Lattice form superstructure topology optimization design method for underwater sound absorption

Abstract

The application relates to the technical field of underwater environment sound absorption and noise reduction, and discloses a dot-matrix form superstructure topology optimization design method for underwater sound absorption. The dot-matrix form superstructure is embedded between two rubber layers as a core wave control component by adopting periodic structure wave fluctuation regulation principles and viscoelastic material sound energy dissipation principles, and an equivalent mathematical model is constructed based on a two-dimensional plane strain assumption. A homogenization method based on an average effect is adopted to equivalently convert the dot-matrix form superstructure into a uniform medium, and an extended transfer matrix method is constructed to calculate the underwater sound absorption material acoustic performance of a sandwich structure composed of rubber layers-dot-matrix form superstructure-rubber layers. A reverse design method based on topology optimization is adopted to automatically optimize the cell topology morphology of the dot-matrix superstructure with broadband acoustic performance as the optimization target, so that the material light weight, high stiffness and optimal sound absorption performance are simultaneously achieved. The dot-matrix form superstructure topology morphology can be efficiently and systematically optimized.

Description

Technical Field

[0001] This invention relates to the field of underwater environment sound absorption and noise reduction technology, and in particular, to a lattice-type superstructure topology optimization design method for underwater sound absorption. Background Technology

[0002] The regulation of acoustic performance in underwater environments is of great significance for applications such as marine resource exploration, underwater communication, and environmental protection. With the increasing frequency of human activities in the ocean, underwater noise control has become one of the key issues urgently needing to be addressed. Underwater sound-absorbing materials, as a type of artificial structure capable of effectively attenuating sound wave energy, have broad application prospects in reducing underwater noise pollution.

[0003] Traditional underwater sound-absorbing materials typically employ methods such as incorporating cavities within a rubber matrix, localized resonant units, or embedded mass blocks to enhance sound wave energy dissipation. However, such design methods heavily rely on empirical construction, making it difficult to systematically coordinate the material's sound absorption performance, lightweight characteristics, and pressure resistance. Especially when pursuing broadband, high-efficiency sound absorption while simultaneously requiring good mechanical properties, traditional design strategies often struggle to achieve both simultaneously.

[0004] In recent years, the emergence of acoustic metastructures has provided new ideas for the design of sound-absorbing materials. Metastructures, through the periodic arrangement of artificially designed microstructural units, can achieve flexible control over the propagation characteristics of sound waves. Compared with traditional materials, underwater sound-absorbing materials based on metastructures show potential in achieving high stiffness, lightweight, and engineering applicability. However, existing research still faces many challenges when applying structural optimization methods to the design of underwater sound-absorbing metastructures, such as low computational efficiency in the optimization process, poor sound absorption performance of the designed structures across a wide frequency band, and the difficulty in simultaneously achieving multiple requirements such as lightweight, high stiffness, and excellent acoustic performance. Summary of the Invention

[0005] This invention provides a topology optimization design method for lattice-type superstructures for underwater sound absorption. It can efficiently and systematically optimize the topology of superstructures, promote the development of new underwater sound-absorbing materials with both excellent sound absorption performance and comprehensive mechanical properties, and solve the technical problem that existing underwater sound-absorbing materials rely on human experience design and are difficult to achieve broadband high-efficiency sound absorption, lightweight and high stiffness at the same time.

[0006] This invention provides a topology optimization design method for a lattice-type superstructure for underwater sound absorption, comprising the following steps: S100, establishing an underwater sound absorption mathematical model, employing the principle of periodic structural wave regulation and the principle of sound energy dissipation of viscoelastic materials, embedding the lattice-type superstructure as the core wave regulation component between two rubber layers, and constructing an equivalent mathematical model based on the two-dimensional plane strain assumption; S200, constructing an acoustic performance calculation method, employing a homogenization method based on average effect to treat the lattice-type superstructure as a homogeneous medium, and constructing an extended transfer matrix method to calculate the acoustic performance of the underwater sound-absorbing material of the sandwich structure composed of rubber layer-lattice-type superstructure-rubber layer; S300, optimizing the sandwich structure, employing a reverse design method based on topology optimization, taking broadband acoustic performance as the optimization objective, automatically optimizing the cell topology of the lattice superstructure, and simultaneously achieving lightweight, high stiffness, and optimal sound absorption performance of the material.

[0007] Further, step S100 includes constructing a basic model: the basic model includes a first layer of rubber, a superstructure layer, and a second layer of rubber sequentially arranged along the incident sound wave in the water. The first layer of rubber, the superstructure layer, and the second layer of rubber constitute an underwater sound-absorbing material, and the sound-absorbing material extends along its upper and lower boundaries. y The direction is a periodic boundary, and the superstructure layer is sealed inside the first and second rubber layers; the first side of the underwater sound-absorbing material is used for coupling with the water area, and the second side of the underwater sound-absorbing material is used to be applied to the steel backing, which is coupled with the air domain; the superstructure layer adopts a lattice-shaped superstructure, which has the characteristic of regulating the propagation of sound waves; both the first and second rubber layers are made of viscoelastic damping materials, which have the ability to dissipate sound waves.

[0008] Furthermore, in underwater sound-absorbing materials, the lattice-shaped superstructure is the core component for wave modulation. The lattice-shaped superstructure consists of periodically arranged cellular structures and insulating plates. The insulating plates ensure a good connection surface between the lattice-shaped superstructure and the rubber materials of the first and second layers of rubber. The cellular structures are arranged along a local coordinate system. x 1. Direction and y 1. Periodic extension in one direction, rotating the entire cellular structure counterclockwise to correspond to the global coordinate system of the underwater sound-absorbing material. xy The axis forms an angle θ Cellular structure along z Since the direction is infinitely stretched in thickness, underwater sound-absorbing materials are equivalent to a two-dimensional plane strain problem, which simplifies the calculation and analysis.

[0009] Further, step S200 specifically includes: the working state of the lattice-form superstructure does not depend on the local resonant mode, therefore wave propagation in the lattice-form superstructure is controlled by the quasi-static average material properties, and its behavior is characterized by the average effect; when the lattice-form superstructure is homogenized by periodic cells, the equivalent density of the lattice-form superstructure is the average density, and the equivalent elastic constant of the lattice-form superstructure is obtained by the energy homogenization method, thereby constructing the extended transfer matrix method to calculate the acoustic performance of the underwater sound-absorbing material in the sandwich form of the first rubber-lattice-form superstructure-second rubber.

