Three-field floating projection acoustic vibration coupling topological optimization method based on linear material interpolation

Abstract

The invention relates to the technical field of acoustic-structure coupling topological optimization in computational dynamics, in particular to a three-field floating projection acoustic-vibration coupling topological optimization method based on linear material interpolation. According to the technical scheme, the method comprises the following four steps that 1, an initial geometric model, including the size, the sound domain and the solid domain, of the sound-vibration coupling system is established based on COMSOL multi-physics field software, and 2, finite element analysis calculation is conducted in combination with COMSOL and Matlab, values of a target function and a constraint function and sensitivity of corresponding units are obtained, and the initial geometric model of the sound-vibration coupling system is obtained. In the third step, OC is used for updating design variables and convergence judgment of the optimization model according to sensitivity information, and in the fourth step, the design variables are linearly interpolated into the whole design domain, and pixel base design is obtained. According to the method, the optimal material layout of the structure is searched, so that the active design of the acoustic-structure coupling structure is realized, and specific acoustic performance requirements are met.

Description

Technical Field

[0001] The invention relates to the technical field of acoustic-solid coupling topology optimization in computational dynamics, and in particular to a three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation. Background Art

[0002] With the continuous development of engineering technology, especially in the fields of aerospace, automobile manufacturing and construction, the design structure must not only meet the requirements of mechanical strength and stiffness, but also must comprehensively consider the acoustic performance and vibration characteristics. Traditional design methods usually focus on the optimization of a single performance and are difficult to deal with complex acoustic-vibration coupling problems. For example, in the aerospace field, the aircraft fuselage must not only withstand external aerodynamic loads, but also effectively control noise and vibration to improve passenger comfort and safety; in the automotive industry, noise, vibration and harshness (NVH) performance directly affects the vehicle's market competitiveness and user experience. Traditional design methods are difficult to meet these high-standard performance requirements, so there is an urgent need for an innovative optimization method to achieve efficient design in the field of acoustic-vibration coupling.

[0003] Topology optimization is a design method that aims to improve the performance of a structure or system by optimizing the distribution of materials. This method is widely used in the fields of mechanics, aerospace, civil engineering, etc., aiming to design lighter, stronger and more economical structures. The core concept of structural topology optimization is to use mathematical models and algorithms to optimize the distribution of materials in a specific design space to meet given performance requirements and constraints. Depending on the design variables, structural topology optimization is mainly divided into two categories: unit-based structural topology optimization and boundary-based structural topology optimization. Currently, the commonly used unit-based topology optimization methods include homogenization method, variable thickness method, variable density method, independent continuous mapping method, progressive structural optimization method, and three-field floating projection topology optimization. These methods all use units as design variables to solve structural optimization problems.

[0004] COMSOL Multiphysics is a multi-physics simulation platform that supports joint simulation of multiple physical phenomena. Matlab can extract the sensitivity analysis results of COMSOL model calculations through code, including the calculation of the objective function and the response of the constraint function to the design variables. Then, Matlab returns the updated variables (i.e., the new material distribution) after topology optimization to COMSOL through code for new finite element calculations.

[0005] To this end, we propose a three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation to solve the existing problems. Summary of the invention

[0006] The purpose of the present invention is to address the problems existing in the background technology and to propose a three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation.

[0007] To achieve the above-mentioned purpose, the present invention provides the following technical solutions: a three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation, comprising the following four steps, the step 1: establishing an initial geometric model of an acoustic-vibration coupling system based on COMSOL multi-physics field software, including size, acoustic domain and solid domain, the step 2: combining COMSOL and Matlab to perform finite element analysis and calculation to obtain the values ​​of objective function and constraint function and the sensitivity of corresponding units, the step 3: using OC to update the convergence judgment of design variables and optimization model according to sensitivity information, the step 4: linearly interpolating the design variables into the entire design domain to obtain a pixel-based design.

[0008] Preferably, the step 1 is optimization initialization, establishing a finite element model of the acoustic-vibration coupling system for the geometric model, including physical fields, boundary conditions, load conditions and design domain definitions, meshing the model, and defining the loads and fixed boundary constraints of the model.

[0009] Preferably, the boundary conditions include: plane wave radiation boundary, fixed boundary, acoustic-solid coupling boundary, hard acoustic wall boundary, and optimized frequency.

