theory and engineering. The primary challenge is the inherent contradiction between pressure resistance and acoustic performance.

[0004] In recent years, research on five-mode metamaterials has mainly focused on microstructure design based on transformation acoustic principles and the generalized Snell's law. This design process induces deformation and anisotropy in the cellular structure, thereby disrupting the spatial symmetry of the five-mode metamaterial. This disruption of spatial symmetry brings novel properties to the five-mode metamaterial. For topological states in phononic crystals, these can be achieved not only through temporal and spatial modulation, acoustic ring coupling, and lattice symmetry design, but also by applying a background flow field. Therefore, for five-mode metamaterials with phononic crystal properties, research on designing underwater acoustic three-dimensional five-mode metamaterials based on acoustic transformation principles requires considering not only the influence of anisotropy but also reducing their operating frequency. Therefore, this invention, based on Bragg's theorem and the spring-mass theorem, designs a three-layer composite isotropic three-dimensional five-mode cellular structure based on symmetric bipyramidal units, proposes a method for suppressing internal acoustic waves, and utilizes the principle of local resonance to reduce the corresponding operating frequency. This allows for the design of acoustic devices for efficient transmission and control of underwater acoustic waves. Meanwhile, the high pressure resistance and material universality of the material were verified through material parameter and stress theory analysis, which can then be used to design acoustic devices for efficient transmission and control of underwater sound waves. Summary of the Invention

[0005] The purpose of this invention is to provide a three-dimensional, three-layer, isotropic five-mode metamaterial. The second and third objectives of this invention are to provide a method for obtaining it and its application in low-frequency underwater acoustic modulation.

[0006] To achieve the first objective of this invention, the following technical solution is adopted:

[0007] A three-dimensional, three-layered, isotropic five-mode metamaterial is disclosed. The cell structure of the metamaterial is composed of 16 symmetrical three-dimensional five-mode symmetrical bipyramids intersecting to form a face-centered cubic lattice. Each symmetrical three-dimensional five-mode symmetrical bipyramid has a three-layer nested bipyramid structure. The three-layer nested bipyramid structure is composed of bipyramids made of three different materials. From the outer layer to the inner layer, the layers are a first hard material layer, a soft material layer, and a second hard material layer.

[0008] Furthermore, the first hard material layer is a resin layer, the soft material layer is a rubber layer, and the second hard material layer is a lead layer.

[0009] Furthermore, in the three-dimensional five-mode symmetrical bipyramidal structure, the bottom wide diameter, top narrow diameter, and height of the resin layer cone are D1, d1, and H1, respectively; the corresponding parameters for the rubber layer cone are D2, d2, and H2; and the corresponding parameters for the lead layer cone are D3, d3, and H3. The lattice constant is a, which is fixed at 37.3 mm. The values ​​are d1 = 0.55 mm, D1 = 3 mm, and H1 = 16.15 mm. The range of d2 / d3 is (0.69, 0.94), the range of D2 / D3 is (0.35, 0.725), and the range of H2 / H3 is (0.51, 0.65).

[0010] To achieve the second objective of this invention, this invention further discloses a method for obtaining a low-frequency, pressure-resistant, three-dimensional, three-layer, isotropic five-mode metamaterial for underwater acoustics, comprising the following steps:

[0011] 1) Obtain the structural and material parameters of the original isotropic five-mode metamaterial;

[0012] 2) By analogy with the isotropic five-mode metamaterial of step 1), three kinds of isotropic five-mode metamaterial bicones with different diameters and heights are obtained. Three different materials are selected, namely resin material, rubber material and lead material from the outside to the inside. They are then combined in order of volume to finally obtain a three-dimensional three-layer isotropic five-mode metamaterial.

[0013] 3) Run the three-dimensional three-layer isotropic five-mode metamaterial from step 2) in a simulation environment, update the structural information data, and calculate the band frequency value using the finite element method;

[0014] 4) The band frequency values ​​obtained in step 3) are used to calculate the quality factor, single mode and first bandgap and other acoustic characteristic parameters of the three-dimensional three-layer isotropic five-mode metamaterial by finite element algorithm.

[0015] 5) Compare the acoustic characteristic parameters obtained in step 4) to obtain the three-dimensional three-layer isotropic five-mode metamaterial with the best low-frequency performance.