[0010] Furthermore, the transfer matrix method is extended, specifically including: in a multilayer elastic medium of finite length, when an elastic wave propagates, it will form a reflected wave within the medium. The displacement field in a certain layer of the elastic medium can be represented as a product of vectors: (1) in This represents the displacement amplitude of different wave modes. P , N x The matrices are the polarization vector matrix and the phase matrix, respectively, specifically: (2) (3) in P x 、P y 、Q x 、Q y They are respectively longitudinal waves and transverse waves. x direction and y Directional polarization components, k qL and k qT These are the wave numbers of the longitudinal and transverse waves, respectively. To describe the propagation relationship of elastic waves in different elastic medium layers, field quantities are defined. Describing the velocity field in an elastic medium , and stress field quantity , This field quantity is described based on the displacement field: (4) Where the L matrix is (5) For the same elastic medium, any x and x + d Location, field volume Satisfying transitivity: (6) Where T d For length is d The transfer matrix in the elastic medium layer.

[0011] Furthermore, when elastic waves propagate in different media, the interface between the two elastic media satisfies the continuity condition, namely, velocity continuity and stress continuity. At this point, the field quantities on both sides of the interface... Since they are equal, based on the leftmost field quantity in the multilayer elastic medium, the field quantity corresponding to the rightmost interface can be further obtained through the transfer relation. This transfer relation is written as... (7) in T t This is the total transfer matrix.

[0012] Furthermore, when calculating the sound absorption performance of underwater sound-absorbing materials, the presence of sound pressure amplitude in the water area is considered. P i along +x A plane wave propagates in a specific direction; when excited by this sound wave, it interacts with underwater sound-absorbing materials to generate a sound pressure amplitude of [value missing]. P r The reflected sound waves generate a sound pressure amplitude of [value missing] in the air domain. P t The transmitted sound waves; according to the acoustic-structure interaction boundary conditions, the interface between the elastic body and the fluid satisfies continuous normal stress, continuous normal particle velocity, and zero tangential stress. Therefore, at the interface between the fluid and the underwater sound-absorbing material... For along +x Plane waves propagating in a specific direction For along -x Plane waves propagating in a specific direction Z The corresponding characteristic impedance in the fluid domain is, in water, the corresponding value is: Z w In the air domain Z a Define vector Based on the fluid-structure interaction relationship, we can obtain the following: (8) For underwater sound-absorbing materials, the transmission relationship between the incident s- interface and the transmitting s+ interface is established. (9) Where T d1 ~T d6 These represent the transfer matrices from the first rubber layer to the steel backing, obtained according to formula (6); The amplitude of the incident sound wave is pi = At 1 Pa, since the water and air domains are semi-infinite, at the interface... s- and s+ There is a boundary condition vector. (10) Substituting formula (10) into (9), the reflection coefficient in the water can be calculated: (11) in H lm For matrix H elements, l and m The value can be 1, 2, 3 or 4; substituting the calculation result into formula (9) can further obtain the transmitted sound wave. When calculating the sound absorption performance, the first fluid domain is water, and the second fluid domain is air. Because the characteristic impedance of the air domain differs greatly from that of the steel backing, the amplitude of the transmitted sound wave pressure is extremely small and can be ignored in the calculation of the sound absorption coefficient. Therefore, the sound absorption coefficient is: (12).

[0013] Furthermore, step S300 specifically includes: addressing the underwater sound absorption problem, optimizing directly based on the established extended transfer matrix method to meet acoustic performance requirements; and, to achieve on-demand reverse design of periodic unit cells in the superstructure, designing the unit cell structure as a square lattice and discretizing it into 100×100 elements, assigning design variables to each element based on an isotropic material penalty function model. ( To improve numerical stability during the optimization process, global design variables were adjusted. Density filtering and Heaviside projection are performed, and a robust formula is used to ensure the stability of the optimization results.

[0014] Furthermore, based on the isotropic material penalty function model, the Young's modulus within each mesh element... E and density ρ Described as physical variables The interpolation function, namely: (13) in ρ 1 and E 1 represents the Young's modulus and density of the matrix material. ρ 0=10 -6 ρ 1 and E 0=10 -6 E 1 represents an artificial variable parameter. p1 and p 2 is the penalty factor for the density and Young's modulus interpolation function.

[0015] Furthermore, to achieve on-demand design of sound-absorbing materials, based on the reflection coefficient... To optimize the target, since the characteristic impedance difference between the steel backing and the air domain is large, the transmitted sound wave can be ignored. The smaller the reflection coefficient, the larger the sound absorption coefficient; the average reflection coefficient within the target frequency band is Since the robustness formula leads to three different design implementations during the optimization process, the objective function of the optimization problem is transformed into a typical max-min problem, i.e.: (14) in Represents the reflection coefficient R The table below shows e, b, and d, which represent the three solutions generated in the robustness formula. To ensure that the optimization results have good load-bearing capacity and structural connectivity, the equivalent Young's modulus ( ) is used. E x 、 E y Constraining the optimization results and applying density constraints, we obtain the following optimization formula: (15) in, and for x direction and y Equivalent Young's modulus in the direction. E t The constraint threshold for the equivalent Young's modulus. The volume fraction of the physical structure. V t The volume fraction constraint threshold is used to ensure that the formed physical structure has good connectivity and roughly reflects the load-bearing capacity of the structure through the equivalent Young's modulus constraint; the volume fraction constraint can ensure the lightweight requirement.

[0016] The present invention has the following beneficial effects: 1. In terms of modeling: By combining the principle of periodic structural wave modulation with the principle of acoustic energy dissipation of viscoelastic materials, and using the two-dimensional plane strain assumption to construct an equivalent mathematical model, the complex physical problem of two-dimensional lattice superstructure can be simplified into a computable model. This ensures that the constructed model can simultaneously reflect the two core physical mechanisms of acoustic wave modulation and material dissipation, providing a correct theoretical starting point for subsequent accurate performance calculations and effective topology optimization, and avoiding deviations in optimization results caused by model distortion.

