CN107357992B - Composite structure correction method for finite element model based on cluster analysis - Google Patents

Abstract

The invention provides a method for correcting the finite element model of composite material structure based on cluster analysis, establishes the initial finite element analysis model, measures the experimental modal frequency and mode shape of the structure, and calculates the relative sensitivity matrix of the parameters to be corrected, Use the hierarchical clustering algorithm to group the parameters to be corrected, and then analyze the relative sensitivity of the clustering parameters, select the clustering parameter with the largest relative sensitivity average value among the parameters for correction, and construct the modal frequency and measured mode of the analysis model The residual vector of the frequency is used to establish the objective function required for the correction of the analytical model, and the optimization inverse problem of the objective function is constructed to correct the finite element model of the composite material structure. The present invention combines numerical simulation, test and optimization technology, adopts the relative sensitivity matrix of parameters to carry out cluster analysis, reduces the number of parameters to be corrected, improves the stability of the correction program, and provides an accurate method based on numerical simulation, test and optimization for engineering applications. Combination of equivalent finite element model parameter correction method for composite materials.

Description

Modification Method of Finite Element Model of Composite Material Structure Based on Cluster Analysis

technical field

The invention relates to a composite material structure, in particular to a correction method for a finite element model of a composite material structure.

Background technique

Composite material structures have excellent properties such as high specific strength, high specific stiffness, and fatigue resistance, and are widely used in aerospace, automobile, and shipbuilding industries. But at the same time, due to its various material components and complex manufacturing process, the mesoscopic material parameters of the composite material structure are different. In the process of structural analysis and design, if the mesoscopic analysis model of the composite material structure is used, the difficulty of modeling and the time cost of the product design stage will be greatly increased. Therefore, it is of great significance to establish the finite element analysis model of composite material structure and modify it to improve the accuracy of composite material structure analysis.

The finite element model of composite material structure usually adopts orthotropic materials. For multilayer composite material structure, the number of parameters to be corrected will be far more than the experimental data, resulting in ill-conditioned problems in the correction process. How to obtain an accurate and efficient finite element analysis model of composite material structure while reducing the parameters in the correction process in the case of incomplete measured data has become a practical engineering problem to be solved urgently.

Contents of the invention

Purpose of the invention: the purpose of the present invention is to address the deficiencies in the prior art, to provide a method for correcting the finite element model of composite material structure based on cluster analysis, which uses the relative sensitivity matrix of the parameters to be corrected to carry out cluster analysis on the parameters, through incomplete The method of modifying the parameters of the multi-parameter equivalent composite finite element model improves the accuracy and efficiency of parameter modification.

Technical solution: The present invention provides a method for correcting the composite material structure finite element model based on cluster analysis, comprising the following steps:

(1) According to the geometric characteristics and component composition of the composite material structure, the actual composite material structure is modeled using the orthotropic material relationship, the detailed components are simplified, and only the macroscopic configuration of the composite material structure is considered to establish an equivalent The initial finite element analysis model of ;

(2) According to the actual geometric parameters of the composite material structure, the experimental model is established, and the experimental modal frequency and mode shape of the structure are measured by using the dynamic modal experiment technology;

(3) Calculate the relative sensitivity matrix of the parameters to be corrected, use the hierarchical clustering algorithm to group the parameters to be corrected, and then analyze the relative sensitivity of the clustering parameters, and select the clustering parameter with the largest relative sensitivity average among each parameter for correction ;

(4) Construct the modal frequency of the analytical model and the residual vector of the measured modal frequency, establish the objective function required for the correction of the analytical model, and construct the optimization inverse problem of the objective function to correct the finite element model of the composite material structure.

Further, step (2) obtains the process of experimental mode frequency and mode shape and comprises the following steps:

2.1) Establish its experimental model according to the geometric parameters of the composite material structure;

2.2) Structural measurement point layout, select the vibration pickup point at the structure boundary away from the modal stagnation point;

2.3) Suspend the composite material structure with a rubber rope, define the direction perpendicular to the suspension surface as the Z-axis direction, and make it in a free-free state, and fix the acceleration sensor at the selected vibration pickup point;

2.4) Connect the force hammer and the acceleration sensor to the signal acquisition instrument respectively;

2.5) Use the modal analysis module of the dynamic test system to set the modal analysis parameters;

2.6) Use a force hammer to sequentially apply pulse force to the excitation points on the structure along the Z-axis direction, and sequentially collect the input force of each measuring point when it is excited by the pulse and the output acceleration signal at the vibration pickup point;

2.7) Use the signal analyzer of the test system to correct the modal parameters of the input and output signals, and obtain the measured modal frequency and mode shape of the composite material structure through the aggregation, fitting and correction of the frequency response functions of each point.

