A topology optimization method considering stability and frequency of structural gravity

Abstract

The application provides a topology optimization method considering stability and frequency of structural gravity, constructs a buckling eigenvalue equation considering influence of self-weight of a structure based on a small displacement gradient assumption of structural instability; constructs a multi-objective structure topology optimization model aiming at minimizing flexibility and maximizing critical natural frequency of the structure based on the buckling eigenvalue equation; introduces a limit variable not directly related to flexibility and natural frequency of the structure, constructs an action relationship of flexibility and natural frequency of the structure and the limit variable, and couples natural frequency of the structure and flexibility by taking the limit variable as an intermediate medium, converts the multi-objective structure topology optimization model into a single-objective structure topology optimization model taking the limit variable as an optimization target; and solves the single-objective structure topology optimization model and iterates multiple times until the model converges. The application can improve the calculation accuracy.

Description

Technical Field

[0001] This application belongs to the field of structural optimization design technology, specifically relating to a topology optimization method that considers the stability and frequency of structural gravity. Background Technology

[0002] Structural topology optimization methods seek the optimal material distribution within a given design domain, thereby obtaining a structural topology with optimal target performance while satisfying a series of constraints. In existing studies considering buckling, to simplify the research model and reduce complexity, the coupling between buckling and other influencing factors (such as the structure's own weight) is usually not considered. However, in cases where the structure's gravity accounts for a large proportion of the total load, ignoring the influence of the structure's own gravity load can lead to inaccurate calculation results and fail to meet optimization requirements.

[0003] Therefore, it is necessary to provide a topology optimization method that considers the stability and frequency of structural gravity in order to solve the problems mentioned in the background art. Summary of the Invention

[0004] This application provides a topology optimization method that considers the stability and frequency of structural gravity. By comprehensively considering buckling and structural self-weight, it can achieve lightweight structural design in engineering design, while taking into account static and dynamic characteristics and structural stability. It reduces the parasitic effect of gravity load in the optimization process and can improve the accuracy of calculation.

[0005] To solve the above-mentioned technical problems, this application is implemented as follows:

[0006] A topology optimization method considering structural gravity stability and frequency includes the following steps:

[0007] S1: Establish a finite element model of the structure to be optimized based on hybrid stress elements;

[0008] S2: Based on the assumption of small displacement gradient in structural instability, a buckling eigenvalue equation considering the influence of the structure's self-weight is constructed.

[0009] S3: Based on the buckling eigenvalue equation, construct a multi-objective structural topology optimization model with the objectives of minimizing compliance and maximizing the critical natural frequency of the structure.

[0010] S4: Introduce a boundary variable that is not directly related to compliance and structural natural frequency, construct the interaction relationship between compliance and structural natural frequency and the boundary variable, and use the boundary variable as an intermediate medium to couple structural natural frequency and compliance, and transform the multi-objective structural topology optimization model in step S3 into a single-objective structural topology optimization model with the boundary variable as the optimization objective.

[0011] S5: Perform buckling and frequency analysis on the formed single-objective structural topology optimization model to obtain the sensitivity analysis results. Combine the sensitivity analysis with the solution of the single-objective structural topology optimization model and iterate multiple times until the model converges.

[0012] Preferably, step S1 specifically includes:

[0013] A smooth penalty function is constructed to reasonably match the element stiffness matrix, stress stiffness matrix, and element density, expressed as:

[0014]

[0015] In the formula, k e，0 and k e These are the natural stiffness matrix and stiffness matrix of the e-th element, respectively. Know These are the inherent stress stiffness matrix and stress stiffness matrix of the e-th element, respectively. For the physical variable of unit e, its lower limit h is the parameter of the stiffness penalty polynomial function, taken as h = 1 / 16; p and q are penalty factors, taken as p = 4 and q = 2; ρ e Let ρ be the density of the e-th element. 0 The density of the substrate;

[0016] The inherent stiffness matrix k of the e-th element e，0 and inherent stress stiffness matrix The calculation process is as follows:

[0017] In the formula, s and r are the local natural coordinate variables of the e-th element; t represents the element thickness; |J| represents the Jacobian determinant; g represents the partial derivative matrix of the displacement shape function; H -1 S and Θ(σ) e ) represent the elastic matrix, strain-displacement matrix, and stress matrix of the e-th element, respectively; T represents matrix transpose;

[0018] H -1 S and Θ(σ) e The expression is as follows:

[0019]

[0020]

[0021]

[0022] In the formula, a and b are each half the side length of the unit; E0 and μ are the elastic modulus and Poisson's ratio of the substrate, respectively. These are the stress components along the x-axis, the stress components along the y-axis, and the shear stress components at a point in element e, respectively.