[0010] Preferably, the material parameters of the solid domain and the acoustic domain include: density, Young's modulus, Poisson's ratio, and sound velocity; the material parameters of the acoustic domain include: density and sound velocity.

[0011] Preferably, the design domain definition includes defining topology optimization design variables in the design domain: defining topology optimization design variables based on the sensitivity module (Mathematics>Optimization and Sensitivity>Sensitivity) under COMSOL multi-physics software, establishing an objective function, taking the minimum flexibility of the design domain as the objective function, and initializing and defining optimization parameters in Matlab, including volume fraction, control factor for controlling the steepness of the Heaviside function, and filter radius r. min , maximum number of iterations, and convergence tolerance.

[0012] Preferably, in step 2, finite element analysis calculation of the acoustic-vibration coupling problem is performed in COMSOL, a topology optimization problem description is established, a linear material interpolation model is established, the COMSOL calculation objective function and constraint function are extracted using Matlab, and the unit sensitivity calculated by COMSOL is extracted using Matlab, thereby smoothing the sensitivity based on the historical iteration information of the sensitivity.

[0013] Preferably, the step 3 is to perform iterative update in Matlab, calculate the sensitivity of the objective function and the constraint function, update the design variables according to the OC, define the upper / lower boundary constraints of the design variables, and in order to eliminate the grid dependency and checkerboard problem in the optimization process, the design variables are further processed by applying implicit filtering constraints. Subsequently, the filter field variables are subjected to substitution projection processing based on the Heaviside function, and the design variables are updated using floating projection constraints. Subsequently, in order to drive the design variables to tend to a 0 / 1 distribution, the design variables are subjected to projection function constraint processing based on the Heaviside function, and then the design variables are added with movement restrictions, the constraints are updated, and finally it is determined whether the inner loop and the outer loop converge.

[0014] Preferably, the method for determining whether the inner loop converges is: using the Newton-Raphson method to define the initial value of the Lagrange multiplier and repeatedly updating the Lagrange multiplier according to the Newton-Raphson method. When the change value before and after is small enough (such as 1.0-9), the inner loop will stop converging, thereby obtaining the exact value of the Lagrange multiplier that meets the convergence condition.

[0015] Preferably, the method for judging whether the outer loop converges is as follows: in order to obtain a clear topological configuration, the design variables need to be pushed to a 0 / 1 distribution. According to the convergence criterion, the numerical values ​​of the target values ​​for nearly 10 iterations are taken for convergence judgment calculation. Once the convergence criterion is met, the iteration stops, otherwise the iteration continues.

[0016] Preferably, step 4 is a smoothing representation of the optimization result, in which the design variables are linearly interpolated into the entire design domain to obtain a pixel-based design, i.e., a smooth design solution. Finally, once the convergence criterion is satisfied, it indicates that the smooth design solution obtained by the 0 / 1 pixel-based design is equivalent to the initial unit-based design solution. This also means that a smooth design solution that satisfies the convergence condition is obtained. At this time, the floating projection constraint control factor β no longer increases and remains unchanged until the entire iteration converges and stops.

[0017] Compared with the prior art, the present invention has the following beneficial effects:

[0018] Optimized configuration boundary smoothing: Through the application of linear material interpolation and level set function, this method can effectively avoid the jagged boundary problem common in traditional topology optimization methods and obtain a design configuration with clear and smooth boundaries;

[0019] Simplified coupling interface tracking: Since this method does not need to explicitly track the coupling interface, it can simplify the optimization process, improve computational efficiency, and avoid errors caused by inaccurate interface tracking;

[0020] Local mode suppression in low-density areas: Through floating projection constraints and design variable movement restrictions, this method can effectively suppress local modes in low-density areas and obtain a more stable and reliable design configuration;

[0021] Multi-physics coupling analysis capability: This method is based on the COMSOL Multiphysics platform and can perform multi-physics coupling analysis of acoustic-solid coupling problems, thereby more comprehensively evaluating the acoustic performance and vibration characteristics of the design configuration;

[0022] Improved optimization efficiency: The combined use of Matlab and COMSOL can automate the optimization process and use Matlab's powerful programming capabilities to develop algorithms and optimize parameters, thereby improving optimization efficiency.