Further, step (3) includes the following steps:

3.1) Calculate the relative sensitivity matrix of the modal frequency relative to the elastic parameters according to the relative sensitivity calculation formula:

where S _r is the relative sensitivity matrix, f is the output modal frequency vector, and p is the elastic parameter vector to be corrected;

3.2) Calculate the distance d between the relative sensitivity column vectors g _α and g _β :

Utilize the hierarchical clustering algorithm to classify the parameters with close relative sensitivity distances to obtain the hierarchical tree expression of the elastic parameters, and then use the distance threshold 0.2 as the parameter grouping standard to group the elastic parameters p={p ₁ ; p ₂ ; ...; p _j }, n ₁ , n ₂ ,...,n _j are the number of parameters in each group respectively, and n ₁ +n ₂ +...+n _j =N, N represents the total number of elastic parameters;

3.3) Define the clustering parameter θ _j as the relative change of the elastic parameter of the jth group:

In the formula, p _j is the jth group of elastic parameter vectors, Indicates the initial value of the j-th group of elastic parameters, and θ _j is the clustering parameter corresponding to the j-th group of elastic parameters;

Use the sensitivity calculation formula of the clustering parameters to select the clustering parameters and determine the clustering parameters to be corrected:

In the formula, S _c represents the sensitivity matrix of clustering parameters, θ is a clustering parameter vector, and the elements of this vector are composed of clustering parameters θ _j .

Further, step (4) includes the following steps:

4.1) For the obtained experimental modal frequency of the composite material structure, use the modal confidence MAC to determine the corresponding analytical modal frequency of each order of the experimental modal shape, and perform the modal shape matching:

In the formula, M represents the modal confidence matrix, Φ _a and Φ _e represent the analytical and experimental mode shapes respectively, and finally determine the following objective optimization function:

In the formula, p is the elastic parameter vector to be corrected, J(p) represents the objective function, ε(p) is the residual vector constructed by modal frequency f _a and modal frequency f _e and ε(p)=f _e -f _a (p), Wε=round(max(f _e ) diag(f _e )), represents the weighting matrix obtained according to the experimental modal frequency values, round(·), max(·) and diag(·) represent Rounding, maximum value operation and diagonal matrix operation; the physical meaning of the objective function is: within the variation range of the parameters [p _l , p _u ], find the optimal parameters so that the experimental modal frequency and the analytical modal frequency vector The two-norm of the difference is the smallest;

4.3) Construct an optimization inverse problem based on the constructed objective function (12) to modify the elastic parameters of the composite finite element model to obtain effective elastic parameters.

Beneficial effects: the present invention provides a method for grouping and selecting parameters to be corrected based on the relative sensitivity matrix of the parameters based on cluster analysis, establishes the finite element initial analysis model and experimental model of the composite material plate, and simultaneously constructs the modal frequency target optimization function , the finite element model of the equivalent composite plate structure is revised, which has very important engineering application value.

The present invention combines numerical simulation, test and optimization technology to correct the material parameters of the multi-parameter composite material equivalent finite element model, considering that the equivalent finite element model selects more material constitutive relation parameters, resulting in the accuracy of parameter correction For lower problems, the relative sensitivity matrix of parameters is used for cluster analysis, which reduces the number of parameters to be corrected, improves the stability of the correction program, and provides an accurate composite material based on the combination of numerical simulation, experiment and optimization for engineering applications. Effective finite element model parameter correction method.

Description of drawings

Fig. 1 is the equivalent composite material plate structure in the embodiment;

Figure 2 is a relative sensitivity diagram of the parameters to be corrected;

Fig. 3 is a layered dendrogram of parameters to be corrected;

Figure 4 is a sensitivity analysis diagram of clustering parameters;

Fig. 5 is a flowchart of the method of the present invention.

Detailed ways

The technical solutions of the present invention will be described in detail below, but the protection scope of the present invention is not limited to the embodiments.

Embodiment: A method for correcting the composite material structure finite element model based on cluster analysis, as shown in Figure 5, the specific process is as follows:

Step 1, use shell elements and solid elements to model the composite plate structure to obtain an equivalent finite element model, as shown in Figure 1, where 1 represents the shell element of the upper panel, 2 represents the solid element of the core layer, and 3 represents the lower panel Shell element, the initial elastic parameter value of the structure is

Step 2, using the dynamic modal experimental technique to obtain the structural experimental modal frequency and mode shape:

2.1) Establish its experimental model according to the geometric parameters of the composite material structure;

2.2) Structural measurement point layout, select the vibration pickup point at the structure boundary away from the modal stagnation point;

2.3) Suspend the composite material structure with a rubber rope, define the direction perpendicular to the suspension surface as the Z-axis direction, and make it in a free-free state; use glue to bond the acceleration sensor to the selected boundary vibration pickup position;

2.4) Connect the force hammer and the acceleration sensor to the corresponding interface of the signal acquisition instrument with the connection line;

2.5) Use the modal analysis module of the dynamic test system to set the modal analysis parameters;

2.6) Use a force hammer to sequentially apply pulse force to the excitation points on the structure along the Z-axis direction, and sequentially collect the input force of each measuring point when it is excited by the pulse and the output acceleration signal at the vibration pickup point;

2.7) Use the signal analyzer of the test system to correct the modal parameters of the input and output signals, and obtain the measured modal frequency f _e and mode shape Φ of the composite material structure through aggregation, fitting and correction of the frequency response functions of each point _e .

