A method for optimizing the multi-shape dimensions of hydrophones based on eigenfunction expansion

Abstract

This invention belongs to the field of underwater acoustic detection and underwater acoustic transducer technology, and discloses a multi-shape size optimization method for hydrophones based on eigenfunction expansion. This method addresses the problems in existing hydrophone sensitive element design, such as the reliance on empirical or numerical fitting for mechanical domain equivalent parameters, the lack of a unified analytical modeling framework for different geometries, and the high dependence of structural size optimization on finite element parameter scanning with a lack of clear size-performance mapping relationships. The proposed multi-shape size optimization method for hydrophones based on eigenfunction expansion reduces reliance on large-scale finite element parameter scanning and purely empirical design, thereby improving the efficiency and interpretability of structural design. Furthermore, it achieves integrated modeling and optimization of sensitive elements with various geometries, providing a unified mechanical modeling foundation and reliable size design basis for the serial design of hydrophones and MEMS hydrophones, equivalent circuit parameter extraction, and subsequent array mutual coupling analysis.

Description

Technical Field

[0001] This invention belongs to the field of underwater acoustic detection and underwater acoustic transducer technology, specifically relating to a method for optimizing the multi-shape size of hydrophones based on eigenfunction expansion. Background Technology

[0002] In underwater acoustic measurement systems, hydrophones are the core components for acquiring underwater sound pressure. Their sensitivity, bandwidth, directivity, and structural reliability directly determine the overall detection performance and application scenarios. As underwater acoustic detection evolves towards wider bandwidth, lower noise, array-based design, and miniaturization, MEMS hydrophones and novel vector hydrophones have gradually become research hotspots. Under limited chip area and fabrication constraints, refining the shape and size of the sensing element has become one of the key issues in current research and engineering applications. Traditional large-size piezoelectric ceramic hydrophones rely heavily on empirical formulas combined with experimental calibration for design. However, microstructures, thin-film, or cantilever sensing elements require precise characterization of vibration and electromechanical conversion characteristics under liquid loading environments, placing higher demands on modeling theories and optimization methods.

[0003] Existing modeling methods mainly fall into two categories: equivalent circuit modeling and finite element numerical analysis. The former, based on equivalent circuit models such as Mason, KLM, and BVD, can quickly analyze frequency response and sensitivity; the latter relies on finite element analysis (FEA) or multiphysics simulation to determine reasonable values ​​for design variables such as diaphragm thickness, cavity size, and support structure. While the eigenfunction method offers high accuracy and analytical performance for solving vibrations in regular plate-like structures, current research primarily focuses on the modal characteristics and hydroelastic effects of engineering plate and shell components. It lacks a comprehensive design system for hydrophone sensitive elements, encompassing electromechanical coupling analysis, equivalent parameter extraction, and dimensional optimization, and also lacks a unified modeling method adaptable to various geometries.

[0004] Existing structural optimization methods suffer from common shortcomings: equivalent circuit modeling requires pre-defined lumped parameters such as equivalent mass, stiffness, and radiation impedance; parameter values ​​depend on experience, experiments, or simulation fitting; the intrinsic mapping relationship between geometric dimensions, mechanical parameters, and acoustic-electric performance is ambiguous, making it difficult to support quantitative comparison and systematic dimensional optimization of sensitive units with different shapes. While joint optimization using finite element method and intelligent algorithms has versatility, it requires separate modeling for each structure, resulting in high computational costs and long design cycles for multiple rounds of parameter scanning; furthermore, the optimization results are mostly discrete numerical data, lacking directly reusable analytical design criteria, and failing to reveal the impact mechanism of dimensional changes on key performance indicators.

[0005] Existing plate-liquid vibration theories based on eigenfunctions or modal expansions are mostly for single ideal structures. The models are mostly limited to modal frequency and mode shape analysis, and have not extracted mechanical domain equivalent parameters that can be embedded in hydrophone equivalent circuits or acoustic transfer models. Furthermore, they often derive eigenfunctions for single shapes or specific boundaries, and lack a systematic generalization that covers multiple sensitive unit shapes under a unified framework.

[0006] In summary, current technologies lack analytical derivation methods for the equivalent parameters of sensitive elements that take into account liquid loading effects and adapt to various geometries, and also lack size optimization criteria with clear analytical forms. How to construct a unified design framework based on the eigenfunction method, thereby reducing over-reliance on large-scale parameter sweeps of the finite element method and overcoming the limitations of a single analytical model, is a technological gap that urgently needs to be filled. Summary of the Invention