[0023] The present invention searches for the optimal material layout of the structure, thereby realizing the active design of the acoustic-solid coupling structure to meet specific acoustic performance requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 It is a flow chart of topology optimization design of the present invention;

[0025] Figure 2 The model and boundary conditions in the embodiment of the present invention;

[0026] Figure 3 It is a schematic diagram of the smoothing characterization of the present invention;

[0027] Figure 4 It is a topology optimization iteration diagram in the embodiment;

[0028] Figure 5 This is the basic solution of the design domain unit after topology optimization in the embodiment;

[0029] Figure 6 This is the smooth solution of the design domain after topology optimization in the embodiment;

[0030] Figure 7 This is the stress distribution of the model after topology optimization in the embodiment. DETAILED DESCRIPTION

[0031] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0032] Embodiment 1

[0033] like Figure 1-Figure 7 As shown, the present invention proposes a three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation, comprising the following steps:

[0034] Step 1: Optimize initialization

[0035] Step 1.1 Establish the initial geometric model of the acoustic-vibration coupling system based on COMSOL multiphysics software, including the size, acoustic domain (white area) and solid domain (gray area).

[0036] Step 1.2: Establish a finite element model of the acoustic-vibration coupling system for the geometric model of step 1.1, including physical fields, boundary conditions, and load conditions; mesh the model, and define the load and fixed boundary constraints of the model.

[0037] Step 1.3 boundary conditions include: plane wave radiation boundary, fixed boundary, acoustic-solid coupling boundary, hard acoustic wall boundary, and optimized frequency f = 300 Hz;

[0038] Step 1.4: Determine the material parameters of the solid domain and the acoustic domain including density, Young's modulus, Poisson's ratio, and acoustic velocity;

[0039] Step 1.5 determines the sound field material parameters including: density, sound velocity;

[0040] Step 1.7 Define topology optimization design variables in the design domain: Define topology optimization design variables based on the sensitivity module under COMSOL multiphysics software (Mathematics > Optimization and Sensitivity > Sensitivity);

[0041] Step 1.6: Establish the objective function, taking the minimum flexibility of the design domain as the objective function;

[0042] Step 1.8 Initialize and define optimization parameters in Matlab, such as volume fraction Vo, control factors β and β1 that control the steepness of the Heaviside function, and filter radius r min , maximum number of iterations, convergence tolerance∈, etc.

[0043] Step 2: Combine COMSOL & Matlab to perform finite element analysis

[0044] Step 2.1 Perform finite element analysis on the acoustic-vibration coupling problem in COMSOL

[0045] ZU=f1

[0046] f, Z and U represent the force vector, stiffness matrix and displacement vector respectively.

[0047] Step 2.2 Establish the topology optimization problem description

[0048] The mathematical model of topology optimization problem is expressed as

[0049]

[0050] Where x={x1,x2,…x i ,…xN} T is the unit design variable, N is the total number of units in the design domain. J(x,U(x)) is the objective function, V i is the volume of the ith unit. V f and V0 are the target volume and its constraint value for optimization respectively. i =1 means that the i-th unit is all solid material, and the corresponding i =0 indicates acoustic medium.

[0051] Step 2.3 Establish a linear material interpolation model

[0052] This method uses a linear material interpolation model to calculate the bulk modulus, shear modulus, and density of each unit based on its design variable x. i Interpolation is performed, that is

[0053]

[0054] The subscripts 's' and 'a' in the formula represent the solid material and the acoustic medium respectively.

[0055] Step 2.4 Use Matlab to extract COMSOL calculation objective function and constraint function

[0056] By introducing the Lagrange multiplier λ, the objective function in equation (2) can be transformed into

[0057] J=J(x,U(x))+λ T (ZU-f) 4

[0058] Since the equilibrium equation can be satisfied during the optimization process, that is, ZU-f = 0, equation (4) is equivalent to the original objective function.