[0023] Preferably, the buckling eigenvalue equation is expressed as:

[0024]

[0025] In the formula, λ i Let λ be the critical buckling load factor of order i, i∈[1,n], and 0<λ1≤λ2≤…≤λ n ; K represents the buckling mode; K is the overall elastic stiffness matrix of the structure; K σ (x, U) is the overall geometric stiffness matrix of the structure, where U is the structural displacement vector;

[0026] The loads on the structure include density-dependent gravity loads and external loads, and the load vector F can be expressed as:

[0027] F = G + P

[0028] In the formula, P is the external load applied to the structure; G is the structural gravity load vector. In the formula, τ represents a node, and N Node The total number of nodes;

[0029] The gravity load of each element is evenly distributed to each node of the element, such that the equivalent gravity load borne by each node is the sum of the gravity components of all elements connected to it, that is:

[0030]

[0031] In the formula, Let g be the set of element numbers connected to the τ-th node; g is the acceleration due to gravity; v e Let e ​​be the volume of the e-th unit;

[0032] Let μ i Buckling load factor λ i The reciprocal of μ i =1 / λ i The buckling eigenvalue equation can be rewritten as:

[0033] Consider the first n d The influence of the first-order true buckling mode on the structure is determined by constructing the following buckling constraint expression:

[0034]

[0035] In the formula, R c The specified lower limit of the buckling load factor; α is a correction factor. The KS condensation function is the reciprocal of the buckling load factor. In the formula, η is the cohesion factor, taken as η = 8, n d =8; This is an eigenvalue transformed by the interval factor χ to solve the problem of repeated eigenvalues. The transformation formula is: Let χ = ​​1.01.

[0036] Preferably, the multi-objective structural topology optimization model is expressed as:

[0037]

[0038] In the formula, C represents the compliance of the structure; ω represents the natural frequency of the structure. j γ is the j-th natural frequency of the structure; M is the total mass matrix of the structure; j Let V be the regularized mode shape vector corresponding to the j-th natural frequency; x is a variable, V f The optimized structural volume is given by θ, which is the relaxation parameter for the lower limit of the volume, taken as θ = 0.004; V (0) The structural volume when the design domain is filled with material; The n represents the intrinsic volume of the e-th element; N is the total number of elements; n t Let n be the order of the natural frequency to be intercepted. t =3.

[0039] Preferably, the relationship h1(x) between compliance and the boundary variable is expressed as:

[0040]

[0041] In the formula, C represents compliance; C0 represents the initial compliance value obtained in the first step of the iterative solution; Indicates a boundary variable;

[0042] The relationship between the structure's natural frequency and the aforementioned limit variable, h2(x), is expressed as:

[0043]

[0044] In the formula, Represents the norm condensation function, ξ represents the cohesion factor, and we take ξ = 6; This is the initial value of the norm condensation function obtained in the first step of the iterative solution.

[0045] Preferably, the single-objective structural topology optimization model is expressed as:

[0046]

[0047] This application provides a topology optimization method considering structural stability and frequency under gravity. A finite element model is constructed using hybrid stress elements. A buckling eigenvalue equation considering the structural self-weight is established. Based on this buckling eigenvalue equation, a multi-objective structural topology optimization model is constructed, aiming to minimize compliance and maximize the critical natural frequency of the structure. A boundary variable unrelated to compliance and structural natural frequency is introduced. The interaction relationship between compliance, structural natural frequency, and the boundary variable is established. The boundary variable acts as an intermediary to couple the structural natural frequency and compliance, transforming the multi-objective structural topology optimization model into a single-objective structural topology optimization model with the boundary variable as the optimization objective, thus simplifying the solution process. The topology optimization method provided in this application comprehensively considers buckling and structural self-weight, enabling lightweight structural designs in engineering, while also taking into account static and dynamic characteristics and structural stability. It reduces the parasitic effects of gravity loads during the optimization process and improves computational accuracy. Attached Figure Description

[0048] Figure 1 This is a schematic diagram showing the initial design domain in Embodiment 1.

