Viscoelastic structure broadband topological optimization method based on high-order Krylov subspace method

Abstract

The invention belongs to the technical field of structure topological optimization, and particularly relates to a viscoelastic structure broadband topological optimization method based on a high-order Krylov subspace method. According to the method, aiming at the nonlinear frequency response problem of the viscous-elastic material, a reduced-order subspace is constructed through high-order Taylor expansion and moment matching orthogonalization, and the solving precision and the frequency band range are remarkably improved. And based on the reduced-order model, combining the displacement square sum objective function and the volume constraint, and utilizing a gradient optimization algorithm to realize efficient topological optimization. According to the method, high-precision frequency response analysis is realized through high-order Taylor expansion and a correction term, the calculation cost is greatly reduced by more than 80% under a small amount of basis vectors, collaborative optimization is carried out through a multi-frequency-point logarithmic weighting target function, the broadband dynamic performance is remarkably improved, and the method is suitable for complex engineering scenes such as submarines and MEMS devices.

Description

Technical Field

[0001] The invention belongs to the technical field of structural topology optimization, and in particular relates to a wide-band topology optimization method for a viscoelastic structure based on a high-order Krylov subspace method. Background Art

[0002] With the rapid development of high-end equipment and high-precision devices (such as submarines and MEMS devices), the requirements for dynamic performance such as vibration reduction, noise reduction, wave transmission and wave guiding have increased significantly. Topology optimization technology has become a key means to design high-performance structures due to its innovative ability and the completeness of its mathematical framework. However, in dynamic optimization, it is necessary to repeatedly solve the frequency response equation at discrete frequency points. When the number of frequency points is large, the stiffness matrix and damping matrix of viscoelastic materials are both nonlinear functions of frequency (caused by the frequency-dependent characteristics of the shear modulus of the material), which makes it impossible to pre-decompose the equation and the calculation scale explodes. In addition, the optimization iteration process further increases the computational complexity, making it difficult for traditional methods to meet the actual engineering needs.

[0003] At present, the original high-dimensional system is projected into a low-dimensional subspace to reduce the amount of calculation. There are mainly three methods: (1) modal superposition method: constructing a reduced-order model based on the linear combination of structural vibration modes; (2) Padé expansion method: approximating the frequency response characteristics through rational functions; (3) second-order Krylov subspace method (such as SOAR_k0, SOAR_k2): constructing a reduced-order subspace by matching the first two moments, which has the advantages of stability and computational efficiency.

[0004] However, existing order reduction methods have the following problems when dealing with viscoelastic materials: (1) Ignoring high-order nonlinear terms: The traditional second-order Krylov subspace method only matches the first two moments and ignores the high-order frequency dependence of the stiffness matrix and the damping matrix, resulting in large solution errors and a narrow frequency band; (2) Poor frequency band adaptability: In wide-band optimization scenarios, traditional methods need to significantly increase the number of basis vectors to maintain accuracy, which in turn offsets the computational advantages of order reduction; (3) Insufficient accuracy of the adjoint equation: The existing methods are also limited by the low-order expansion when solving the adjoint equation, and the accumulated errors in the sensitivity analysis cause the optimization results to deviate from the full-order model. Summary of the invention

[0005] The purpose of the present invention is to provide a broadband topology optimization method for viscoelastic structures based on a high-order Krylov subspace method to overcome the shortcomings of the prior art.

[0006] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is: A first aspect of an embodiment of the present invention provides a method for broadband topology optimization of a viscoelastic structure based on a high-order Krylov subspace method, comprising: Establish the frequency response equation and adjoint equation of viscoelastic structure; At the selected frequency expansion point, the left-hand matrix and the right-hand load vector of the frequency response equation are expanded into high-order Taylor series respectively, and the initial basis vectors are generated after truncating the high-order terms; The initial basis vectors are iteratively corrected, orthogonalized in combination with the correction terms, and the orthogonalized basis vector matrix is ​​output; Project the orthogonalized basis vector matrix to the frequency response equation, construct a low-dimensional frequency response equation and a low-dimensional adjoint equation, and solve the reduced-order displacement response vector and the reduced-order adjoint variable; Taking the minimization of the weighted logarithm sum of the square sum of the reduced-order displacement response vector in a wide frequency band as the objective function, a topology optimization model is established in combination with volume constraints, and the sensitivity of the objective function to the design variables is calculated based on the reduced-order adjoint variables. The gradient optimization algorithm is used to iteratively update the design variables until the convergence conditions are met and the optimal topological configuration is obtained.

