Structure vibration reduction topological optimization design method based on anti-resonance frequency constraint

Abstract

The invention discloses a structural vibration reduction topological optimization design method based on anti-resonant frequency constraint. The method comprises the following steps: constructing a finite element model corresponding to structure parameters; obtaining an initial structure and an optimization column formula according to the finite element model; obtaining an objective function value through finite element analysis; obtaining an anti-resonance frequency correlation characteristic value through generalized asymmetric characteristic value analysis; and the design variables are updated in combination with optimization column constraints, and a topological optimization design configuration is obtained after convergence judgment. According to the method, explicit calculation and precise control of the dynamic compliance anti-resonant frequency are achieved, optimization and advanced convergence are avoided through anti-resonant frequency constraint, a clear vibration reduction topological configuration can be obtained under high-frequency excitation without additional mass, the vibration reduction performance and the static force bearing capacity of the structure are both considered, and the method is suitable for large-scale popularization and application. And the design requirements for light weight and high vibration attenuation of the structure in the fields of aerospace and the like are met.

Description

Technical Field

[0001] This application relates to the field of structural vibration reduction topology optimization technology, and in particular to a structural vibration reduction topology optimization design method based on anti-resonance frequency constraints. Background Technology

[0002] With the continuous development of aerospace equipment technology, the structural design of large military and civilian aircraft, launch vehicles, aerospace engines, and other equipment faces increasingly stringent requirements for vibration reduction performance and lightweighting. These types of equipment operate in vibration environments for extended periods, resulting in significant noise and structural fatigue problems caused by mechanical vibration. Structural topology optimization, as an important tool in the initial conceptual design phase, can leverage the structural design potential by adjusting the material distribution within the design domain, becoming a key method for improving the vibration reduction performance of aerospace structures. Furthermore, anti-resonance theory, due to its ability to suppress vibrations at specific frequencies, is gradually being applied to the field of structural vibration reduction design.

[0003] In current methods, anti-resonance theory is mainly applied to the parameter optimization of active vibration reduction devices such as dynamic vibration absorbers and anti-resonance vibration isolators. It artificially generates anti-resonance frequencies by introducing additional mass-stiffness substructures to suppress the vibration of the main structure. Meanwhile, structural dynamic compliance topology optimization, as a representative method of vibration reduction optimization design, mostly aims to minimize dynamic compliance. It uses gradient algorithms to carry out dynamic response topology optimization and attempts to combine anti-resonance theory to improve the vibration reduction effect. When solving for the anti-resonance frequency, it mostly relies on graphical methods with frequency resolution. Some eigenvalue methods have also been tried for the calculation of the anti-resonance frequency.

[0004] However, these methods have many shortcomings: 1. Dynamic response topology optimization with the goal of minimizing dynamic compliance will have a large number of intermediate design variables when the excitation frequency is higher than the fundamental frequency of the initial structure, and a clear topology configuration cannot be obtained. This is because traditional dynamic compliance topology optimization based on gradient algorithm gets stuck at the anti-resonance point in the early stage of optimization and converges prematurely.

[0005] 2. There is a lack of explicit calculation methods for anti-resonance frequencies suitable for topology optimization of dynamic compliance of structures. Traditional eigenvalue methods not only cannot explicitly solve for anti-resonance frequencies, but also have difficulty adapting to the sensitivity calculation of large-scale design variables in dynamic compliance topology optimization.

[0006] 3. The vibration reduction effect of active vibration damping devices is highly positively correlated with the added mass, requiring the introduction of additional mass into the main structure, occupying design space, and making it difficult to meet the design requirements of the aerospace field for extreme lightweight structures.

[0007] Therefore, how to improve the vibration reduction performance of a structure and achieve lightweight structural design without introducing additional mass is an urgent problem to be solved in the field. Summary of the Invention

[0008] In view of this, the structural vibration reduction topology optimization design method based on anti-resonance frequency constraints provided in this application can avoid premature convergence of the optimization through anti-resonance frequency constraints, and obtain a clear vibration reduction topology configuration under high-frequency excitation without adding mass. It balances structural vibration reduction performance and static load-bearing capacity, and is suitable for the design requirements of lightweight and high vibration reduction in fields such as aerospace. The structural vibration reduction topology optimization design method based on anti-resonance frequency constraints provided in this application is implemented as follows: This application provides a structural vibration reduction topology optimization design method based on anti-resonance frequency constraints, including: Construct a finite element model corresponding to the structural parameters, and obtain the initial structure and optimization formula based on the finite element model; Finite element analysis is performed on the optimized formulation and the initial structure to obtain the objective function value; Generalized asymmetric eigenvalue analysis is performed on the dynamic compliance frequency response function corresponding to the objective function value to obtain anti-resonance frequency related eigenvalues. The anti-resonance frequency is obtained by tracking the eigenvector corresponding to the anti-resonance frequency related eigenvalues. Sensitivity analysis is performed on the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data. The design variables corresponding to the sensitivity data and the constraints in the optimization formula are updated to obtain the updated design variables. The iteration results corresponding to the updated design variables and the constraints of the optimized formula are subjected to convergence judgment. If the convergence judgment result meets the preset convergence threshold, the topology optimization design configuration is obtained.

[0009] In some embodiments, constructing a finite element model corresponding to the structural parameters, and obtaining an initial structure and optimized formulation based on the finite element model, includes: The structural parameters are integrated and modeled to obtain a parametric finite element model, wherein the structural parameters include geometric parameters, material parameters, mesh generation parameters, and boundary condition parameters; Information extraction processing is performed on the parametric finite element model to obtain system information, which includes system stiffness matrix, mass matrix, element node information and coordinate data. The system information is initialized using design variable fields to obtain the initial structure; The initial structure is subjected to frequency response analysis to obtain the initial anti-resonance frequency; Based on the constraints of the initial anti-resonance frequency and the dynamic relaxation factor, the static compliance and dynamic compliance are weighted and normalized to obtain the optimized formulation.

[0010] In some embodiments, performing finite element analysis on the optimized formulation and the initial structure to obtain the objective function value includes: The initial structure is subjected to finite element analysis to obtain the current static compliance and the current dynamic compliance. The current dynamic compliance is converted to an absolute value to obtain a non-negative dynamic compliance. Based on the weighted normalization rule of the optimized column, the current static compliance and non-negative dynamic compliance are subjected to weight factor matching processing to obtain weight parameters; The current static compliance and the non-negative dynamic compliance are weighted and normalized based on the weight parameters to obtain the objective function value.

[0011] In some embodiments, the sensitivity analysis processing of the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data includes: The sensitivity of the objective function is obtained by performing projection variable sensitivity derivation on the current static compliance and non-negative dynamic compliance corresponding to the objective function. The design variable sensitivity is derived by performing a design variable sensitivity derivation process on the anti-resonance frequency corresponding to the constraint function to obtain the constraint function sensitivity; The sensitivity of the objective function and the sensitivity of the constraint function are integrated to obtain sensitivity data.

[0012] In some embodiments, the generalized asymmetric eigenvalue analysis of the dynamic compliance frequency response function corresponding to the objective function value to obtain anti-resonance frequency-related eigenvalues ​​includes: Obtain the steady-state equation of motion; The dynamic compliance characteristics of the structure and the steady-state motion equations are analyzed and combined to obtain the fitting equations. The fitting equations are processed by introducing a null space matrix to obtain intermediate equations; The intermediate equation is transformed to obtain asymmetric generalized eigenvalues; The asymmetric generalized eigenvalues ​​are solved and filtered to obtain anti-resonance frequency related eigenvalues.