[0007] The purpose of this invention is to propose a multi-shape size optimization method for hydrophones based on eigenfunction expansion. This method can not only reduce the reliance on large-scale finite element parameter scanning and pure empirical design, and improve the efficiency and interpretability of structural design, but also realize the integrated modeling and optimization of sensitive elements with various geometric forms. This provides a unified mechanical modeling basis and reliable size design basis for the serial design of hydrophones, the extraction of equivalent circuit parameters, and subsequent array mutual coupling analysis.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: A method for optimizing the dimensions of hydrophones of various shapes based on eigenfunction expansion includes the following steps: Step 1. Establish the vibration control model of the hydrophone sensing element, i.e., the thin plate, under the action of external sound pressure, and obtain the non-homogeneous partial differential equation of the forced vibration of the thin plate, i.e., the control equation of the forced vibration of the thin plate. Step 2. Use the eigenfunction method to obtain the solution of the governing equation for forced vibration of thin plate; based on the governing equation for forced vibration of thin plate, obtain the parametric expressions for equivalent mass, equivalent stiffness and acoustic transduction coefficient by normalizing the characteristic length and the amplitude of the mode shape function. Step 3. Establish the parametric expressions for electromechanical conductance coefficient and static capacitance. Based on the parametric expressions for equivalent mass, equivalent stiffness and acoustic conductance coefficient obtained in Step 2, establish the quantitative relationship between the characteristic size of the sensitive unit, the thickness of the piezoelectric layer, the area of ​​the upper electrode and the electrical input impedance, static acoustic pressure sensitivity and resonant frequency, and then determine the optimized structural size parameters that meet the target performance indicators.

[0009] Furthermore, based on the above-mentioned method for optimizing the multi-shape size of hydrophones based on eigenfunction expansion, this invention also proposes a computer device, which includes a memory and one or more processors. The memory stores executable code, and when the processor executes the executable code, it implements the steps of the hydrophone multi-shape size optimization method based on eigenfunction expansion mentioned above.

[0010] Furthermore, based on the aforementioned method for optimizing the multi-shape size of hydrophones based on eigenfunction expansion, this invention also proposes a computer-readable storage medium storing a program thereon; when executed by a processor, this program is used to implement the steps of the aforementioned method for optimizing the multi-shape size of hydrophones based on eigenfunction expansion.

[0011] The present invention has the following advantages: As described above, this invention discloses a multi-shape size optimization method for hydrophones based on eigenfunction expansion. This method addresses the problems in existing hydrophone sensing element design, such as the reliance on empirical or numerical fitting for mechanical domain equivalent parameters, the lack of a unified analytical modeling framework for different geometries, and the high dependence of structural size optimization on finite element parameter scanning with a lack of clear size-performance mapping relationships. Using the eigenfunction method as its core, this method constructs analytical eigenfunctions applicable to hydrophone sensing elements with different geometries and combines them with normalized coordinate transformation to achieve unified modeling, parameter solving, and size optimization for sensing elements with various geometric forms. Furthermore, key parameters such as equivalent mass, equivalent stiffness, electromechanical coupling coefficient, and static sound pressure sensitivity can be obtained analytically, and a quantitative relationship between structural dimensions and performance indicators can be established accordingly. Therefore, this invention reduces the reliance on large-scale finite element parameter scanning and empirical trial and error, improves design efficiency, and provides a unified mechanical modeling foundation and reliable size design basis for the serial design of hydrophones and MEMS hydrophones, equivalent circuit parameter extraction, and subsequent array mutual coupling analysis. Attached Figure Description

[0012] Figure 1 This is a flowchart of a hydrophone multi-shape size optimization method based on eigenfunction expansion in an embodiment of the present invention.

[0013] Figure 2 This is a schematic diagram of the overall structure of the hydrophone unit in an embodiment of the present invention.

[0014] Figure 3 This is a schematic diagram of the layered explosion structure of the hydrophone unit in an embodiment of the present invention.

[0015] Figure 4 This is a schematic diagram comparing the input impedance of the lumped parameter model and the finite element simulation in an embodiment of the present invention.

[0016] Figure 5 This is a schematic diagram showing the relationship between the feature size of the sensitive unit and the resonant frequency in an embodiment of the present invention.

[0017] Figure 6This is a schematic diagram showing the relationship between the piezoelectric layer thickness and the resonant frequency in an embodiment of the present invention.

[0018] Figure 7 This is a schematic diagram showing the relationship between the thickness of the bottom electrode layer and the resonant frequency in an embodiment of the present invention.

[0019] Figure 8 This is a schematic diagram showing the relationship between device layer thickness and resonant frequency in an embodiment of the present invention.

[0020] Figure 9 This is a schematic diagram illustrating the relationship between the thickness of the buried oxide layer and the resonant frequency in an embodiment of the present invention.

[0021] Figure 10 This is a schematic diagram showing the relationship between the feature size of the sensitive unit, the thickness of the piezoelectric layer, and the static sensitivity level in an embodiment of the present invention.

[0022] Figure 11 This is a schematic diagram illustrating the influence and optimization of the piezoelectric layer thickness on the static sensitivity level under different shaped units in this embodiment of the invention.