[0007] Alternatively, the frequency response equation is: ;in, represents the i-th load frequency, represents the overall stiffness matrix at the i-th load frequency, represents the damping matrix at the i-th load frequency, represents the mass matrix, represents the displacement response vector at the i-th load frequency, represents the load vector at the i-th loading frequency.

[0008] Optionally, the adjoint equation is: ;in, represents the i-th load frequency, represents the overall stiffness matrix at the i-th load frequency, represents the damping matrix at the i-th load frequency, represents the mass matrix, represents the adjoint variable at the i-th load frequency, represents the degree of freedom selection matrix, Represents the conjugate vector of the displacement response at the i-th loading frequency.

[0009] Optionally, the initial basis vectors are iteratively corrected, orthogonalized in combination with the correction terms, and the orthogonalized basis vector matrix is ​​output, including: Normalize the initial basis vectors; By dynamically superimposing the high-order derivatives of the stiffness matrix and the load, subsequent basis vectors are generated, and correction terms are introduced to adjust the direction of the basis vectors; The Gram-Schmidt orthogonalization method is used to gradually eliminate the projection components between basis vectors, and the orthogonalized basis vectors are normalized. Arrange all normalized and orthogonal basis vectors in the order of generation to form an orthogonal basis matrix.

[0010] Optionally, the correction term is: ;in, Represents a correction item, Indicates the starting index of the correction term, represents the index of the basis vector, represents the cutoff order of Taylor expansion, Represents multiplication of index variables, Represents the multiplication operator.

[0011] Alternatively, the low-dimensional frequency response equation is: ;in, represents the i-th load frequency, represents the reduced-order mass matrix, represents the reduced-order damping matrix, represents the reduced-order stiffness matrix, represents the reduced-order displacement response vector at the i-th load frequency, represents the reduced-order load vector at the i-th load frequency; The low-dimensional adjoint equation is: ;in, represents the reduced-order adjoint vector at the i-th load frequency, represents the reduced-order weighting matrix, Represents the conjugate vector of the reduced-order displacement response at the i-th loading frequency.

[0012] Optionally, the topology optimization model is: ;in, represents the objective function maximization column form, Represents the unit pseudo-density design variable, with a value range of [0, 1], represents the objective function, represents the total number of load frequencies, represents the i-th load frequency, represents the objective function at the i-th load frequency, represents the displacement response vector at the i-th load frequency, represents the degree of freedom selection matrix, represents the conjugate vector, represents the number of target degrees of freedom, Represents the total number of units, Indicates the unit number, represents the density of the e-th unit after filtering and projection, represents the e-th unit volume, represents the total volume of the design domain, represents the upper limit of volume constraint, represents a dynamic matrix, represents the load vector at the i-th loading frequency.

[0013] Optionally, the sensitivity calculation rule is as follows: ;in, , represents the derivative of the objective function with respect to the element pseudo-density design variable, represents the objective function, Represents the unit pseudo-density design variable, with a value range of [0, 1], Indicates the number of discretized frequency points in the frequency interval, represents the i-th load frequency, Represents the single frequency response index, represents the derivative of the objective function at the i-th load frequency with respect to the element pseudo-density design variable, represents taking the real part of a complex number, represents the transpose of a matrix or vector, represents the adjoint variable at the i-th load frequency, represents a dynamic matrix, represents the derivative of the dynamic matrix with respect to the element pseudo-density design variable, represents the displacement response vector at the i-th load frequency.

[0014] Optionally, the gradient optimization algorithm is a moving asymptote method.

[0015] The present invention has the following beneficial effects: (1) High precision and wide-band adaptation: The high-order Krylov subspace method (HOMMG) is used to introduce high-order Taylor expansion and correction terms, which significantly improves the accuracy and frequency band adaptability of the frequency response analysis of viscoelastic materials, overcoming the problems of narrow frequency band and large error caused by the traditional second-order method that ignores nonlinear terms.

[0016] (2) Improved computational efficiency: The HOMMG method only requires a small number of basis vectors in topology optimization to achieve optimization results consistent with the full-order model. It also reduces the frequency sweep calculation cost by more than 80%, solving the problem that traditional order reduction methods require a large number of basis vectors and produce unstable optimization results.