[0017] 2. In terms of performance calculation: A homogenization method based on the average effect is adopted to equate the lattice-shaped superstructure to a homogeneous medium, and then the extended transfer matrix method is constructed to calculate the overall acoustic performance of the sandwich structure, achieving an important balance between computational efficiency and accuracy. The homogenization method avoids the time-consuming full-scale finite element simulation of cells with complex microscopic topology, while the extended transfer matrix method can efficiently and accurately predict the acoustic wave transmission and absorption characteristics of multilayer structures in a wide frequency band. Together, these methods solve the problem of long optimization design process and provide computational feasibility for introducing topology optimization into such complex acoustic problems.

[0018] 3. In terms of optimization design: A topology optimization reverse design method with broadband acoustic performance as the direct optimization target is adopted to automatically optimize the cellular topology of the lattice superstructure. This achieves synchronous and collaborative design of multiple performance targets. Broadband acoustic performance is used as the optimization target, while lightweight and high stiffness of the material are used as intrinsic design constraints or synchronous optimization targets. This overcomes the shortcomings of traditional empirical design, which is difficult to take multiple performances into account. It can systematically and automatically search for topological configurations that simultaneously meet the requirements of excellent sound absorption, lightweight and high stiffness. Through this reverse design process of topology optimization, the limitations of traditional empirical design can be broken through to obtain the optimal topological morphology, thereby improving the final design results and achieving a comprehensive performance level of broadband high-efficiency sound absorption, lightweight and high stiffness.

[0019] In addition to the objectives, features, and advantages described above, the present invention has other objectives, features, and advantages. The invention will now be described in further detail with reference to the figures. Attached Figure Description

[0020] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings: Figure 1 This is a schematic diagram of the underwater sound-absorbing material according to a preferred embodiment of the present invention; Figure 2 This is a schematic diagram of the acoustic-structure coupling transmission relationship of the underwater sound-absorbing material according to a preferred embodiment of the present invention; Figure 3 This is a diagram illustrating the iterative process of topology optimization design according to a preferred embodiment of the present invention, wherein... Figure 3 (a) is the graph of the objective function. Figure 3 (b) is a diagram of the iterative process of physical variables; Figure 4 This is a diagram showing the result of topology optimization design according to a preferred embodiment of the present invention. Figure 4 (a) shows the lattice-type superstructure cell UC1 and the corresponding underwater sound-absorbing material M1. Figure 4(b) is a graph of the sound absorption coefficient of the optimization results. Detailed Implementation

[0021] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings. However, the present invention can be implemented in many different ways as defined and covered below.

[0022] The preferred embodiment of the present invention provides a topology optimization design method for a lattice-type superstructure for underwater sound absorption, comprising the following steps: S100, establishing an underwater sound absorption mathematical model, employing the principle of periodic structural wave regulation and the principle of sound energy dissipation of viscoelastic materials, embedding the lattice-type superstructure as the core wave regulation component between two rubber layers, and constructing an equivalent mathematical model based on the two-dimensional plane strain assumption; S200, constructing an acoustic performance calculation method, employing a homogenization method based on the average effect, treating the lattice-type superstructure as a homogeneous medium, and constructing an extended transfer matrix method to calculate the acoustic performance of the underwater sound-absorbing material of the sandwich structure composed of rubber layer-lattice-type superstructure-rubber layer; S300, optimizing the sandwich structure, employing a reverse design method based on topology optimization, taking broadband acoustic performance as the optimization objective, automatically optimizing the cell topology of the lattice superstructure, and simultaneously achieving lightweight, high stiffness, and optimal sound absorption performance of the material. This invention presents a topology optimization design method for lattice-type superstructures used for underwater sound absorption. Through a systematic "modeling-computation-optimization" design process, it achieves a paradigm shift from parametric design to automated, optimal design of lattice-type superstructures. By combining the principles of periodic structural wave modulation and the sound energy dissipation of viscoelastic materials, and employing a two-dimensional plane strain assumption to construct an equivalent mathematical model, the complex physical problems of two-dimensional lattice superstructures can be simplified into a computable model. This ensures that the constructed model simultaneously reflects both the sound wave modulation and material dissipation mechanisms, providing a correct theoretical starting point for subsequent accurate performance calculations and effective topology optimization, and avoiding deviations in optimization results caused by model distortion. A homogenization method based on the average effect is adopted to treat the lattice-shaped superstructure as a homogeneous medium, and then an extended transfer matrix method is constructed to calculate the overall acoustic performance of the sandwich structure, achieving an important balance between computational efficiency and accuracy. The homogenization method avoids the time-consuming full-scale finite element simulation of cells with complex microscopic topology, while the extended transfer matrix method can efficiently and accurately predict the acoustic wave transmission and absorption characteristics of multilayer structures in a wide frequency band. Together, these methods solve the problem of long optimization design process and provide computational feasibility for introducing topology optimization into such complex acoustic problems. A topology optimization reverse design method with broadband acoustic performance as the direct optimization target is adopted to automatically optimize the cellular topology of lattice superstructures. This method achieves synchronous and collaborative design of multiple performance targets. Broadband acoustic performance is used as the optimization target, while lightweight and high stiffness of materials are used as intrinsic design constraints or synchronous optimization targets. This overcomes the shortcomings of traditional empirical design, which is unable to take multiple performances into account. It can systematically and automatically search for topological configurations that simultaneously meet the requirements of excellent sound absorption, lightweight and high stiffness. Through this reverse design process of topology optimization, the limitations of traditional empirical design can be broken through to obtain the optimal topological morphology, thereby improving the final design results and achieving a comprehensive performance level of broadband high-efficiency sound absorption, lightweight and high stiffness.This invention presents a topology optimization design method for lattice-type superstructures used for underwater sound absorption. By establishing an equivalent model that integrates the principles of sound wave modulation and dissipation, employing homogenization and extended transfer matrix methods to achieve efficient performance calculations, and finally applying topology optimization centered on broadband sound absorption, a complete, efficient, and automated design closed loop is formed. Ultimately, it can systematically and automatically design underwater sound-absorbing superstructures that possess high-efficiency sound absorption, lightweight, and high stiffness characteristics over a wide frequency range, achieving an optimal balance of multi-objective performance. This effectively overcomes the shortcomings of traditional design methods that rely on experience and struggle to consider multiple performance indicators simultaneously.