[0023] Among them, 1-molybdenum upper electrode, 2-aluminum nitride piezoelectric film, 3-molybdenum lower electrode, 4-silicon device layer, 5-buried oxide layer, 6-silicon substrate, and 7-cavity. Detailed Implementation

[0024] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments: Example 1 This embodiment discloses a method for modeling mechanical parameters and optimizing the size of hydrophone sensing units based on the eigenfunction method. This method introduces the eigenfunction expansion method into the vibration modeling and parameter solving process of hydrophone sensing units with different geometries, realizing unified analysis of sensing units with multiple shapes. It can also optimize the structural size based on the analytical results. It is especially suitable for the design and performance optimization of hydrophone units with different geometries (including circles, rectangles, etc.) and can be extended to the structural design of micro underwater acoustic sensors such as MEMS hydrophones.

[0025] Before providing a detailed description of the method of the present invention, let me first introduce the general idea of ​​the method.

[0026] The purpose of this invention is to address the following problems in the design of existing hydrophone sensing elements: mechanical domain equivalent parameters largely rely on experience or numerical fitting; different geometric shapes lack a unified analytical modeling framework; and structural size optimization highly depends on finite element parameter scanning and lacks a clear size-performance mapping relationship. This invention proposes a method for modeling mechanical parameters and optimizing structural dimensions of hydrophone sensing elements based on the eigenfunction method. By establishing vibration control equations considering liquid loading in both polar and rectangular coordinate systems, eigenfunction systems satisfying different boundary conditions are constructed. Based on this, analytical or semi-analytical expressions for mechanical domain equivalent mass, equivalent stiffness, and related mechanical impedance parameters applicable to sensing elements with various geometric shapes such as circles, rings, and rectangles are derived, establishing a clear functional relationship between the geometric dimensions of the sensing element and key performance indicators.

[0027] Based on the above analytical expression of mechanical parameters, this invention further constructs a structural dimension optimization model for sensitive units of hydrophones with different shapes. Using target operating frequency band, sensitivity, bandwidth, and modal isolation as constraints or optimization objectives, it forms a general design process from inputting material parameters and performance requirements to outputting the optimal or near-optimal combination of structural dimensions that meets design specifications. This technical solution reduces reliance on large-scale finite element parameter scanning and purely empirical design, improving the efficiency and interpretability of structural design. Furthermore, it enables integrated modeling and optimization of sensitive units with various geometric forms, providing a unified mechanical modeling foundation and reliable dimensional design basis for the serial design of hydrophones and MEMS hydrophones, the extraction of equivalent circuit parameters, and subsequent array mutual coupling analysis.

[0028] This invention primarily considers lumped parameter modeling of the MEMS piezoelectric hydrophone unit structure, such as... Figure 2 and Figure 3 As shown, the hydrophone unit in this embodiment includes a molybdenum upper electrode 1, an aluminum nitride piezoelectric film 2, a molybdenum lower electrode 3, a silicon device layer 4, a buried oxide layer 5, and a silicon substrate 6, wherein a cavity 7 is formed on the silicon substrate 6. Specifically, the Mo / AlN / Mo stack structure consists of an AlN piezoelectric layer (i.e., aluminum nitride piezoelectric film 2) sandwiched between Mo electrodes (i.e., molybdenum upper electrode 1 and molybdenum lower electrode 3). The SOI substrate consists of a base silicon layer (i.e., silicon substrate 6), an oxide layer (i.e., buried oxide layer 5), and a silicon device layer 4, and includes a cavity structure (i.e., cavity 7). The shape of the PMUT diaphragm in the base silicon layer is approximately the same as the shape of the cavity, and the boundary conditions are fixed boundaries.

[0029] The method of the present invention will be described in detail below.

[0030] like Figure 1 As shown, the hydrophone multi-shape size optimization method based on eigenfunction expansion specifically includes the following steps: Step 1. Establish the vibration control model of the hydrophone sensing element, i.e., the thin plate, under the action of external sound pressure, and obtain the non-homogeneous partial differential equation of the forced vibration of the thin plate, i.e., the control equation of the forced vibration of the thin plate.

[0031] The equivalent structural model of the multilayer sensing unit is defined: For the structural parameters, material parameters and external sound pressure load of the multilayer sensing unit of the MEMS piezoelectric hydrophone, the overall modeling method of multilayer plate energy equivalence, fixed boundary constraints and variational method is adopted to derive the kinetic energy, potential energy and bending stiffness of the multilayer diaphragm, and the external sound pressure load is introduced into the control equation. Finally, a non-homogeneous partial differential equation applicable to the forced vibration of a thin plate without holes with arbitrary fixed boundary is obtained.

[0032] The vibration control equations output in step 1 will serve as the basic input for the eigenfunction expansion and modal solution in step 2.

[0033] In step 1 of this embodiment, establishing a vibration control model for the hydrophone sensing unit under applied sound pressure includes calculating the total kinetic and potential energy of the multilayer board. This process is specifically as follows: The expression for the total kinetic energy of a multilayer plate is: .

