JP5239660B2 - Structure design method and program - Google Patents

Description

  The present invention relates to a structure design method, and more particularly to a topology optimum design method using beam elements.

  CAE is widely applied to the development of products such as automobiles. There, a calculation model exceeding one million degrees of freedom is used to guarantee high accuracy, and it is possible to reduce the number of prototype vehicles, thereby contributing to cost and time reduction. However, if the calculation result does not satisfy the target, it takes a lot of time and effort to identify the cause and create an effective countermeasure plan. In particular, in the case of a structure vibration problem, finding a countermeasure site and a countermeasure structure with a minimum mass increase is an important issue. In general, optimization problems can be classified into size optimization problems, shape optimization problems, and topology optimization problems. In the vibration problem, the application of shape optimization and phase optimization remains at the component level. In the vibration problem of a large-scale structure, the dimension optimization problem is used.

  Researches that apply topology optimization methods to vibration problems include eigenvalue problems and frequency response problems for basic structures such as flat plates.

  On the other hand, topology optimal design problems using beam elements of discretized structural elements are also handled as stiffness problems and eigenvalue problems. In Patent Document 1 below, a design model in which adjacent nodes and non-adjacent nodes are connected by beam elements having six-axis rigidity is created and applied as an average compliance, which is an objective function that optimizes the design model. The inner product of force and displacement is set, and while satisfying the predetermined volume constraint condition or weight constraint condition, the stiffness of the beam element in the design model is changed, the sensitivity for the average compliance is detected, and the detected sensitivity Thus, there is disclosed a technique for optimizing the shape of the design model so as to maximize the rigidity by minimizing the average compliance.

  In addition, in order to easily solve a large-scale problem, a technique for reducing the degree of freedom by a degeneracy technique is also known. There are two main types: the degeneration method using a modal coordinate system and the degeneration method while maintaining the physical coordinate system. Patent Document 2 below discloses the latter method.

Japanese Patent No. 3551910 Japanese Patent Laid-Open No. 2001-126087

  In the dimension optimization problem, since design variables are limited due to calculation restrictions, there is a problem that the optimum value is not considered in consideration of all possibilities in the design space.

  In addition, the topology optimal design method using continuum elements requires a lot of processing time when applied to a large-scale model such as a vehicle body structure, and thus has a problem that it is not a practical tool.

  On the other hand, the topology optimal design method using beam elements is applied to the stiffness problem and the eigenvalue problem, but is not applied to the frequency response problem at all. In addition, the basic structure is mainly used, and although it is an effective method at the conceptual design stage, there is a problem that is not suitable for the detailed design stage.

  An object of the present invention is to provide a method capable of finding a countermeasure site and a countermeasure structure with a minimum mass increase by analyzing a frequency response characteristic at an arbitrary frequency in a topology optimum design method using a beam element. It is in.

The present invention is a method for designing a structure using a computer, and the structure includes a shell element and a beam element arranged as a reinforcing material at least between adjacent nodes of the shell element. And causing the computer processor to create a dynamic stiffness matrix including mass, damping, and stiffness as characteristics of the shell element and the beam element, respectively, and the dynamic stiffness matrix of the beam element and the dynamic stiffness of the shell element A step of creating a whole matrix by superimposing the matrix, a step of deriving a motion equation of the structure using the whole matrix, and exciting an evaluation node of the structure at a predetermined frequency based on the motion equation And a step of deriving the amplitude as an objective function, the shell element as a non-design element, the beam element as a design element, and the objective function as Characterized in that to perform the step of optimizing the total volume of constraint conditions of the beam the beam element the arrangement of elements to a small reduction.

Further, the present invention is a program for causing a computer to design a model of a structure, the structure including a shell element and a beam element disposed as a reinforcing material at least between adjacent nodes of the shell element. And the program causes a computer processor to create a dynamic stiffness matrix including mass, damping, and stiffness as characteristics of the shell element and the beam element, and the dynamic stiffness matrix of the beam element and the A step of creating an overall matrix by superimposing a dynamic stiffness matrix of a shell element; a step of deriving a motion equation of the structure using the overall matrix; and a predetermined evaluation node of the structure based on the motion equation A step of deriving an amplitude as an objective function when vibrating at a frequency of a non-design element, the beam element As a design element, characterized in that to perform the step of optimizing the total volume of constraint conditions of the beam element arrangement of said beam elements so as to minimize the objective function.