[0023] like Figure 1 As shown, in this embodiment, step S100 includes constructing a basic model: the basic model includes a first layer of rubber, a superstructure layer, and a second layer of rubber sequentially arranged along the incident sound wave in the water. The first layer of rubber, the superstructure layer, and the second layer of rubber constitute an underwater sound-absorbing material. The sound-absorbing material extends along its upper and lower boundaries... y The direction (upper and lower boundaries) is a periodic boundary. The superstructure layer is sealed within the first and second rubber layers. The first side of the underwater sound-absorbing material is used for coupling with the water area, and the second side is used to attach to the steel backing, which couples with the air domain. The superstructure layer adopts a lattice-type superstructure (also referred to as superstructure), which has the characteristic of modulating the propagation of sound waves. Both the first and second rubber layers are made of viscoelastic damping materials, which have the ability to dissipate sound waves. The "rubber-superstructure-rubber" sandwich configuration seals the lattice-type superstructure layer. Between the two rubber layers, a functionally distinct sandwich structure is formed, achieving the physical separation and efficient integration of sound wave modulation and sound energy dissipation functions. The middle superstructure layer is dedicated to modulating propagating sound waves (e.g., through local resonance, wave mode conversion, etc.), while the two viscoelastic rubber layers on either side of the superstructure layer are dedicated to dissipating sound energy. This clearly defined configuration avoids the performance trade-offs that occur when a single material undertakes multiple functions simultaneously, providing a clear structural platform for optimizing each component and ultimately synergistically improving overall performance. The setting of the periodic boundary places the model within... yThe direction (i.e., the upper and lower boundaries within the plane perpendicular to the direction of sound wave propagation) is set as a periodic boundary, which is an acoustic metamaterial / metasurface based on the assumption of an infinitely large plane, simplifying the calculation. Its sound absorption performance originates from the periodic arrangement of the cellular structure, ensuring that the optimized single-cell structure can maintain the acoustic performance predicted by the calculation when extended to actual engineering materials, thus enhancing the scalability and engineering practicality of the design results. The multi-layer coupling boundary effect of the water-underwater sound-absorbing material-steel backing-air domain, the boundary condition setting (the first side couples to the water, and the second side couples to the air through the steel backing) accurately simulates the typical practical application scenario of underwater sound-absorbing materials. This ensures that any performance calculation and optimization design based on this model considers the complex reflection, transmission and coupling process of sound waves from water to the material and then to the rigid backing, ensuring that the optimization target is directly aimed at the acoustic performance under real working conditions, thereby significantly improving the practicality and effectiveness of the final design results and avoiding the problem of performance distortion of the structure optimized under ideal boundary conditions in actual applications.

[0024] In this embodiment, the lattice-shaped superstructure is the core component for wave modulation in the underwater sound-absorbing material. The lattice-shaped superstructure consists of periodically arranged cellular structures and an isolation plate. The isolation plate ensures a good connection surface between the lattice-shaped superstructure and the rubber materials of the first and second layers of rubber. The cellular structure (e.g., cellular structure...) Figure 1 (As shown in the enlarged diagram below) along the local coordinate system x 1. Direction and y 1. Periodic extension in one direction, rotating the entire cellular structure counterclockwise to correspond to the global coordinate system of the underwater sound-absorbing material. xy The axis forms an angle θ Cellular structure along z With the direction of infinite stretching, underwater sound-absorbing materials are equivalent to a two-dimensional plane strain problem, thus simplifying calculations and analysis. By setting up an isolation plate, an optimized connection interface is formed between the lattice-shaped superstructure and the first and second rubber layers, ensuring effective stress transfer between the superstructure layers and the rubber layers, enhancing the integrity and stability of the sandwich structure. The isolation plate provides a design transition zone, mitigating stress concentration or connection failure issues that may occur when the periodic lattice microstructure is directly combined with the macroscopic uniform rubber layer, thereby contributing to improved durability and reliability of the final product. This is achieved by rotating the cellular structure by a specific angle in the global coordinate system. θThis introduces controllable degrees of freedom, allowing designers to alter the relative relationship between the incident sound wave direction and the superstructure lattice direction simply by adjusting the spatial orientation of the cellular structure without changing its own topology. This enables flexible control of the structure's acoustic bandgap or vibrational mode characteristics, providing the possibility to further broaden the sound absorption band or improve the sound absorption performance at specific frequencies by optimizing the orientation angle under a fixed manufacturing process. By clearly defining the cellular structure in... z The direction is infinitely stretched in thickness, which reasonably simplifies the problem into a two-dimensional plane strain problem, thus facilitating efficient topology optimization. This greatly reduces the dimensionality and complexity of the computational model, transforming the original three-dimensional elastic dynamics problem, which required huge computational resources, into a problem that can be solved in a two-dimensional plane. This reduces the computational cost and time in subsequent acoustic performance calculations and topology optimization iterations, making automated, reverse design strategies based on numerical simulation feasible in engineering practice.

[0025] In this embodiment, the superstructure in the underwater sound-absorbing material is formed by a periodic array of structures in the form of a lattice, such as... Figure 1As shown. Step S200 specifically includes: the working state of the lattice-form superstructure does not depend on local resonant modes; therefore, wave propagation in the lattice-form superstructure is controlled by quasi-static average material properties, and its behavior is characterized by the average effect. When the lattice-form superstructure undergoes periodic cell homogenization, the equivalent density of the lattice-form superstructure is the average density, and the equivalent elastic constant of the lattice-form superstructure is obtained by the energy homogenization method. Then, the extended transfer matrix method is constructed to calculate the acoustic performance of the underwater sound-absorbing material in the sandwich form of the first rubber-lattice-form superstructure-second rubber. Step S200 abandons the analysis path dependent on complex resonant modes and instead adopts a homogenization method based on the average effect. The working state of the lattice-form superstructure does not depend on local resonant modes but is controlled by quasi-static average material properties, allowing the use of the classic and mature mechanical theory of homogenization to characterize the acoustic behavior of the superstructure. This avoids the huge computational overhead and model uncertainty caused by simulating complex, high-frequency local resonances, laying the foundation for establishing an efficient and robust calculation method. By employing average density and obtaining equivalent elastic constants through energy homogenization, lattice superstructures with complex microscopic topologies can be accurately represented as macroscopically homogeneous anisotropic materials. This not only captures the static stiffness characteristics of the structure but also implicitly reveals the fundamental features of its dynamic response, ensuring the accuracy of subsequent acoustic performance calculations. Furthermore, transforming the microscopically complex heterogeneous structure into a homogeneous medium greatly simplifies the computational model, creating conditions for rapid acoustic analysis. Based on the obtained equivalent homogeneous material parameters, an extended transfer matrix method is constructed to calculate the acoustic performance of sandwich structures, achieving an optimal balance between computational efficiency and engineering practicality. The transfer matrix method itself is well-suited for handling plane wave propagation problems in multilayer media, offering fast computation speed. Extending the transfer matrix method to be compatible with anisotropic superstructure layers obtained by homogenization methods creates a highly customized and efficient computational tool specifically for sandwich structures. This enables rapid acoustic performance evaluation during the iterative loop of topology optimization, solving the problem of excessively time-consuming optimization processes and enabling the practical implementation of automated reverse engineering.