[0049] Figure 2 Represents the optimal topology configuration without buckling constraints;

[0050] Figures 3-10 These represent the optimized structures obtained by using the topology optimization method of this application at iteration steps 70, 100, 180, 310, 350, 370, 390, and 401, respectively. Detailed Implementation

[0051] 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 embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0052] This invention provides a topology optimization method that considers the stability and frequency of structural gravity, comprising the following steps:

[0053] S1: Establish a finite element model of the structure to be optimized based on the hybrid stress element.

[0054] In the establishment of the finite element model, hybrid stress elements were selected. In existing technologies, conventional linear elements are widely used in the finite element analysis of continuous structure optimization design, but they cannot accurately describe the characteristics of elements under pure bending. When the structure undergoes linear bending, the approximate model using linear elements exhibits shear locking behavior (SLB), causing a deviation between the calculated bending moment and the correct value, resulting in large errors and low accuracy in the design. The hybrid stress element effectively overcomes the stress stiffening problem of the element and provides more accurate stress analysis results.

[0055] Topology optimization considering self-weight and buckling constraints often suffers from problems such as pseudo-buckling modes caused by the aggregation of low-density elements and parasitic effects in low-density regions due to self-weight. To reduce the number of low-order pseudo-buckling modes in the optimized structure and avoid parasitic effects caused by the self-weight of low-density elements, a smooth penalty function is constructed that reasonably matches the element stiffness matrix, stress stiffness matrix, and element density, expressed as:

[0056]

[0057] In the formula, k e，0 and k e These are the natural stiffness matrix and stiffness matrix of the e-th element, respectively. Know These are the inherent stress stiffness matrix and stress stiffness matrix of the e-th element, respectively. For the physical variable of unit e, its lower limit h is the parameter of the stiffness penalty polynomial function, taken as h = 1 / 16; p and q are penalty factors, taken as p = 4 and q = 2; ρ e Let ρ be the density of the e-th element. 0 This refers to the density of the substrate.

[0058] The inherent stiffness matrix k of the e-th element e，0 and inherent stress stiffness matrix The calculation process is as follows:

[0059] In the formula, s and r are the local natural coordinate variables of the e-th element; t represents the element thickness; |J| represents the Jacobian determinant; g represents the partial derivative matrix of the displacement shape function; H -1 S and Θ(σ) e ) represent the elastic matrix, strain-displacement matrix, and stress matrix of the e-th element, respectively; T represents matrix transpose.

[0060] H -1 S and Θ(σ) e The expression is as follows:

[0061]

[0062]

[0063]

[0064] In the formula, a and b are each half the side length of the unit; E0 and μ are the elastic modulus and Poisson's ratio of the substrate, respectively. These are the stress components along the x-axis, the stress components along the y-axis, and the shear stress components at a point in element e, respectively.

[0065] S2: Based on the assumption of small displacement gradient in structural instability, a buckling eigenvalue equation considering the influence of the structure's self-weight is constructed.

[0066] The buckling eigenvalue equation is expressed as:

[0067]

[0068] In the formula, λ i Let λ be the critical buckling load factor of order i, i∈[1,n], and 0<λ1≤λ2≤…≤λ n ; K represents the buckling mode; K is the overall elastic stiffness matrix of the structure; K σ (x, U) is the overall geometric stiffness matrix of the structure, where U is the structural displacement vector.

[0069] The loads on the structure include density-dependent gravity loads and external loads. The load vector F can be expressed as: F = G + P;

[0070] In the formula, P is the external load applied to the structure; G is the structural gravity load vector. In the formula, τ represents a node, and N Node This represents the total number of nodes.