  According to the present invention, the frequency response characteristic can be obtained by using the dynamic rigidity of the target structure. Further, according to the present invention, a countermeasure site and its structural characteristics that can achieve the maximum effect with a specified increase in mass can be obtained. Further, according to the present invention, a stable and high-speed optimization calculation can be performed by using a beam element as a design element. Further, according to the present invention, by making the main structure a non-design element, it is possible to improve only an arbitrary frequency characteristic while maintaining the overall rigidity.

  Hereinafter, embodiments of the present invention will be described with reference to the drawings.

  FIG. 1 shows a target area 1 composed of structural elements in the present embodiment. The target area is composed of a shell element 10 and a beam element 12. The beam element 12 connects between adjacent nodes and non-adjacent nodes and has six-axis rigidity. A certain node (node) 14 is vibrated, and the amplitude w of the evaluation node 16 for an arbitrary frequency is used as an objective function. A fixed area 18 is set in the target area 1 as necessary. The discretized equation of motion of this structure is

  Here, {F} and {U} are an excitation vector and a displacement vector, respectively, [M], [C], and [K] are mass, damping, and stiffness matrices, respectively, [B] is a dynamic stiffness matrix, ω represents the angular frequency. Using the mutual average compliance l, the response of the evaluation node 16 is given by

The objective function w (ω) is given by

Here, {G} and {V} respectively indicate a unit excitation vector of the evaluation node 16 and a displacement vector when the force is applied. L ^* represents a conjugate value of l.

  Next, the beam element 12 is arranged based on the ground structure method in a fixed design region where the shell element 10 as shown in FIG. In the ground structure method, all beam elements 12 are arranged between adjacent nodes. If necessary, the beam elements 12 can be arranged so as to extend between non-adjacent nodes. In topology optimization, in order to determine the setting of a fixed design region and the presence or absence of a structural element to be arranged in the fixed design region, it is based on a characteristic function of the following equation.

  Here, Ω is a fixed design region, Ωd is a region forming an optimal structure selected by optimization from the fixed design region, and x is a position coordinate in the fixed design region Ω. By using this characteristic function, the optimum design problem can be replaced with an element placement problem. When this characteristic function is used, a discretized variable must be handled, and calculational handling becomes impossible. Therefore, the characteristic function is replaced with a continuous function by the following formula based on the density method.

  Here, ρv is a normalized design variable. Ρ is a parameter that gives a penalty to ρv. By using the above equation, the optimum design problem can be replaced with a continuous distribution problem of elements within the fixed design region Ω. In the present embodiment, only the beam element 12 is handled as a design element, and the shell element 10 is a non-design element. If the above equation is applied to the beam element 12 having the maximum cross-sectional area Amax and length L, the volume V becomes the following equation, and the arrangement of the beam element 12 can be determined by the value of ρv. Using this normalized area density ρv as a design variable for each element, an optimization problem is set.

  FIG. 2 shows a beam element 12 having a thin cross section. Compared with the solid cross section, the thin cross section can have higher bending rigidity and torsional rigidity. Therefore, when the contribution of bending and torsional rigidity is large, a necessary reduction effect may be obtained with a small additional mass. The relationship between the diameter and the wall thickness is defined by the following equation based on the wall thickness ratio γ.

  Here, γ = 1 indicates a solid section. Using the above equation, the cross-sectional area A, the cross-sectional secondary moments Iy and Iz, the cross-sectional secondary pole moment J, and the mass m are defined by the following equations.

  Here, ρm represents the element mass density. A dynamic stiffness matrix is created from the cross-sectional characteristics of these beams. Further, an overall matrix is created by superimposing the dynamic stiffness matrices of the shell elements 10 as the main structure, and the value of the objective function is derived from the frequency response analysis.

  When the dynamic stiffness matrix of the formula (1) is represented by the dynamic stiffness matrices [B0] and [Ba] of the base shell element 10 and the beam element 12, the following formula is obtained.

