Wide-band Topology Optimization Method for Viscoelastic Structures Based on the High-order Krylov Subspace Method

Abstract

The present invention belongs to the technical field of structural topology optimization, and specifically relates to a wide-band topology optimization method for viscoelastic structures based on the high-order Krylov subspace method. This method addresses the non-linear frequency response problem of viscoelastic materials by constructing a reduced-order subspace through high-order Taylor expansion and moment matching orthogonalization, significantly improving the solution accuracy and frequency band range. Based on the reduced-order model, combined with the displacement squared sum objective function and volume constraint, a gradient optimization algorithm is used to achieve efficient topology optimization. The present invention achieves high-precision frequency response analysis through high-order Taylor expansion and correction terms, significantly reducing the computational cost by more than 80% with a small number of basis vectors, and co-optimizing with a multi-frequency point logarithmic weighted objective function to significantly improve the wide-band dynamic performance, which is applicable to complex engineering scenarios such as submarines and MEMS devices.

Description

Technical Field

[0001] The present invention belongs to the technical field of structural topology optimization, and particularly relates to a broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method. Background Art

[0002] With the rapid development of high-end equipment and high-precision devices (such as submarines, MEMS devices), the requirements for dynamic performance such as vibration reduction, noise reduction, wave propagation, and waveguide are significantly improved. Due to its innovation ability and the completeness of the mathematical framework, topology optimization technology has become a key means for designing high-performance structures. 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, since the stiffness matrix and damping matrix of viscoelastic materials are both non-linear functions of frequency (caused by the frequency-dependent characteristics of the material shear modulus), the equation cannot be pre-decomposed, and the computational scale increases explosively. In addition, the optimization iteration process further exacerbates the computational complexity, making it difficult for traditional methods to meet the actual engineering requirements.

[0003] Currently, there are mainly three methods to reduce the computational amount by projecting the original high-dimensional system into a low-dimensional subspace: (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, the existing reduced-order methods have the following problems when dealing with viscoelastic materials: (1) Ignoring high-order non-linear terms: The traditional second-order Krylov subspace method only matches the first two moments, ignoring the high-order frequency dependence of the stiffness matrix and damping matrix, resulting in large solution errors and narrow frequency bands; (2) Poor frequency band adaptability: In the broadband optimization scenario, traditional methods need to significantly increase the number of basis vectors to maintain accuracy, which instead offsets the computational advantages of the reduced order; (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 cumulative error of sensitivity analysis leads to the deviation of the optimization result 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 the high-order Krylov subspace method to solve the disadvantages in the prior art.

[0006] To achieve the above-mentioned invention purpose, the technical solution adopted by the present invention is:

[0007] In the first aspect of the embodiment of the present invention, a broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method is provided, including:

[0008] Establish the frequency response equation and adjoint equation of the viscoelastic structure;

[0009] At the selected frequency expansion point, expand the left matrix and the right load vector of the frequency response equation into high-order Taylor series respectively, and generate the initial basis vectors after truncating the high-order terms;

[0010] Iteratively correct the initial basis vectors, perform orthogonalization processing in combination with the correction terms, and output the orthogonalized basis vector matrix;

[0011] Project the orthogonalized basis vector matrix onto the frequency response equation, construct the low-dimensional frequency response equation and low-dimensional adjoint equation, and solve the reduced-order displacement response vector and reduced-order adjoint variables;

[0012] Taking the weighted logarithmic sum of the squares of the reduced-order displacement response vectors in the wide frequency band as the objective function, establish a topology optimization model in combination with volume constraints, and calculate the sensitivity of the objective function to the design variables based on the reduced-order adjoint variables;

[0013] Adopt the gradient optimization algorithm to iteratively update the design variables until the convergence condition is satisfied to obtain the optimal topology configuration.

[0014] Optionally, the frequency response equation is:

[0015] ; where 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 load frequency.

[0016] Optionally, the adjoint equation is:

[0017] ; where 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 load frequency.

[0018] Optionally, iteratively correcting the initial basis vectors, performing orthogonalization processing in combination with the correction terms, and outputting the orthogonalized basis vector matrix includes:

[0019] Normalize the initial basis vectors;

[0020] Generate subsequent basis vectors by dynamically superimposing the stiffness matrix and the higher-order derivative terms of the load, and introduce correction terms to adjust the direction of the basis vectors;

[0021] Use the Gram - Schmidt orthogonalization method to gradually eliminate the projection components between the basis vectors, and normalize the orthogonalized basis vectors;

[0022] Arrange all the normalized and orthogonal basis vectors in the order of generation to form an orthogonal basis matrix.

