Optimization method for transcendental eigenvalue problems based on modal count and heuristic algorithm
By combining modal counting and heuristic algorithms, the problem of large computational load and low efficiency in eigenvalue optimization is solved, achieving efficient structural optimization that is applicable to various engineering design needs, especially the vibration mode and buckling stability problems of large-scale structures.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- CENT SOUTH UNIV
- Filing Date
- 2023-11-22
- Publication Date
- 2026-07-03
AI Technical Summary
Existing technologies suffer from high computational cost and low solution efficiency in eigenvalue optimization, making it difficult to handle vibration modes and buckling stability problems of large-scale structures. In particular, topology optimization is computationally demanding, and traditional methods struggle to address complex issues such as repetitive frequencies and spurious local modes.
A method combining modal counting and heuristic algorithms is adopted. By using dynamic stiffness modeling, modal counting optimization, and heuristic algorithms, an optimization model is constructed. This avoids direct calculation of eigenvalues and uses modal counting functions as constraints or objective functions. The Wittrick-Williams algorithm is combined to improve computational efficiency and robustness.
It achieves efficient structure optimization, can handle eigenvalue problems of large-scale structures, avoids repeated eigenvalues and local modes, improves computational efficiency and applicability, and is suitable for various lightweight structural designs and eigenvalue maximization problems.
Smart Images
Figure CN117494576B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of structural optimization design analysis, and in particular to an optimization method for transcendental eigenvalue problems based on modal counting and heuristic algorithms. Background Technology
[0002] In the field of structural optimization, eigenvalue considerations are crucial, primarily focusing on two aspects: structural vibration modes and buckling stability. Optimizing structural vibration modes aims to increase the structure's natural frequencies to avoid performance degradation or structural damage due to frequency resonance. This type of optimization is widely applied in areas such as repetition rate control, local modal analysis, and natural frequency maximization. On the other hand, buckling stability is a key indicator for evaluating the performance of engineering structures, focusing on preventing destructive deformation that could lead to engineering accidents when the critical buckling load is exceeded.
[0003] The challenge of eigenvalue optimization lies in its high computational cost and low solution efficiency. Unlike simple structural dynamic response problems, eigenvalue optimization cannot be simply reduced to an iterative problem of dynamic response. Especially in the context of large-scale topology optimization, developing efficient solution methods has become an important research direction for expanding the scale of structural analysis and optimization.
[0004] Eigenvalue optimization plays a crucial role in engineering shape design, encompassing the intersection of multiple disciplines including structural finite element method, structural dynamics, mathematical programming, and numerical computation. As it is a computationally intensive nonlinear problem requiring multiple solutions, its numerical optimization computational efficiency is typically low. In large-scale dynamic topology optimization, the computational burden primarily stems from dynamic analysis. Eigenvalue optimization is essentially a nested loop problem; the inner loop involves iteratively solving a generalized eigenvalue problem, while the outer loop iteratively optimizes the design variables. In this process, the computational burden of the inner loop is usually much higher than that of the outer loop. Although iterative methods with controllable accuracy are generally used to solve for eigenvalues, exact solutions increase computational costs, while approximate solutions may affect the optimization results. Therefore, seeking efficient eigenvalue optimization methods is crucial for expanding the analysis and design scale of dynamic optimization.
[0005] The main challenges of eigenvalue optimization in engineering include: large structural scale, numerous variables, and long modeling and computation time, leading to low computational efficiency; difficulty in converging to the global optimum; complex gradient information calculation in traditional optimization methods, with different gradient formulas for different structures, lacking universality; and existing methods' inability to handle complex problems such as frequency repetition and spurious local modes. Therefore, there is an urgent need for an eigenvalue optimization method with broad applicability, high efficiency and stability, and the ability to solve complex problems such as frequency repetition and local modes. Summary of the Invention
[0006] This invention aims to solve the optimization challenges of the eigenvalue problem in the prior art by providing an innovative optimization method that transcends the eigenvalue problem by combining modality counting and heuristic algorithms. The algorithm steps specifically include:
[0007] S1) Optimization Model Construction: Construct an optimization model for the structural eigenvalue problem, which includes considering multiple aspects such as size, shape, and topology. Based on this, clearly define the objective function, design variables, and constraints to enable customized optimization for specific engineering needs.
[0008] S2) Dynamic Stiffness Modeling: Dynamic stiffness modeling methods are applied to construct analytical models for analyzing complex composite structures. These models can accurately simulate key structural features such as free vibration, buckling, and wave propagation, and have relatively few degrees of freedom, thus improving computational efficiency. The dynamic stiffness model includes material and geometric parameters, achieving parametric modeling.