[0026] In this embodiment, the extended transfer matrix method specifically includes: In a multilayer elastic medium of finite length, when an elastic wave propagates, it will form a reflected wave within the medium. The displacement field in a certain layer of the elastic medium is represented as a product of vectors: (1) in This represents the displacement amplitude of different wave modes. P , N x The matrices are the polarization vector matrix and the phase matrix, respectively, specifically: (2) (3) in P x 、P y 、Q x 、Q y These are respectively (quasi) longitudinal waves and (quasi) transverse waves. x direction and y Directional polarization components, k qL and k qT These are the wave numbers of (quasi) longitudinal waves and (quasi) transverse waves, respectively. To describe the propagation relationship of elastic waves in different elastic medium layers, field quantities are defined. Describing the velocity field in an elastic medium , and stress field quantity , This field quantity is described based on the displacement field: (4) Where the L matrix is (5) For the same elastic medium, any x and x + d Location, Field Volume Satisfying transitivity: (6) Where T d For length is d The transfer matrix in the elastic medium layer. Furthermore, when elastic waves propagate in different media, the interface between the two elastic media satisfies the continuity condition, i.e., velocity continuity and stress continuity. At this time, the field quantities on both sides of the interface... Since they are equal, based on the leftmost field quantity in the multilayer elastic medium, the field quantity corresponding to the rightmost interface can be further obtained through the transfer relation. This transfer relation is written as... (7) in T t This is the total transfer matrix.

[0027] When calculating the sound absorption performance of underwater sound-absorbing materials, the sound pressure amplitude in the water area is taken into account. P i along +x Plane waves propagating in a specific direction, such as Figure 2 As shown; under the excitation of this sound wave, after interacting with the underwater sound-absorbing material, a sound pressure amplitude of [value missing] will be generated. Pr The reflected sound waves generate a sound pressure amplitude of [value missing] in the air domain. P t The transmitted sound waves; according to the acoustic-structure interaction boundary conditions, the interface between the elastic body and the fluid satisfies continuous normal stress, continuous normal particle velocity, and zero tangential stress. Therefore, at the interface between the fluid and the underwater sound-absorbing material... For along +x Plane waves (reflected sound waves) propagate in a specific direction. For along -x Plane waves propagating in a specific direction Z The corresponding characteristic impedance in the fluid domain is, in water, the corresponding value is: Z w In the air domain Z a Define vector Based on the fluid-structure interaction relationship, the formula can be derived. (8) For underwater sound-absorbing materials, the transmission relationship between the incident s- interface and the transmitting s+ interface is established. (9) Where T d1 ~T d6 These represent the transfer matrices from the first rubber layer to the steel backing, obtained according to formula (6); The amplitude of the incident sound wave is p i = At 1 Pa, since the water and air domains are semi-infinite, at the interface... s- and s+ There is a boundary condition vector. (10) Substituting formula (10) into (9), the reflection coefficient in the water can be calculated: (11) in H lm For matrix H elements, l and m The value can be 1, 2, 3 or 4; substituting the calculation result into formula (9) can further obtain the transmitted sound wave. When calculating the sound absorption performance, the first fluid domain is water, and the second fluid domain is air. Because the characteristic impedance of the air domain differs greatly from that of the steel backing, the amplitude of the transmitted sound wave pressure is extremely small and can be ignored in the calculation of the sound absorption coefficient. Therefore, the sound absorption coefficient is: (12) By defining a feature including sound pressure amplitude (p i ,p r ), normal particle velocity field ( v x and stress field quantity ( , The field vector of the acoustic-solid coupling between the fluid and solid domains was calculated, and a precise mathematical description was given of the boundary conditions (continuous normal stress, continuous normal particle velocity, and zero tangential stress) between the fluid and solid domains. This comprehensively and accurately depicted the transmission, reflection, transmission, and dissipation of sound wave energy along the complex path of "water-sound-absorbing material-steel backing-air domain," ensuring the high physical realism and reliability of the calculated sound absorption coefficient and providing a precise theoretical basis for performance evaluation and optimization. A system transfer matrix connecting the incident and transmission ends was constructed. H This matrix consists of the interface coupling matrix ( G ) and the transfer matrix of each layer (T) d1 , T d2 By chaining and multiplying, the complex multi-layered wave problem is transformed into a series of standardized matrix operations, establishing a highly systematic and modular computational framework. This framework not only has clear physical meaning but also greatly facilitates the implementation and iterative computation of computer programs, making the evaluation of the acoustic performance of complex sandwich structures fast and efficient. It provides a key computational feasibility guarantee for embedding these structures into automated loops of topology optimization for large-scale and rapid performance evaluation.