[0023] Optionally, the correction term is:

[0024] ; where, represents the correction term, represents the starting index of the correction term, represents the index of the basis vector, represents the truncation order of the Taylor expansion, represents the product index variable, represents the product operator.

[0025] Optionally, the low - dimensional frequency response equation is:

[0026] ; where, 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;

[0027] The low - dimensional adjoint equation is:

[0028] ; where, represents the reduced - order adjoint vector at the i - th load frequency, represents the reduced - order weighted matrix, represents the conjugate vector of the reduced - order displacement response at the i - th load frequency.

[0029] Optionally, the topology optimization model is:

[0030] ; where, represents the maximization formulation of the objective function, represents the element pseudo - density design variable, with a value range of [0, 1], denotes the objective function, denotes the total number of load frequencies, denotes the i-th load frequency, denotes the objective function at the i-th load frequency, denotes the displacement response vector at the i-th load frequency, denotes the degree-of-freedom selection matrix, denotes the conjugate vector, denotes the number of target degrees of freedom, denotes the total number of elements, denotes the element number, denotes the e-th element density after filtering and projection, denotes the volume of the e-th element, denotes the total volume of the design domain, denotes the upper limit of the volume constraint, denotes the dynamic matrix, denotes the load vector at the i-th load frequency.

[0031] Optionally, the sensitivity calculation rule is as follows:

[0032] ; where, , denotes the derivative of the objective function with respect to the element pseudo-density design variable, denotes the objective function, denotes the element pseudo-density design variable, with a value range of [0, 1], denotes the number of discretized frequency points within the frequency interval, denotes the i-th load frequency, denotes the single-frequency point response index, denotes the derivative of the objective function with respect to the element pseudo-density design variable at the i-th load frequency, denotes taking the real part of a complex number, denotes the transpose of a matrix or vector, denotes the adjoint variable at the i-th load frequency, denotes the dynamic matrix, denotes the derivative of the dynamic matrix with respect to the element pseudo-density design variable, denotes the displacement response vector at the i-th load frequency.

[0033] Optionally, the gradient optimization algorithm is the moving asymptotes method.

[0034] The present invention has the following beneficial effects:

[0035] (1) High-precision and wide-bandwidth adaptation: Through the Higher-Order Krylov Subspace Method (HOMMG), introducing high-order Taylor expansion and correction terms, significantly improves the accuracy and bandwidth adaptability of the frequency response analysis of viscoelastic materials, overcoming the problems of narrow bandwidth and large errors caused by the traditional second-order method ignoring nonlinear terms.

[0036] (2) Improve computational efficiency: The HOMMG method only requires a small number of basis vectors in topology optimization to achieve the same optimization results as the full-order model, while reducing the swept-frequency calculation cost by more than 80%, solving the defects of traditional reduced-order methods that require a large number of basis vectors and unstable optimization results.

[0037] (3) Multi-frequency point collaborative optimization: Taking the logarithmically weighted multi-frequency point displacement response as the objective function, combined with the high-order correction of the adjoint equation, realizes the global optimization of dynamic performance within a wide bandwidth. The optimized configuration (such as a lattice structure) effectively converges wave energy, and the frequency response amplitude is improved throughout the frequency band, suitable for complex engineering scenarios such as submarines and MEMS devices. Description of the Drawings

[0038] Figure 1 It is a schematic flow chart of the wide-bandwidth topology optimization method for viscoelastic structures based on the Higher-Order Krylov Subspace Method provided by the embodiment of the present application;

[0039] Figure 2 It is a schematic flow chart of the wide-bandwidth topology optimization method for viscoelastic structures based on the Higher-Order Krylov Subspace Method provided by another embodiment of the present application;

[0040] Figure 3 It is a schematic diagram of the dual-material topology optimization design for elastic wave conduction provided by the embodiment of the present application, where Figure 3 (a) is a schematic diagram of the design domain and boundary conditions of the elastic wave conduction optimization problem, Figure 3 (b) is a schematic diagram of the structural topology of elastic materials and viscoelastic materials;

[0041] Figure 4 It is a schematic diagram of the relative error comparison of solving the state equation by different reduced-order models provided by the embodiment of the present application;