[0059] Step 2.5 Extract the unit sensitivity calculated by COMSOL using Matlab

[0060] The sensitivity of the objective function can be calculated by the adjoint method. According to the chain rule, the objective function J has a dependence on the design variable x. i The partial derivative of is:

[0061]

[0062] Since the applied load is independent of the design variables, we have Then formula (5) can be transformed into

[0063]

[0064] In order to avoid calculating the partial derivatives of the displacement with respect to the design variables Let its coefficient matrix be 0, then the Lagrange multiplier λ can be calculated as:

[0065]

[0066] Among them, U(U=U re +iU im , subscripts 're' and 'im' represent the real and imaginary parts respectively) are state variables. Then we have It can be written as:

[0067]

[0068] Due to the symmetry of the diagonal matrix, substituting equation (8) into equation (7) yields:

[0069]

[0070] From formula (9), the Lagrange multiplier λ can be obtained, that is, formula (6) can be calculated as:

[0071]

[0072] Among them, Re(·) represents the real part of the complex number. Further combined with the linear material interpolation model, It can be calculated as:

[0073]

[0074] in,

[0075] In addition, the sensitivity of the design domain volume fraction to the design variables can be directly expressed as:

[0076]

[0077] Step 2.6 Smoothing sensitivity based on historical iteration information of sensitivity (k>1)

[0078] Step 3: Perform iterative updates in Matlab

[0079] Step 3.1 Update design variables

[0080] By calculating the sensitivity of the objective function and the constraint function, the design variable x is updated according to the OC optimization criterion (Optimality CriteriaMethod) 1,i .

[0081] In order to solve the topology optimization problem in equation (2), Lagrange multipliers are introduced to calculate all constraints, and the objective function can be changed to

[0082]

[0083] In the formula, gj and are the corresponding constraints on the design variables and their constraint values. Because the original problem in equation (2) contains a large number of constraints on the design variables close to 0 / 1, it is difficult to calculate mathematically. In order to calculate these design variable constraints with a large number of 0 / 1 characteristics (that is, including explicit upper and lower bound constraints on the design variables, implicit density filtering and floating projection constraints), these constraints are embedded in the update strategy to ensure that they are strictly satisfied during the iterative optimization process. Therefore, the last term of equation (13) can be ignored. ψ≥0 is the Lagrange multiplier of the volume constraint. Therefore, the above equation (13) can be equivalent to solving a boundary Lagrange multiplier Λ problem only for volume constraints, that is

[0084]

[0085] Where 0≤Λ<1. Then the optimization criterion for the above minimization problem can be expressed as

[0086]

[0087] In order to enhance the stability of the iterative update process of the design variables, the unit sensitivity is averaged based on the sensitivity information of the previous iteration step, that is,

[0088]

[0089] In the formula, the subscripts 'k' and 'k-1' represent the current and previous iterations, respectively. Then the optimization criterion in formula (15) can be further expressed as

[0090]

[0091] Then the design variables can be updated and iterated according to the following criteria:

[0092]

[0093] Compared with the traditional OC iterative update criterion, the above update strategy can be applied to problems with positive or negative sensitivity.

[0094] Step 3.2 Define upper / lower boundary constraints for design variables

[0095] In order to realize the upper and lower bound constraints of the design variables, the design variables can be changed to

[0096]

[0097] Step 3.3 Filter

[0098] In order to eliminate the grid dependency and checkerboard problem in the optimization process, the design variables are further processed by applying implicit filtering constraints as follows:

[0099]

[0100] Among them, r ij represents the distance between the i-th unit and the j-th unit, w(r ij ) is the weight factor.

[0101] Then there is

[0102]

[0103] Among them, r min is the filter radius.

[0104] Step 3.4 Projection

[0105] Then, the filtered field variables are projected based on the Heaviside function to overcome the "grayscale" units caused by the filtering scheme and convert the variables in the physical field Convert to x i .

[0106]

[0107] Set η = 0.5, β1>0 as the control factor for controlling the steepness of the Heaviside function, such as Figure 1 As shown in Figure 1, the inner loop that computes the Lagrange multipliers stops when the physics variables satisfy the specified volume constraints.

[0108] Step 3.5 Floating projection constraints

[0109] The design variables are updated using floating projection constraints. Subsequently, in order to drive the design variables toward a 0 / 1 distribution, projection function constraints are applied to the design variables based on the Heaviside function. Specifically,

[0110]

[0111] Among them, β>0 is the control factor that controls the steepness of the Heaviside function. k is a floating threshold value, which can be calculated by dichotomy to ensure that the total volume of the design variables before and after projection remains consistent, that is,

[0112] Step 3.6 Design variables to add movement restrictions

[0113] Updated design variables It will be used for the next finite element analysis. In each iteration, the movement limit value δ of each iteration needs to be set in advance, that is, the design variable change is In the present invention, δ=0.02.