[0009] S3) Modal counting optimization: The optimization method is based on modal counting functions, rather than directly on eigenvalues, thus avoiding time-consuming eigenvalue calculations. By using monotonically increasing modal counting functions with integer properties to construct constraints or objective functions, the need for bisection method to calculate eigenvalues is eliminated, and computational efficiency and robustness are significantly improved.
[0010] S4) Construction of Constraints or Objective Functions: For various types of optimization problems, methods for constructing constraints or objective functions based on modal counting are proposed. These methods combine normalization techniques and set reasonable weights to achieve single-objective or multi-objective optimization.
[0011] S5) Application of heuristic algorithms: Without needing to calculate eigenvalues, heuristic algorithms are used as optimization tools to effectively find the global optimum. This method does not rely on the differentiability of the modal counting function, thus improving the algorithm's adaptability and efficiency.
[0012] Furthermore, in step S1, the selection of variables can be achieved by selecting the cross-sectional dimensions of beams and slabs to optimize the structure's dimensions, selecting the coordinates of structural connection points to optimize the shape, and selecting the presence or absence of beam structural units to optimize the topology.
[0013] Furthermore, the eigenvalue optimization problem defined in step S1 includes lightweight structural design under eigenvalue upper and lower bound constraints, specific eigenvalue constraints, or bandgap width constraints, and eigenvalue maximization or bandgap width maximization problems under mass constraints, effectively avoiding common problems in traditional structural eigenvalue optimization calculations such as repeated eigenvalues, local modes, and mode switching. Furthermore, the structural eigenvalue modeling in step S2 uses the dynamic stiffness method to derive vibration mode problems or buckling models for different beam-plate composite structures, including:
[0014] The time-domain control differential equation of the structure is obtained based on Hamilton's and Newton's theorems or the Lagrange equation.
[0015] L(u,t)=0
[0016] It is a linear differential operator, u is the displacement vector, and t is time; then based on the formula u(x,t)=φe iωt The harmonic oscillation assumption is given, where φ is the displacement amplitude vector and ω is the natural frequency; then it is transformed into the governing equations in the frequency domain.
[0017] L1(φ,ω)=0
[0018] It is a linear differential operator in the frequency domain, from which the analytical general solution is obtained as follows:
[0019] φ=C(ω)A
[0020] Where C is the exact shape function and A is the unknown constant vector; the unit nodal displacement can be written as d = BA, and the nodal force vector can be written as F = DA. Eliminating the matrix yields the relationship between the force and displacement vectors.
[0021] F=DB -1 d = K * d
[0022] If it is a vibration modal problem, then Kw in the equation is the dynamic stiffness matrix of the structure containing frequency ω, and there exists...
[0023] K * (ω)d=0
[0024] If it is a buckling problem, then Kw in the equation is the dynamic stiffness matrix of the structure containing the buckling load factor λ, and there exists...
[0025] K * (λ)d=F
[0026] Furthermore, the optimization method in step S3 is based on a modal counting function rather than eigenvalues, including:
[0027] Given any eigenvalue λ * The expression represented by J
[0028]
[0029] Based on the Wittrick-Williams algorithm, calculate the mode counts for all eigenvalues that transcend the structure's eigenvalues.
[0030]
[0031] In the formula J0(λ) *) is the total number of eigenvalues of the structure that are lower than λ when both ends of each unit are fixed, s{K * (λ * )} represents the process of decomposing K using Gaussian elimination or LDLT. * (λ * The number of negative elements on the main diagonal after transforming into an upper triangular matrix.
[0032] Then, further step S3, based on the modal counting function, allows different eigenvalue optimization problem models to be modified as follows:
[0033] The traditional formulation for lightweight design problems under eigenvalue lower bound constraints is:
[0034] Find x∈R n
[0035]
[0036]
[0037] K * (λ1,x)d1=0
[0038] x l ≤x≤x u
[0039] The formula for this method can be written as follows:
[0040] Find x∈R n
[0041]
[0042]
[0043] x l ≤x≤x u
[0044] The traditional formulation for lightweight design problems under specified eigenvalue constraints is:
[0045] Find x∈R n
[0046]
[0047]
[0048] K * (λ i ,x)d i =0, i=1,2,...