[0042] Figure 5 It is a schematic diagram of the relative error comparison of solving the adjoint equation by different reduced-order models provided by the embodiment of the present application;

[0043] Figure 6 It is a schematic diagram of the curve of the shear modulus and damping ratio of viscoelastic materials changing with frequency provided by the embodiment of the present application;

[0044] Figure 7 It is a schematic diagram of the topology optimization result of the low-frequency guided wave structure provided by the embodiment of the present application;

[0045] Figure 8 Schematic diagram for comparing frequency response curves of optimization results provided by embodiments of the present application

[0046] Figure 9 Schematic diagram for comparing topology optimization results of different order reduction methods under different numbers of basis vectors provided by embodiments of the present application Specific embodiments

[0047] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying Figures 1-9 drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Unless otherwise specified, the technical means used in the embodiments are conventional means well known to those skilled in the art

[0048] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other

[0049] It should be noted that similar reference numerals and letters denote similar items in the following drawings. Therefore, once an item is defined in one drawing, it does not need to be further defined and explained in subsequent drawings

[0050] Referring to Figure 1 and Figure 2 , embodiments of the present invention provide a broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method. The method includes the following steps

[0051] S100, establishing a frequency response equation and an adjoint equation for the viscoelastic structure

[0052] In one embodiment, the frequency response equation is ; where 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 load frequency

[0053] It can be found that the frequency response equation of the viscoelastic material is no longer a quadratic function of the frequency , but a non-linear 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

[0054] In one embodiment, the adjoint equation is ; where 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 load frequency.

[0055] S200, at the selected frequency expansion point, expand the left-end matrix and the right-end load vector of the frequency response equation into high-order Taylor series respectively, and generate the initial basis vectors after truncating the high-order terms;

[0056] Specifically, at the expansion point, expand both the left-end term matrix and the right-end term load vector of the frequency response equation into Taylor series form:

[0057] ; where represents the order of the Taylor expansion of the coefficient matrix, represents the index of the order of the Taylor expansion of the coefficient matrix, represents the i-th load frequency, represents the frequency at the basis vector expansion point, represents the factorial of, represents the order derivative of the coefficient matrix, represents the displacement vector at the i-th load frequency, represents the order of the Taylor expansion of the load vector, represents the index of the order of the Taylor expansion of the load vector, represents the factorial of, represents the order derivative of the load vector at the i-th load frequency.

[0058] S300, through iterative correction of the initial basis vectors, combine the correction terms for orthogonalization processing, and output the orthogonalized basis vector matrix;

[0059] In one embodiment, step S300 specifically includes:

[0060] S310, perform normalization processing on the initial basis vectors;

[0061] S320, generate subsequent basis vectors by dynamically superimposing the high-order derivative terms of the stiffness matrix and the load, and introduce correction terms to adjust the direction of the basis vectors;

[0062] Specifically, the correction term is: ; where Indicates a correction term, Indicates the starting index of the correction term, Indicates the index of the basis vector, Indicates the truncation order of the Taylor expansion, Indicates the product index variable, Indicates the product operator.

[0063] Specifically, = 1 or 2, product order , matrix Indicates the matrix of to rows, to column block.

[0064] S330, Use the Gram - Schmidt orthogonalization method to gradually eliminate the projection components between the basis vectors and normalize the orthogonalized basis vectors;

[0065] S340, Arrange all the normalized and orthogonal basis vectors in the generation order to form an orthogonal basis matrix.

[0066] S400, Project the orthogonalized basis vector matrix onto the frequency response equation, construct the low - dimensional frequency response equation and the low - dimensional adjoint equation, and solve for the reduced - order displacement response vector and the reduced - order adjoint variable;

[0067] Specifically, the low - dimensional frequency response equation is: ; where, Indicates the i - th load frequency, Indicates the reduced - order mass matrix, Indicates the reduced - order damping matrix, Indicates the reduced - order stiffness matrix, Indicates the reduced - order displacement response vector at the i - th load frequency, Indicates the reduced - order load vector at the i - th load frequency.

[0068] Specifically, the low - dimensional adjoint equation is: ; where, Indicates the reduced - order adjoint vector at the i - th load frequency, Indicates the reduced - order weighted matrix, Indicates the conjugate vector of the reduced - order displacement response at the i - th load frequency.

[0069] , , , , .