[0071] The gravity load of each element is evenly distributed to each node of the element, such that the equivalent gravity load borne by each node is the sum of the gravity components of all elements connected to it, that is:

[0072]

[0073] In the formula, Let g be the set of element numbers connected to the τ-th node; g is the acceleration due to gravity; v e Let e ​​be the volume of the e-th unit.

[0074] To ensure the validity of the Lanczos iteration, let μ i Buckling load factor λ i The reciprocal of μ i=1 / λ i The buckling eigenvalue equation can be rewritten as:

[0075]

[0076] To improve the buckling resistance of the structure under the influence of its own weight and external loads, and taking into account both the calculation results and the calculation cost, this application considers the first n... d The influence of the first-order true buckling mode on the structure is determined, and the buckling constraint expression is constructed as follows:

[0077]

[0078] In the formula, R c The specified lower limit of the buckling load factor; α is a correction factor. The KS condensation function is the reciprocal of the buckling load factor. In the formula, η is the cohesion factor, taken as η = 8, n d =8; This is an eigenvalue transformed by the interval factor χ to solve the problem of repeated eigenvalues. The transformation formula is: Let χ = ​​1.01.

[0079] S3: Based on the buckling eigenvalue equation, construct a multi-objective structural topology optimization model with the objectives of minimizing compliance and maximizing the critical natural frequency of the structure.

[0080] The multi-objective structural topology optimization model is expressed as follows:

[0081]

[0082] In the formula, C represents the compliance of the structure; ω represents the natural frequency of the structure. j γ is the j-th natural frequency of the structure; M is the total mass matrix of the structure; j Let V be the regularized mode shape vector corresponding to the j-th natural frequency; x is a variable, V f The optimized structural volume is given by θ, which is the relaxation parameter for the lower limit of the volume, taken as θ = 0.004; V (0) The structural volume when the design domain is filled with material; The n represents the intrinsic volume of the e-th element; N is the total number of elements; n t Let n be the order of the natural frequency to be intercepted. t =3.

[0083] S4: Introduce a boundary variable that is not directly related to compliance and structural natural frequency, construct the interaction relationship between compliance and structural natural frequency and the boundary variable, and use the boundary variable as an intermediate medium to couple structural natural frequency and compliance, thereby converting the multi-objective structural topology optimization model in step S3 into a single-objective structural topology optimization model with the boundary variable as the optimization objective.

[0084] The relationship between compliance and the aforementioned boundary variable, h1(x), is expressed as:

[0085]

[0086] In the formula, C represents compliance; C0 represents the initial compliance value obtained in the first step of the iterative solution;

[0087] The relationship between the structure's natural frequency and the aforementioned limit variable, h2(x), is expressed as:

[0088]

[0089] In the formula, Represents the norm condensation function, j = 1, 2, ..., n t , ξ represents the cohesion factor, and we take ξ = 6; This is the initial value of the norm condensation function obtained in the first step of the iterative solution.

[0090] Choosing the initial compliance value C0 obtained in the first step of the iterative solution as the parameter for compliance normalization can normalize compliance, and then pass the boundary variable... Coupling with the structure's natural frequencies. Considering the influence of multiple natural frequencies on the structure, the objective of maximizing the critical natural frequency of the structure is transformed by introducing a norm condensation function. This allows the objectives of maximizing the critical natural frequency and minimizing compliance to be better coupled using the same boundary.

[0091] The single-objective structural topology optimization model is expressed as follows:

[0092]

[0093] S5: Perform buckling and frequency analysis on the formed single-objective structural topology optimization model to obtain the sensitivity analysis results. Combine the sensitivity analysis with the solution of the single-objective structural topology optimization model and iterate multiple times until the model converges.

[0094] Sensitivity analysis specifically includes the following process:

[0095] Gravity along the negative y-axis affects physical variables Differentiating the equations yields the effect of the equivalent gravitational load on the physical variables at the nodes. The derivative is:

[0096]

[0097] In the formula, Let γ, w, and ρ represent the set of global node numbers for the e-th element. 0 These represent the global node index, global node index, and intrinsic density of the substrate for the e-th element, respectively.