[0013] In some embodiments, the step of tracking the feature vector corresponding to the anti-resonance frequency-related feature value to obtain the anti-resonance frequency includes: The modal confidence criterion is extended to obtain the eigenvector tracking criterion corresponding to the anti-resonance frequency; The anti-resonance frequency related feature values ​​are subjected to feature vector extraction processing to obtain a feature vector set; The feature vector set and the first target feature vector are compared for similarity to obtain the modal confidence criterion value. The modal confidence criterion values ​​are filtered to obtain the second target feature vector; The second target feature vector is subjected to frequency correlation processing to obtain the anti-resonance frequency.

[0014] This application provides a structural vibration reduction topology optimization design device based on anti-resonance frequency constraints, comprising: The construction module is used to construct the finite element model corresponding to the structural parameters, and to obtain the initial structure and optimized formula based on the finite element model. The processing module is used to perform finite element analysis on the optimized formula and the initial structure to obtain the objective function value; The processing module is also used to perform generalized asymmetric eigenvalue analysis on the dynamic compliance frequency response function corresponding to the objective function value to obtain anti-resonance frequency related eigenvalues. The processing module is also used to track the feature vector corresponding to the anti-resonance frequency related feature value to obtain the anti-resonance frequency; The processing module is further configured to perform sensitivity analysis on the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data; The processing module is also used to update the design variables corresponding to the sensitivity data and the constraints in the optimization formula to obtain the updated design variables. The judgment module is also used to perform convergence judgment on the iteration results corresponding to the updated design variables and the constraints of the optimization formula. If the convergence judgment result meets the preset convergence threshold, the topology optimization design configuration is obtained.

[0015] The computer device provided in this application includes a memory and a processor. The memory stores a computer program that can run on the processor. When the processor executes the program, it implements the method described in this application.

[0016] The computer-readable storage medium provided in this application embodiment stores a computer program thereon, which, when executed by a processor, implements the method described in this application embodiment.

[0017] This application provides a structural vibration reduction topology optimization design method based on anti-resonance frequency constraints. It constructs a finite element model corresponding to the structural parameters, obtains the initial structure and optimization formula from the finite element model, obtains the objective function value through finite element analysis, obtains the anti-resonance frequency-related eigenvalues ​​through generalized asymmetric eigenvalue analysis, obtains the anti-resonance frequency through eigenvector tracking, obtains sensitivity data through sensitivity analysis, updates the design variables by combining the optimization formula constraints, and obtains the topology optimization design configuration after convergence judgment. This method avoids premature convergence of the optimization through anti-resonance frequency constraints, obtains a clear vibration reduction topology configuration under high-frequency excitation without adding mass, balances structural vibration reduction performance and static load-bearing capacity, and adapts to the design requirements of lightweight and high vibration reduction in aerospace and other fields, solving the technical problems mentioned in the background art. Attached Figure Description

[0018] To more clearly illustrate the technical solutions of the embodiments of this application, the drawings used in the description of the embodiments of this application or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0019] Figure 1 A schematic diagram illustrating the implementation process of a structural vibration reduction topology optimization design method based on anti-resonance frequency constraints provided in this application embodiment; Figure 2 This application provides a schematic diagram of an implementation process for obtaining an initial structure and optimizing the column format. Figure 3 A schematic diagram of a structural vibration reduction topology optimization design device based on anti-resonance frequency constraints provided in this application embodiment; Figure 4 This is a schematic diagram of the initial structure of a planar cantilever beam provided in an embodiment of this application; Figure 5 A schematic diagram of the initial dynamic compliance frequency response curve of a planar cantilever beam structure provided in an embodiment of this application; Figure 6 A schematic diagram of an optimal configuration at an excitation frequency of 800 rad / s provided for an embodiment of this application; Figure 7 A schematic diagram of an optimal configuration without anti-resonance frequency constraint at an excitation frequency of 800 rad / s provided in this application embodiment; Figure 8 A schematic diagram of an optimal configuration at an excitation frequency of 2700 rad / s provided for an embodiment of this application; Figure 9A schematic diagram of an optimal configuration without anti-resonance frequency constraint at an excitation frequency of 2700 rad / s provided in this application embodiment; Figure 10 A schematic diagram of an optimal configuration at an excitation frequency of 3600 rad / s provided for an embodiment of this application; Figure 11 A schematic diagram of an optimal configuration without anti-resonance frequency constraint at an excitation frequency of 3600 rad / s provided in this application embodiment; Figure 12 The dynamic compliance frequency response curves of an optimal configuration under different excitation frequencies are provided for embodiments of this application. Detailed Implementation

[0020] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0021] The following description of some technologies involved in the embodiments of this application is provided to aid understanding and should be considered merely exemplary. Therefore, those skilled in the art should recognize that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this application. Similarly, for clarity and brevity, some descriptions of well-known functions and structures are omitted in the following description.

[0022] Figure 1 This is a schematic diagram illustrating the implementation flow of a structural vibration reduction topology optimization design method based on anti-resonance frequency constraints provided in an embodiment of this application, including steps 101 to 107. Wherein, Figure 1 This is merely one execution order shown in the embodiments of this application, and does not represent the only execution order of a structural vibration reduction topology optimization design method based on anti-resonance frequency constraints. Where the final result can be achieved, Figure 1 The steps shown can be performed in parallel or in reverse order.

[0023] Step 101: Construct the finite element model corresponding to the structural parameters, and obtain the initial structure and optimization formula based on the finite element model.

[0024] In this embodiment, the structural parameters include geometric parameters, material parameters, mesh generation parameters, and boundary condition parameters. First, the structural parameters are integrated and modeled to create a parametric finite element model using a programming script. Information extraction is performed on the parametric finite element model to obtain system information including the system stiffness matrix, mass matrix, element node information, and coordinate data. Initial values ​​for design variables are uniformly set for all elements within the design domain. The extracted system information is then combined with design variable field initialization processing. After reconstructing and coupling the finite element model, an initial structure is obtained. Frequency response analysis is performed on the initial structure, and a dynamic compliance frequency response curve is plotted. The position of the excitation frequency in the frequency response curve is observed to obtain the initial anti-resonance frequency within the same resonance interval. Based on the initial anti-resonance frequency, corresponding anti-resonance frequency constraints are set. Simultaneously, a dynamic relaxation factor is introduced, and the static and dynamic compliance are weighted and normalized to establish an optimized formula. Among them, the optimization formula sets a lower limit for design variables to avoid matrix singularity problems, converts dynamic compliance into absolute value form, avoids optimization failure caused by non-positive definite dynamic stiffness matrix under high frequency harmonic excitation, and the weighted normalization rule in the optimization formula includes weight factors for adjusting the static and dynamic performance of the structure, which can be adjusted according to actual design requirements.

[0025] Specifically, the optimized formula is as follows: .