[0016] Furthermore, the three-dimensional three-layer isotropic five-mode metamaterial in step 2) is composed of 16 intersecting symmetrical bipyramids forming a face-centered cubic lattice. Each symmetrical three-dimensional five-mode symmetrical bipyramid has a three-layer nested bipyramid structure. The three-layer nested bipyramid structure is composed of bipyramids made of three different materials, and from the outer layer to the inner layer, they are the first hard material layer, the soft material layer, and the second hard material layer.

[0017] Furthermore, the first hard material layer is a resin layer, the soft material layer is a rubber layer, and the second hard material layer is a lead layer.

[0018] Furthermore, in step 3), the calculation of information data combines two basic principles, namely Bragg's theorem and the spring-mass theorem. Boundary conditions and environmental parameters are set according to the basic theorems during the calculation process.

[0019] Furthermore, the finite element algorithm described in step 3) includes the following steps:

[0020] a) Data preprocessing;

[0021] b) Iterative training: By dynamically updating the strategy and Adam optimizer, the loss function is calculated in each training iteration, and then the model parameters are updated through backpropagation;

[0022] c) Model evaluation: The trained model is evaluated using methods such as validation sets or cross-validation to check its performance and generalization ability.

[0023] d) Model application: Predict the structural parameters of a three-dimensional, three-layer, isotropic, five-mode metamaterial using trained and validated models.

[0024] Further, in step a) data preprocessing: the parameter data of the insulator structure is cleaned and normalized, and then divided into training set and test set.

[0025] The third objective of this invention is to provide a new application for the aforementioned three-dimensional, three-layer, isotropic, five-mode metamaterial, namely, its application in the low-frequency modulation of underwater acoustic waves.

[0026] The three-dimensional, three-layer, isotropic five-mode metamaterial of this invention, the corresponding anisotropic unit cell structure, and the materials used are as follows: Figure 1 As shown. This invention achieves its objective by employing an isotropic five-mode metamaterial with a symmetrical biconical structure. The inventive concept of the three-dimensional, three-layer isotropic five-mode metamaterial is as follows: First, based on a single-layer isotropic five-mode metamaterial biconical structure, additional material is added, and then the bicones are nested sequentially according to their radii and heights from largest to smallest to form a three-dimensional, three-layer isotropic five-mode metamaterial. Next, boundary topological conditions are set using the finite element method, and the five-mode metamaterial structure is meshed and parametrically scanned. Finally, the obtained parameters are processed using MATLAB, and the acoustic properties are analyzed and compared through finite element numerical simulation to determine the impact of anisotropy on the low-frequency characteristics of the five-mode metamaterial.

[0027] Compared with the prior art, the present invention has the following beneficial effects:

[0028] 1) This invention selects the finite element method (FEM) for numerical analysis. The FEM is a numerical method used to solve complex engineering and mathematical physics problems. The FEM can conveniently consider various material properties. For linear materials, it can accurately calculate deformation relationships based on fundamental theories such as the spring-mass law. Simultaneously, boundary conditions are also a crucial factor in the numerical calculation of five-mode metamaterials. The FEM can flexibly apply various boundary conditions, such as displacement boundary conditions, force boundary conditions, and thermal boundary conditions. When simulating the anisotropy introduced by deformation in five-mode metamaterials, appropriate displacement and force boundary conditions are applied to analyze stress distribution and deformation.

[0029] 2) In the study of low-frequency acoustic properties, this invention constructs three isotropic five-mode metamaterial symmetrical biconical structures of different sizes and materials, nesting them in a specific order to form a three-dimensional, three-layer five-mode metamaterial biconical structure, and then performs simulation numerical analysis. In three-dimensional geometric space, the ratios of the narrow diameter d, wide diameter D, and height H of the rubber and lead biconical materials are varied. Through comparative analysis, it is found that the isotropic five-mode metamaterials exhibit differences in acoustic properties at low frequencies after changing the structural parameters. Each type of metamaterial may also have unique microstructures and physical mechanisms, and these differences will be reflected in the low-frequency acoustic properties. In addition, through experimental measurement and comparative analysis of the three-dimensional, three-layer five-mode metamaterial, the theoretical model is verified and corrected, making it more accurately reflect the physical behavior of actual materials and improving the reliability and predictive ability of the theoretical model.