[0034] in, Indicates total kinetic energy; It is a hierarchical index. , This represents the total number of layers in the multilayer board system. Indicates the region where the thin plate is located. This indicates the density of a single-layer board. Indicates displacement. Indicates time, , , Represents the vibration coordinates. This indicates the operating frequency. This indicates the total density of the multilayer board. Indicates the area of ​​the thin plate. Indicates the thin plate in coordinates Displacement The square of.

[0035] According to the strain energy formula for linear elastic materials, the expression for the total potential energy of a multilayer plate is: .

[0036] in, Represents the total potential energy. and They represent the first Layer material in Xianghe Normal stress in the direction, and They represent the first Layer material in Xianghe Positive strain in the direction, Indicates the first Layer material in In-plane shear stress Indicates the first Layer material in Shear strain in the plane.

[0037] Substituting the strain and stress components, the expression for the total potential energy of the multilayer plate is expanded as follows: .

[0038] in, Indicates the first The center coordinates of the layer material Indicates the first The center coordinates of the layer material Indicates the first The center coordinates of the layer material Indicates the first Young's modulus of the layered material. It represents Poisson's ratio.

[0039] In this embodiment, the process of obtaining the control equation for forced vibration of the thin plate in step 1 is as follows: Based on the total potential energy expression of a multilayer plate, fixed boundary conditions are further introduced, and the higher-order derivative terms are simplified using Green's formula or the double integral integration by parts method, resulting in a simplified expression for the potential energy under boundary constraints that is more suitable for analytical derivation: .

[0040] in, This represents the equivalent bending stiffness of the multilayer board. Describes the differential operator. Right now .

[0041] Equivalent bending stiffness of multilayer boards The following expression exists: .

[0042] Based on the total kinetic and potential energy results, the Lagrange density expression for the thin plate system is constructed using the Lagrange modeling method: .

[0043] in, This represents the Lagrange density of the thin-plate system.

[0044] Using the Euler-Lagrange equations, a variational solution is performed on the multilayer plate system to obtain the governing equations for the free vibration of the thin plate: .

[0045] in, Indicates the displacement of the thin plate. Right now .

[0046] Further increase the external sound pressure Introducing the external excitation term into the model yields the non-homogeneous partial differential equations for the forced vibration of the thin plate, which are then used as the original governing equations for subsequent eigenfunction expansion and solution: .

[0047] Step 2. Use the eigenfunction method to obtain the solution of the governing equation for forced vibration of the thin plate; based on the governing equation for forced vibration of the thin plate, obtain the parametric expressions for equivalent mass, equivalent stiffness and acoustic transduction coefficient by normalizing the characteristic length and the amplitude of the mode shape function.

[0048] Due to the external sound pressure in step 1 Since the equations are not in a separated variable form, solving non-homogeneous partial differential equations using the method of separation of variables will encounter difficulties. To address this, we employ eigenfunction expansion, inner product projection, self-adjoint analysis, orthogonality simplification, and frequency domain transformation to convert the original partial differential equations into a form solvable modally, thus obtaining the analytical expression for the modal response of the thin plate under applied sound pressure.

[0049] In step 2 of this embodiment, the eigenfunction expansion process is specifically as follows: Similar to the method of separation of variables, let the solution of the governing equation for the forced vibration of a thin plate be... It must meet the following form: .

[0050] in, The summation index for modal expansion. , Indicates the first The mode shape function after amplitude normalization of the first order; Represents a function with respect to time.

[0051] The inner product form is: .

[0052] in, Indicates the first The modal shape function of the first order, The test mode order is the one corresponding to the inner product projection. Right now .

[0053] By taking the first By performing an inner product of the first-order eigenfunctions with respect to the governing equations, and utilizing the orthogonality of the eigenfunctions, the original coupled equations are simplified to those with respect to the second-order eigenfunctions. Independent equations for the first modal coordinates.

[0054] Rewritten as the eigenvalue equations in a Cartesian coordinate system: .

[0055] in, For eigenvalues, These are eigenfunctions.

[0056] For the rewritten eigenfunction equations, Green's formula and boundary condition analysis are used to prove that the relevant differential operators are self-adjoint operators under fixed or simply supported boundary conditions, thus providing a theoretical basis for the orthogonality of eigenfunctions.

[0057] Based on the self-adjoint property of differential operators, and further utilizing the orthogonality of eigenfunctions corresponding to different eigenvalues, the coupling terms after modal projection are simplified to obtain mutually independent modal equations: .

[0058] in, Indicates the first Modal mass of the first order, Indicates the first Modal coordinates of order, Indicates the first Modal stiffness of the first order, express Time of the first The load of the step.

[0059] In this embodiment, the process of obtaining the solution to the control equation of forced vibration of the thin plate in step 2 is as follows: For the second-order vibration equation, i.e., the control equation for forced vibration of a thin plate, Fourier transform is used to convert it to the frequency domain for solution and express it as a convolution integral: .

[0060] in, Indicates the first The frequency of the order, express Time of the first The load of the step.