[0049] x l≤x≤x u
[0050] There are K specified eigenvalue ranges Constraints, where ε k For the precision tolerance of the k-th specified eigenvalue, the formula of this method can be written as follows:
[0051] Find x∈R n
[0052]
[0053]
[0054] x l ≤x≤x u
[0055] The traditional formulation for the eigenvalue maximization design problem under quality constraints is:
[0056] Find x∈R n
[0057] Minimize λ1
[0058]
[0059] K * (λ1,x)d1=0
[0060] x l ≤x≤x u
[0061] The formula for this method can be written as follows:
[0062] Find x∈R n
[0063] Minimize λ1
[0064]
[0065] J(λ1)=0
[0066] x l ≤x≤x u
[0067] The traditional formulation for lightweight design problems under bandwidth-free constraints is as follows:
[0068] Find x∈R n
[0069]
[0070]
[0071]
[0072] K * (ω i ,x)d i =0, i=1,2,...
[0073] x l ≤x≤x u
[0074] The formula for this method can be written as follows:
[0075] Find x∈R n
[0076]
[0077]
[0078]
[0079] J(ω1 l ) < 1
[0080] x l ≤x≤x u
[0081] The traditional formulation for the bandwidth maximization design problem under quality constraints is:
[0082] Find x∈R n
[0083]
[0084]
[0085] K * (λ1,x)d1=0
[0086]
[0087]
[0088]
[0089] x l ≤x≤x u
[0090] Where, Δ k To the k-th specified intermediate frequency The bandwidth between, w k These are the weights of different bandwidths in the objective function. The formula for this method can be written as follows:
[0091] Find x∈R n
[0092]
[0093]
[0094]
[0095]
[0096]
[0097] x l ≤x≤x u
[0098] Furthermore, the optimization method in step S3 is based on a modal counting function rather than eigenvalues, which can selectively consider the difficulties that may arise in eigenvalue optimization problems, including avoiding complex situations such as repeated eigenvalues, local modes, and mode switching.
[0099] In eigenfrequency optimization, due to the symmetry of the structure and its dependence on multiple design parameters, repetitive eigenvalues are difficult to avoid in optimization designs with multiple degrees of freedom. Modal counting constraints can be used to avoid the problem of repetitive eigenvalues. For example, in the specified eigenvalue constraint problem, to avoid the occurrence of repetitive eigenvalues, the following modal counting constraint can be added to prevent the occurrence of repetitive eigenvalues in the specified eigenvalue constraint problem.
[0100]
[0101] ε k It represents a very small range of eigenvalues;
[0102] For problems where the structure needs to specify m (m>1) repeated eigenvalues, add constraints.
[0103]
[0104] This method does not require calculating the derivative of the repetition frequency; it can accurately identify the order of the natural frequency simply by adding appropriate equation constraints.
[0105] Local modal problems often exist during natural frequency optimization. In this case, the fundamental frequency of the optimal structure may be a local vibration in a low-density region of the feasible domain, rather than the true vibration frequency of the actual structure. To avoid the generation of local modal problems, the physical properties of J0 can be utilized by adding the following constraints to the optimization model.
[0106] J 0m (λ)=0
[0107] For the mode switching problem, since the present invention uses a mode counting function that combines heuristic algorithms instead of model gradient information, the mode switching problem that is usually involved in gradient-based optimization problems will not occur.
[0108] Furthermore, the method for constructing constraints or objective functions based on modal counting in step S4, combined with a normalization method and setting reasonable weights, can be expressed in the following forms according to different eigenvalue optimization problems:
[0109] The objective function for a single eigenvalue lower bound constraint optimization problem is constructed as follows:
[0110]
[0111] The construction expression for the objective function of an optimization problem with multiple eigenvalue lower bound constraints is as follows:
[0112]
[0113] The constructive expression for the objective function of a given eigenvalue-constrained optimization problem is as follows:
[0114]
[0115] The constructive expression for the objective function of the eigenvalue maximization optimization problem is as follows:
[0116]
[0117] The objective function for the band-limited constraint optimization problem is constructed as follows:
[0118]
[0119] The constructive expression for the objective function of the bandwidth maximization optimization problem is as follows:
[0120]
[0121]
[0122] Compared with the prior art, the present invention has the following beneficial effects:
[0123] (1) Analytical modeling and parametric design: The dynamic stiffness matrix is used as the analytical modeling matrix, which enables the algorithm to extract an infinite number of modes from a small-scale transcendental function matrix. This invention integrates design variables such as material parameters, structural dimensions, shape variables, and topology variables into parameters of the dynamic stiffness model, realizing parametric modeling of structural optimization without mesh generation, which is applicable to multiple fields such as size, shape and topology optimization.