[0098] Compliance C versus physical variables The derivative is:

[0099]

[0100] In the formula, U e G represents the displacement of the e-th element. e This represents the gravity load vector of the e-th element.

[0101] Buckling constraint f(x) on physical variables The derivative is:

[0102]

[0103] In the formula, the accompanying variable Obtained from the following equation:

[0104]

[0105]

[0106] h1(x) for any physical variable The derivative is:

[0107]

[0108] The eigenvalue equations for the structural dynamic characteristics can be expressed as:

[0109]

[0110] Multiply both sides of the first equation of the structural dynamic characteristics by the left side. have to:

[0111]

[0112] Simultaneously apply physical variables to both sides of the above equation Differentiating and rearranging, we get

[0113]

[0114] Combining the second equation of the eigenvalue equation of the structural dynamic characteristics, the square of the structure's natural frequency can be obtained. For physical variables The derivative is:

[0115]

[0116] Convert the natural frequency Substituting into the above equation, we get:

[0117]

[0118] If it is a constant value, then h2(x) depends on the physical variable. The derivative can be expressed as:

[0119]

[0120] According to the chain rule, the derivatives of the objective function and constraint functions with respect to the design variables can be easily obtained. The MMA algorithm is used for iterative solution. The iteration terminates when the optimization model satisfies the convergence condition, which is expressed as:

[0121]

[0122] Where ε1 = 0.001, ε2 = 0.05.

[0123] Example 1

[0124] Please see Figures 1-10 Initial design domain such as Figure 4 As shown, its dimensions are 0.6m × 1.2m, and its thickness is 0.002m. The bottom surface is fixed, and the load P is 5.5 × 10⁻⁶ m. 3 N / m, directed downwards, uniformly distributed in region d, with a length of 0.1m.

[0125] The structural material has an elastic modulus of 200 GPa and a Poisson's ratio of 0.3. The initial design domain is divided into 7200 plane strain elements of equal size, and a target volume ratio θ = 0.30 is specified. Figure 2 The optimized topology obtained without buckling constraints is shown, with a compliance of 0.32 N·m, a critical natural frequency of 44.5 Hz, and a critical buckling load factor of 90.

[0126] Please see Figures 3-10 When the critical buckling load factor R c =170, Figures 3-10 The optimized structures are shown for iterations at steps 70, 100, 180, 310, 350, 370, 390, and 401, respectively. The topology of the optimized structure obtained using the method of this application is shown below. Figure 10The final optimized structure has a structural compliance of 0.5 N·m, a critical natural frequency of 33.7 Hz, a critical buckling load factor of 201.9, and a critical buckling load enhancement ratio of 124.3.

[0127] The optimization iteration process revealed that this method effectively suppresses parasitic effects and improves computational accuracy. Optimization data shows that the target volume is also well controlled. The strain energy of the upper and lower support rods in the topology-optimized configuration is higher than that in the middle region, leading to changes primarily occurring in the upper and lower support rods during optimization to enhance structural stability.

[0128] The embodiments of this application have been described above with reference to the accompanying drawings. However, this application is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of this application without departing from the spirit and scope of the claims, and all of these forms are within the protection scope of this application.

Claims

1. A topology optimization method considering the stability and frequency of structural gravity, characterized in that, Includes the following steps: S1: Establish a finite element model of the structure to be optimized based on hybrid stress elements; S2: Based on the assumption of small displacement gradient in structural instability, a buckling eigenvalue equation considering the influence of the structure's self-weight is constructed. S3: Based on the buckling eigenvalue equation, construct a multi-objective structural topology optimization model with the objectives of minimizing compliance and maximizing the critical natural frequency of the structure. S4: Introduce a boundary variable that is not directly related to compliance and structural natural frequency, construct the interaction relationship between compliance and structural natural frequency and the boundary variable, and use the boundary variable as an intermediate medium to couple structural natural frequency and compliance, and transform the multi-objective structural topology optimization model in step S3 into a single-objective structural topology optimization model with the boundary variable as the optimization objective. S5: Perform buckling and frequency analysis on the formed single-objective structural topology optimization model to obtain the sensitivity analysis results. Combine the sensitivity analysis with the solution of the single-objective structural topology optimization model and iterate multiple times until the model converges. The relationship between compliance and the aforementioned limit variable Represented as: ； In the formula, Indicates flexibility; This represents the initial compliance value obtained in the first step of the iterative solution; Indicates a boundary variable; The relationship between the structure's natural frequency and the aforementioned limit variable Represented as: ； In the formula, Represents the norm condensation function, ; ; Represents the cohesion factor, taking ; This is the initial value of the norm condensation function obtained in the first step of the iterative solution.