[0026] Where find means to search, and x represents the topology design variable vector; represents the topology design variables of the i-th element; N represents the total number of topology design variables, which is the same as the number of design domain elements; T represents the transpose sign; st represents the constraint condition. This represents the volume of the i-th unit; and These represent the volume fraction and the total volume of the structure, respectively. This represents the lower bound of the design variables set to avoid matrix singularity problems. min represents minimization; f represents a weighting function consisting of a linear combination of the static and dynamic performance of the structure. Considering that the actual structure has a certain static bearing capacity and to avoid structural separation, it is necessary to introduce static compliance as a weighted objective, and convert the weighted static compliance and weighted dynamic compliance into dimensionless forms respectively; α represents a weighting factor, and a higher value means that more importance is attached to the dynamic performance of the structure. Indicates the static compliance of the structure. It is the structural displacement under static load, which can be solved by solving the static equilibrium equations. get; S Represents static load; and These refer to the dynamic compliance and static compliance of the initial structural design. Under high-frequency harmonic excitation, the dynamic stiffness matrix may be non-positive definite, in which case the dynamic compliance... Since the value is negative, minimizing it will result in failure (tending towards negative infinity). Therefore, the dynamic compliance function is set to its absolute value. ; d Represents dynamic load; The frequency represents the l-th anti-resonance frequency in the dynamic compliance function. The inequality in which this frequency is located introduces an anti-resonance frequency constraint, the purpose of which is to avoid... The problem of premature convergence caused by the early overlap of Ω; This represents anti-resonance; simultaneously, based on the fact that the excitation frequency Ω is located in the elastic region of the initial design (i.e., to the right of the anti-resonance frequency, there exists...), ) and the inertial region (i.e., located to the left of the anti-resonance frequency, where there exists Depending on the position of ), D is defined as follows: .

[0027] Furthermore, η(k) is the dynamic relaxation factor, where k represents the current iteration step, and has the following power function form: .

[0028] in, It is a power exponent; =5 is a suitable parameter obtained through extensive numerical experiments. During the iteration process, the dynamic relaxation factor exhibits the following characteristics: initially remaining constant to prevent the excitation frequency from coinciding with the anti-resonance frequency, and then gradually approaching 1 to transform the constraint into an active constraint. =Ω. The dynamic relaxation factor is chosen in this way to ensure that the initial point is within the feasible region, avoiding meaningless iterations outside the feasible region, thereby improving the stability and efficiency of topology optimization design, and achieving the goal of ensuring that the optimal configuration's dynamic compliance falls within the anti-resonance point at a given excitation frequency.

[0029] Step 102: Perform finite element analysis on the optimized formula and the initial structure to obtain the objective function value.

[0030] In this embodiment, static / dynamic finite element analysis is performed on the initial structure to calculate the current static compliance and current dynamic compliance. The current dynamic compliance is then converted to an absolute value to obtain a non-negative dynamic compliance, avoiding interference from negative dynamic compliance in subsequent calculations. Based on the weighted normalization rules in the optimization formula, corresponding weighting factors are matched to the current static compliance and non-negative dynamic compliance to obtain weighting parameters. The values ​​of these weighting parameters determine the degree of emphasis in optimizing the static load-bearing capacity and dynamic vibration reduction performance of the structure. Based on these weighting parameters, a weighted normalization calculation is performed on the current static compliance and non-negative dynamic compliance to obtain the objective function value. In subsequent iterative optimization processes, the structure corresponding to the updated design variables is used as the new initial structure, and the finite element analysis and calculation steps are repeated to obtain the objective function value for the corresponding number of iterations.

[0031] Step 103: Perform generalized asymmetric eigenvalue analysis on the dynamic compliance frequency response function corresponding to the objective function value to obtain the anti-resonance frequency related eigenvalues.

[0032] In this embodiment, the undamped forced vibration under harmonic load excitation is first discretized using finite element method to establish a steady-state motion equation. The dynamic compliance characteristics of the structure under anti-resonance frequency excitation are analyzed and processed. After normalizing the external force vector, it is combined with the steady-state motion equation to obtain the fitting equation. The fitting equation is processed by introducing a null space matrix, and the right-hand side of the equation is eliminated to obtain an intermediate equation. The intermediate equation is transformed to obtain an asymmetric generalized eigenvalue problem. The asymmetric generalized eigenvalue problem is solved, and positive real eigenvalues ​​that do not contain system eigenvalues ​​are selected. These positive real eigenvalues ​​are the anti-resonance frequency related eigenvalues, and the eigenvalues ​​are the squares of the anti-resonance frequency.

[0033] Specifically, the steady-state motion equation of undamped forced vibration under harmonic load excitation, discretized using the finite element method, is as follows: .

[0034] Where F and U are the magnitude vectors of the harmonic load and displacement response, respectively. K Here is the structural stiffness matrix. M This is the structural quality matrix.

[0035] For the structural dynamic compliance under anti-resonance frequency excitation, the following equation holds: .

[0036] The steady-state response R of a certain degree of freedom of the structure can be expressed as a linear combination of displacement magnitude vectors, i.e. , α yes The column vectors have 1 degree of freedom in the area of ​​interest and 0 degrees of freedom elsewhere; the external force vectors can be... Consider as displacement amplitude The coefficient vector, for Normalization is performed to obtain the vector. Combining this with the steady-state equations of motion, we obtain the following equation:

[0037] This can then be rewritten as:

[0038] In the formula, , , .

[0039] To eliminate the right-hand side of the equation, we introduce... The null space matrix has .

[0040] Where T satisfies the relation is an (n+1)×n matrix. Null() represents the operator for solving the null space of the matrix.

[0041] Multiply both sides of the above equation by the left side. ,get: .

[0042] The above equation can be rewritten as an asymmetric generalized eigenvalue problem: .

[0043] in, , By solving the asymmetric generalized eigenvalue problem shown in the above equation, the anti-resonance frequency of the dynamic compliance function can be obtained. Due to the asymmetry of the matrix, its eigenvalues ​​include real numbers, complex numbers, and infinity. Only the positive real eigenvalues ​​that do not contain system eigenvalues ​​are the desired result, i.e., the square of the dynamic compliance anti-resonance frequency.

[0044] Step 104: Track the eigenvectors corresponding to the relevant eigenvalues ​​of the anti-resonance frequency to obtain the anti-resonance frequency.

[0045] In this embodiment, the modal confidence criterion is first extended to obtain a feature vector tracking criterion applicable to asymmetric eigenvalue problems; feature vector extraction is performed on the anti-resonance frequency related eigenvalues ​​to obtain the feature vector set for the current iteration; the feature vector set for the current iteration is compared with the target feature vector of the previous iteration to calculate the modal confidence criterion value; in the first iteration, the feature vector corresponding to the initial anti-resonance frequency is used as the first target feature vector; the modal confidence criterion value is filtered to select the feature vector with the highest similarity as the target feature vector for this iteration; frequency correlation processing is performed on the target feature vector to determine and obtain the anti-resonance frequency for the current iteration.

[0046] Specifically, the expression for the extended Modal Confidence Criterion (MAC) function is as follows: .

[0047] Where k represents the current iteration step; It is the target feature vector corresponding to the l-th order anti-resonance frequency in the previous iteration; It is the s-th feature vector extracted in the current iteration, which originates from the feature vector matrix formed by the feature vectors corresponding to the anti-resonance frequencies in the feature value sequence of the current iteration.

[0048] The MAC value varies between 0 and 1. The closer the MAC value is to 1, the more similar the two feature vectors are. The feature vector corresponding to the maximum MAC value is the target feature vector, that is, the target feature vector corresponding to the l-th order anti-resonance frequency in the current iteration.

[0049] Step 105: Perform sensitivity analysis on the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data.

[0050] In this embodiment, for the current static compliance and non-negative dynamic compliance corresponding to the objective function, the adjoint method is used to derive the sensitivity of the projected variables to obtain the sensitivity of the objective function; for the anti-resonance frequency corresponding to the constraint function, the sensitivity of the design variables is derived by combining the asymmetric eigenvalue problem to obtain the sensitivity of the constraint function; the sensitivity of the objective function and the sensitivity of the constraint function are integrated to obtain the comprehensive sensitivity data used for subsequent design variable updates, which is the sensitivity data.