[0061] Will Substitute into the equation The solution to the governing equation for forced vibration of a thin plate is obtained: .

[0062] in, For the first The frequency of the first order is the resonant frequency. For the first Modal mass of the first order, for Time of the first The load of the step.

[0063] At this point, the solution to the governing equation for forced vibration of a thin plate has been obtained using the eigenfunction method. If If the signal is causal, then the lower limit of integration is changed to 0. In this case, the solution result obtained by the eigenfunction method is consistent with the result obtained by applying the homogenization principle.

[0064] The process of obtaining the equivalent mass, equivalent stiffness, and acoustic transduction coefficient in step 2 is described below.

[0065] Based on the control equation for forced vibration of a thin plate, it is assumed that the sound pressure p on the hydrophone unit region A is uniformly distributed and can be considered a constant. In the study of the lumped parameter equivalent circuit, only the fundamental frequency corresponding to the operating frequency of the MEMS hydrophone, i.e., the q=1 mode, is usually considered. After normalization of the characteristic length and the amplitude normalization of the mode shape function, the equivalent mass is obtained. Equivalent stiffness Harmony force transduction coefficient The expression is: .

[0066] .

[0067] .

[0068] in, This indicates the characteristic length, which is the characteristic size of the sensing element; for a circular diaphragm, this means a circular sensing element. The radius is the element radius; for non-circular sensitive elements, The equivalent feature length is used to characterize its planar scale. For the force region in normalized coordinates, This represents the first-order normalized mode shape function.

[0069] This step corresponds to the simplified disk equation of the equivalent circuit expression derived through the energy equivalence method. This not only verifies the correctness of the derivation process but also yields a more general conclusion. The derivation process shows that the theoretical basis of the energy equivalence method is the orthogonality of mode shapes. For a thin plate of arbitrary shape, under fixed boundary conditions… It is a self-adjoint operator, therefore the lumped parameter modeling method is applicable to hydrophone units of various shapes.

[0070] Step 3. Establish the parametric expressions for electromechanical conductance coefficient and static capacitance. Based on the parametric expressions for equivalent mass, equivalent stiffness and acoustic conductance coefficient obtained in Step 2, establish the quantitative relationship between the characteristic size of the sensitive unit, the thickness of the piezoelectric layer, the area of ​​the upper electrode and the electrical input impedance, static acoustic pressure sensitivity and resonant frequency, and then determine the optimized structural size parameters that meet the target performance indicators.

[0071] Specifically, in step 3 of this embodiment, based on the modal response and equivalent parameters obtained in step 2, a quantitative relationship is further established between the input impedance, static acoustic pressure sensitivity, and resonant frequency and the characteristic dimensions of the sensitive unit, the thickness of each functional layer, and the area of ​​the upper electrode. Based on this, optimized structural dimension parameters that meet the target operating frequency band and sensitivity requirements are determined. The thickness of each functional layer specifically includes the piezoelectric layer thickness, electrode layer thickness, device layer thickness, and buried oxide layer thickness.

[0072] In this embodiment, the process of establishing the parametric expressions for the electromechanical conductance coefficient and the static capacitance in step 3 is as follows: Based on the piezoelectric constitutive relation and the definition of static capacitance, the electromechanical transduction coefficient is... and static capacitor The parametric expression is transformed to a Cartesian coordinate system, and combined with feature length normalization, to obtain the following results: .

[0073] .

[0074] in, Indicates the piezoelectric coefficient. This represents the distance from the center of the piezoelectric layer to the neutral plane. Indicates the area of ​​the upper electrode. It represents the dielectric constant under constant stress. This represents the normalized area of ​​the upper electrode. Indicates the thickness of the piezoelectric layer. This represents the electromechanical coupling coefficient.

[0075] The process of obtaining the analytical expression for static sound pressure sensitivity in step 3 is described below.

[0076] The equivalent mass obtained Equivalent stiffness Harmony force transduction coefficient Substituting the expression into the input impedance equation, we obtain the expression for electrical input impedance as follows: .

[0077] in, Indicates electrical input impedance. Represents the equivalent mechanical impedance of a mechanical branch; It is the imaginary unit.

[0078] The theoretical results of the frequency-impedance spectrum are obtained, and the comparison between the model calculation and simulation results is as follows: Figure 4 As shown, it exhibits good overall consistency.

[0079] The process of obtaining the analytical expression for static sound pressure sensitivity in step 3 is described below.

[0080] The static sensitivity of a hydrophone is usually defined as the sensitivity of the device at its operating frequency. Far below its main resonant frequency At this point, the ratio between the output voltage and the incident sound pressure is considered. Under this low-frequency excitation condition, the wavelength of the sound wave is much larger than the size of the hydrophone's sensitive structure, making the sound pressure distribution across the entire device surface seem uniform. In this case, the hydrophone's sensitivity response tends to a constant plateau region independent of frequency. This characteristic makes static sensitivity an important parameter for evaluating the low-frequency performance of hydrophones, simplifying theoretical models, and guiding the design of low-frequency devices such as MEMS hydrophones.