[0070] In one embodiment, the ability of the present invention to solve the state equation and the adjoint equation in a wide frequency range was 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 following is the specific process:

[0071] Step 1: To evaluate the solution of the frequency response and plot the frequency response diagram, the amplitude of the structure was defined and the adjoint vector The 2-norm of is:

[0072] Step 2: To evaluate the accuracy of various reduced-order models relative to the full model, a relative error index was introduced:

[0073] ; where ROM and FULL represent the solutions obtained from the reduced-order model and the full model, respectively.

[0074] Step 3: For the frequency response problem of the wide-band guided wave structure of the present invention, using Figure 3 The topology of the bi-material structure of the elastic material and the viscoelastic material shown in (b), the accuracy of various reduced-order methods was tested, and the solution model was Figure 3 The elastic wave conduction optimization problem shown in (a) was given, and the accuracy, efficiency and stability of different reduced-order methods were given.

[0075] Step 4: The SOAR_k0, SOAR_k2 and HOMMG methods were used to solve the state equation. For ease of description, considering ( ) of ( )-order expansion of the HOMMG method is denoted as . The considered frequency range is , and the frequency interval was discretized into 401 frequency points for sweep frequency analysis.

[0076] Step 5: To further evaluate the solution accuracy of various reduced-order methods, the present invention used the frequency response displacement calculated by the full-order model as the reference solution, and calculated the relative errors of various reduced-order methods in solving the state equation when the number of basis vectors in the subspace was 40, 80 and 160, as shown in Figure 4 . It can be clearly seen from the table that the HOMMG method proposed in this paper has smaller solution errors compared with the traditional SOAR_k0 and SOAR_k2 methods, and can obtain high-precision solutions in a relatively large frequency band near the expansion point. In this regard, the ranges in which various reduced-order methods can obtain high-precision solutions were compared and analyzed in this paper. An acceptable error limit was set, and it was set that the solution relative error was less than The frequency range is the solvable bandwidth of the reduction method, as shown in Table 1. The proposed HOMMG method in this paper has a larger solvable bandwidth compared with the traditional reduction methods. Moreover, as the number of basis vectors increases, the solvable ranges of various reduction methods gradually expand. Among them, the solvable range of the HOMMG method can quickly cover the full frequency range, and accurate solutions for the entire frequency band can be achieved with fewer basis vectors, which will greatly reduce the computational time. In addition, the proposed HOMMG method in this paper expands the left - hand side term of the state equation into the form of a Taylor series, and explores the influence of high - order expansion on the solution accuracy and bandwidth of the reduction method. Compared with , it has a smaller relative error and a larger solvable bandwidth, which also proves the effectiveness of high - order expansion.

[0077] Table 1 Comparison table of solvable bandwidths for solving the state equation by different reduced - order models

[0078]

[0079] Step 6: In this paper, a test model is used to evaluate the accuracy of various reduction methods for solving the adjoint equation. The relative errors and solvable bandwidths of various reduction methods are as Figure 5 and shown in Table 2. It can be seen from the table that the proposed HOMMG method in this paper still has high accuracy and stability when solving the adjoint equation. By means of high - order expansion of the left - hand side term of the adjoint equation, the HOMMG method can effectively improve the solution accuracy of the reduction method, and high - precision solutions within the full frequency band can be obtained through a small number of basis vectors. Moreover, it is worth noting that when solving the adjoint equation, in this paper, the right - hand side term of the adjoint equation is also expanded into the form of a Taylor series, and the influence of high - order expansion of the right - hand side term on the solution accuracy of the HOMMG method for solving the adjoint equation is explored. The test results show that compared with , it can greatly improve the solution accuracy near the expansion point and obtain local high - precision solutions near the expansion point.

[0080] Table 2 Comparison table of solvable bandwidths for solving the adjoint equation by different reduced - order models

[0081]

[0082] S500, taking the weighted logarithmic sum minimization of the sum of squares of the reduced - order displacement response vectors in the wide frequency band as the objective function, establishing a topology optimization model combined with volume constraints, and calculating the sensitivity of the objective function to the design variables based on the reduced - order adjoint variables;

[0083] Specifically, the topology optimization model is: ; where, represents the maximization column of the objective function, Denote the unit pseudo - density design variable, with the value range [0, 1]. Denote the objective function. Denote the total number of load frequencies. Denote the i - th load frequency. Denote the objective function at the i - th load frequency. Denote the displacement response vector at the i - th load frequency. Denote the degree - of - freedom selection matrix. Denote the conjugate vector. Denote the number of target degrees of freedom. Denote the total number of elements. Denote the element number. Denote the density of the e - th element after filtering and projection. Denote the volume of the e - th element. Denote the total volume of the design domain. Denote the upper limit of the volume constraint. Denote the dynamic matrix. Denote the load vector at the i - th load frequency.