[0028] The extended transfer matrix method, through rigorous mathematical modeling, transforms the complex propagation behavior of elastic waves in multilayer media into computable matrix operations; by defining a method that includes velocity field quantities ( , ) and stress field quantity ( , The field vector f x This study provides a complete physical description of wave states in elastic media, considering not only the vibration characteristics (velocity) of the medium but, more importantly, the mechanical interactions (stress) during wave propagation. This ensures a realistic and accurate simulation of the reflection, transmission, and coupling behavior of sound waves at the interface between adjacent layers (such as a rubber layer and an equivalent layer of a superstructure), providing a theoretical foundation for calculating the sound absorption coefficient, an indicator directly related to energy transfer. This is achieved by introducing a polarization vector matrix. P To distinguish between (quasi) longitudinal waves and (quasi) transverse waves and their propagation directions, and using a phase matrix. N xTo accurately describe the phase changes of waves propagating in space, this modeling method strictly adheres to the fundamental principles of elastic dynamics. It can simultaneously handle multiple wave types (quasi-longitudinal waves, quasi-transverse waves) and their reflected waves generated at various layers, rather than being limited to simple sound pressure waves. This makes the extended transfer matrix method applicable to analyzing complex multilayer systems composed of rubber (viscoelastic bodies) and equivalent homogeneous superstructures (anisotropic elastic bodies), and can analyze the sound wave transmission relationship under fluid-structure interaction, overcoming the limitations of traditional transfer matrix methods that may be limited to fluid or simple medium layers. The displacement field u and field vector f are used to... x Ultimately, all of these are expressed in matrix form related to the magnitude vector A (Formula 1, Formula 4), establishing a highly systematic and standardized computational framework; through the matrix L By associating displacements with field vectors, the connection conditions between different layers (displacement continuity, stress continuity) can be transformed into a chain operation of the transfer matrix. This formalized expression greatly facilitates the implementation and iterative calculation of computer programs, providing operability and computational efficiency guarantees for seamlessly embedding this method into the automated loop of topology optimization and conducting thousands of rapid performance evaluations.

[0029] In this embodiment, step S300 specifically includes: addressing the underwater sound absorption problem, optimizing directly based on the established extended transfer matrix method to meet acoustic performance requirements; to achieve on-demand reverse design of periodic unit cells in the superstructure, the unit cell structure is designed as a square lattice and discretized into 100×100 elements, and design variables are assigned to each element based on the isotropic material penalty function model. ( To improve numerical stability during the optimization process, global design variables were adjusted. Density filtering and Heaviside projection are performed, and robustness formulas are used to ensure the stability of the optimization results. Based on the established extended transfer matrix method, optimization is carried out directly targeting acoustic performance requirements, establishing a reverse design paradigm driven directly by acoustic performance. The optimization objective is transformed from traditional, indirect mechanical properties (such as maximizing stiffness) to direct acoustic indicators such as sound absorption coefficient, ensuring that the final output of the entire optimization process accurately meets the core engineering requirement of underwater sound absorption, guaranteeing the functional relevance and effectiveness of the optimization results from the design source. By defining the cell structure as a square lattice and discretizing it into a finite number of elements, and assigning continuous design variables to each element, the optimization problem of topology is transformed into a standard, computer-processable mathematical optimization problem, realizing highly parameterized and automated design of the microstructure topology. The design variables define the presence or absence of material in each grid element, enabling the optimization algorithm to automatically search for the optimal material distribution in a huge design space, fundamentally breaking through the limitations of traditional design relying on trial and error based on human experience. Introducing density filtering, Heaviside projection, and robustness formulas are key measures to address common numerical problems in topology optimization (such as mesh dependency and checkerboard patterns) and ensure the manufacturability of the results. Density filtering, by introducing minimum size control, effectively eliminates the dependence of the optimization results on the finite element mesh and avoids non-physical checkerboard patterns. Heaviside projection helps to polarize intermediate density elements towards 0 or 1, thereby obtaining a clear and well-defined topology. Robustness formulas, by considering manufacturing errors (such as dimensional over-corrosion or under-corrosion), aim to optimize for robust designs insensitive to small perturbations. These measures work together to significantly improve the numerical stability and convergence of the optimization process, ultimately ensuring that the optimized structural topology is clear, reliable, and easy to manufacture in engineering.

[0030] In this embodiment, based on the isotropic material penalty function model, the Young's modulus within each mesh cell is... E and density ρ Described as physical variables The interpolation function, namely: (13) in ρ 1 and E 1 represents the Young's modulus and density of the matrix material. ρ 0=10 -6 ρ 1 and E 0=10 -6 E 1 represents an artificial variable parameter. p 1 and p 2 is the penalty factor for the density and Young's modulus interpolation function.

[0031] In this embodiment, to achieve on-demand design of sound-absorbing materials, the reflection coefficient is used as the basis. To optimize the target, since the characteristic impedance difference between the steel backing and the air domain is large, the transmitted sound wave can be ignored. The smaller the reflection coefficient, the larger the sound absorption coefficient; the average reflection coefficient within the target frequency band is Since the robustness formula leads to three different design implementations during the optimization process, the objective function of the optimization problem is transformed into a typical max-min problem, i.e.: (14) in Represents the reflection coefficient R The table below shows e, b, and d, which represent the three solutions generated in the robustness formula. To ensure that the optimization results have good load-bearing capacity and structural connectivity, the equivalent Young's modulus ( ) is used. E x 、 E y Constraining the optimization results and applying density constraints, we obtain the following optimization formula: (15) in, and for x direction and y Equivalent Young's modulus in the direction. E t The constraint threshold for the equivalent Young's modulus. The volume fraction of the physical structure. V t The volume fraction constraint threshold is used to ensure that the formed physical structure has good connectivity and roughly reflects the load-bearing capacity of the structure through the equivalent Young's modulus constraint; the volume fraction constraint can ensure the lightweight requirement.