Step 3, use cluster analysis to group and select the parameters to be corrected:

3.1) According to the relative sensitivity calculation formula, calculate the relative sensitivity matrix of the modal frequency relative to the elastic parameters. The relative sensitivity matrix calculation formula is as follows:

In the formula, S _r is the relative sensitivity matrix, f is the output modal frequency vector, p is the elastic parameter vector, and the relative sensitivity of the obtained elastic parameters is shown in Figure 2;

3.2) cluster analysis, according to the elastic parameter relative sensitivity matrix obtained in step 3.1, calculate the relative sensitivity column vector g _α , the distance d of g _β :

The hierarchical clustering algorithm is used to classify the parameters with close relative sensitivity distances, so as to obtain the hierarchical clustering dendrogram of elastic parameters, as shown in Figure 3, and then use the distance threshold of 0.2 as the parameter grouping standard, as shown by the dotted line in Figure 2 It can be seen from Figure 3 that the grouping is as follows: p={p ₁ ;p ₂ ;...;p _j }={{p ₁ ,p ₁₁ };{p ₂ ,p ₃ } {p ₄ , p ₁₀ }; {p ₅ }; {p ₆ , p ₉ , _{p 12} }; _{{ p 7} , p _{13 }} ; , n ₂ =2, n ₃ =2, n ₄ =1, n ₅ =3, n ₆ =2, n ₇ =3;

3.3) Define the clustering parameter θ _j as the relative change of the elastic parameter of the jth group:

In the formula, p _j is the jth group of elastic parameter vectors, Indicates the initial value of the j-th group of elastic parameters, θ _j is the clustering parameter corresponding to the j-th group of elastic parameters; use the sensitivity calculation formula of the clustering parameter to select the clustering parameter and determine the clustering parameter to be corrected:

In the formula, S _c represents the sensitivity matrix of clustering parameters, θ is a clustering parameter vector, and the elements of this vector are composed of clustering parameters θ _j ;

According to the value of relative sensitivity, select the clustering parameter with the largest average value of relative sensitivity in each parameter for correction, and finally obtain the sensitivity of clustering parameters as shown in Figure 4. The sensitivity values of clustering parameters θ ₂ and θ ₄ are in magnitude The other five clustering parameters are relatively low, so the final parameters with correction are reduced from 15 to 5, which reduces the ill-conditionedness in the correction process.

Step 4, use the optimization method to modify the parameters of the equivalent composite finite element model:

4.1) According to the modal frequency f _a obtained by the finite element analysis model and the measured modal frequency f _e , the residual error vector ε(p)=f _e −f _a (p) is constructed;

4.2) Determine the target optimization function: According to the experimental modal frequency and modal shape obtained in step 2, use the modal confidence MAC to perform modal shape matching, and determine the corresponding analysis modal frequencies of each order of the experimental modal shape ,

In the formula, M represents the modal confidence matrix, Φ _a and Φ _e represent the analytical and experimental mode shapes respectively, and finally determine the following objective optimization function:

In the formula, W=round(max(f _e ) diag(f _e )) represents the weighting matrix obtained according to the experimental modal frequency values, and round(·), max(·) and diag(·) respectively represent rounding , the maximum value operation and diagonal matrix operation, p represents the parameter vector to be corrected, the physical meaning of the objective function is: within the parameter range [p _l , p _u ], find the optimal parameters so that the experimental modal frequency and analysis The two-norm of the modal frequency vector difference is the smallest;

4.3) Construct an optimization inverse problem based on the constructed objective function (18) to modify the elastic parameters of the finite element model of the composite material to obtain an accurate and effective finite element analysis model.