[0084] Specifically, the sensitivity calculation rule is as follows: ; Among them, , Denote the derivative of the objective function with respect to the unit pseudo - density design variable. Denote the objective function. Denote the unit pseudo - density design variable, with the value range [0, 1]. Denote the number of discretized frequency points in the frequency interval. Denote the i - th load frequency. Denote the single - frequency response index. Denote the derivative of the objective function with respect to the unit pseudo - density design variable at the i - th load frequency. Denote taking the real part of a complex number. Denote the transpose of a matrix or vector. Denote the adjoint variable at the i - th load frequency. Denote the dynamic matrix. Denote the derivative of the dynamic matrix with respect to the unit pseudo - density design variable. Denote the displacement response vector at the i - th load frequency.

[0085] S600, adopt the gradient optimization algorithm to iteratively update the design variables until the convergence condition is met, and obtain the optimal topological configuration.

[0086] Specifically, for the optimization process of giving the optimal topological configuration, the following are the specific steps:

[0087] Step 1: For the design domain shown in Figure 3 (a), topological optimization of the elastic wave conduction problem is carried out, and the optimization goal is to maximize the amplitude at the dotted line of the right boundary. The design domain is discretized using the mesh, and the design variable is the element pseudo-density. The filtering radius is 10 times the element size, and the parameter in the Heaviside filter has an initial value of 1, which doubles every 50 steps, i.e., , and the maximum value is . The number of iteration steps is 440. The optimization problem is a two-material layout optimization, and Table 3 lists the material properties of the two materials. Among them, Material 1 is an elastic model with high stiffness and low damping, and the structural damping adopts the Rayleigh damping model. Material 2 is a viscoelastic material, and the GHM model is used for the shear model. Five terms are taken for the series expansion. The specific model parameters are shown in Table 4, and the relationship curves of the shear modulus, damping ratio and frequency are as shown in Figure 6 .

[0088] In the optimization model, the volume fraction of Material 1 is set to 0.4, and topological optimization design is carried out for the low-frequency problem ( )( ). During the calculation process, 401 frequency points are used to uniformly discretize the considered frequency range.

[0089] Table 3 Material properties of elastic and viscoelastic materials

[0090]

[0091] Table 4 Model parameter table of GHM viscoelastic material

[0092]

[0093] Step 2: The topological optimization results of the low-frequency problem are shown in Figure 7 . It can be seen that the distributions of the elastic material and the viscoelastic material are in a grid-like distribution and converge to the central point on the right side. In addition, grid structures are arranged in the upper left corner and the lower right corner of the optimization result to isolate wave propagation, so that the energy of the wave converges to the central point on the right side. The swept-frequency curve of this optimization result is shown in Figure 8 . The blue line represents the initial solution frequency response curve; the yellow line represents the frequency response curve of the optimization result. It can be seen that the frequency response amplitude after optimization is improved in the full frequency range, and the objective function value of this optimization result is 836.8713.

[0094] Step 3: The topological optimization configurations based on various reduced-order methods are shown in Figure 9As shown. Among them, the traditional order reduction methods (SOAR_k0 and SOAR_k2) have large prediction errors for the dynamic performance of materials because they ignore the high-order expansion terms of the complex shear modulus of viscoelastic materials. It can be seen from the table that both methods need to construct 30 basis vector subspaces to converge to a stable topology optimization result, and there is still a large difference from the optimization result of the full-order model. The HOMMG method can obtain a stable topology configuration when the number of basis vectors is 20. In particular, the optimization result of the HOMMG_k4f2 method is highly consistent with the full-order model. Therefore, by performing high-order expansion on the state equation and the adjoint equation using the HOMMG method, constructing a high-order Krylov subspace method, a high-precision approximate solution and a topology optimization configuration can be obtained using fewer basis vectors, greatly reducing the time-consuming of the frequency sweep calculation and reducing the computational cost of the frequency response topology optimization.

[0095] In one embodiment, the gradient optimization algorithm is the moving asymptote method.