[0051] Specifically, the sensitivity of static and dynamic compliance amplitudes to projected variables is derived using the adjoint method, and the derivation formula for the sensitivity of the anti-resonance frequency to design variables is as follows: .

[0052] in, For projection variables, For the filtered design variables, For eigenvalues, and All of these are eigenvectors associated with the eigenvalue problem.

[0053] Step 106: Update the design variables corresponding to the sensitivity data and the constraints in the optimization formula to obtain the updated design variables.

[0054] In this embodiment, to address numerical instability issues such as checkerboard patterns and grid dependence that are prone to occur during the optimization of design variables, a three-field filtering method consisting of density filtering, sensitivity filtering, and projection methods is adopted to obtain stabilization rules. Based on sensitivity data and all constraints in the optimization formula, such as anti-resonance frequency constraints, volume fraction constraints, and lower limit constraints of design variables, the MMA optimization algorithm is used to iteratively update the design variables to obtain initially updated design variables. The initially updated design variables are then substituted into the stabilization rules for verification and optimization to ensure that the design variables meet all constraints of the optimization formula and are free from the aforementioned numerical instability issues, ultimately yielding the updated design variables.

[0055] Step 107: Perform convergence judgment processing on the iteration results corresponding to the updated design variables and the constraints of the optimized formula. If the convergence judgment result meets the preset convergence threshold, the topology optimization design configuration is obtained.

[0056] In this embodiment, the maximum change in design variables between two consecutive iterations is calculated, and the change is compared with a preset convergence threshold. Simultaneously, the current iteration step count is recorded. If the maximum change in design variables is less than the preset convergence threshold, or the current iteration step count reaches the preset maximum iteration step count, the convergence judgment result is determined to meet the requirements. The updated design variables are then transformed to obtain the final topology optimization design configuration. If the convergence judgment result does not meet the requirements, the structure corresponding to the updated design variables is used as the new initial structure, and the process returns to step 102 of this application to continue the subsequent iterative optimization process until the iteration result meets the convergence judgment requirements.

[0057] In this application, the preset convergence threshold is set to be less than 1 × 10⁻⁶ for the maximum change in design variables between two consecutive iterations. - ³, the preset maximum number of iteration steps is set to 210. By setting the parameter, the computation time of iterative optimization can be controlled while ensuring the accuracy of the topology optimization design configuration.

[0058] This application's embodiment solves the core problems in existing technologies by constructing a complete closed-loop optimization process—including finite element model construction, objective function calculation, anti-resonance frequency correlation eigenvalue analysis, anti-resonance frequency tracking, sensitivity analysis, design variable update, and convergence judgment—namely, the lack of explicit calculation methods for dynamic compliance anti-resonance frequencies, topological configuration ambiguity when the excitation frequency is higher than the initial structural fundamental frequency, and the requirement for additional mass in active vibration damping devices, which cannot meet the lightweight requirements of aerospace. It eliminates the need to introduce additional mass-stiffness substructures into the main structure, achieving accurate calculation and full-process control of dynamic compliance anti-resonance frequencies. This effectively avoids premature convergence to non-0-1 intermediate solutions during the optimization process. Even under conditions where the excitation frequency is higher than the initial structural fundamental frequency, a clear topology optimization design configuration suitable for engineering manufacturing can still be obtained. Simultaneously, the optimization process considers both static and dynamic performance of the structure, ultimately minimizing the dynamic compliance of the structure, significantly improving its vibration damping performance, and adapting to the dual design requirements of extreme lightweighting and high vibration damping in aerospace and other fields. The optimization iteration process is stable and efficient, demonstrating strong engineering practicality.

[0059] In the above Figure 1 Based on the above, this application embodiment also provides a schematic diagram of the implementation process for obtaining the initial structure and optimizing the column format. For example... Figure 2 As shown, steps 201 to 205 are included: Step 201: Integrate and model the structural parameters to obtain a parametric finite element model.

[0060] In this embodiment, the structural parameters include geometric parameters, material parameters, mesh generation parameters, and boundary condition parameters. Based on the actual vibration reduction and lightweight design requirements of aerospace structures, the specific values ​​of the geometric parameters, material parameters, mesh generation parameters, and boundary condition parameters are determined. The four types of structural parameters are integrated and modeled using MATLAB scripts. The model is created according to preset structural geometry, material physical properties, mesh generation accuracy, and physical field and boundary condition requirements, resulting in a parametric finite element model that can be used for subsequent analysis and data extraction.

[0061] Step 202: Extract information from the parametric finite element model to obtain system information.

[0062] In this embodiment, the system information includes the system stiffness matrix, mass matrix, element node information, and coordinate data. From the constructed parametric finite element model, core system information for subsequent structural analysis and optimization is accurately extracted, specifically including the system stiffness matrix and mass matrix. Simultaneously, element node information and coordinate data are extracted to ensure the completeness and accuracy of the extracted system information, providing data support for subsequent design variable field initialization.

[0063] Step 203: Initialize the system information using design variable fields to obtain the initial structure.

[0064] In this embodiment, based on the extracted system stiffness matrix, mass matrix, element node information, and coordinate data, a given material volume fraction is uniformly set as the initial value of the design variable for all elements in the design domain in MATLAB. The design variable field is fully initialized according to the initial value of the design variable. After the configuration is completed, the finite element model is reconstructed and coupled. After coupling is completed, an initial structure that can be used for subsequent frequency response analysis is obtained.

[0065] Step 204: Perform frequency response analysis on the initial structure to obtain the initial anti-resonance frequency.

[0066] In this embodiment of the application, frequency response simulation analysis is performed on the initial structure after reconfiguration and coupling. Based on the results of the analysis and calculation, the dynamic compliance frequency response curve of the initial structure is plotted. The specific position of the excitation frequency is determined and observed in the dynamic compliance frequency response curve. Based on the position of the excitation frequency, the initial anti-resonance frequency that is in the same resonance range as the excitation frequency is specified.

[0067] Step 205: Based on the constraint of the initial anti-resonance frequency and the dynamic relaxation factor, the static compliance and dynamic compliance are weighted and normalized to obtain the optimized formula.

[0068] In this embodiment, based on the specified initial anti-resonance frequency, a corresponding anti-resonance frequency constraint is set to avoid premature convergence of the excitation frequency and the anti-resonance frequency. Simultaneously, a dynamic relaxation factor in the form of a power function is introduced, with the exponent set to 5 to ensure the initial optimization point is within the feasible region. To balance the static bearing capacity and dynamic vibration reduction performance of the structure and avoid structural separation during optimization, static compliance and dynamic compliance are weighted and normalized, converting them to dimensionless forms. Weighting factors are set to adjust the optimization emphasis on static and dynamic performance. Considering the possibility of non-positive definite dynamic stiffness matrix under high-frequency harmonic excitation, the dynamic compliance function is set to absolute value form to avoid optimization failure due to negative dynamic compliance. A lower limit for design variables is also set in the formula to avoid matrix singularity. Combining the initial anti-resonance frequency constraint, dynamic relaxation factor, weighted normalization rules for static and dynamic compliance, and various anti-optimization failure settings, the overall optimization rules and parameters are defined, ultimately establishing the optimization formula.