[0017] (3) Multi-frequency collaborative optimization: Taking the logarithmically weighted multi-frequency displacement response as the objective function and combining it with the high-order correction of the adjoint equation, the global optimization of the dynamic performance within a wide bandwidth is achieved. The optimized configuration (such as the grid structure) effectively converges wave energy and improves the frequency response amplitude across the entire frequency band. This is suitable for complex engineering scenarios such as submarines and MEMS devices. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 A schematic diagram of a flow chart of a method for broadband topology optimization of viscoelastic structures based on a high-order Krylov subspace method provided in an embodiment of the present application; Figure 2 A schematic diagram of a flow chart of a method for broadband topology optimization of viscoelastic structures based on a high-order Krylov subspace method provided in another embodiment of the present application; Figure 3 A schematic diagram of a dual-material topology optimization design for elastic wave conduction provided in an embodiment of the present application, wherein: Figure 3 (a) Schematic diagram of the design domain and boundary conditions for the elastic wave conduction optimization problem. Figure 3 (b) Schematic diagram of the structural topology of elastic materials and viscoelastic materials; Figure 4 A schematic diagram showing a comparison of relative errors of solving state equations using different reduced-order models provided in an embodiment of the present application; Figure 5 A schematic diagram showing a comparison of relative errors of solving adjoint equations using different reduced-order models provided in an embodiment of the present application; Figure 6 A schematic diagram of a curve showing a change in shear modulus and damping ratio of a viscoelastic material with frequency provided in an embodiment of the present application; Figure 7 A schematic diagram of the topology optimization results of the low-frequency waveguide structure provided in an embodiment of the present application; Figure 8 A schematic diagram showing a comparison of frequency response curves of optimization results provided in an embodiment of the present application; Fig. 9 A schematic diagram comparing topology optimization results of different order reduction methods under different numbers of basis vectors provided in the embodiments of the present application. DETAILED DESCRIPTION

[0019] The following will be combined with the attached embodiment of the present invention Figure 1-9 , the technical solutions in the embodiments of the present invention are clearly and completely described. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of them. Unless otherwise specified, the technical means used in the embodiments are conventional means well known to those skilled in the art.

[0020] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features of the embodiments may be combined with each other.

[0021] It should be noted that similar reference numerals and letters denote similar items in the following drawings, and therefore, once an item is defined in one drawing, further definition and explanation thereof is not required in subsequent drawings.

[0022] Reference Figure 1 and Figure 2 The embodiment of the present invention provides a method for broadband topology optimization of viscoelastic structures based on a high-order Krylov subspace method, the method comprising the following steps: S100, establish the frequency response equation and adjoint equation of the viscoelastic structure; In one embodiment, the frequency response equation is: ;in, represents the i-th load frequency, represents the overall stiffness matrix at the i-th load frequency, represents the damping matrix at the i-th load frequency, represents the mass matrix, represents the displacement response vector at the i-th load frequency, represents the load vector at the i-th loading frequency.

[0023] It can be found that the frequency response equation of viscoelastic materials is no longer It is not a quadratic function, but a nonlinear function. The traditional second-order Krylov subspace method is difficult to solve this frequency response equation. The present invention establishes the HOMMG method to achieve fast frequency response analysis of viscoelastic structures.

[0024] In one embodiment, the adjoint equation is: ;in, represents the i-th load frequency, represents the overall stiffness matrix at the i-th load frequency, represents the damping matrix at the i-th load frequency, represents the mass matrix, represents the adjoint variable at the i-th load frequency, represents the degree of freedom selection matrix, Represents the conjugate vector of the displacement response at the i-th loading frequency.

[0025] S200, at the selected frequency expansion point, the left-hand matrix and the right-hand load vector of the frequency response equation are respectively expanded into high-order Taylor series, and the initial basis vectors are generated after truncating the high-order terms; Specifically, at the expansion point, the left-hand matrix and the right-hand load vector of the frequency response equation are expanded into Taylor series form: ;in, represents the Taylor expansion order of the coefficient matrix, represents the index of the Taylor expansion order of the coefficient matrix, represents the i-th load frequency, represents the frequency at the expansion point of the basis vector, express The factorial of Represents the coefficient matrix Derivatives, represents the displacement vector at the i-th load frequency, represents the Taylor expansion order of the load vector, represents the Taylor expansion order index of the load vector, express The factorial of represents the load vector at the i-th load frequency Derivative of order.