[0030] 3) In constructing the anisotropic five-mode metamaterial, this invention employs composite materials for its biconical structure, namely soft materials (rubber materials) and hard materials (lead and resin materials). Soft and hard materials possess distinctly different physical properties. Rubber materials typically have a low elastic modulus, enabling them to undergo significant deformation when subjected to sound waves. This large deformation characteristic allows rubber materials to exhibit unique vibration modes and energy dissipation mechanisms at low frequencies. Hard materials, on the other hand, have a high elastic modulus and low deformation capacity. In low-frequency acoustics, they may be more inclined to reflect and scatter sound waves. Combining the two allows for the coverage of a wider range of acoustic properties, including sound wave absorption, reflection, scattering, and wave propagation modes. During the propagation of low-frequency sound waves, the hard material can act as a structural framework, providing stable support and a reflective interface, while the soft material can adhere to the surface of the hard material or fill its internal voids, utilizing its viscoelasticity to absorb the energy of low-frequency sound waves. In addition, the composite structure proposed in this invention can be broadly described as a "hard-soft-hard" structure. This structure conforms to the spring-mass theorem and can effectively analyze the influence of structural parameters on the acoustic properties of the five-mode metamaterial during simulation experiments.

[0031] 4) This invention not only analyzes the acoustic performance of the three-dimensional, three-layer, five-mode metamaterial bicone, but also conducts material variation and stress analysis on its pressure resistance underwater. Previous studies often featured simple structural designs for five-mode metamaterials, and when designed for low-frequency bands, they lacked suitable structural dimensions, making practical applications difficult. Furthermore, in low-frequency environments such as underwater, they failed to maintain good acoustic performance or experienced severe deformation, resulting in the loss of their original acoustic characteristics. Through a three-dimensional, three-layer design, the performance of the five-mode metamaterial can be tailored to specific low-frequency acoustic requirements. By continuously adjusting material parameters, the sound absorption coefficient can be improved, reflectivity reduced, or wave propagation modes improved. These parameters allow the five-mode metamaterial to meet these diverse needs even in complex real-world environments. Results show a wider single-mode region, with a maximum single-mode frequency difference of 1679.25 Hz. Moreover, the quality factor consistently remains above 150 throughout the entire operating range. Attached Figure Description

[0032] Figure 1 This invention relates to a three-dimensional five-mode metamaterial cell structure and a three-layer bipyramidal internal structure; wherein the basic parameters are determined as follows: a = 37.3 mm, d1 = 0.55 mm, d2 = 0.45 mm, d3 = 0.35 mm, D1 = 3 mm, D2 = 2 mm, D3 = 1 mm, H1 = 16.15 mm, H2 = 11.15 mm, H3 = 6.15 mm.

[0033] Figure 2 The effect of changes in structural parameter ratios on the band structure of a three-dimensional five-mode metamaterial: (a) d2 / d3 = 0.83, (b) d2 / d3 = 0.84, (c) D2 / D3 = 0.6 and (d) H2 / H3 = 0.6.

[0034] Figure 3 The effect of changes in the ratio of structural parameters on the quality factor of the three-dimensional five-mode metamaterial; (a) d2 / d3 = 0.83; (b) d2 / d3 = 0.84; (c) D2 / D3 = 0.6; (d) H2 / H3 = 0.6; where the gray area represents the single-mode region of the five-mode metamaterial and the black area represents the first bandgap region of the five-mode metamaterial;

[0035] Figure 4 The influence of changes in the ratio of structural parameters on the upper and lower bounds of the single-mode frequency of a three-dimensional five-mode metamaterial; (a) changes in d2, (b) changes in d3, (c) changes in D2, (d) changes in D3, (e) changes in H2 and (f) changes in H3;

[0036] Figure 5The influence of changes in the ratio of structural parameters on the upper and lower bounds of the first bandgap frequency of a three-dimensional five-mode metamaterial; (a) changes in d2, (b) changes in d3, (c) changes in D2, (d) changes in D3, (e) changes in H2 and (f) changes in H3;

[0037] Figure 6 The effect of changes in the ratio of structural parameters on the relative bandwidth of a three-dimensional five-mode metamaterial; (a) changes in d2, (b) changes in d3, (c) changes in D2, (d) changes in D3, (e) changes in H2 and (f) changes in H3;

[0038] Figure 7 The effect of changes in the ratio of structural parameters on the band structure of a three-dimensional five-mode metamaterial after changes in the basic material parameters; (a) E3 = 4.1826 × 10 10 Pa and ρ3=11700kg / m 3 (b)E3=4.2826×10 10 Pa and ρ3=11800kg / m 3 (c)E2=1.275×10 5 Pa and ρ2=1400kg / m 3 (d)E2=1.375×10 5 Pa and ρ2=1500kg / m 3 . Detailed Implementation

[0039] To further illustrate the effects of the present invention, the present invention will be further described below in conjunction with specific embodiments and accompanying drawings.