[0069] This application's embodiments integrate structural parameters such as geometry, materials, mesh generation, and boundary conditions into a single model, ensuring the integrity and accuracy of the parametric finite element model. The extracted system stiffness matrix, mass matrix, and other system information provide core data support for subsequent design variable field initialization and structural optimization analysis. Unified initialization of the design variable field ensures the consistency and comparability of the initial structure. Precise acquisition of the initial anti-resonance frequency through frequency response analysis provides a direct basis for setting anti-resonance frequency constraints. Simultaneously, based on the initial anti-resonance frequency constraints and dynamic relaxation factors, weighted normalization of static and dynamic compliance is used to construct an optimization formula. This avoids matrix singularity problems and optimization failures caused by negative dynamic compliance under high-frequency harmonic excitation. It also prevents premature convergence of optimization caused by premature overlap between the excitation frequency and the anti-resonance frequency. Furthermore, by adjusting the weights, it balances the static bearing capacity and dynamic vibration reduction performance of the structure, ensuring that the initial optimization point is within the feasible region. This provides a scientific and reasonable formula basis for subsequent full-process optimization iterations, improving the stability of the topology optimization design.

[0070] In some embodiments, finite element analysis is performed on the optimized formulation and the initial structure to obtain the objective function value, including: performing finite element analysis on the initial structure to obtain the current static compliance and the current dynamic compliance.

[0071] Specifically, based on the parameter definitions of the optimized formula, a static and dynamic finite element analysis is carried out on the initial structure. In the first iteration, the analysis and calculation are completed based on the stiffness matrix and mass matrix of the initial structure. In subsequent iterations, the analysis is performed based on the stiffness matrix and mass matrix of the updated structure, and the current static compliance and current dynamic compliance are obtained by solving them respectively. The current static compliance is the structural displacement under static load, which is obtained by solving the static equilibrium equation. The current dynamic compliance is the dynamic compliance characteristic parameter of the structure under vibration excitation. The two together reflect the static bearing capacity and dynamic vibration reduction foundation performance of the structure.

[0072] Furthermore, the current dynamic compliance is converted to an absolute value to obtain a non-negative dynamic compliance.

[0073] Specifically, considering that the dynamic stiffness matrix of the structure may be non-positive definite under high-frequency harmonic excitation, the calculated current dynamic compliance will be negative. If it is directly used for subsequent optimization calculations, the minimization of dynamic compliance will lead to the result tending to negative infinity, causing optimization failure. Therefore, the current dynamic compliance is converted into an absolute value to avoid the interference of negative values ​​on subsequent weighted calculations and optimization iterations, thus ensuring the effectiveness of the calculation process.

[0074] Furthermore, based on the weighted normalization rule of the optimized column, the current static compliance and non-negative dynamic compliance are subjected to weight factor matching processing to obtain weight parameters.

[0075] Specifically, the optimization formula pre-sets a weighted normalization rule for static compliance and dynamic compliance. The weighted normalization rule introduces a weight factor to adjust the optimization emphasis on the static bearing capacity and dynamic vibration reduction performance of the structure. According to the actual design requirements of the structure, the weight factor value is matched with the current static compliance and non-negative dynamic compliance based on the weighted normalization rule to obtain the weight parameter. The higher the value of the weight parameter, the more emphasis is placed on improving the dynamic vibration reduction performance of the structure during the optimization process. At the same time, the setting of this rule can also take into account the static bearing capacity of the structure and avoid the problem of structural separation during the optimization process.

[0076] Furthermore, the current static compliance and non-negative dynamic compliance are weighted and normalized based on the weight parameters to obtain the objective function value.

[0077] Specifically, following the weighted normalization rules of the optimized formula, the current static compliance and non-negative dynamic compliance are first converted into dimensionless forms to eliminate calculation biases caused by dimensional differences. Then, the weight parameters are substituted, and a linear combination calculation is performed on the dimensionless current static compliance and non-negative dynamic compliance to obtain the final objective function value. The objective function value comprehensively reflects the overall performance of the structure in terms of static load-bearing capacity and dynamic vibration reduction. It serves as the foundational data for subsequent dynamic compliance frequency response function analysis, anti-resonance frequency solution, and design variable sensitivity analysis. During the optimization iteration process, this value is recalculated after each structural update, serving as an important basis for judging the optimization progress.

[0078] This application's embodiments accurately obtain the current static compliance and current dynamic compliance, reflecting the static and dynamic foundation performance of the structure, through finite element analysis of the initial structure. The absolute value conversion of the current dynamic compliance effectively avoids the problem of negative dynamic compliance and optimization results tending towards negative infinity caused by the non-positive definite dynamic stiffness matrix under high-frequency harmonic excitation, ensuring the effectiveness of the calculation process. Weight factor matching based on the weighted normalization rule of the optimization formula allows for flexible adjustment of the optimization emphasis on static bearing capacity and dynamic vibration reduction performance according to actual engineering needs, adapting to different design scenarios. The weighted normalization calculation of static and dynamic compliance eliminates the dimensional differences between the two, avoiding optimization deviations caused by different numerical magnitudes. The obtained objective function value comprehensively and accurately reflects the overall static and dynamic performance of the structure, providing accurate and reliable core data for subsequent anti-resonance frequency analysis and sensitivity analysis, ensuring the effectiveness and accuracy of subsequent optimization steps.

[0079] In some embodiments, sensitivity analysis is performed on the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data, including: performing projection variable sensitivity derivation on the current static compliance and non-negative dynamic compliance corresponding to the objective function to obtain the objective function sensitivity.

[0080] Specifically, for the current static compliance, which characterizes the static bearing performance of the structure, and the non-negative dynamic compliance, which characterizes the dynamic vibration reduction performance of the structure, the adjoint method is used to derive and calculate the sensitivity of the projected variables. The sensitivity of the current static compliance to the projected variables and the sensitivity of the non-negative dynamic compliance to the projected variables are solved separately. The two are then integrated to obtain the objective function sensitivity that can reflect the response characteristics of the static and dynamic performance parameters of the structure to changes in the projected variables.

[0081] Furthermore, the design variable sensitivity is derived by processing the anti-resonance frequency corresponding to the constraint function to obtain the constraint function sensitivity.

[0082] Specifically, taking the anti-resonance frequency obtained from generalized asymmetric eigenvalue analysis as the core, and combining the solution results of the asymmetric eigenvalue problem, we carry out the derivation and calculation of the sensitivity of the anti-resonance frequency to the design variables. In the derivation process, two eigenvectors of the accompanying eigenvalue problem are introduced to participate in the calculation, and finally the constraint function sensitivity is obtained. The sensitivity reflects the response characteristics of the constraint parameter of anti-resonance frequency to changes in design variables, and is an important basis for updating design variables under the constraint of anti-resonance frequency.

[0083] Furthermore, the sensitivity of the objective function and the sensitivity of the constraint function are integrated to obtain sensitivity data.

[0084] Specifically, the obtained objective function sensitivity and constraint function sensitivity are integrated and processed in a unified manner to form comprehensive sensitivity data. The sensitivity data reflects the response characteristics of the objective function and constraint function to changes in the corresponding variables. It provides a complete and accurate sensitivity basis for subsequent updates of design variables based on the constraint requirements in the optimization formula, ensuring that the subsequent optimization algorithm can efficiently and accurately complete the iterative updates of design variables based on this data.