[0026] S300, iteratively correcting the initial basis vectors, performing orthogonalization processing in combination with the correction terms, and outputting an orthogonalized basis vector matrix; In one embodiment, step S300 specifically includes: S310, normalizing the initial basis vectors; S320, generating subsequent basis vectors by dynamically superimposing high-order derivatives of the stiffness matrix and the load, and introducing correction terms to adjust the directions of the basis vectors; Specifically, the amendments are: ;in, Represents a correction item, Indicates the starting index of the correction term, represents the index of the basis vector, represents the cutoff order of Taylor expansion, Represents multiplication of index variables, Represents the multiplication operator.

[0027] Specifically, =1 or 2, the order of multiplication ,matrix Representation Matrix of to OK, to Column blocks.

[0028] S330, using a Gram-Schmidt orthogonalization method to gradually eliminate projection components between basis vectors, and normalizing the orthogonalized basis vectors; S340, arranging all normalized and orthogonal basis vectors in a generated order to form an orthogonal basis matrix.

[0029] S400, projecting the orthogonalized basis vector matrix to the frequency response equation, constructing a low-dimensional frequency response equation and a low-dimensional adjoint equation, and solving the reduced-order displacement response vector and the reduced-order adjoint variable; Specifically, the low-dimensional frequency response equation is: ;in, represents the i-th load frequency, represents the reduced-order mass matrix, represents the reduced-order damping matrix, represents the reduced-order stiffness matrix, represents the reduced-order displacement response vector at the i-th load frequency, represents the reduced-order load vector at the i-th loading frequency.

[0030] Specifically, the low-dimensional adjoint equation is: ;in, represents the reduced-order adjoint vector at the i-th load frequency, represents the reduced-order weighting matrix, Represents the conjugate vector of the reduced-order displacement response at the i-th loading frequency.

[0031] , , , , .

[0032] In one embodiment, the ability of the present invention to solve the state equation and the adjoint equation in a wide frequency range is evaluated, and compared with the traditional second-order Arnoldi method (denoted as SOAR_k0) and the improved second-order Arnoldi method (denoted as SOAR_k2). The specific process is as follows: Step 1: To evaluate the frequency response solution and plot the frequency response, define the amplitude of the structure and the adjoint vector The 2-norm of is:

[0033] Step 2: In order to evaluate the accuracy of various reduced-order models relative to the full model, the relative error index is introduced: ; where ROM and FULL represent the solutions obtained by the reduced-order model and the full model, respectively.

[0034] Step 3: The present invention aims at the frequency response problem of the broadband waveguide structure by using Figure 3 (b) shows the dual-material structure topology of elastic material and viscoelastic material. The accuracy of various order reduction methods is tested and the solution model is Figure 3 (a) The elastic wave propagation optimization problem shown in Figure 2 shows the accuracy, efficiency, and stability of different order reduction methods.

[0035] Step 4: Solve the state equation using SOAR_k0, SOAR_k2 and HOMMG methods. For ease of description, we will consider ( )of ( The HOMMG method of the order expansion is denoted as The frequency range considered is The frequency interval is discretized into 401 frequency points for sweep frequency analysis.

[0036] Step 5: In order to further evaluate the solution accuracy of various order reduction methods, the present invention uses the frequency response displacement calculated by the full-order model as the reference solution and calculates the relative errors of various order reduction methods in solving the state equation when the number of basis vectors in the subspace is 40, 80 and 160, respectively. Figure 4 As shown in the table, it can be clearly seen that the HOMMG method proposed in this paper has a smaller solution error than the traditional SOAR_k0 and SOAR_k2 methods, and can obtain high-precision solutions in a larger frequency band near the expansion point. In this regard, this paper compares and analyzes the range of high-precision solutions that can be obtained by various reduced-order methods. Set an acceptable error limit , set the relative error of the solution to be less than The frequency range is the solvable bandwidth of the reduced-order method, as shown in Table 1. The HOMMG method proposed in this paper has a larger solvable bandwidth than the traditional reduced-order method. Moreover, with the increase in the number of basis vectors, the solvable range of various reduced-order methods gradually expands. Among them, the solution range of the HOMMG method can quickly cover the entire frequency domain, and accurate solutions to the entire frequency band can be achieved using fewer basis vectors, which will greatly reduce the calculation time. In addition, the HOMMG method proposed in this paper expands the left-hand side of the state equation into a Taylor series form, and explores the influence of high-order expansion on the solution accuracy and bandwidth of the reduced-order method. Compared to , There is a smaller relative error and a larger solvable bandwidth, which also proves the effectiveness of the higher-order expansion.