[0040] Example

[0041] A three-dimensional, three-layered, isotropic five-mode metamaterial is disclosed. The metamaterial exhibits a three-layered nested bipyramidal structure, consisting of 16 intersecting symmetrical three-dimensional five-mode symmetrical bipyramids forming a face-centered cubic lattice. Each three-dimensional five-mode symmetrical bipyramid is composed of bipyramids made of three different materials. Considering future practical engineering applications, the structure needs to be as lightweight as possible and easy to fabricate; therefore, the outermost bipyramid is made of resin. For the two inner cones, to verify the effect of the spring-mass formula in the three-dimensional five-mode metamaterial, rubber and lead were selected respectively. In the three-dimensional five-mode symmetrical bipyramid, the bottom wide diameter, top narrow diameter, and height of the resin cone are D1, d1, and H1, respectively. The corresponding parameters for the rubber cone are D2, d2, and H2, and for the lead cone, D3, d3, and H3. The lattice constant is a. The corresponding three-dimensional five-mode symmetrical bipyramidal cell structure is shown below. Figure 1 As shown.

[0042] Quality factor and frequency variation are key indicators for evaluating the acoustic performance of five-mode metamaterials, especially in the low-frequency range. This invention provides a rigorous theoretical framework and quantitative analysis of these properties for the proposed three-dimensional three-layer symmetric biconical structure. To calculate the phonon band structure, this invention uses the finite element simulation software Comsol MultipHysics to numerically calculate the unit cell of the proposed three-dimensional five-mode metamaterial under Bloch boundary conditions, systematically studying its performance. This invention sets some relevant basic parameters for the proposed three-dimensional five-mode metamaterial: a = 37.3 mm, d1 = 0.55 mm, d2 = 0.45 mm, d3 = 0.35 mm, D1 = 3 mm, D2 = 2 mm, D3 = 1 mm. Simultaneously, the coordinates of the narrow diameter intersection point P of the lattice unit cell are defined as (0.25a, ​​0.25a, ​​0.25a), i.e., m1 = m2 = m3 = 0.25. H1 can be calculated by moving the point p, and then H2 and H3 can be obtained by decreasing H1. The following are the specific formulas for H1, H2, and H3. Substituting these values ​​into the formulas, we get the initial values: H1 = 16.15 mm, H2 = 11.15 mm, and H3 = 6.15 mm.

[0043]

[0044] H2=ΔH1=H1-5mm (2)

[0045] H3=ΔH2=H1-10mm (3)

[0046] To investigate the band structure and transport characteristics of the designed structure, this invention utilizes the finite element analysis software COMSOL MultipHysics to numerically study the band structure and transport characteristics of the phononic crystal. The lattice constant a = 37.3 mm and the air density ρ = 1.12 kg / m³ are fixed. 3 The sound velocity is c = 343 m / s. In the numerical simulation using COMSOL MultipHysics, boundary conditions are applied to the anisotropic five-mode metamaterial, and 21 band structures are calculated. After obtaining the current band parameters, the band parameters are processed using the finite element method in Matlab to obtain various acoustic properties of the five-mode metamaterial, such as band structure, quality factor, single-mode characteristics, and first bandgap characteristics. To investigate the acoustic properties of this three-dimensional, three-layer five-mode metamaterial, we selected and varied the ratios of the structural parameters of lead biconical and rubber biconical materials based on the initial structural parameters. Simultaneously, while changing the ratios of the same structural parameters, other structural parameters were kept constant to investigate the influence of each structural parameter on the acoustic properties of the five-mode metamaterial.

[0047] In this invention, a series of studies were conducted on the band structure of three-dimensional five-mode metamaterials, and some results are as follows: Figure 2As shown, the horizontal and vertical coordinates correspond to the entire approximate Brillouin zone and the frequency boundary, respectively. The black area represents the first bandgap region, and the gray area represents the single-mode region. The quality factor (FOM) of the three-dimensional five-mode metamaterial can be obtained from... Figure 2 The two lines in the middle (C) B and C G The ratio of the slopes of the five-mode metamaterials is obtained. The FOM characterizes the ability of the five-mode metamaterial to suppress shear waves and allow compression waves to propagate in the single-mode region. Therefore, its magnitude represents the ability of the five-mode metamaterial to decouple compression and shear waves. In the expression based on the elastic constant, C 11 C 12 and C 44 These are the three independent elastic coefficients of the elastic matrix of a three-dimensional five-mode metamaterial. Where C... B It is the relative velocity of the five-mode metamaterial compression wave, C G It is the relative velocity of the shear wave. The quality factor of the anisotropic five-mode metamaterial can be obtained by considering the two line segments (C) in Figure (2). B and C G The slope ratio is calculated. Propagation of both compression and shear waves is suppressed within the first bandgap range. Within the frequency range of the single-mode region, compression and shear waves are decoupled. That is, only compression waves can propagate, while shear waves are suppressed. The quality factor represents the ability of the five-mode metamaterial to decouple compression and shear waves. The quality factor has the following proportional relationship with the ratio of the compression and shear wave phase velocities:

[0048] G=C 44 (4)

[0049] B = (C 11 +2C 12 ) / 3 (5)

[0050]

[0051] The above formula also reflects that the quality factor is determined by the slopes of the compression and shear waves of the phonon bandgap structure.

[0052] exist Figure 2 In (a), the parameter ratio of the rubber and lead materials was varied by setting d2 / d3 = 0.83. The five-mode metamaterial opened its first bandgap between 1953 Hz and 1963 Hz, corresponding to a relative bandwidth of 0.0015. Furthermore, the lower and upper bound frequencies of the single-mode were 541.1 Hz and 1936.6 Hz, respectively. Figure 2 In (b), with d2 / d3 = 0.84, the five-mode metamaterial opens a very narrow first bandgap between 1941 Hz and 1945 Hz, with a relative bandwidth of 0.0017. The lower and upper bound frequencies for the single mode are 537.73 Hz and 1918.9 Hz, respectively. Figure 2 In (c), setting D2 / D3 = 0.6 opens the first bandgap between 1963.9 Hz and 1968.9 Hz, with a relative bandwidth of 0.0025. Furthermore, the lower and upper bound frequencies for single-mode are 514.9 Hz and 1931.5 Hz, respectively. Figure 2 In (d), the height variation of the bicone was studied by setting H2 / H3 = 0.6, revealing a narrow first bandgap between 1938 Hz and 1940 Hz, with a relative bandwidth of 0.0007. The single-mode lower and upper bound frequencies were 534.8 Hz and 1923.5 Hz, respectively. Figure 2 The study reveals that the proposed three-dimensional five-mode metamaterial possesses a relatively wide single-mode bandwidth, indicating that its mechanical response closely resembles that of an ideal fluid. Furthermore, it maintains compressive stiffness during structural changes, ensuring efficient propagation of compression waves and a certain degree of compressive resistance. These findings provide some theoretical basis for the vibration reduction and noise reduction applications of five-mode metamaterials in complex environments.

[0053] To investigate the impact of structural parameter variations on Form of Internal Membrane (FOM), this invention will explore the influence of three-dimensional five-mode metamaterials on FOM by adjusting the ratios of identical structural parameters in different materials. The effects of varying the ratios (d, D, H) of structural parameters (bi-conical in rubber and bi-conical in lead) on FOM are shown below. Figure 3 As shown, the performance index of the quality factor remains above 150 throughout the entire working range. Because the rubber and lead materials in the biconical structure of the three-dimensional pentamode metamaterial deform when the structural parameters change, this in turn affects the band structure of the pentamode metamaterial in different ways.

[0054] exist Figure 3 As can be observed in (a), when d2 increases, the FOM of the three-dimensional five-mode metamaterial exhibits a trend similar to a linear function, with the maximum value of FOM reaching 172.3. Figure 3 As can be observed in (b), when d3 increases by this ratio, FOM does not show any significant numerical change, fluctuating generally around 170.4. Figure 3 In (c), when D2 increases through a ratio transformation, FOM exhibits an increasing trend similar to a quadratic function, at which point FOM can reach a maximum of 175.3. Figure 3 In (d), increasing D3 has no significant impact on the FOM of the three-dimensional five-mode metamaterial, with the overall fluctuation remaining around 170.5. Figure 3 As can be observed in (e), when H2 increases, FOM exhibits a decreasing trend similar to a linear function, at which point the maximum value of FOM can reach 173.7. Figure 3In (f), as H3 increases, the FOM of the three-dimensional pentamodal metamaterial does not change significantly, with the overall value remaining around 170.3. This indicates that when the diameter of the bipyramid changes, lead material has no significant effect on the FOM of the pentamodal metamaterial, while the influence of rubber material on the FOM increases.

