CN-116525031-B - Topological optimization method considering stability and frequency of structural gravity

CN116525031BCN 116525031 BCN116525031 BCN 116525031BCN-116525031-B

Abstract

The application provides a topological optimization method considering stability and frequency of structural gravity, which is characterized by constructing a buckling eigenvalue equation considering influence of structural dead weight based on a small displacement gradient assumption of structural instability, constructing a multi-target structural topological optimization model taking minimized compliance and maximized structural critical natural frequency as targets based on the buckling eigenvalue equation, introducing a limit variable which is not directly related to the compliance and the structural natural frequency, constructing an action relation between the compliance and the structural natural frequency and the limit variable, coupling the structural natural frequency and the compliance by taking the limit variable as an intermediate medium, converting the multi-target structural topological optimization model into a single-target structural topological optimization model taking the limit variable as an optimization target, and solving the single-target structural topological optimization model for multiple iterations until the model converges. The application can improve the calculation precision.

Inventors

YI JIJUN
RONG JIANHUA
FENG JUNJIE
ZHONG YU
ZHAO LEI
ZHAO ZHIJUN

Assignees

长沙理工大学

Dates

Publication Date: 20260508
Application Date: 20230314

Claims (5)

1. A topology optimization method considering stability and frequency of structural gravity, comprising the steps of: s1, establishing a finite element model of a structure to be optimized based on a mixed stress unit; S2, constructing a buckling characteristic value equation considering the influence of the dead weight of the structure based on a small displacement gradient assumption of the structural instability; S3, constructing a multi-objective structure topology optimization model with the aim of minimizing compliance and maximizing critical natural frequency of the structure based on the buckling eigenvalue equation; S4, introducing a limit variable which is not directly related to the flexibility and the structure natural frequency, constructing the action relation between the flexibility and the structure natural frequency and the limit variable, coupling the structure natural frequency and the flexibility by taking the limit variable as an intermediate medium, and converting the multi-objective structure topology optimization model in the step S3 into a single-objective structure topology optimization model taking the limit variable as an optimization target; S5, executing buckling and frequency analysis aiming at the formed single-target structure topological optimization model to obtain a sensitivity analysis result, solving the single-target structure topological optimization model by combining with the sensitivity analysis, and iterating for a plurality of times until the model converges; Action relation of compliance and said limit variable Expressed as: ; in the formula, The softness is represented; Representing the initial value of the flexibility obtained in the first step of iterative solution; Represents a limit variable; Functional relation of natural frequency of structure and limit variable Expressed as: ; in the formula, Representing the function of the norm agglomeration, ; ; Representing the coacervation factor, taking ; And (5) solving the initial value of the norm condensation function obtained in the first step for iteration.
2. The method for topological optimization taking into account the stability and frequency of structural gravity according to claim 1, wherein said step S1 is specifically: Constructing a smooth penalty function reasonably matched with the unit stiffness matrix and the stress stiffness matrix and the unit density, wherein the smooth penalty function is expressed as follows: ; in the formula, And Respectively the first The intrinsic stiffness matrix and stiffness matrix of the number element, And Respectively the first The intrinsic stress stiffness matrix and the stress stiffness matrix of the number cell, Is the first Physical variable of number element, its lower limit ; Taking the parameters of the stiffness penalty polynomial function ; And Taking as penalty factor , ; Is the first The density of the number units is determined, Is the density of the substrate; First, the Matrix of intrinsic stiffness of number elements And an intrinsic stress stiffness matrix The calculation process of (1) is as follows: ; in the formula, And Respectively the first A local natural coordinate variable of the number unit; the thickness of the cell is indicated as such, Representing jacobian; Representing a displacement shape function bias matrix; 、 And Respectively represent the first An elastic matrix, a strain displacement matrix and a stress matrix of the number unit; representing a matrix transpose; 、 And The expression is as follows: ; ; ; in the formula, 、 Taking half of the side length of the unit respectively; 、 the elastic modulus and poisson ratio of the substrate are respectively; 、、 Respectively the first Some point edge in the number unit Axial stress component, edge An axial stress component and a shear stress component.
3. The method of topological optimization taking into account the stability and frequency of structural gravity according to claim 2, wherein the buckling eigenvalue equation is expressed as: ; in the formula, Is the first The critical buckling load factor of the order, And (2) and ; Is a buckling mode; Is the overall elastic stiffness matrix of the structure; Is an overall geometric stiffness matrix of the structure, wherein Is a structural displacement vector; the load on the structure comprises gravity load and external load which depend on density, and the load vector Can be expressed as: in the formula, For an external load on the structure G is the structure gravity load vector, In which, in the process, The node is represented by a set of nodes, Is the total number of nodes; The gravity load of each unit is evenly distributed to each node of the unit, so that the equivalent gravity load borne by each node is the sum of the gravity components of all units connected with the node, namely: ; in the formula, Is the first to A set of unit numbers connected by the individual nodes; gravitational acceleration; Is the first The volume of the number unit; Order the Is a buckling load factor Inverse of (i.e.) The buckling eigenvalue equation is rewritten as: ; Before consideration of The effect of the order real buckling mode on the structure is constructed as the following buckling constraint expression: ; in the formula, Is a prescribed buckling load factor lower limit value; In order to correct the coefficient of the coefficient, ; As KS condensation function of the inverse buckling load factor, In the formula, Is taken as a coacervation factor , ; Is to introduce spacing factors for solving the problem of heavy eigenvalue The transformed eigenvalue is transformed into Taking out 。
4. A method of topological optimization taking into account the stability and frequency of structural gravity according to claim 3, wherein the multi-objective structural topological optimization model is expressed as: ; in the formula, Is the flexibility of the structure; As the natural frequency of the structure, Is the first of the structure The order natural frequency; Is the total mass matrix of the structure; Is the first Regularized vibration mode vector corresponding to the order natural frequency; as a variable which is to be taken as a result, In order to optimize the volume of the structure, Relaxation parameter is the lower limit of volume, take ; The structural volume when the design domain is filled with material; Represent the first Number cell intrinsic volume; Is the total number of units; to extract the natural frequency order 。
5. The topological optimization method considering the stability and the frequency of the structural gravity according to claim 4, wherein the single-objective structural topological optimization model is expressed as: 。