[0037] Table 1 Comparison of solvable bandwidths of different reduced-order models for solving state equations

[0038] Step 6: This paper uses the test model to evaluate the accuracy of various order reduction methods in solving the adjoint equation. The relative errors and solvable bandwidths of various order reduction methods are as follows: Figure 5As shown in Table 2. As can be seen from the table, the HOMMG method proposed in this paper still has high accuracy and stability when solving the adjoint equation. The HOMMG method can effectively improve the accuracy of the reduced-order method by expanding the left-hand side of the adjoint equation in a high-order manner. A high-precision solution in the full frequency band can be obtained with a small number of basis vectors. Moreover, it is worth noting that when solving the adjoint equation, this paper also considers expanding the right-hand side of the adjoint equation. The results show that compared with the conventional method, the high-order expansion of the right-hand side term has a great influence on the accuracy of solving the adjoint equation by the HOMMG method. , It can greatly improve the solution accuracy near the expansion point and obtain a local high-precision solution near the expansion point.

[0039] Table 2 Comparison of solvable bandwidths of different reduced-order models for solving adjoint equations

[0040] S500, with the objective function of minimizing the weighted logarithm sum of the square sum of the reduced-order displacement response vector in a wide frequency band, a topology optimization model is established in combination with volume constraints, and the sensitivity of the objective function to the design variables is calculated based on the reduced-order adjoint variables; Specifically, the topology optimization model is: ;in, represents the objective function maximization column form, Represents the unit pseudo-density design variable, with a value range of [0, 1], represents the objective function, represents the total number of load frequencies, represents the i-th load frequency, represents the objective function at the i-th load frequency, represents the displacement response vector at the i-th load frequency, represents the degree of freedom selection matrix, represents the conjugate vector, represents the number of target degrees of freedom, Represents the total number of units, Indicates the unit number, represents the density of the e-th unit after filtering and projection, represents the e-th unit volume, represents the total volume of the design domain, represents the upper limit of volume constraint, represents a dynamic matrix, represents the load vector at the i-th loading frequency.

[0041] Specifically, the sensitivity calculation rules are as follows: ;in, , represents the derivative of the objective function with respect to the element pseudo-density design variable, represents the objective function, Represents the unit pseudo-density design variable, with a value range of [0, 1], Indicates the number of discretized frequency points in the frequency interval, represents the i-th load frequency, Represents the single frequency response index, represents the derivative of the objective function at the i-th load frequency with respect to the element pseudo-density design variable, represents taking the real part of a complex number, represents the transpose of a matrix or vector, represents the adjoint variable at the i-th load frequency, represents a dynamic matrix, represents the derivative of the dynamic matrix with respect to the element pseudo-density design variable, represents the displacement response vector at the i-th load frequency.

[0042] S600 uses a gradient optimization algorithm to iteratively update the design variables until the convergence conditions are met and the optimal topology configuration is obtained.

[0043] Specifically, the optimization process of the optimal topological configuration is given, and the following are the specific steps: The first step: the present invention is directed to Figure 3 The topology optimization of the elastic wave propagation problem is carried out in the design domain shown in (a), and the optimization objective is to maximize the amplitude at the right boundary dashed line. The design domain is discretized by the grid, and the design variable is the element pseudo density. The filter radius is 10 times the element size, and the parameter in Heaviside filtering is The initial value is 1, and it doubles every 50 steps. , up to , the number of iterations is 440. The optimization problem is dual material layout optimization. Table 3 lists the material properties of the two materials. Material 1 is an elastic model with high stiffness and low damping. The structural damping adopts the Rayleigh damping model. Material 2 is a viscoelastic material. The shear model adopts the GHM model. The series expansion takes 5 items. The specific model parameters are shown in Table 4. The relationship curve between the shear modulus, damping ratio and frequency is shown in Figure 6 shown.