[0055] The effects of changing the ratio of the structural parameters of the rubber biconical and lead biconical materials on the first band gap are as follows: Figure 4 As shown in (a)-(f). In Figure 4 In (a) and (b), it can be observed that the upper and lower bound frequencies of the single-mode do not change significantly, remaining around 1900Hz and 400Hz, respectively. Figure 4 In (c), as D2 increases, the upper limit frequency of the single-mode signal gradually increases as a linear function, reaching a maximum of 1980.3 Hz, while the lower limit frequency of the single-mode signal does not change significantly, remaining around 500 Hz. Figure 4 In (d), it can be observed that the upper bound frequency of the single-mode signal exhibits a trend similar to a quadratic function, first decreasing and then increasing, with a maximum frequency of 2093.4 Hz. Conversely, the lower bound frequency of the single-mode signal shows a slightly decreasing trend, similar to a main function, with a maximum frequency of 593.6 Hz. Figure 4 As can be seen in (e), when H2 changes proportionally, it has no significant effect on the upper and lower bound frequencies of the single-mode signal. Figure 4 In (f), when H3 is changed proportionally, the upper limit frequency of the single-mode mode shows an upward trend up to 2136Hz, while the lower limit frequency of the single-mode mode does not change significantly. This comparison shows that when H3 is changed proportionally, the single-mode region is the widest, with the maximum single-mode frequency difference being 1679.25Hz.

[0056] The frequency range of the first bandgap determines the ability of the three-dimensional five-mode metamaterial to manipulate low-frequency sound waves. The effects of changing the ratio of the structural parameters of the rubber biconical and lead biconical materials on the first bandgap are as follows: Figure 5 As shown in (a)-(f), the effect of varying structural parameter ratios between rubber-based and lead-based bipyramidal structures on the upper and lower bound frequencies of the first bandgap is illustrated. Figure 5 As shown in (a), with the increase of d2, the upper and lower bound frequencies of the first bandgap of the three-dimensional five-mode metamaterial exhibit an increasing trend with a linear function, reaching a maximum frequency of 1961.6 Hz and 1957.3 Hz, respectively. Figure 5 In (b), as d3 increases proportionally, the first bandgap exhibits a decreasing trend similar to a linear function. However, when the ratio increases to 0.88 and 0.89, the structural symmetry of the three-dimensional five-mode metamaterial is disrupted, leading to nonlinear effects in the material and causing drastic changes in the equivalent stiffness, resulting in frequency jumps and discontinuous variations. The highest frequencies at these points are 1944.2 Hz and 1947.4 Hz, respectively. Figure 5As shown in (c), with the increase of D2, the resulting first bandgap frequency band becomes narrower, and the overall change exhibits a linear upward trend. The maximum frequencies of the upper and lower boundaries of the first bandgap frequency band are 1980.7 Hz and 1978.7 Hz, respectively. Figure 5 In (d), as D3 increases, the upper and lower bounds of the first bandgap frequency of the five-mode metamaterial both exhibit a similar quadratic function trend of first decreasing and then increasing, with maximum frequencies of 2110.7 Hz and 2101.3 Hz, respectively. On the other hand, Figure 5 (e) and (f) show that when the height of the three-dimensional five-mode metamaterial changes, both exhibit a similar quadratic function trend of first decreasing and then increasing. When H2 increases, the maximum frequencies of the upper and lower bounds of the first bandgap are 1948.9 Hz and 1946.4 Hz, respectively; when H3 increases, the maximum frequencies of the upper and lower bounds of the first bandgap are 2138.1 Hz and 2129.6 Hz, respectively.

[0057] The effects of changing the ratio of structural parameters on the relative bandwidth of single-mode and first bandgap are as follows: Figure 6 As shown in (a)-(f). Through Figure 6 It can be observed that as the ratio of structural parameters increases, these changes do not affect the relative bandwidth of the single mode and the first bandgap of the three-dimensional five-mode metamaterial. On the contrary, by changing the D2, D3 and H3 parameters, the relative bandwidth of the single mode shows a slight upward trend, with the maximum relative bandwidth values ​​being 1.1312, 1.2358 and 1.1939, respectively.