[0081] To accurately derive the analytical expression for static sensitivity, this invention first considers the vibration response of the hydrophone under a simple harmonic sound pressure p, and introduces the following integral expression and substitutes it into... get: .

[0082] Where p is ; Indicates the first The acoustic transduction coefficients corresponding to the first mode; when only the fundamental frequency is considered... Modal time, Degenerate into .

[0083] For hydrophones, the electrical ports are considered open circuits, i.e., I=0. The static sound pressure sensitivity is obtained by using current. Definition: .

[0084] in, Indicates the output voltage. This represents the displacement amplitude of the first-order mode.

[0085] Substituting the displacement amplitude of the first mode The expression for static sound pressure sensitivity is obtained by retrieving the expression for the static sound pressure sensitivity: .

[0086] The bandwidth of a hydrophone defines the frequency range within which it can effectively respond to and convert acoustic signals, and is one of the key performance indicators for measuring its ability to capture and process underwater acoustic information. The resonant frequency, as a crucial parameter determining the bandwidth of a hydrophone, is closely related to the device's sensitivity. Increasing the resonant frequency can extend the bandwidth of the hydrophone, but this comes at the cost of reduced sensitivity; this trade-off is particularly prominent in practical design. The resonant frequency of a hydrophone unit is mainly determined by material properties and structural parameters, with diaphragm thickness and radius playing a decisive role. By optimizing these key parameters, an ideal resonant frequency can be customized for specific application scenarios, thereby balancing bandwidth and sensitivity and improving the overall performance of the hydrophone.

[0087] The relationship between the characteristic size of the sensitive element and the resonant frequency is as follows: Figure 5 As shown, the relationship between the thickness of each layer in the multi-layer unit structure and the resonant frequency is presented in... Figures 6 to 9 The data in the figure compares the results for circular, square, and regular hexagonal units, revealing similar patterns of variation among the three geometric configurations.

[0088] To analyze the static sensitivity, i.e., static sound pressure sensitivity and the characteristic size of the sensing element. piezoelectric layer thickness and the area of ​​the upper electrode The relationship allows us to write the static sound pressure sensitivity expression as a product of four terms: .

[0089] in, This represents the normalized mode shape function. In the static acoustic pressure sensitivity expression, the first term is mainly determined by the piezoelectric and dielectric material parameters, the second term mainly reflects the influence of dimensional parameters such as the piezoelectric layer thickness and the characteristic dimensions of the sensitive unit, the third term characterizes the influence of the overall shape of the unit on the sensitivity, and the fourth term characterizes the influence of the shape and coverage of the top electrode on the sensitivity.

[0090] The process of dimensional parameter optimization analysis in step 3 is described below.

[0091] According to the analysis, the optimized structural parameters for static sensitivity are applicable to hydrophone units of different shapes, except for the top electrode, which greatly simplifies the complexity of the problem. Figure 10 and Figure 11The relationship between static receiver sensitivity and the feature size of the sensing element and the thickness of the piezoelectric layer is shown. Considering the similarity of static sensitivity results for elements of different shapes, only data for circular elements are presented here. The continuous curves in the figure represent the predicted values ​​from the lumped-parameter model, while the blue scatter dots represent the simulation results obtained from parametric scanning using COMSOL software, further confirming the accuracy of the model. Except for the piezoelectric layer thickness, the static sensitivity decreases monotonically with increasing layer thickness. According to... That is, static sensitivity is directly proportional to the sensing area, and is consistent with the logarithmic coordinate system. Figure 10 and Figure 11 This is consistent. The figure also shows the effect of piezoelectric layer thickness on static sensitivity. As the piezoelectric layer thickness increases, the static sensitivity exhibits a non-monotonic trend of first increasing and then decreasing, with a maximum point. This can be addressed by optimizing the ratio. Determine the optimal piezoelectric layer thickness .

[0092] This invention utilizes the eigenfunction method as its core. By constructing analytical eigenfunctions applicable to hydrophone sensing elements of different geometries and combining them with normalized coordinate transformation, it achieves unified modeling, parameter solving, and dimensional optimization for sensing elements of various shapes. Furthermore, it can analytically obtain key parameters such as equivalent mass, equivalent stiffness, electromechanical coupling coefficient, and static sensitivity, thereby establishing a quantitative relationship between structural dimensions and performance indicators. Therefore, this invention reduces the reliance on repeated parameter sweeps and empirical trial-and-error in large-scale finite element methods, improving design efficiency.

[0093] Example 2 This embodiment 2 describes a computer device that includes a memory and one or more processors.

[0094] The memory stores executable code, which, when executed by the processor, is used to implement the steps of the hydrophone multi-shape size optimization method based on eigenfunction expansion in Embodiment 1 above.

[0095] In this embodiment, the computer device can be any device or apparatus with data processing capabilities, and will not be described in detail here.