[0114] Step 3.7 Update constraints

[0115] At the same time, the Lagrange multiplier Λ (0≤Λ<1) can be obtained from the volume constraint of the kth iteration, that is, g k It can be calculated as

[0116]

[0117] where g k-1 is the volume constraint value of the previous iteration. Constraint function g k It can also be expanded to be written according to different design variable processing methods

[0118]

[0119] Substituting equations (12) and (16) into equation (25), the gradient of the constraint function with respect to the Lagrange multiplier Λ is:

[0120]

[0121] Then the Lagrange multiplier Λ can be calculated according to the Newton-Raphson method as

[0122]

[0123] Where l is the current iteration number of the inner loop.

[0124] Step 3.8: Determine if the inner loop converges

[0125] The Newton-Raphson method is used to define the initial value of the Lagrange multiplier Λ 0 =0.5, and then the Lagrange multiplier is repeatedly updated according to formula (27). When the change value before and after is small enough (such as 1.0-9), the inner loop will stop converging, thereby obtaining the exact value of the Lagrange multiplier that meets the convergence condition.

[0126] Step 3.9: Determine if the outer loop converges

[0127] In order to obtain a clear topological configuration, the design variables need to be pushed to a 0 / 1 distribution. Therefore, the initial value of the control factor β of the floating projection constraint is a sufficiently small positive number (such as 10-6), and then once the convergence criterion in the following formula (28) is met, Δβ is increased each time. The specific convergence criterion is expressed as

[0128]

[0129] Where, ∈ is the allowed convergence tolerance. M is the number of iterations for the convergence criterion calculation. The present invention takes M=5, indicating that the values ​​of the target value for the last 10 iterations are used for convergence judgment calculation. Once the convergence criterion is met, the iteration stops, otherwise it continues. The steepness of the final floating projection is controlled by the control factor β, and if the corresponding convergence criterion is met, the control factor β will no longer increase. The specific convergence criterion details are detailed in step 4. Finally, when the strict convergence criterion ∈≤10 -4 The whole optimization process converges and stops.

[0130] Step 4: Smooth characterization of optimization results

[0131] In order to smooth the obtained unit-based design solution, the design variable x k Linear interpolation to the entire design domain, that is, x k (x,y). k (x, y) can be defined as a pixel-based design, that is, 20×20 pixels are inserted into each unit, and then the level set function is used To reconstruct the boundary of the obtained topological configuration, we can obtain the corresponding smooth design solution with clear and smooth boundaries divided by the zero-dimensional level set, that is,

[0132]

[0133] This ultimately results in a 0 / 1 distributed pixel-based design, i.e., a smooth design solution. However, since the above level set function is purely based on image processing, it is necessary to calculate the difference between the obtained smooth design solution and the original cell-based design solution. Therefore, the 0 / 1 pixel-based design is projected back onto the original fixed grid and the total volume fraction of all cells is calculated. So with x k With v k The difference between the two is measured by the mean square error between

[0134]

[0135] Finally, once the convergence criterion (30) is satisfied, it means that the smooth design solution obtained by the 0 / 1 pixel basis design is equivalent to the initial unit basis design solution. At the same time, it also means that a smooth design solution that satisfies the convergence condition is obtained. At this time, the floating projection constraint control factor β no longer increases and remains unchanged until the entire iteration converges.

[0136] The above-mentioned specific embodiments are only several preferred embodiments of the present invention. Based on the technical solutions of the present invention and the relevant inspirations of the above-mentioned embodiments, those skilled in the art can make various alternative improvements and combinations to the above-mentioned specific embodiments.

[0137] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the present invention can be implemented in other specific forms without departing from the spirit or essential features of the present invention. Therefore, the embodiments should be considered exemplary and non-restrictive in all respects, and the scope of the present invention is defined by the appended claims rather than the above description, and it is intended that all changes falling within the meaning and scope of the equivalent elements of the claims be included in the present invention.