[0058] After obtaining the low-frequency characteristics of the three-dimensional pentamode metamaterial, this invention aims to address the issue of its inability to simultaneously achieve both low-frequency performance and pressure resistance. The invention modulates the Young's modulus (E) and density (ρ) of the rubber and lead materials, obtaining the effect on the band structure of the three-dimensional pentamode metamaterial as follows: Figure 7 As shown in (a)-(d). In Figure 7 In (a), both E and ρ of the lead material are increased, specifically E3 = 4.1826 × 10 10 Pa and ρ3 = 11700 kg / m 3 The first negative bandgap was observed to open in the 1940.3Hz to 1943.7Hz range, corresponding to a relative bandwidth of 0.0018. Furthermore, the lower and upper bounds of the single-mode region are 539.81Hz and 1918Hz, respectively. Figure 7 (b) sets E3 = 4.2826 × 10 10 Pa and ρ3 = 11800 kg / m 3 The material exhibits a narrow first bandgap between 1925 Hz and 1926 Hz, with a relative bandwidth of 0.0006. The lower and upper bounds of the single-mode region are 519.56 Hz and 1904.6 Hz, respectively. Figure 7(c) To study the elastic modulus E and density ρ of rubber materials, E² = 1.275 × 10⁻⁶. 5 Pa, ρ2 = 1400 kg / m 3 The first bandgap opens in the range of 1982.6 Hz to 1984.4 Hz, with a relative bandwidth of 0.0011. Furthermore, the lower and upper bounds of the single-mode region are 499.52 Hz and 1957.4 Hz, respectively. Figure 7 In (d), E2 is set to 1.375 × 10 5 Pa and ρ2 = 1500 kg / m 3 The first bandgap opens between 1927.8 Hz and 1932.8 Hz, with a relative bandwidth of 0.0026. The lower and upper bound frequencies of the single-mode region are 520.57 Hz and 1900.5 Hz, respectively.

[0059] Will Figure 7 and Figure 2 Comparison of the results reveals that even with changes in material parameters, the proposed three-dimensional, three-layered, five-mode metamaterial bicone can still open the first bandgap in the low-frequency range and obtain a relatively wide single-mode region. This three-dimensional five-mode metamaterial exhibits significant insensitivity to changes in the constituent material parameters. This result emphasizes that the low-frequency characteristics of the five-mode metamaterial are primarily determined by its layered topology, rather than relying on inherent material properties. This three-dimensional five-mode metamaterial forms a low-shear stress path from the outer to the inner layers, and its face-centered cubic symmetry allows for destructive interference of shear waves without relying on material damping. Therefore, this structure is versatile, allowing future researchers to freely replace material components while maintaining ideal low-frequency five-mode characteristics. This discovery provides diverse options for the fabrication of five-mode metamaterials in vibration damping and noise reduction applications.

[0060] To ensure the practical applicability of the proposed three-layer PM in low-frequency and pressure-resistant environments, its structural stability under critical compressive load must be evaluated. According to Euler's elastic rod buckling theory, the critical force P of the fixed-end compression rod... α Given by the following formula:

[0061]

[0062] Here, E represents the elastic model of the material, l is the length of the rod, and I is the moment of inertia of the rod section relative to the central principal axis of the column. In the proposed three-layer symmetrical biconical structure, the rubber material is more prone to deformation under compression, and this deformation usually occurs first. Therefore, we focus on analyzing the stability of the rubber cone. For a circular cross-section with diameter D, its moment of inertia is:

[0063]

[0064] Based on the structural parameters and material properties of a symmetrical biconical rubber structure, its buckling resistance can be evaluated. Compared to an asymmetrical biconical structure, the symmetrical three-layer design significantly improves stability. This structure, with its nested conical geometry and reliable radial support, achieves a larger effective moment of inertia, thus obtaining a higher critical buckling force P. α This structure ensures that the proposed five-mode metamaterial maintains its mechanical integrity and does not buckle or deform significantly, even under typical hydrostatic pressure or low-frequency dynamic loads in underwater applications. This effectively prevents drastic structural changes in the five-mode metamaterial during underwater applications, thus maintaining good acoustic properties under load.

Claims

1. A three-dimensional, three-layer, isotropic, five-mode metamaterial, characterized in that, The cell structure of the metamaterial is composed of 16 symmetrical three-dimensional five-mode symmetrical bipyramids intersecting to form a face-centered cubic lattice. Each symmetrical three-dimensional five-mode symmetrical bipyramid has a three-layer nested bipyramid structure. The three-layer nested bipyramid structure is composed of bipyramids made of three different materials. From the outer layer to the inner layer, they are the first hard material layer, the soft material layer, and the second hard material layer.