[0032] Using physical variables The interpolation function (Equation 13) is used to associate the material properties (Young's modulus) of the element. E and density ρ To achieve 0-1 binary design guidance, a penalty factor is set. p 1, p 2. By penalizing intermediate density values, this model can effectively encourage design variables to be optimized. During the optimization process, convergence towards 0 (representing "no material" or weak material) or 1 (representing "matrix material") is beneficial for obtaining clear and well-defined topological morphology, enhancing the manufacturability of the results, and ensuring the stability of numerical computation; introducing minimal artificial variable parameters ( ρ 0 ,E Replacing absolute zero with 0) avoids numerical problems such as singular stiffness matrices in finite element calculations, ensuring the stability and convergence of the entire optimization iteration process. Directly using the reflection coefficient... R As the optimization target, and based on the high reflectivity of the steel backing (negligible transmission), the sound absorption coefficient was... α Association is 1 R This ensures that the optimization objective directly corresponds to the core evaluation indicators of acoustic performance, guaranteeing that all efforts in the optimization process are directly used to improve the final sound absorption efficiency, achieving a precise alignment between function and objective. Three design implementations are generated through a robustness formula ( The optimization problem is transformed into a minimization-max problem (Equation 14). This optimization strategy not only pursues the optimal performance under nominal conditions, but also requires that the optimization result be insensitive to small dimensional deviations that may occur during the manufacturing process (such as over-corrosion or under-corrosion), thus improving the engineering practicality and reliability of the final design scheme and ensuring that the laboratory design can maintain the expected excellent performance under real manufacturing tolerances. Equivalent Young's modulus constraints and volume fraction constraints are systematically applied in the optimization model (Equation 15). E x ,E y The optimization process mandates that the optimized topology possess a certain load-bearing stiffness on a macroscopic scale. This constraint not only initially reflects the structure's load-bearing capacity but also ensures the connectivity and integrity of the material at the microscopic scale. This avoids isolated, load-transfer-incapable floating island structures in the optimization results, thus guaranteeing the mechanical feasibility of the superstructure as a load-bearing unit. The volume fraction constraint directly limits the amount of material used, ensuring that the optimization results naturally meet lightweight design requirements. By constructing a complete mathematical model integrating penalized material interpolation, acoustic performance as the direct objective, robust optimization strategies, and the application of mechanical and lightweight constraints, topology optimization is transformed from an abstract concept into a powerful design tool capable of systematically and automatically balancing acoustic performance, structural stiffness, manufacturing robustness, and lightweight requirements. This tool can efficiently and reliably generate lattice superstructure topologies that possess broadband high-efficiency sound absorption, good structural load-bearing characteristics, are insensitive to manufacturing tolerances, and meet lightweight requirements. This systematically solves the inherent problem of traditional empirical design struggling to balance multiple performance indicators.

[0033] Example 1: Optimization Result Example To design a broadband sound-absorbing material with high stiffness and lightweight properties, this paper conducts topology optimization of the structural design. First, optimization parameters are preset, and the cellular structure is discretized as follows: N d=100×100 quadrilateral mesh elements. In the constraint threshold, set... E t =150MPa, V t =0.55. Subsequently, the basic unit cell is set as an orthogonal anisotropic structure. The thickness of the first rubber layer is set. h r1 =10mm, thickness of the first isolation plate h c1 =1mm, lattice-type ultrastructure thickness h l =28mm, thickness of the second isolation plate h c2 =1mm, thickness of the second rubber layer h r1 =10mm and steel backing thickness h s =20mm.

[0034] Figure 3 This is a topology optimization solution process with a target frequency band of 0.5kHz to 5kHz and a frequency interval of 0.5kHz. The initial solution is a semi-cylindrical perforation with a radius of 0.2a embedded in the center of each of the four sides. Figure 3 (b) Shows the physical variables from generation 1 to generation 300, which correspond to... Figure 3 (a) The area circled in blue. From the solution process, it can be observed that as the iterations progress, the cellular structure gradually transforms from its initial form into a physical structure with clear boundaries, and the objective function value gradually decreases. Ultimately, the obtained average reflection coefficient of the sound-absorbing material is 0.136, the volume fraction is 0.40 (corresponding to an equivalent density of 798 kg / m³), and the equivalent stiffness is... E x =213MPa, E y =674MPa. This indicates that the optimized result is lightweight and high-stiffness. Although its density is lower than that of water, its deformation under 4MPa pressure is roughly estimated to be about 2% based on the stiffness modulus, demonstrating good load-bearing capacity. During the optimization process, each iteration took 3-5 seconds. The code is entirely implemented in MATLAB, running on a 13th Gen Intel(R) Core(TM) i9-13900H processor with 32.0 GB of RAM.

[0035] The underwater sound-absorbing material containing the optimized lattice-shaped superstructure is denoted as M1, and its sound absorption coefficient curve is shown in the figure. Figure 4As shown, the average sound absorption coefficient of underwater sound-absorbing material M1 is calculated to be 0.924 across a 100Hz interval from 0.5kHz to 5kHz. Therefore, through the optimized design of the lattice-shaped superstructure, underwater sound-absorbing materials can exhibit highly efficient sound absorption capabilities while also possessing lightweight and high rigidity characteristics.

[0036] Matters not covered in this invention are common knowledge.

[0037] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0038] The embodiments described above are merely examples of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

[0039] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A topology optimization design method for lattice-type superstructures for underwater sound absorption, characterized in that, Includes the following steps: S100. Establish an underwater sound absorption mathematical model. Adopt the principle of periodic structural wave control and the principle of sound energy dissipation of viscoelastic materials. Embed the lattice-shaped superstructure as the core wave control component between two rubber layers. Based on the assumption of two-dimensional plane strain, construct an equivalent mathematical model. S200. Construct an acoustic performance calculation method. Adopt a homogenization method based on average effect to treat the lattice superstructure as a homogeneous medium. Construct an extended transfer matrix method to calculate the acoustic performance of underwater sound-absorbing materials with a core structure consisting of a rubber layer, a lattice superstructure, and a rubber layer. The S300 features an optimized sandwich structure design, employing a reverse design method based on topology optimization. With broadband acoustic performance as the optimization target, it automatically optimizes the cell topology of the lattice superstructure, simultaneously achieving lightweight, high stiffness, and optimal sound absorption performance of the material.

2. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 1, characterized in that, Step S100 includes building the basic model: The basic model consists of a first layer of rubber, a superstructure layer, and a second layer of rubber sequentially arranged along the incident sound wave in the water. The first layer of rubber, the superstructure layer, and the second layer of rubber constitute the underwater sound-absorbing material. The sound-absorbing material extends along its upper and lower boundaries. y The direction is the periodic boundary, and the superstructure layer is sealed within the first and second rubber layers; The first side of the underwater sound-absorbing material is used for coupling with the water area, and the second side of the underwater sound-absorbing material is used to be attached to the steel backing, which is coupled with the air domain. The superstructure layer adopts a lattice-shaped superstructure, which has the characteristic of modulating the propagation of sound waves; Both the first and second layers of rubber are made of viscoelastic damping materials, which have the ability to dissipate sound waves.

3. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 2, characterized in that, In underwater sound-absorbing materials, lattice-shaped superstructures are the core components of wave modulation. The lattice-shaped superstructure is composed of periodically arranged cell structures and isolation plates. The isolation plates enable the lattice-shaped superstructure to form a good connection surface with the rubber materials of the first and second layers of rubber. Cellular structure along local coordinate system x 1. Direction and y 1. Periodic extension in one direction, rotating the entire cellular structure counterclockwise to correspond to the global coordinate system of the underwater sound-absorbing material. xy The axis forms an angle θ Cellular structure along z Since the direction is infinitely stretched in thickness, underwater sound-absorbing materials are equivalent to a two-dimensional plane strain problem, which simplifies the calculation and analysis.

4. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 1, characterized in that, Step S200 specifically includes: The working state of the lattice superstructure does not depend on the local resonant mode. Therefore, wave propagation in the lattice superstructure is controlled by the quasi-static average material properties, and its behavior is characterized by the average effect. When the periodic cells of the lattice-type superstructure are homogenized, the equivalent density of the lattice-type superstructure is the average density, and the equivalent elastic constant of the lattice-type superstructure is obtained by the energy homogenization method. Then, the extended transfer matrix method is constructed to calculate the acoustic performance of the underwater sound-absorbing material in the form of a sandwich of first rubber-lattice-type superstructure-second rubber.

5. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 4, characterized in that, The extended transfer matrix method specifically includes: In a multilayered elastic medium of finite length, elastic waves propagate and generate reflected waves within the medium. The displacement field in a certain layer of the elastic medium can be represented as a product of vectors: (1) in This represents the displacement amplitude of different wave modes. P , N x The matrices are the polarization vector matrix and the phase matrix, respectively, specifically: (2) (3) in P x 、P y 、Q x 、Q y They are respectively longitudinal waves and transverse waves. x direction and y Directional polarization components, k qL and k qT These are the wave numbers of the longitudinal and transverse waves, respectively. To describe the propagation relationship of elastic waves in different elastic medium layers, field quantities are defined. Describing the velocity field in an elastic medium , and stress field quantity , This field quantity is described based on the displacement field: (4) Where the L matrix is (5) For the same elastic medium, any x and x + d Location, field volume Satisfying transitivity: (6) Where T d For length is d The transfer matrix in the elastic medium layer.

6. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 5, characterized in that, When an elastic wave propagates in different media, the interface between the two elastic media satisfies the continuity condition, namely, velocity continuity and stress continuity. At this time, the field quantities on both sides of the interface... Since they are equal, based on the leftmost field quantity in the multilayer elastic medium, the field quantity corresponding to the rightmost interface can be further obtained through the transfer relation. This transfer relation is written as... (7) in T t This is the total transfer matrix.

7. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 6, characterized in that, When calculating the sound absorption performance of underwater sound-absorbing materials, the sound pressure amplitude in the water area is taken into account. P i along +x Plane waves propagate in a specific direction; Excited by this sound wave, it will generate a sound pressure amplitude of [value missing] after interacting with underwater sound-absorbing materials. P r The reflected sound waves generate a sound pressure amplitude of [value missing] in the air domain. P t Transmitted sound waves; According to the acoustic-structure interaction boundary conditions, the interface between the elastic body and the fluid satisfies continuous normal stress, continuous normal particle velocity, and zero tangential stress. Therefore, at the interface between the fluid and the underwater sound-absorbing material... For along +x Plane waves propagating in a specific direction For along -x Plane waves propagating in a specific direction Z The corresponding characteristic impedance in the fluid domain is, in water, the corresponding value is: Z w In the air domain Z a ; Define vector Based on the fluid-structure interaction relationship, the following is obtained: (8) For underwater sound-absorbing materials, the transmission relationship between the incident s- interface and the transmitting s+ interface is established. (9) Where T d1 ~T d6 These represent the transfer matrices from the first rubber layer to the steel backing, obtained according to formula (6); The amplitude of the incident sound wave is p i = At 1 Pa, since the water and air domains are semi-infinite, at the interface... s- and s+ There is a boundary condition vector. (10) Substituting formula (10) into (9), the reflection coefficient in the water can be calculated: (11) in H lm For matrix H elements, l and m The value can be 1, 2, 3 or 4; substituting the calculation result into formula (9) can further obtain the transmitted sound wave. When calculating the sound absorption performance, the first fluid domain is water, and the second fluid domain is air. Because the characteristic impedance of the air domain differs greatly from that of the steel backing, the amplitude of the transmitted sound wave pressure is extremely small and can be ignored in the calculation of the sound absorption coefficient. Therefore, the sound absorption coefficient is: (12)。 8. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 1, characterized in that, Step S300 specifically includes: To address the underwater sound absorption problem, optimization is carried out directly based on the established extended transfer matrix method, targeting acoustic performance requirements. To achieve on-demand reverse design of periodic unit cells in superstructures, the unit cell structure is designed as a square lattice and discretized into 100×100 elements. Based on the isotropic material penalty function model, design variables are assigned to each element. , ; To improve the numerical stability during the optimization process, global design variables are adjusted. Density filtering and Heaviside projection are performed, and a robust formula is used to ensure the stability of the optimization results.

9. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 8, characterized in that, Based on the penalty function model for isotropic materials, the Young's modulus within each mesh element E and density ρ Described as physical variables The interpolation function, namely: (13) in ρ 1 and E 1 represents the Young's modulus and density of the matrix material. ρ 0=10 -6 ρ 1 and E 0=10 -6 E 1 represents an artificial variable parameter. p 1 and p 2 is the penalty factor for the density and Young's modulus interpolation function.

10. The lattice-type superstructure topology optimization design method for underwater sound absorption according to claim 9, characterized in that, To achieve on-demand design of sound-absorbing materials, based on the reflection coefficient To optimize the target, since the characteristic impedance difference between the steel backing and the air domain is large, the transmitted sound wave can be ignored. The smaller the reflection coefficient, the larger the sound absorption coefficient; the average reflection coefficient within the target frequency band is ; Since the robustness formula leads to three different design implementations during the optimization process, the objective function of the optimization problem is transformed into a typical max-min problem, i.e.: (14) in Represents the reflection coefficient R The table below shows e, b, and d, which represent the three solutions generated in the robustness formula. To ensure that the optimization results have good load-bearing capacity and structural continuity, the equivalent Young's modulus is used. E x 、E y Constraining the optimization results and applying density constraints, we obtain the following optimization formula: (15) in, and for x direction and y Equivalent Young's modulus in the direction. E t The constraint threshold for the equivalent Young's modulus. The volume fraction of the physical structure. V t The volume fraction constraint threshold is used to ensure that the formed physical structure has good connectivity and roughly reflects the load-bearing capacity of the structure through the equivalent Young's modulus constraint; the volume fraction constraint can ensure the lightweight requirement.

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