[0085] This application's embodiments accurately capture the response characteristics of the objective function to changes in projected variables by deriving sensitivity for projected variables of the current static compliance and non-negative dynamic compliance. The derivation of sensitivity for design variables at the anti-resonance frequency enables a quantitative characterization of the constraint function's response to changes in design variables, and the derivation method is adapted to the large-scale design variable solution requirements of dynamic compliance topology optimization. The sensitivity data obtained by integrating the objective function sensitivity and constraint function sensitivity simultaneously reflects the combined static and dynamic performance of the structure and the sensitivity characteristics of the anti-resonance frequency constraint. This provides a comprehensive and accurate gradient basis for subsequent design variable optimization updates, adapting to gradient-based optimization algorithms and ensuring the efficiency and accuracy of subsequent MMA optimization algorithm updates of design variables, thus promoting the practical application of dynamic topology optimization problems in engineering.

[0086] In some embodiments, the dynamic compliance frequency response function corresponding to the objective function value is subjected to generalized asymmetric eigenvalue analysis to obtain anti-resonance frequency related eigenvalues, including: obtaining the steady-state motion equation.

[0087] Specifically, the undamped forced vibration under harmonic load excitation is discretized using the finite element method. Based on the operational rules of finite element discretization, a steady-state motion equation related to the amplitude of the harmonic load and the displacement response is established. This equation provides the basis for the subsequent analysis and derivation of the dynamic compliance frequency response function.

[0088] Furthermore, the dynamic compliance characteristics of the structure and the steady-state motion equations are analyzed and combined to obtain the fitting equations.

[0089] Specifically, the dynamic compliance characteristics of the structure under anti-resonance frequency excitation are first analyzed, and it is clarified that the steady-state response of a certain degree of freedom of the structure can be expressed as a linear combination of displacement amplitude vectors. The external force vector is regarded as the coefficient vector of displacement amplitude and normalized to obtain the normalized vector. Then, the normalized vector is combined with the obtained steady-state motion equation, and the fusion process of the two is completed by rewriting the equation, and finally the fitting equation is obtained.

[0090] Furthermore, the fitting equations are processed by introducing a null space matrix to obtain intermediate equations.

[0091] Specifically, for the mathematical form of the fitting equation, a corresponding null space matrix is ​​introduced. The null space matrix satisfies specific matrix operation relations. After substituting it into the fitting equation, the null space matrix is ​​introduced. At the same time, terms on the right side of the equation in the fitting equation are eliminated, thus obtaining the intermediate equation.

[0092] Furthermore, the intermediate equation is transformed to obtain asymmetric generalized eigenvalues.

[0093] Specifically, by multiplying both sides of the intermediate equation by the corresponding matrix on the left, and transforming and rewriting the intermediate equation according to the rules of matrix operations, the equation is derived into the standard form of the asymmetric generalized eigenvalue problem, and finally the asymmetric generalized eigenvalue is obtained.

[0094] Furthermore, the asymmetric generalized eigenvalues ​​are solved and filtered to obtain the anti-resonance frequency related eigenvalues.

[0095] Specifically, the asymmetric generalized eigenvalue problem is first solved to obtain a set of eigenvalues ​​containing real numbers, complex numbers, and infinity. Then, the system eigenvalues ​​are filtered out from this set, and only the positive real eigenvalues ​​that meet the requirements are retained. The positive real eigenvalues ​​are the anti-resonance frequency related eigenvalues, and the eigenvalues ​​are the square of the anti-resonance frequency of the structure's dynamic compliance.

[0096] This application extends the null-space-based eigenvalue method to solve for the anti-resonance frequency of dynamic compliance, overcoming the limitations of traditional graphical methods that rely on frequency resolution and existing eigenvalue methods that cannot explicitly solve for the anti-resonance frequency of dynamic compliance. It achieves, for the first time, the explicit calculation of the anti-resonance frequency of dynamic compliance. Through the derivation of steady-state motion equations, the introduction of null-space matrices, the transformation of asymmetric generalized eigenvalues, and the filtering of solutions, complex, infinite, and system eigenvalues ​​in the asymmetric eigenvalues ​​are precisely eliminated, retaining only valid positive real eigenvalues. These eigenvalues ​​are the squares of the anti-resonance frequency, providing a precise mathematical basis for obtaining the anti-resonance frequency.

[0097] In some embodiments, tracking the eigenvectors corresponding to the anti-resonance frequency-related eigenvalues ​​to obtain the anti-resonance frequency includes: extending the modal confidence criterion to obtain the eigenvector tracking criterion corresponding to the anti-resonance frequency.

[0098] Specifically, the modal confidence criterion is a commonly used modal tracking technique for tracking target vibration modes in the optimization process. By extending the modal confidence criterion to be applicable to eigenvector tracking in asymmetric eigenvalue problems, the eigenvectors corresponding to the eigenvalues ​​related to the anti-resonance frequency are regarded as special vibration modes. Based on this, the traditional modal confidence criterion is specifically extended to establish similarity judgment and tracking rules applicable to eigenvectors corresponding to eigenvalues ​​related to the anti-resonance frequency, thus obtaining the eigenvector tracking criterion, which provides a judgment basis for subsequent similarity comparison of eigenvectors and selection of target eigenvectors.

[0099] Furthermore, feature vector extraction is performed on the eigenvalues ​​related to the anti-resonance frequency to obtain a set of feature vectors.

[0100] Specifically, for the anti-resonance frequency related eigenvalues ​​obtained from the generalized asymmetric eigenvalue analysis, all eigenvectors corresponding to the anti-resonance frequency related eigenvalues ​​are accurately extracted from the overall solution results of the asymmetric eigenvalue problem. All extracted eigenvectors are then integrated and summarized to form an eigenvector set.

[0101] Furthermore, a similarity comparison is performed between the feature vector set and the first target feature vector to obtain the modal confidence criterion value.

[0102] Specifically, the selection of the first target feature vector changes with the iteration process. In the first iteration, the first target feature vector is the feature vector corresponding to the initial anti-resonance frequency. In subsequent iterations, the first target feature vector is the second target feature vector determined in the previous iteration. Based on the extended feature vector tracking criterion, each feature vector in the feature vector set is compared with the first target feature vector one-to-one to obtain the modal confidence criterion value corresponding to each feature vector. The value ranges from 0 to 1. The closer the value is to 1, the higher the similarity between the two feature vectors.

[0103] Furthermore, the modal confidence criterion values ​​are filtered to obtain the second target feature vector.

[0104] Specifically, the modal confidence criterion values ​​are selected from all modal confidence criterion values, and the modal confidence criterion value with the largest value is selected. The feature vector corresponding to the maximum value is determined as the second target feature vector for this iteration. The feature vector is the target feature vector that matches the anti-resonance frequency in the current iteration process. This ensures that the feature vector corresponding to the anti-resonance frequency can still be accurately tracked when the order of the feature value sequence changes with the optimization iteration.

[0105] Furthermore, frequency correlation processing is performed on the second target feature vector to obtain the anti-resonance frequency.

[0106] Specifically, based on the solution results of generalized asymmetric eigenvalue analysis, the anti-resonance frequency related eigenvalue is the square of the anti-resonance frequency. The second target eigenvector obtained in this iteration is associated and matched with its corresponding anti-resonance frequency related eigenvalue. According to the mathematical correspondence between the two, the anti-resonance frequency of this iteration is calculated through the anti-resonance frequency related eigenvalue. This anti-resonance frequency provides accurate parameter basis for the subsequent construction of constraint functions and sensitivity analysis.