[0044] The volume fraction of material 1 is set to 0.4 in the optimization model. )( ) was used to carry out topology optimization design. During the calculation process, 401 frequency points were used to evenly discretize the frequency range considered.

[0045] Table 3 Elastic and viscoelastic material properties

[0046] Table 4 GHM viscoelastic material model parameters

[0047] Step 2: Low frequency issues The topology optimization results are as follows: Figure 7 As shown in the figure, it can be seen that the distribution of elastic materials and viscoelasticity is a grid-like distribution, and converges to the center point on the right. In addition, grid structures are arranged in the upper left corner and lower right corner of the optimization result to isolate the wave propagation, so that the wave energy converges to the center point on the right. The sweep frequency curve of the optimization result is shown in the figure. Figure 8 As shown in the figure, the blue line represents the frequency response curve of the initial solution, and the yellow line represents the frequency response curve of the optimization result. It can be seen that after optimization, the frequency response amplitude is improved in the entire frequency domain, and the objective function value of the optimization result is 836.8713.

[0048] Step 3: Topology optimization based on various order reduction methods such as Fig. 9 As shown. Among them, the traditional order reduction methods (SOAR_k0 and SOAR_k2) have large prediction errors for the dynamic properties of materials due to the neglect of the high-order expansion terms of the complex shear modulus of viscoelastic materials. It can be seen from the table that the two methods need to construct 30 basis vector subspaces to converge to stable topology optimization results, and there is still a big difference with the optimization results of the full-order model. The HOMMG method can obtain a stable topological configuration when the basis vectors are 20, especially the optimization results of the HOMMG_k4f2 method are highly consistent with the full-order model. Therefore, the HOMMG method is used to perform high-order expansion of the state equation and the adjoint equation, and a high-order Krylov subspace method is constructed. High-precision approximate solutions and topology optimization configurations can be obtained using fewer basis vectors, which greatly reduces the time consumption of frequency sweep calculations and reduces the computational cost of frequency response topology optimization.

[0049] In one embodiment, the gradient optimization algorithm is a moving asymptote method.

[0050] The present invention has the following beneficial effects: (1) High precision and wide-band adaptation: The high-order Krylov subspace method (HOMMG) is used to introduce high-order Taylor expansion and correction terms, which significantly improves the accuracy and frequency band adaptability of the frequency response analysis of viscoelastic materials, overcoming the problems of narrow frequency band and large error caused by the traditional second-order method that ignores nonlinear terms.

[0051] (2) Improved computational efficiency: The HOMMG method only requires a small number of basis vectors in topology optimization to achieve optimization results consistent with the full-order model. It also reduces the frequency sweep calculation cost by more than 80%, solving the problem that traditional order reduction methods require a large number of basis vectors and produce unstable optimization results.

[0052] (3) Multi-frequency collaborative optimization: Taking the logarithmically weighted multi-frequency displacement response as the objective function and combining it with the high-order correction of the adjoint equation, the global optimization of the dynamic performance within a wide bandwidth is achieved. The optimized configuration (such as the grid structure) effectively converges wave energy and improves the frequency response amplitude across the entire frequency band. This is suitable for complex engineering scenarios such as submarines and MEMS devices.

[0053] The above embodiments are only descriptions of the preferred modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various deformations, modifications, and substitutions made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A broadband topology optimization method for viscoelastic structures based on a high-order Krylov subspace method, characterized in that: include: Establish the frequency response equation and adjoint equation of viscoelastic structure; At the selected frequency expansion point, the left-hand matrix and the right-hand load vector of the frequency response equation are expanded into high-order Taylor series respectively, and the initial basis vectors are generated after truncating the high-order terms; The initial basis vectors are iteratively corrected, orthogonalized in combination with the correction terms, and the orthogonalized basis vector matrix is ​​output; Project the orthogonalized basis vector matrix to the frequency response equation, construct a low-dimensional frequency response equation and a low-dimensional adjoint equation, and solve the reduced-order displacement response vector and the reduced-order adjoint variable; Taking the minimization of the weighted logarithm sum of the square sum of the reduced-order displacement response vector in a wide frequency band as the objective function, a topology optimization model is established in combination with volume constraints, and the sensitivity of the objective function to the design variables is calculated based on the reduced-order adjoint variables. The gradient optimization algorithm is used to iteratively update the design variables until the convergence conditions are met and the optimal topological configuration is obtained.