The sensitivity of mutual average compliance to the normalized design variable ρv is given by

This sensitivity can be derived from the sensitivity of the design variable with respect to the dynamic rigidity Ba of the beam element 12 and the two types of response vectors {V} and {U}. Further, the sensitivity to the amplitude is expressed by the following equation.

  This sensitivity is applied to the optimization algorithm, and the optimal solution is derived by updating the design variables. It is also possible to use a degenerated matrix as the base dynamic stiffness matrix. Degeneration by a transfer function is used as a method for degenerating the dynamic stiffness matrix. This is because the effect of adding rigidity can be directly evaluated and the accuracy of the output result of the vibration response is guaranteed even after degeneration. If the transfer function between the degrees of freedom to be reduced is maintained, the relationship between the input and output before and after the reduction does not depend on the number of degrees of freedom and there is no difference other than numerical error. Since the sensitivity of the rigidity value is obtained from the output result of the vibration response, the accuracy of the output result after degeneration is important in performing the optimization calculation.

  The dynamic stiffness matrix is divided into a region O of degree of freedom to degenerate and a region b of degree of freedom to erase, and is expressed as the following equation.

Since no external force acts on the region in the above equation, the degenerated dynamic stiffness matrix is expressed by the following equation.

For the fixed region shown in FIG. 1, the amplitude value of the evaluation node 16 at the angular frequency ωj is ωj. The objective function W is expressed as a weighted linear sum of m-line frequencies to be considered. c ^j represents a weighting factor for each frequency. The shell element 10 is a non-design element, and the n beam elements 12 are design elements. The formulation of the optimization problem is shown below. The evaluation function W is minimized.

However,

It is. Here, VΩ and V ^U are the total volume of the beam element 12 and the upper limit value of the volume constraint, respectively. If {F} ^j and {G} ^j are the same, the objective function degenerates to average compliance and becomes the amplitude value of the excitation point response.

  FIG. 3 is a flowchart showing the optimization procedure in this embodiment. First, the initial values of the beam elements 12 and the arrangement of the non-design element shell elements 10 are given to the fixed region Ω as shown in FIG. 1, and a dynamic stiffness matrix [B] for an arbitrary frequency ωj is created (S101). Alternatively, a degenerated dynamic stiffness matrix may be created.

  Next, the sensitivity of the dynamic stiffness matrix with respect to the design variable ρv, i is calculated (S102). Here, the parameter p for giving a penalty is assumed to be an initial value of 1.

Next, at any frequency, the balance equation is solved to obtain response vectors {U} ^j and {V} ^j (S103).

Next, values of mutual average compliance l ^j , objective function W, and volume constraint VΩ are obtained (S104).

  After calculating the objective function, it is determined whether or not the calculated objective function has converged, that is, whether or not the minimum value has been reached (S105). Whether or not it has converged can be determined by calculating a difference from the previously calculated value and determining whether or not the difference value is smaller than a predetermined allowable value.

  If the objective function has converged, the process is terminated at that point. On the other hand, if the objective function has not converged, the sensitivity to the mutual average compliance, the objective function, and the volume-constrained design variable ρv, i is calculated (S106).

  After calculating the sensitivity, the design variable ρv, i is updated using the optimization method based on the obtained sensitivity (S107), and the processing from S101 is repeated. The processing after S101 is repeatedly executed until it is determined that convergence has occurred at S105.

  In the topology optimization problem, an optimization criterion method, a sequential linear meter, a sequential convex function approximation method, or the like is used as an optimization method. In particular, CONLIN (Convex Linearization), which is one of successive convex function approximation methods, is applied to the stiffness problem, and the stability and fast convergence of the solution have been demonstrated. The basic concept of CONLIN is to approximate a function h that is an objective function or a constraint condition using a parameter ηi of the following equation. Also in this embodiment, it is preferable to use this COLIN as an optimization method.

  In the stiffness problem, the sensitivity of the objective function to the design variable ρv, i is negative. For this reason, the second equation above is applied. On the other hand, with respect to the design variables ρv, i, the objective function is a function close to inverse proportion. That is, since the approximation is close to linear with respect to the parameters, high convergence can be realized.

  Hereinafter, the process of this embodiment is shown concretely.