[0096] The present invention has the following beneficial effects:

[0097] (1) High-precision and wide-band adaptation: Through the high-order Krylov subspace method (HOMMG), introducing high-order Taylor expansion and correction terms, the accuracy and frequency-band adaptability of the frequency response analysis of viscoelastic materials are significantly improved, overcoming the problems of narrow frequency band and large error caused by the traditional second-order method ignoring nonlinear terms.

[0098] (2) Improving computational efficiency: The HOMMG method only requires a small number of basis vectors in topology optimization to achieve an optimization result consistent with the full-order model, while reducing the frequency sweep calculation cost by more than 80%, solving the defect that the traditional order reduction method requires a large number of basis vectors and the optimization result is unstable.

[0099] (3) Multi-frequency point collaborative optimization: Taking the logarithmically weighted multi-frequency point displacement response as the objective function, combined with the high-order correction of the adjoint equation, global optimization of the dynamic performance within a wide frequency band is realized. The optimized configuration (such as a grid structure) effectively converges wave energy, and the frequency response amplitude is improved throughout the frequency band, which is applicable to complex engineering scenarios such as submarines and MEMS devices.

[0100] The above embodiments are only used to describe the preferred mode of the present invention, and do not limit the scope of the present invention. Without departing from the design spirit of the present invention, various deformations, variations, modifications, and substitutions made by those of ordinary skill in the art to the technical solutions of the present invention should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A wide - band topology optimization method for viscoelastic structures based on the high - order Krylov subspace method, characterized in that, Including: Establish the frequency response equation and adjoint equation of the viscoelastic structure; At the selected frequency expansion point, expand the left-end matrix and right-end load vector of the frequency response equation into high-order Taylor series respectively, and generate the initial basis vectors after truncating the high-order terms; Iteratively correct the initial basis vectors, perform orthogonalization processing in combination with the correction terms, and output the orthogonalized basis vector matrix; Project the orthogonalized basis vector matrix onto the frequency response equation, construct the low-dimensional frequency response equation and low-dimensional adjoint equation, and solve the reduced-order displacement response vector and reduced-order adjoint variables; Taking the weighted logarithmic sum minimization of the sum of squares of the reduced-order displacement response vectors in the wide frequency band as the objective function, establish a topology optimization model in combination with volume constraints, and calculate the sensitivity of the objective function to the design variables based on the reduced-order adjoint variables; Adopt the gradient optimization algorithm to iteratively update the design variables until the convergence condition is met to obtain the optimal topology configuration.

2. The broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method according to claim 1, characterized in that The frequency response equation is: ; where, 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 load frequency.

3. The broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method according to claim 1, characterized in that, The adjoint equation is: ; among which, 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 load frequency.

4. The broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method according to claim 1, characterized in that By iteratively correcting the initial basis vectors, performing orthogonalization processing in combination with the correction terms, and outputting the orthogonalized basis vector matrix, including: Normalize the initial basis vectors; Generate subsequent basis vectors by dynamically superimposing the high-order derivative terms of the stiffness matrix and the load, and introduce correction terms to adjust the direction of the basis vectors; Use the Gram-Schmidt orthogonalization method to gradually eliminate the projection components between the basis vectors, and normalize the orthogonalized basis vectors; Arrange all the normalized and orthogonal basis vectors in the generation order to form an orthogonal basis matrix.

5. The broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method according to claim 4, characterized in that The correction term is: ; among which, represents the correction term, represents the starting index of the correction term, represents the index of the basis vector, represents the truncation order of the Taylor expansion, represents the product index variable, represents the product operator.

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

7. The broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method according to claim 1, wherein , The topology optimization model is: ; among which, represents the maximization formulation of the objective function, represents the unit pseudo-density design variable, with the value range [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 elements, represents the element number, represents the density of the e-th element after filtering and projection, represents the volume of the e-th element, represents the total volume of the design domain, represents the upper limit of the volume constraint, represents the dynamic matrix, represents the load vector at the i-th load frequency.

8. The broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method according to claim 1, characterized in that , The sensitivity calculation rule is as follows: ; among them, , represents the derivative of the objective function with respect to the element pseudo-density design variable, represents the objective function, represents the element pseudo-density design variable, and its value range is [0, 1], 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 derivative of the objective function with respect to the element pseudo-density design variable at the i-th load frequency, 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 the 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 broadband topology optimization method for viscoelastic structures based on the high-order Krylov subspace method according to claim 1, characterized in that , The gradient optimization algorithm is the moving asymptotes method.

Images