Description

Topological optimization method considering stability and frequency of structural gravity Technical Field The application belongs to the technical field of structural optimization design, and particularly relates to a topological optimization method considering stability and frequency of structural gravity. Background The structural topology optimization (Structural topology optimization) method can seek the best distribution of materials in a given design domain to achieve a structural topology with optimal target performance while satisfying a range of constraints. In the existing study considering buckling, in order to simplify a study model and reduce complexity, the coupling condition of buckling and other influencing factors (such as the gravity of the structure) is not considered generally, but under the condition that the gravity of some structures occupies a large proportion of the total load, the influence of the gravity load of the structure is ignored, so that the inaccuracy of a calculation result can be caused, and the optimization requirement cannot be met. Therefore, it is necessary to provide a topology optimization method considering stability and frequency of structural gravity, so as to solve the problems in the prior art. Disclosure of Invention The application provides a topological optimization method considering stability and frequency of structure gravity, buckling and structure dead weight are comprehensively considered, a structure light-weight design can be obtained in engineering design, meanwhile, static and dynamic characteristics and structure stability are considered, the parasitic effect of gravity load in the optimization process is reduced, and the calculation precision can be improved. In order to solve the technical problems, the application is realized as follows: A topological optimization method considering stability and frequency of structural gravity comprises the following steps: s1, establishing a finite element model of a structure to be optimized based on a mixed stress unit; S2, constructing a buckling characteristic value equation considering the influence of the dead weight of the structure based on a small displacement gradient assumption of the structural instability; S3, constructing a multi-objective structure topology optimization model with the aim of minimizing compliance and maximizing critical natural frequency of the structure based on the buckling eigenvalue equation; S4, introducing a limit variable which is not directly related to the flexibility and the structure natural frequency, constructing the action relation between the flexibility and the structure natural frequency and the limit variable, coupling the structure natural frequency and the flexibility by taking the limit variable as an intermediate medium, and converting the multi-objective structure topology optimization model in the step S3 into a single-objective structure topology optimization model taking the limit variable as an optimization target; and S5, executing buckling and frequency analysis aiming at the formed single-target structure topological optimization model to obtain a sensitivity analysis result, solving the single-target structure topological optimization model by combining with the sensitivity analysis, and iterating for a plurality of times until the model converges. Preferably, the step S1 specifically includes: Constructing a smooth penalty function reasonably matched with the unit stiffness matrix and the stress stiffness matrix and the unit density, wherein the smooth penalty function is expressed as follows: Where k e,0 and k e are the intrinsic stiffness matrix and the stiffness matrix of the element e respectively, It is known thatThe intrinsic stress stiffness matrix and the stress stiffness matrix of the e-th cell respectively,Is the physical variable of the No. e element, its lower limitH is a parameter of a stiffness punishment polynomial function, h=1/16, p and q are punishment factors, p=4 and q=2, ρ e is the density of an e-th unit, and ρ 0 is the substrate density; Inherent stiffness matrix k e,0 and inherent stress stiffness matrix of cell No. e The calculation process of (1) is as follows: Wherein S and r are local natural coordinate variables of the e-th unit respectively, T represents the thickness of the unit, |J| represents jacobian, g represents a displacement shape function partial guide matrix, H -1, S and Θ (sigma e) represent an elastic matrix, a strain displacement matrix and a stress matrix of the e-th unit respectively, and T represents matrix transposition; The expressions H -1, S, and Θ (σ e) are as follows: Wherein a and b are half of the side length of the unit respectively, E 0 and mu are the elastic modulus and Poisson's ratio of the base material respectively; the stress component along the x-axis, the stress component along the y-axis and the shear stress component at a certain point in the e-th unit respectively. Preferably,