Claims

1. A three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation, comprising the following four steps, characterized in that: The step 1 is to establish an initial geometric model of the acoustic-vibration coupling system based on COMSOL multi-physics field software, including size, acoustic domain and solid domain; the step 2 is to perform finite element analysis calculations in combination with COMSOL and Matlab to obtain the values ​​of the objective function and constraint function and the sensitivity of the corresponding units; the step 3 is to use OC to update the convergence judgment of the design variables and the optimization model according to the sensitivity information; the step 4 is to linearly interpolate the design variables into the entire design domain to obtain a pixel-based design.

2. According to claim 1, a three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation is characterized by: The step 1 is optimization initialization, establishing a finite element model of the acoustic-vibration coupling system for the geometric model, including physical fields, boundary conditions, load conditions and design domain definitions, meshing the model, and defining the load and fixed boundary constraints of the model.

3. The three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation according to claim 2 is characterized in that: The boundary conditions include: plane wave radiation boundary, fixed boundary, acoustic-solid coupling boundary, hard acoustic wall boundary, and optimized frequency.

4. The three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation according to claim 1 is characterized in that: The material parameters of the solid domain and the acoustic domain include: density, Young's modulus, Poisson's ratio, and sound velocity; the material parameters of the acoustic domain include: density and sound velocity.

5. The three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation according to claim 2 is characterized in that: The design domain definition includes defining topology optimization design variables in the design domain: defining topology optimization design variables based on the sensitivity module (Mathematics>Optimization and Sensitivity>Sensitivity) under COMSOL multi-physics software, establishing an objective function, taking the minimum flexibility of the design domain as the objective function, and initializing and defining optimization parameters in Matlab, including volume fraction, control factor for controlling the steepness of the Heaviside function, and filter radius r. min , maximum number of iterations, and convergence tolerance.

6. The three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation according to claim 1 is characterized in that: The step 2 performs finite element analysis calculation of the acoustic-vibration coupling problem in COMSOL, establishes a topology optimization problem description, establishes a linear material interpolation model, uses Matlab to extract the COMSOL calculation objective function and constraint function, and uses Matlab to extract the unit sensitivity calculated by COMSOL, thereby smoothing the sensitivity based on the historical iteration information of the sensitivity.

7. The three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation according to claim 1 is characterized in that: The step 3 is to perform iterative updates in Matlab, calculate the sensitivity of the objective function and the constraint function, update the design variables according to the OC, define the upper / lower boundary constraints of the design variables, and in order to eliminate the grid dependency and checkerboard problem in the optimization process, the design variables are further processed by applying implicit filtering constraints. Subsequently, the filter field variables are subjected to substitution projection processing based on the Heaviside function, and the design variables are updated using floating projection constraints. Subsequently, in order to drive the design variables to tend to a 0 / 1 distribution, the design variables are subjected to projection function constraint processing based on the Heaviside function, and then the design variables are added with movement restrictions, the constraints are updated, and finally it is determined whether the inner loop and the outer loop converge.

8. The three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation according to claim 7 is characterized in that: The method for determining whether the inner loop converges is as follows: the initial value of the Lagrange multiplier is defined by the Newton-Raphson method, and the Lagrange multiplier is repeatedly updated according to the Newton-Raphson method. When the change value before and after is small enough (such as 1.0-9), the inner loop will stop converging, thereby obtaining the exact value of the Lagrange multiplier that meets the convergence condition.

9. The three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation according to claim 7 is characterized in that: The method for judging whether the outer loop converges is as follows: in order to obtain a clear topological configuration, the design variables need to be pushed to a 0 / 1 distribution. According to the convergence criterion, the numerical values ​​of the target values ​​for nearly 10 iterations are taken for convergence judgment calculation. Once the convergence criterion is met, the iteration stops, otherwise the iteration continues.

10. The three-field floating projection acoustic-vibration coupling topology optimization method based on linear material interpolation according to claim 1, characterized in that: The step 4 is a smoothing representation of the optimization result. The design variables are linearly interpolated into the entire design domain to obtain a pixel-based design, that is, a smooth design solution. Finally, once the convergence criterion is met, it means that the smooth design solution obtained by the 0 / 1 pixel-based design is equivalent to the initial unit-based design solution. At the same time, it also means that a smooth design solution that meets the convergence conditions is obtained. At this time, the floating projection constraint control factor no longer increases and keeps the value unchanged until the entire iteration converges and stops.

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