[0107] This application's embodiments, by extending the modal confidence criterion, derive a feature vector tracking criterion adapted to asymmetric eigenvalue problems, solving the problem that traditional modal tracking techniques cannot be applied to feature vector tracking corresponding to asymmetric eigenvalues. Through feature vector extraction, similarity comparison, and modal confidence criterion value filtering, it accurately addresses the core issues of eigenvalue sequence changes and difficulty in tracking feature vectors corresponding to anti-resonance frequencies caused by structural material layout variations during optimization iterations. It can accurately select target feature vectors matching the anti-resonance frequency in each iteration. Finally, through the correlation processing between feature vectors and frequencies, it accurately obtains the anti-resonance frequency of the current iteration, ensuring accurate and stable tracking of the anti-resonance frequency during optimization iterations. This ensures the effective execution of anti-resonance frequency constraints, allowing the optimization process to gradually approach the anti-resonance point as designed, avoiding premature convergence at the anti-resonance point in the early stages of optimization, and guaranteeing the smooth operation of optimization iterations. This provides accurate frequency parameter basis for ultimately minimizing structural dynamic compliance.

[0108] The following detailed explanation of the specific implementation steps of this application is provided through a typical example of dynamic compliance topology optimization for a planar cantilever beam: The first step is to establish a planar cantilever beam example. The geometric model and boundary conditions of the example are as follows: Figure 4As shown. The dimensions of the planar cantilever beam are 1m × 0.5m × 0.001m, and its material properties are: elastic modulus E = 210 GPa, Poisson's ratio υ = 0.3, and density ρ = 7860 kg / m³. The left side of the structure is fixed, and a vertically downward harmonic excitation force is applied near the midpoint of the right side. Furthermore, to avoid structural separation near the load when the excitation frequency is in the inertial region, the area near the load is designated as a non-design domain. Figure 4 The region with a length of 0.03m and a width of 0.02m represents the non-design domain. The entire structure is divided into 100 × 50 = 5000 four-node plane stress elements, with an element size of 0.01m. The design variables for the design domain elements in the initial structure are uniformly set to 0.5. Frequency response analysis is performed on the planar cantilever beam structure to obtain the initial structure's dynamic compliance frequency response curve, as shown below. Figure 5 As shown. Subsequently, a topology optimization model was established, and using the aforementioned optimization formula, an excitation frequency of 800 rad / s, which is higher than the initial design fundamental frequency of 782 Hz, was selected for dynamic compliance topology optimization design. The anti-resonance frequency order corresponding to this excitation frequency is 1st order.

[0109] The second step is to perform static / dynamic finite element analysis to calculate the initial static compliance, initial dynamic compliance, and static compliance and dynamic compliance of the planar cantilever beam during subsequent iterations. Given a weighting factor α of 0.95, the objective function is determined.

[0110] The third step involves conducting generalized asymmetric eigenvalue analysis to predict the anti-resonance frequency in the dynamic compliance frequency response function of the planar cantilever beam structure. Simultaneously, during the iteration process, the modal confidence criterion (MAC) is used to calculate the maximum MAC value to track the eigenvector corresponding to the first-order anti-resonance frequency eigenvalue, accurately obtaining the first-order anti-resonance frequency for each iteration.

[0111] Fourth, based on the above optimization formula, derive the sensitivity information of the objective function and constraint functions, and update the design variables using the MMA optimization algorithm and the three-field filtering method. If the design variables meet the preset convergence condition, i.e., the maximum change in design variables between two consecutive iterations is less than 1 × 10⁻⁶, then the design variables are considered to have converged. - ³ Or, if the maximum number of iterations (210) is reached, the optimal design variables are output; if the preset convergence condition is not met, the design variables are modified and iterations are performed to finally obtain the topology optimization result, such as... Figure 6 As shown. The topology optimization design results without applying anti-resonance frequency constraints are as follows. Figure 7 As shown.

[0112] The fifth step further provides the topology optimization design results under excitation frequencies of 2700 rad / s and 3600 rad / s, as shown below. Figure 8 , 10As shown, the anti-resonance frequency order corresponding to the excitation frequency of 2700 rad / s is 1st order, and the anti-resonance frequency order corresponding to the excitation frequency of 3600 rad / s is 2nd order. This is compared with the topology optimization design results under the same excitation frequency without applying anti-resonance frequency constraints (e.g., ...). Figure 9 , 11 (As shown) for comparison. The dynamic compliance frequency response curves of the optimal configuration under different excitation frequencies are shown in the figure. Figure 12 As shown, and in combination Figure 6 , 7 As can be seen from 8, 9, 10, and 11, the dynamic compliance of all optimal topological configurations falls into the anti-resonance point, which means that the dynamic compliance of the structure reaches the minimum under different excitation frequencies. Under the premise of obtaining a clear configuration, this application achieves excellent vibration reduction performance design, proving the effectiveness of the design method proposed in this application.

[0113] While this application provides the method operation steps as described in the embodiments or flowcharts, more or fewer operation steps may be included based on conventional or non-inventive labor. The order of steps listed in this embodiment is merely one possible execution order among many and does not represent the only execution order. In actual device or client product execution, the methods shown in this embodiment or the accompanying drawings can be executed sequentially or in parallel (e.g., in a parallel processor or multi-threaded processing environment).

[0114] like Figure 3 As shown in the illustration, this application also provides a structural vibration reduction topology optimization design device 300 based on anti-resonance frequency constraints. The device includes: Module 301 is used to construct the finite element model corresponding to the structural parameters, and to obtain the initial structure and optimized formula based on the finite element model.

[0115] The processing module 302 is used to perform finite element analysis on the optimized formula and the initial structure to obtain the objective function value.

[0116] The processing module 302 is also used to perform generalized asymmetric eigenvalue analysis on the dynamic compliance frequency response function corresponding to the objective function value to obtain anti-resonance frequency related eigenvalues.

[0117] The processing module 302 is also used to track the eigenvectors corresponding to the eigenvalues ​​related to the anti-resonance frequency to obtain the anti-resonance frequency.

[0118] The processing module 302 is also used to perform sensitivity analysis on the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data.

[0119] The processing module 302 is also used to update the design variables corresponding to the sensitivity data and the constraints in the optimization formula to obtain the updated design variables.

[0120] The judgment module 303 is also used to perform convergence judgment processing on the iteration results corresponding to the constraints of the updated design variables and optimization formulas. If the convergence judgment result meets the preset convergence threshold, the topology optimization design configuration is obtained.

[0121] Some modules in the apparatus described in this application can be described in the general context of computer-executable instructions that are executed by a computer, such as program modules. Generally, program modules include routines, programs, objects, components, data structures, classes, etc., that perform a specific task or implement a specific abstract data type. This application can also be practiced in distributed computing environments where tasks are performed by remote processing devices connected via a communication network. In distributed computing environments, program modules can reside in local and remote computer storage media, including storage devices.

[0122] The apparatus or module described in the above embodiments can be implemented by a computer chip or physical entity, or by a product with a certain function. For ease of description, the above apparatus is described by dividing it into various modules according to their functions. When implementing the embodiments of this application, the functions of each module can be implemented in one or more software and / or hardware. Of course, a module that implements a certain function can also be implemented by combining multiple sub-modules or sub-units.