[0124] (2) Optimization method based on modal counting: The optimization method of this invention is based on modal counting function rather than directly on eigenvalue, thereby avoiding time-consuming eigenvalue calculation; using modal counting based on the monotonically increasing integer property of Wittrick-Williams algorithm as a constraint, the requirement for bisection calculation of eigenvalue is eliminated, significantly reducing the calculation time, and enhancing the robustness of the algorithm.
[0125] (3) Efficient global optimization using heuristic algorithms: Without calculating eigenvalues, this invention uses heuristic algorithms to efficiently search for the global optimal solution; this method does not depend on the differentiability of the modal counting function, thus making it more flexible and efficient in the design of optimization methods.
[0126] (4) Wide applicability to engineering applications: The method of the present invention is applicable to various lightweight structural designs, including upper and lower limit constraints of eigenvalues, specific eigenvalue constraints or bandgap width constraints; it is also applicable to optimization problems that maximize eigenvalues or maximize bandgap width under mass constraints; in addition, since there is no need to calculate the gradient information of structural eigenvalues, this method effectively avoids the common problems of repeated eigenvalues, local modes and mode switching in traditional structural eigenvalue optimization problems. Attached Figure Description
[0127] Figure 1 This is a schematic diagram of the modal counting function in related technologies;
[0128] Figure 2 This is a flowchart of eigenvalue optimization under the heuristic optimization method based on modal counting constraints in related technologies;
[0129] Figure 3 This is a flowchart of the eigenvalue optimization process under the immune genetic algorithm based on modality counting constraints in related technologies.
[0130] Figure 4 This is a schematic diagram of a corrugated sandwich panel as an application embodiment of the present invention;
[0131] Figure 5 The design variables are optimized based on the results of applying the present invention to corrugated sandwich panels;
[0132] Figure 6 The first-order mode shape of the corrugated sandwich panel before optimization in an application embodiment of the present invention;
[0133] Figure 7 The first-order mode shape of the corrugated sandwich panel is optimized according to an application embodiment of the present invention.
[0134] Figure 8 This is a flowchart of the optimization method for the transcendental eigenvalue problem based on modal counting and heuristic algorithms of the present invention. Detailed Implementation
[0135] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings. However, the present invention can be implemented in many different ways as defined and covered by the claims.
[0136] This embodiment uses a simple numerical example to illustrate the specific application of the present invention in eigenvalue problem optimization.
[0137] See the flowchart for the heuristic optimization method. Figure 2 This embodiment uses Figure 4 Taking a corrugated sandwich panel consisting of 61 nodes and 118 elements as an example, structural eigenvalue optimization is performed. The structural material parameters are E = 71 GPa, v = 0.3, and ρ = 2700 kg / m². 3 For a plate thickness of t = 0.02 m, an immune genetic algorithm is used as the optimization method. The flowchart is as follows: Figure 3 This includes the following steps:
[0138] 1) Establish an optimization model for plate composite structures.
[0139] With lightweighting as the objective and the fundamental frequency lower limit as the constraint, the design variables are the thickness of 118 plate elements and the coordinates of 57 nodes. The node coordinates are the horizontal coordinates of the middle nodes, and the movement range of the middle nodes is within 0.025m. The four nodes at both ends of the plate are fixed nodes. The optimization model is established as follows:
[0140] Find x∈R n
[0141]
[0142]
[0143] K * (λ1,x)d1=0
[0144] x l ≤x≤x u
[0145] 2) Based on the theory of dynamic stiffness of plates, the dynamic stiffness matrix K(ω) is derived.
[0146]
[0147] in,
[0148]
[0149]
[0150]
[0151]
[0152]
[0153] k 22 =(r1δ1-r2δ2)(C h1 S h2 δ1-C h2 S h1 δ2)
[0154] k 24 =(r2δ2-r1δ1)(S h2 δ1-S h1 δ2)
[0155] C hi =cosh(r i b),S hi =sinh(r i b)
[0156]
[0157]
[0158]
[0159] 3) Utilizing the WW algorithm
[0160]
[0161] The first natural frequency ω1 of the initial structure was calculated. l =315.6Hz, calculate the initial mass. 0 =367.5kg.
[0162] 4) Transform the original constraints With K * (λ1,x)d1=0 is
[0163] Construct a normalized global multivariate objective function by combining the constraints and the objective function.
[0164]
[0165] 5) Optimize the design variables based on the immune genetic algorithm, including: randomly generating the initial population; calculating affinity, antibody concentration and activation degree based on the normalized multinomial objective function; immunization operation; population refresh; loop termination; outputting the optimal design variables.