As shown in FIG. 4, the case where the reinforcing material is arranged on the flat plate under the volume restriction is targeted. Four vertices are simply supported (indicated by triangular points in the figure), and a unit periodic external force in the out-of-plane direction is applied to the center (indicated by arrows in the figure). Assuming iron, Young's modulus is 2.1 × 10 ⁻⁵ N / mm ² , Poisson's ratio is 0.3, and mass density is 7.86 × 10 ⁻⁶ kg / mm ³ . The thickness of the base shell element is 5 mm, and the beam element 12 disposed thereon is used as a design element. The initial value of the cross-sectional area of the normalized beam, which is a design variable, is set to ρvi = 10 ⁻⁴ . Further, the thin-wall ratio γ = 0.2 and the maximum diameter dmax = 10 mm. The number of shell elements is 16, and the number of beam elements is 72. The amplitude of the excitation point is an objective function at an arbitrary frequency of 500 Hz or less. The volume of the beam element 12 to be added is set as a constraint, and is 25% or less of the maximum volume. In addition, although dynamic rigidity including rigidity, mass, and damping of the beam element 12 is used, the maximum mass of the beam element alone is 28 g with respect to the total mass of 4.7 kg formed by the shell elements, which is less than 1%. In addition, the eigenvalue of each element is as high as 10 KHz or higher, that is, it is considered that the influence on the frequency characteristics of 500 Hz or less is small. Therefore, in the calculation, only the rigidity of the beam element 12 is considered, and the mass of the beam element 12 can be substantially ignored.

  FIG. 5 shows the excitation point compliance in the initial state. Peaks occur at 68 Hz and 371 Hz. 6A, 6B, and 6C show the deformation modes of the static deformation mode (0 Hz), the mode 1 (68 Hz) and the mode 2 (371 Hz) that form a peak, together with the strain energy distribution.

  6A is a static mode, FIG. 6B is a mode 1, and FIG. 6C is a mode 2 strain energy distribution. It can be seen that the deformations of the static mode and mode 1 are equivalent, but the strain energy distribution is different. The secondary peak is different in both mode and strain energy. The solution of the optimization problem was searched for the frequencies of the three modes shown in FIG.

  The optimization results for 0 Hz, 68 Hz, and 371 Hz are Opt1, Opt2, and Opt3, and FIG. 7 shows the optimization results for each frequency. 8 shows a transition diagram of Opt1, FIG. 9 shows a transition diagram of Opt2, and FIG. 10 shows a transition diagram of Opt3. In these drawings, only the beam element 12 having ρvi of 0.95 or more is shown.

  FIG. 11 shows a change in peak frequency based on the optimization result. FIG. 12 shows the normalized peak frequency with respect to the initial value. Whether the peak level is reduced or not can be expressed by a change in peak frequency. FIG. 13 shows peak values based on the optimum results. FIG. 14 shows a normalized peak value with respect to the initial value. The result of the optimal arrangement corresponds to a region with a high strain energy distribution, and appropriate reinforcement is given in each mode. In Opt1 and Opt2, the optimization results are partially different, but the peak frequencies are the same. This is considered that there was no big difference in deformation mode and strain energy distribution, and the same effect was obtained. Opt3 has the highest peak frequency in mode 2, but mode 1 has the least change. From these results, the optimum arrangement is different at each frequency, and it is necessary to perform optimization at the target frequency in order to obtain a high reduction effect. Each optimization result corresponds to the strain energy distribution and is considered to be a mechanically reasonable result.

  As described above, in this embodiment, by using the dynamic stiffness characteristics of the target structure, frequency response calculation can be performed in the same manner as static response calculation. In addition, based on the ground structure method, all topologies (layouts) in the design space can be candidates for design changes, thereby optimizing considering all possibilities from the conventional optimization with limited design variables. Is possible.

  In the present embodiment, the main topology (layout) is maintained by setting the shell element as the main structure as a non-design variable. Thereby, stable optimization calculation is possible. If even the main structure is a design variable, extreme topology changes disturb the stability of the solution, and an appropriate solution cannot be obtained. In addition, the degree of freedom increases by the square and the third power by using shell elements and solid elements even in the same design space. Compared with these, the beam element of a one-dimensional element can make a freedom degree small. Further, by using the successive convex function approximation method, high-speed convergence can be realized.