[0123] The methods, apparatus, or modules described in this application can be implemented in a computer-readable program code manner. The controller can be implemented in any suitable manner, such as a microprocessor or processor and a computer-readable medium storing computer-readable program code (e.g., software or firmware) executable by the (micro)processor, logic gates, switches, application-specific integrated circuits (ASICs), programmable logic controllers, and embedded microcontrollers. Examples of controllers include, but are not limited to, the following microcontrollers: ARC 625D, Atmel AT91SAM, Microchip PIC18F26K20, and Silicon Labs C8051F320. A memory controller can also be implemented as part of the control logic of a memory. Those skilled in the art will also recognize that, in addition to implementing the controller in purely computer-readable program code manner, the same functionality can be achieved by logically programming the method steps to make the controller take the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, such a controller can be considered a hardware component, and the means included within it for implementing various functions can also be considered as structures within the hardware component. Alternatively, the device used to implement various functions can be viewed as either a software module implementing the method or a structure within a hardware component.

[0124] This application also provides an apparatus, the apparatus comprising: a processor; a memory for storing processor-executable instructions; wherein, when the processor executes the executable instructions, it implements the method described in this application embodiment.

[0125] This application also provides a non-volatile computer-readable storage medium storing a computer program or instructions thereon, which, when executed, enables the method described in this application embodiment to be implemented.

[0126] Furthermore, in the various embodiments of the present invention, each functional module can be integrated into a processing module, or each module can exist independently, or two or more modules can be integrated into a single module.

[0127] The aforementioned storage media include, but are not limited to, Random Access Memory (RAM), Read-Only Memory (ROM), Cache, Hard Disk Drive (HDD), or Memory Card. The memory can be used to store computer program instructions.

[0128] As can be seen from the above description of the embodiments, those skilled in the art can clearly understand that this application can be implemented by means of software plus necessary hardware. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, can be embodied in the form of a software product, or it can be embodied in the process of data migration. The computer software product can be stored in a storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, mobile terminal, server, or network device, etc.) to execute the methods described in various embodiments or some parts of the embodiments of this application.

[0129] The various embodiments described in this specification are presented in a progressive manner. Similar or identical parts between embodiments can be referred to interchangeably. Each embodiment focuses on its differences from other embodiments. All or part of this application can be used in numerous general-purpose or special-purpose computer system environments or configurations. Examples include: personal computers, server computers, handheld or portable devices, tablet devices, mobile communication terminals, multiprocessor systems, microprocessor-based systems, programmable electronic devices, network PCs, minicomputers, mainframe computers, and distributed computing environments including any of the above systems or devices, etc.

[0130] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit this application. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of this application.

Claims

1. A structural vibration reduction topology optimization design method based on anti-resonance frequency constraints, characterized in that, include: Construct a finite element model corresponding to the structural parameters, and obtain the initial structure and optimization formula based on the finite element model; Finite element analysis is performed on the optimized formulation and the initial structure to obtain the objective function value; Generalized asymmetric eigenvalue analysis is performed on the dynamic compliance frequency response function corresponding to the objective function value to obtain anti-resonance frequency related eigenvalues. The anti-resonance frequency is obtained by tracking the eigenvector corresponding to the anti-resonance frequency related eigenvalues. Sensitivity analysis is performed on the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data. The design variables corresponding to the sensitivity data and the constraints in the optimization formula are updated to obtain the updated design variables. The iteration results corresponding to the updated design variables and the constraints of the optimized formula are subjected to convergence judgment. If the convergence judgment result meets the preset convergence threshold, the topology optimization design configuration is obtained.

2. The method according to claim 1, characterized in that, The construction of the finite element model corresponding to the structural parameters, and the obtaining of the initial structure and optimized formulation based on the finite element model, include: The structural parameters are integrated and modeled to obtain a parametric finite element model, wherein the structural parameters include geometric parameters, material parameters, mesh generation parameters, and boundary condition parameters; Information extraction processing is performed on the parametric finite element model to obtain system information, which includes system stiffness matrix, mass matrix, element node information and coordinate data. The system information is initialized using design variable fields to obtain the initial structure; The initial structure is subjected to frequency response analysis to obtain the initial anti-resonance frequency; Based on the constraints of the initial anti-resonance frequency and the dynamic relaxation factor, the static compliance and dynamic compliance are weighted and normalized to obtain the optimized formulation.

3. The method according to claim 1, characterized in that, The step of performing finite element analysis on the optimized formulation and the initial structure to obtain the objective function value includes: The initial structure is subjected to finite element analysis to obtain the current static compliance and the current dynamic compliance. The current dynamic compliance is converted to an absolute value to obtain a non-negative dynamic compliance. Based on the weighted normalization rule of the optimized column, the current static compliance and non-negative dynamic compliance are subjected to weight factor matching processing to obtain weight parameters; The current static compliance and the non-negative dynamic compliance are weighted and normalized based on the weight parameters to obtain the objective function value.

4. The method according to claim 1, characterized in that, The sensitivity analysis process is performed on the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data, including: The sensitivity of the objective function is obtained by performing projection variable sensitivity derivation on the current static compliance and non-negative dynamic compliance corresponding to the objective function. The design variable sensitivity is derived by performing a design variable sensitivity derivation process on the anti-resonance frequency corresponding to the constraint function to obtain the constraint function sensitivity; The sensitivity of the objective function and the sensitivity of the constraint function are integrated to obtain sensitivity data.

5. The method according to claim 1, characterized in that, The generalized asymmetric eigenvalue analysis of the dynamic compliance frequency response function corresponding to the objective function value yields anti-resonance frequency-related eigenvalues, including: Obtain the steady-state equation of motion; The dynamic compliance characteristics of the structure and the steady-state motion equations are analyzed and combined to obtain the fitting equations. The fitting equations are processed by introducing a null space matrix to obtain intermediate equations; The intermediate equation is transformed to obtain asymmetric generalized eigenvalues; The asymmetric generalized eigenvalues ​​are solved and filtered to obtain anti-resonance frequency related eigenvalues.

6. The method according to claim 1, characterized in that, The step of tracking the feature vector corresponding to the anti-resonance frequency-related feature value to obtain the anti-resonance frequency includes: The modal confidence criterion is extended to obtain the eigenvector tracking criterion corresponding to the anti-resonance frequency; The anti-resonance frequency related feature values ​​are subjected to feature vector extraction processing to obtain a feature vector set; The feature vector set and the first target feature vector are compared for similarity to obtain the modal confidence criterion value. The modal confidence criterion values ​​are filtered to obtain the second target feature vector; The second target feature vector is subjected to frequency correlation processing to obtain the anti-resonance frequency.

7. A structural vibration reduction topology optimization design device based on anti-resonance frequency constraints, characterized in that, include: The construction module is used to construct the finite element model corresponding to the structural parameters, and to obtain the initial structure and optimized formula based on the finite element model. The processing module is used to perform finite element analysis on the optimized formula and the initial structure to obtain the objective function value; The processing module is also used to perform generalized asymmetric eigenvalue analysis on the dynamic compliance frequency response function corresponding to the objective function value to obtain anti-resonance frequency related eigenvalues. The processing module is also used to track the feature vector corresponding to the anti-resonance frequency related feature value to obtain the anti-resonance frequency; The processing module is further configured to perform sensitivity analysis on the objective function corresponding to the objective function value and the constraint function corresponding to the anti-resonance frequency to obtain sensitivity data; The processing module is also used to update the design variables corresponding to the sensitivity data and the constraints in the optimization formula to obtain the updated design variables. The judgment module is also used to perform convergence judgment on the iteration results corresponding to the updated design variables and the constraints of the optimization formula. If the convergence judgment result meets the preset convergence threshold, the topology optimization design configuration is obtained.

8. A computer device comprising a memory and a processor, the memory storing a computer program executable on the processor, characterized in that, When the processor executes the program, it implements the steps of the method according to any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method as described in any one of claims 1 to 6.

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