2. The three-dimensional, three-layer, isotropic five-mode metamaterial according to claim 1, characterized in that, The first hard material layer is a resin layer, the soft material layer is a rubber layer, and the second hard material layer is a lead layer.

3. The three-dimensional, three-layer, isotropic, five-mode metamaterial according to claim 1, characterized in that, In the three-dimensional five-mode symmetrical bicone, the base wide diameter, the apex narrow diameter, and the height of the resin layer cone are respectively... D 1, d 1, H 1. Corresponding parameters of the rubber layer cone D 2, d 2, H 2. The parameters corresponding to the lead-layered cone are: D 3, d 3, H 3. The lattice constant is a Fixed lattice constant a = 37.3mm unchanged, d 1 = 0.55 mm, D 1 = 3 mm, H 1 = 16.15 mm; d 2 / d The value of 3 ranges from (0.69, 0.94). D 2 / D The value of 3 ranges from (0.35, 0.725). H 2 / H The value of 3 ranges from (0.51, 0.65).

4. A method for obtaining a low-frequency, pressure-resistant, three-dimensional, three-layer, isotropic five-mode metamaterial for underwater acoustics, characterized in that... Includes the following steps: 1) Obtain the structural and material parameters of the original isotropic five-mode metamaterial; 2) By analogy with the isotropic five-mode metamaterial of step 1), three kinds of isotropic five-mode metamaterial bicones with different diameters and heights were obtained. Three different materials were selected, namely resin material, rubber material and lead material from the outside to the inside. They were combined in order of volume to finally obtain a three-dimensional three-layer isotropic five-mode metamaterial. 3) Run the three-dimensional three-layer isotropic five-mode metamaterial from step 2) in a simulation environment, update the structural information data, and calculate the band frequency value using the finite element method; 4) The band frequency values ​​obtained in step 3) are used to calculate the quality factor, single mode and first bandgap and other acoustic characteristic parameters of the three-dimensional three-layer isotropic five-mode metamaterial by using the finite element algorithm; 5) Compare the acoustic characteristic parameters obtained in step 4) to obtain the three-dimensional three-layer isotropic five-mode metamaterial with the best low-frequency performance.

5. The method for obtaining a low-frequency, pressure-resistant, three-dimensional, three-layer, isotropic five-mode metamaterial for underwater acoustics according to claim 3, characterized in that, Step 2) The three-dimensional three-layer isotropic five-mode metamaterial is composed of 16 intersecting symmetrical bipyramids forming a face-centered cubic lattice. Each symmetrical three-dimensional five-mode symmetrical bipyramid has a three-layer nested bipyramid structure. The three-layer nested bipyramid structure is composed of bipyramids made of three different materials. From the outer layer to the inner layer, they are the first hard material layer, the soft material layer, and the second hard material layer.

6. The method for obtaining a low-frequency, pressure-resistant, three-dimensional, three-layer, isotropic five-mode metamaterial for underwater acoustics according to claim 4, characterized in that, The first hard material layer is a resin layer, the soft material layer is a rubber layer, and the second hard material layer is a lead layer.

7. The method for obtaining a low-frequency, pressure-resistant, three-dimensional, three-layer, isotropic five-mode metamaterial for underwater acoustics according to claim 4, characterized in that, In step 3), the calculation of information data combines two basic principles: Bragg's theorem and the spring-mass theorem. Boundary conditions and environmental parameters are set according to the basic theorems during the calculation process.

8. The method for obtaining a low-frequency, pressure-resistant, three-dimensional, three-layer, isotropic five-mode metamaterial for underwater acoustics according to claim 1, characterized in that, The finite element algorithm described in step 3) includes the following steps: a) Data preprocessing; b) Iterative training: By dynamically updating the strategy and Adam optimizer, the loss function is calculated in each training iteration, and then the model parameters are updated through backpropagation; c) Model evaluation: The trained model is evaluated using methods such as validation sets or cross-validation to check its performance and generalization ability; d) Model application: Predict the structural parameters of a three-dimensional, three-layer, isotropic five-mode metamaterial using trained and validated models.

9. The method for obtaining a low-frequency, pressure-resistant, three-dimensional, three-layer, isotropic five-mode metamaterial for underwater acoustics according to claim 8, characterized in that, Step a) Data preprocessing: The parameter data of the insulator structure is cleaned and normalized, and then divided into training set and test set.

10. The application of the three-dimensional three-layer isotropic five-mode metamaterial according to any one of claims 1-3 in the low-frequency modulation of underwater acoustic waves.

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