[0096] Example 3 This embodiment 3 describes a computer-readable storage medium storing a program that, when executed by a processor, implements the steps of a hydrophone multi-shape size optimization method based on eigenfunction expansion.

[0097] The computer-readable storage medium can be an internal storage unit of any device or apparatus with data processing capabilities, such as a hard disk or memory, or an external storage device of any device with data processing capabilities, such as a plug-in hard disk, smart media card (SMC), SD card, flash card, etc.

[0098] Of course, the above description is only a preferred embodiment of the present invention. The present invention is not limited to the above-described embodiments. It should be noted that any equivalent substitutions or obvious modifications made by those skilled in the art under the guidance of this specification fall within the scope of this specification and should be protected by the present invention.

Claims

1. A method for optimizing the dimensions of hydrophones of various shapes based on eigenfunction expansion, characterized in that, Includes the following steps: Step 1. Establish the vibration control model of the hydrophone sensing element, i.e., the thin plate, under the action of external sound pressure, and obtain the non-homogeneous partial differential equation of the forced vibration of the thin plate, i.e., the control equation of the forced vibration of the thin plate. Step 2. Use the eigenfunction method to obtain the solution of the governing equation for the forced vibration of the thin plate; Based on the control equation of forced vibration of thin plate, the parametric expressions of equivalent mass, equivalent stiffness and acoustic transduction coefficient are obtained by normalizing the characteristic length and the amplitude of the mode shape function. Step 3. Establish the parametric expressions for electromechanical conductance coefficient and static capacitance. Based on the parametric expressions for equivalent mass, equivalent stiffness and acoustic conductance coefficient obtained in Step 2, establish the quantitative relationship between the characteristic size of the sensitive unit, the thickness of the piezoelectric layer, the area of ​​the upper electrode and the electrical input impedance, static acoustic pressure sensitivity and resonant frequency, and then determine the optimized structural size parameters that meet the target performance indicators. In step 1, the process of obtaining the governing equation for the forced vibration of the thin plate is as follows: Based on the total potential energy expression of a multilayer plate, fixed boundary conditions are further introduced, and the higher-order derivative terms are simplified to obtain the simplified expression for the potential energy under boundary constraints: ； in, Represents the total potential energy. This represents the equivalent bending stiffness of the multilayer board. Indicates the area of ​​the thin plate. Describes the differential operator. Indicates the thin plate in coordinates Displacement, Right now ; , , Represents the vibration coordinates; Equivalent bending stiffness of multilayer boards The following expression exists: ； in, It is a hierarchical index. , This represents the total number of layers in the multilayer board system. Indicates the first The center coordinates of the layer material Indicates the first The center coordinates of the layer material Indicates the first The center coordinates of the layer material Indicates the first Young's modulus of the layered material. Indicates Poisson's ratio; Based on the total kinetic and potential energy results, the Lagrange density expression for the thin plate system is constructed using the Lagrange modeling method: ； in, The Lagrangian density of a thin-plate system is represented by... Indicates the total density of the multilayer board; Indicates displacement. Indicates time; Using the Euler-Lagrange equations, a variational solution is performed on the multilayer plate system to obtain the governing equations for the free vibration of the thin plate: ； in, Right now ; Further increase the external sound pressure Introducing the external excitation term into the model yields the non-homogeneous partial differential equations for the forced vibration of the thin plate, which are then used as the original governing equations for subsequent eigenfunction expansion and solution: 。 2. The hydrophone multi-shape size optimization method based on eigenfunction expansion according to claim 1, characterized in that, In step 1, establishing the vibration control model of the hydrophone sensing unit under applied sound pressure includes calculating the total kinetic energy and total potential energy of the multilayer plate. This process is as follows: The expression for the total kinetic energy of a multilayer plate is: ； in, Indicates total kinetic energy; Indicates the region where the thin plate is located. This indicates the density of a single-layer board. Indicates the operating frequency. Indicates the thin plate in coordinates Displacement The square of; According to the strain energy formula for linear elastic materials, the expression for the total potential energy of a multilayer plate is: ； in, and They represent the first Layer material in Xianghe Normal stress in the direction, and They represent the first Layer material in Xianghe Positive strain in the direction, Indicates the first Layer material in In-plane shear stress Indicates the first Layer material in Shear strain in the plane; Substituting the strain and stress components, the expression for the total potential energy of the multilayer plate is expanded as follows: .