[0166] See the design variables for the numerical results. Figure 5The optimized mass decreased from 668.1 kg to 241.9 kg, a reduction of 63.8%. The optimized fundamental frequency was 319.6 Hz. Within the frequency constraint, the entire optimization process took 618.3 s to compute. The first mode shapes before and after optimization are as follows: Figure 6 , Figure 7 As shown.
[0167] Furthermore, the technical solutions described in the embodiments of this invention can be combined arbitrarily without conflict. The above descriptions are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention.
Claims
1. A modal count and heuristic algorithm based transcendental eigenvalue problem optimization method characterized in that, Includes the following steps: S1) Construct an optimization model for the structural eigenvalue problem. Based on different optimization elements such as size, shape, and topology, this model clarifies the functional representation of the objective function, design variables, and constraints under different optimization elements. S2) For structural eigenvalue problems of free vibration, buckling and wave propagation, an analytical model of complex combined structures containing material and geometric parameters is constructed using dynamic stiffness modeling method. This model is used to obtain the transcendental function matrix related to the eigenvalues of frequency or buckling coefficient, thus realizing parametric modeling with a smaller degree of freedom. S3) Use a monotonically increasing modal counting function with integer properties to construct constraints or objective functions, and use efficient and robust modal counting calculations to avoid time-consuming bisection eigenvalue calculations. The optimization method in step S3 is based on the modal counting function, thereby transforming the optimization problem of different eigenvalues into a modal counting optimization problem, including: For lightweight problems under eigenvalue lower bound constraints, traditional optimization methods can be written as follows: The modal counting function can be expressed as: For lightweight problems with specified eigenvalue constraints, traditional optimization methods can be written as follows: The modal counting function can be expressed as: For lightweighting problems under band-limited constraints, traditional optimization methods can be written as follows: The modal counting function can be expressed as: For the bandwidth maximization problem under quality constraints, traditional optimization methods can be written as follows: The modal counting function can be expressed as: ； S4) For various types of optimization problems, the modal counting function is combined with the normalization method and reasonable weights are set so as to transform the constraints or objectives under different eigenvalue optimization problems into specific mathematical representations and construct a global objective function to achieve single or multiple objective optimization. S5) The global objective function is solved by using a heuristic algorithm. The global optimal solution is obtained effectively without considering the differentiability of the modal counting function, and the optimal design variables and sub-objective values are output.
2. The optimization method of claim 1, wherein, The step S1 of constructing an optimization model for the structural eigenvalue problem specifically includes: The defined eigenvalue optimization problems include lightweight structural design under eigenvalue upper and lower bound constraints, specific eigenvalue constraints, or bandgap width constraints, as well as eigenvalue maximization or bandgap width maximization problems under mass constraints. At the same time, it avoids the common problems of repeated eigenvalues, local modes, and mode switching in traditional structural eigenvalue optimization calculations.
3. The optimization method according to claim 1, characterized in that, The dynamic stiffness modeling method in step S2 is used to construct an analytical model of complex combined structures, specifically including: Use the dynamic stiffness method to establish mathematical models for vibration mode problems or buckling problems of different beam and plate composite structures; If it is a vibration modal problem, then the model includes frequencies. The dynamic stiffness matrix of the structure; If it is a buckling problem, the model includes a buckling load factor. The stability stiffness matrix of the structure.
4. The optimization method according to claim 1, characterized in that, The optimization method in step S3 is based on the modal counting function, which describes the structure as exceeding a given eigenvalue among all eigenvalues. Number of Eigenvalue sizes include: Based on the monotonically increasing integer step property of the modal counting function, at any eigenvalue... Represented as The form is as follows The Wittrick-Williams algorithm was used to calculate the mode counts across all eigenvalues of the structure transcendental eigenvalues. In the formula When both ends of each unit are fixed, the eigenvalue of the structure is lower than Total number This indicates that Gaussian elimination or LDLT decomposition is used. The number of negative elements on the main diagonal after transforming into an upper triangular matrix.
5. The optimization method according to claim 1, characterized in that, In step S4, for various types of optimization problems, the modal counting function is combined with a normalization method and appropriate weights are set to transform the constraints or objectives under different eigenvalue optimization problems into specific mathematical representations, including: For the structural mass minimization objective representation Can write ; For single eigenvalue lower bound constraints Can write ; For multiple eigenvalue lower bound constraints Can write , which is a conditional statement; For specified eigenvalue constraints Can write ; For the eigenvalue maximization objective Can write For frequency band restricted areas Can write ; For the goal of maximizing frequency band Can write 。 6. The optimization method according to claim 1, characterized in that, In step S5, an optimization design based on heuristic algorithms is adopted to achieve single-objective or multi-objective optimization and obtain the global optimal solution.