  Moreover, even when applied to a vehicle body structure that exceeds one million degrees of freedom, the degree of freedom can be reduced without degrading accuracy by the reduction technique.

  Furthermore, since the main structure known in the present embodiment is a non-design element, initial rigidity is ensured. On the other hand, since it is solved as an optimal placement problem of the reinforcing material at an arbitrary frequency, it is possible to achieve both rigidity and vibration reduction. In the first place, since the overall structure of a vehicle structure is determined by collision and rigidity, a practical solution can be obtained by treating the vibration problem as an optimal reinforcement placement problem.

  Specifically, the design method of the present embodiment can be executed on a computer by installing a processing program in a computer such as a personal computer or a workstation. Since the design model of this embodiment uses elements based on structural mechanics such as shell elements and beam elements, a large-capacity memory is unnecessary, and is used as a spreadsheet software front end on a personal computer, for example. It is also possible. The processing program can be recorded on any medium that can hold information electromagnetically, optically, or chemically, such as a CD-ROM, DVD, hard disk, or semiconductor memory. The processing program can be installed, for example, by recording the processing program on a CD-ROM and supplying the processing program from the CD-ROM to the hard disk of the computer. Of course, the processing program may be stored in the hard disk or ROM of the computer from the beginning and used as a computer dedicated to the structure model design.

It is explanatory drawing which shows the structural element of embodiment. It is explanatory drawing of a beam element. It is a processing flowchart of an embodiment. It is explanatory drawing in the case of arrange | positioning a reinforcing material to a flat plate. It is an excitation point compliance explanatory drawing of an initial state. It is distortion energy distribution explanatory drawing of a static mode, mode 1, and mode 2. FIG. It is arrangement | positioning explanatory drawing which shows the optimal result of static mode, mode 1, and mode 2. FIG. It is a transition diagram accompanying optimization of a static mode. It is a transition diagram accompanying the optimization of mode 1. It is a transition diagram accompanying the optimization of mode 2. It is a table | surface figure which shows the peak frequency change by the optimization of mode 1 and mode 2. FIG. It is a graph which shows the normalization peak frequency by optimization of mode 1 and mode 2. FIG. It is a table | surface figure which shows the peak level change by the optimization of mode 1 and mode 2. FIG. It is a graph which shows the normalization peak level by optimization of mode 1 and mode 2. FIG.

Explanation of symbols

  10 shell elements, 12 beam elements.

Claims (4)

A method of designing a structure using a computer,
The structure is composed of a shell element and a beam element arranged as a reinforcing material at all between at least adjacent nodes of the shell element,
In the processor of the computer,
Creating a dynamic stiffness matrix including mass, damping, and stiffness as characteristics of each of the shell element and the beam element;
Creating an overall matrix by superimposing the dynamic stiffness matrix of the beam element and the dynamic stiffness matrix of the shell element;
Deriving an equation of motion of the structure using the overall matrix;
Deriving the amplitude when the evaluation node of the structure is vibrated at a predetermined frequency based on the equation of motion as an objective function;
Executing the step of optimizing the arrangement of the beam elements under the constraint condition of the total volume of the beam elements so as to minimize the objective function, using the shell element as a non-design element and the beam element as a design element; A structure design method characterized by the above. The structure design method according to claim 1,
The structure design method , wherein the dynamic stiffness matrix is a degenerated matrix . A program that allows a computer to design a model of a structure,
The structure is composed of a shell element and a beam element arranged as a reinforcing material at all between at least adjacent nodes of the shell element ,
The program is stored in a computer processor .
Creating a dynamic stiffness matrix including mass, damping, and stiffness as characteristics of each of the shell element and the beam element;
Creating an overall matrix by superimposing the dynamic stiffness matrix of the beam element and the dynamic stiffness matrix of the shell element;
Deriving an equation of motion of the structure using the overall matrix;
Deriving the amplitude when the evaluation node of the structure is vibrated at a predetermined frequency based on the equation of motion as an objective function;
Executing the step of optimizing the arrangement of the beam elements under the constraint condition of the total volume of the beam elements so as to minimize the objective function, using the shell element as a non-design element and the beam element as a design element; A program characterized by The program according to claim 3,
  The dynamic stiffness matrix is a degenerated matrix.