3. The hydrophone multi-shape size optimization method based on eigenfunction expansion according to claim 2, characterized in that, In step 2, the eigenfunction expansion process is as follows: Let the solution of the governing equation for forced vibration of a thin plate be given. It must meet the following form: ； in, The summation index for modal expansion. , Indicates the first The mode shape function after amplitude normalization of the first order; A function representing time; The inner product form is: ； in, Indicates the first The modal shape function of the first order, The test mode order is the one corresponding to the inner product projection. Right now ; By taking the first By performing an inner product of the first-order eigenfunctions with respect to the governing equations, and utilizing the orthogonality of the eigenfunctions, the original coupled equations are simplified to those with respect to the second-order eigenfunctions. Independent equations for first-order modal coordinates; Rewritten as the eigenvalue equations in a Cartesian coordinate system: ； in, These are eigenvalues; For the rewritten eigenvalue equations, Green's theorem and boundary conditions are used for analysis to prove that the differential operator is a self-adjoint operator under fixed or simply supported boundary conditions. Based on the self-adjoint property of the differential operator, the orthogonality of eigenfunctions corresponding to different eigenvalues ​​is further utilized to simplify the coupling terms after modal projection, resulting in mutually independent modal equations: ； in, Indicates the first Modal mass of the first order, Indicates the first Modal coordinates of order, Indicates the first Modal stiffness of the first order, express Time of the first The load of the step.

4. The hydrophone multi-shape size optimization method based on eigenfunction expansion according to claim 3, characterized in that, In step 2, the process of obtaining the solution to the control equation of the forced vibration of the thin plate is as follows: For the second-order vibration equation, i.e., the control equation for forced vibration of a thin plate, Fourier transform is used to convert it to the frequency domain for solution and express it as a convolution integral: ； in, Indicates the first The frequency of the order, express Time of the first The load of the step; Will Substitute into the equation The solution to the governing equation for forced vibration of a thin plate is obtained: ； in, For the first The frequency of the first order is the resonant frequency. For the first Modal mass of the first order, for Time of the first The load of the step.

5. The hydrophone multi-shape size optimization method based on eigenfunction expansion according to claim 4, characterized in that, In step 2, after normalizing the characteristic length and the amplitude of the mode shape function, the equivalent mass is obtained. Equivalent stiffness Harmony force transduction coefficient The expression is: ； ； ； in, Indicates the feature size of the sensitive unit; Represents the force region in normalized coordinates. This represents the first-order normalized mode shape function.

6. The hydrophone multi-shape size optimization method based on eigenfunction expansion according to claim 5, characterized in that, In step 3, the process of establishing the parametric expressions for the electromechanical conductance coefficient and the static capacitance is as follows: Based on the piezoelectric constitutive relation and the definition of static capacitance, the electromechanical transduction coefficient is... and static capacitor The parametric expression is transformed to a Cartesian coordinate system, and combined with feature length normalization, to obtain the following results: ； ； in, Indicates the piezoelectric coefficient. This represents the distance from the center of the piezoelectric layer to the neutral plane. Indicates the area of ​​the upper electrode. It represents the dielectric constant under constant stress. This represents the normalized area of ​​the upper electrode. Indicates the thickness of the piezoelectric layer. This represents the electromechanical coupling coefficient.

7. The hydrophone multi-shape size optimization method based on eigenfunction expansion according to claim 6, characterized in that, In step 3, the obtained equivalent mass Equivalent stiffness Harmony force transduction coefficient Substituting the expression into the input impedance equation, we obtain the expression for electrical input impedance as follows: ； in, Indicates electrical input impedance. Represents the equivalent mechanical impedance of a mechanical branch; The imaginary unit; Considering the vibration response of a hydrophone under a simple harmonic sound pressure p, we introduce the following integral expression and substitute it into... get: ； Where p is ; Indicates the first The acoustic transduction coefficients corresponding to the first mode; when only the fundamental frequency is considered... Modal time, Degenerate into ; For hydrophones, the electrical ports are considered open circuits, i.e., I=0. The static sound pressure sensitivity is obtained by using current. Definition: ； in, Indicates the output voltage. This represents the displacement amplitude of the first-order mode; Substituting the displacement amplitude of the first mode The expression for static sound pressure sensitivity is obtained by retrieving the expression for the static sound pressure sensitivity: ； To analyze the static sensitivity, i.e., static sound pressure sensitivity and the characteristic size of the sensing element. piezoelectric layer thickness and the area of ​​the upper electrode The relationship allows us to write the static sound pressure sensitivity expression as a product of four terms: ； in, Represents the normalized mode shape function; Based on the electrical input impedance expression, the static sound pressure sensitivity expression, and the quantitative relationship between the resonant frequency and the characteristic size of the sensitive unit, the piezoelectric layer thickness, and the area of ​​the upper electrode, the characteristic size of the sensitive unit, the piezoelectric layer thickness, and the area of ​​the upper electrode that meet the target operating frequency band, resonant frequency, and sensitivity requirements are selected as the optimized structural size parameters, thus obtaining the optimized structural size parameters that meet the target performance indicators.

8. A computer device comprising a memory and one or more processors, wherein the memory stores executable code, characterized in that, When the processor executes the executable code, it implements the steps of the hydrophone multi-shape size optimization method based on eigenfunction expansion as described in any one of claims 1 to 7.

9. A computer-readable storage medium having a program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the hydrophone multi-shape size optimization method based on eigenfunction expansion as described in any one of claims 1 to 7.

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