2. The topology optimization method considering structural gravity stability and frequency according to claim 1, characterized in that, Step S1 specifically involves: A smooth penalty function is constructed to reasonably match the element stiffness matrix, stress stiffness matrix, and element density, expressed as: ； In the formula, and The first The inherent stiffness matrix and stiffness matrix of element number 1. and The first The inherent stress stiffness matrix and stress stiffness matrix of element number 1. For the first The physical variables of unit number 1, their lower limit ; The parameters of the stiffness-penalized polynomial function are taken as follows: ; and As a penalty factor, take , ; For the first The density of cell number 1 The density of the substrate; No. The inherent stiffness matrix of element number 1 and inherent stress stiffness matrix The calculation process is as follows: ; In the formula, and The first Local natural coordinate variables of cell number; Indicates the unit thickness. Represents the Jacobian determinant; This represents the partial derivative matrix of the displacement shape function; , and They represent the first The elastic matrix, strain-displacement matrix, and stress matrix of element number 1; Indicates matrix transpose; , and The expression is as follows: ； ； ； In the formula, , Take half the side length of each unit; , These are the elastic modulus and Poisson's ratio of the substrate, respectively. , , The first A point in cell number along Axial stress components, along Axial stress components and shear stress components.

3. The topology optimization method considering structural gravity stability and frequency according to claim 2, characterized in that, The buckling eigenvalue equation is expressed as: ； In the formula, For the first First-order critical buckling load factor ,and ; For buckling mode; This is the overall elastic stiffness matrix of the structure; Let be the global geometric stiffness matrix of the structure, where The structural displacement vector; The loads on the structure include density-dependent gravity loads and external loads, and the load vector... It can be represented as: In the formula, G is the external load applied to the structure; G is the structural gravity load vector. In the formula, Represents a node. The total number of nodes; The gravity load of each element is evenly distributed to each node of the element, such that the equivalent gravity load borne by each node is the sum of the gravity components of all elements connected to it, i.e.: ； In the formula, In order to be with the first A set of unit numbers connected to each node; It is the acceleration due to gravity; For the first Volume of unit number 1; make Buckling load factor The reciprocal of, that is The buckling eigenvalue equation can be rewritten as: ； Before considering The influence of the first-order true buckling mode on the structure is determined by constructing the following buckling constraint expression: ； In the formula, The specified lower limit of the buckling load factor; For correction factor, ; The KS condensation function is the reciprocal of the buckling load factor. In the formula, As the cohesion factor, take , ; The margin factor is introduced to solve problems involving repeated eigenvalues. The transformed eigenvalues ​​are expressed as follows: ,Pick .

4. The topology optimization method considering structural gravity stability and frequency according to claim 3, characterized in that, The multi-objective structural topology optimization model is expressed as follows: ； In the formula, For the flexibility of the structure; The natural frequency of the structure, For the structure of the first First natural frequency; The total mass matrix of the structure; For the first The regularized mode shape vector corresponding to the first natural frequency; As variables, For the optimized structural volume, The relaxation parameter is the lower limit of the volume. ; The structural volume when the design domain is filled with material; Indicates the first The inherent volume of unit number 1; The total number of units; To determine the order of the natural frequency being intercepted, take... .

5. The topology optimization method considering structural gravity stability and frequency according to claim 4, characterized in that, The single-objective structural topology optimization model is expressed as follows: 。

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