2. The method for broadband topology optimization of viscoelastic structures based on high-order Krylov subspace method according to claim 1, characterized in that: The frequency response equation is: ;in, represents the i-th load frequency, represents the overall stiffness matrix at the i-th load frequency, represents the damping matrix at the i-th load frequency, represents the mass matrix, represents the displacement response vector at the i-th load frequency, represents the load vector at the i-th loading frequency.

3. The method for broadband topology optimization of viscoelastic structures based on high-order Krylov subspace method according to claim 1, characterized in that: The adjoint equation is: ;in, represents the i-th load frequency, represents the overall stiffness matrix at the i-th load frequency, represents the damping matrix at the i-th load frequency, represents the mass matrix, represents the adjoint variable at the i-th load frequency, represents the degree of freedom selection matrix, Represents the conjugate vector of the displacement response at the i-th loading frequency.

4. The method for broadband topology optimization of viscoelastic structures based on high-order Krylov subspace method according to claim 1, characterized in that: By iteratively correcting the initial basis vectors and combining the correction terms for orthogonalization, the orthogonalized basis vector matrix is ​​output, including: Normalize the initial basis vectors; By dynamically superimposing the high-order derivatives of the stiffness matrix and the load, subsequent basis vectors are generated, and correction terms are introduced to adjust the direction of the basis vectors; The Gram-Schmidt orthogonalization method is used to gradually eliminate the projection components between basis vectors, and the orthogonalized basis vectors are normalized. Arrange all normalized and orthogonal basis vectors in the order of generation to form an orthogonal basis matrix.

5. The method for broadband topology optimization of viscoelastic structures based on high-order Krylov subspace method according to claim 4, characterized in that: The correction items are: ;in, Represents a correction term, Indicates the starting index of the correction term, represents the index of the basis vector, represents the cutoff order of Taylor expansion, Represents multiplication of index variables, Represents the multiplication operator.

6. The method for broadband topology optimization of viscoelastic structures based on high-order Krylov subspace method according to claim 1, characterized in that: The low-dimensional frequency response equation is: ;in, represents the i-th load frequency, represents the reduced-order mass matrix, represents the reduced-order damping matrix, represents the reduced-order stiffness matrix, represents the reduced-order displacement response vector at the i-th load frequency, represents the reduced-order load vector at the i-th load frequency; The low-dimensional adjoint equation is: ;in, represents the reduced-order adjoint vector at the i-th load frequency, represents the reduced-order weighting matrix, Represents the conjugate vector of the reduced-order displacement response at the i-th loading frequency.

7. The method for broadband topology optimization of viscoelastic structures based on high-order Krylov subspace method according to claim 1, characterized in that ,The topology optimization model is: ;in, represents the objective function maximization column form, Represents the unit pseudo-density design variable, with a value range of [0, 1], represents the objective function, represents the total number of load frequencies, represents the i-th load frequency, represents the objective function at the i-th load frequency, represents the displacement response vector at the i-th load frequency, represents the degree of freedom selection matrix, represents the conjugate vector, represents the number of target degrees of freedom, Represents the total number of units, Indicates the unit number, represents the density of the e-th unit after filtering and projection, represents the e-th unit volume, represents the total volume of the design domain, represents the upper limit of volume constraint, represents a dynamic matrix, represents the load vector at the i-th loading frequency.

8. The method for broadband topology optimization of viscoelastic structures based on high-order Krylov subspace method according to claim 1, characterized in that ,The sensitivity calculation rules are as follows: ;in, , represents the derivative of the objective function with respect to the element pseudo-density design variable, represents the objective function, Represents the unit pseudo-density design variable, with a value range of [0, 1], Indicates the number of discretized frequency points in the frequency interval, represents the i-th load frequency, Represents the single frequency response index, represents the derivative of the objective function at the i-th load frequency with respect to the element pseudo-density design variable, represents taking the real part of a complex number, represents the transpose of a matrix or vector, represents the adjoint variable at the i-th load frequency, represents a dynamic matrix, represents the derivative of the dynamic matrix with respect to the element pseudo-density design variable, represents the displacement response vector at the i-th load frequency.

9. The method for broadband topology optimization of viscoelastic structures based on high-order Krylov subspace method according to claim 1, characterized in that ,The gradient optimization algorithm is the moving asymptote method.

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