Claims (3)

1. A composite material structure finite element model correction method based on cluster analysis, is characterized in that: comprise the following steps: (1) According to the geometric characteristics and component composition of the composite material structure, the actual composite material structure is modeled using the orthotropic material relationship, the detailed components are simplified, and only the macroscopic configuration of the composite material structure is considered to establish an equivalent The initial finite element analysis model of ; (2) According to the actual geometric parameters of the composite material structure, the experimental model is established, and the experimental modal frequency and mode shape of the structure are measured by using the dynamic modal experiment technology; (3) Calculate the relative sensitivity matrix of the parameters to be corrected, use the hierarchical clustering algorithm to group the parameters to be corrected, and then analyze the relative sensitivity of the clustering parameters, and select the clustering parameter with the largest relative sensitivity average among each parameter for correction ; (4) Construct the modal frequency of the analytical model and the residual vector of the measured modal frequency, establish the objective function required for the correction of the analytical model, and construct the optimization inverse problem of the objective function to correct the finite element model of the composite material structure; Wherein, step (3) comprises the following steps: 3.1) Calculate the relative sensitivity matrix of the modal frequency relative to the elastic parameters according to the relative sensitivity calculation formula: <mrow><msub><mi>S</mi><mi>r</mi></msub><mo>=</mo><mi>f</mi><mfrac><mrow><mo>&amp;part;</mo><mi>f</mi></mrow><mrow><mo>&amp;part;</mo><mi>p</mi></mrow></mfrac><mi>p</mi><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>1</mo>mn><mo>)</mo></mrow></mrow> where S _r is the relative sensitivity matrix, f is the output modal frequency vector, and p is the elastic parameter vector to be corrected; 3.2) Calculate the distance d between the relative sensitivity column vectors g _α and g _β : <mrow><mi>d</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mrow><msubsup><mi>g</mi><mi>&amp;alpha;</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>&amp;beta;</mi></msub></mrow><msqrt><mrow><mo>(</mo><msubsup><mi>g</mi><mi>&amp;alpha;</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>&amp;alpha;</mi></msub><mo>)</mo><mo>&amp;CenterDot;</mo><mo>(</mo><msubsup><mi>g</mi><mi>&amp;beta;</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>&amp;beta;</mi></msub><mo>)</mo></mrow></msqrt></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow> Utilize the hierarchical clustering algorithm to classify the parameters with close relative sensitivity distances to obtain the hierarchical tree expression of the elastic parameters, and then use the distance threshold 0.2 as the parameter grouping standard to group the elastic parameters p={p ₁ ; p ₂ ; ...; p _j }, n ₁ , n ₂ ,...,n _j are the number of parameters in each group respectively, and n ₁ +n ₂ +...+n _j =N, N represents the total number of elastic parameters; 3.3) Define the clustering parameter θ _j as the relative change of the elastic parameter of the jth group: <mrow><msub><mi>p</mi><mi>j</mi></msub><mo>=</mo><msubsup><mi>p</mi><mi>j</mi><mn>0</mn></msubsup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow> In the formula, p _j is the jth group of elastic parameter vectors, Indicates the initial value of the j-th group of elastic parameters, and θ _j is the clustering parameter corresponding to the j-th group of elastic parameters; Use the sensitivity calculation formula of the clustering parameters to select the clustering parameters and determine the clustering parameters to be corrected: <mrow><msub><mi>S</mi><mi>c</mi></msub><mo>=</mo><mfrac><mrow><mo>&amp;part;</mo><mi>f</mi></mrow><mrow><mo>&amp;part;</mo><mi>p</mi></mrow></mfrac><mfrac><mrow><mo>&amp;part;</mo><mi>p</mi></mrow><mrow><mo>&amp;part;</mo><mi>&amp;theta;</mi></mrow></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow> In the formula, S _c represents the sensitivity matrix of clustering parameters, θ is a clustering parameter vector, and the elements of the clustering parameter vector are composed of clustering parameters θ _j . 2. the composite material structure finite element model correction method based on cluster analysis according to claim 1, is characterized in that: the process that step (2) obtains experimental mode frequency and mode shape comprises the following steps: 2.1) Establish its experimental model according to the geometric parameters of the composite material structure; 2.2) Structural measurement point layout, select the vibration pickup point at the structure boundary away from the modal stagnation point; 2.3) Suspend the composite material structure with a rubber rope, define the direction perpendicular to the suspension surface as the Z-axis direction, and make it in a free-free state, and fix the acceleration sensor at the selected vibration pickup point; 2.4) Connect the force hammer and the acceleration sensor to the signal acquisition instrument respectively; 2.5) Use the modal analysis module of the dynamic test system to set the modal analysis parameters; 2.6) Use a force hammer to sequentially apply pulse force to the excitation points on the structure along the Z-axis direction, and sequentially collect the input force of each measuring point when it is excited by the pulse and the output acceleration signal at the vibration pickup point; 2.7) Use the signal analyzer of the test system to correct the modal parameters of the input and output signals, and obtain the measured modal frequency and mode shape of the composite material structure through the aggregation, fitting and correction of the frequency response functions of each point. 3. the composite material structure finite element model correction method based on cluster analysis according to claim 1, is characterized in that: step (4) comprises the following steps: 4.1) For the obtained experimental modal frequency of the composite material structure, use the modal confidence MAC to determine the corresponding analytical modal frequency of each order of the experimental modal shape, and perform the modal shape matching: <mrow><mi>M</mi><mo>=</mo><mfrac><mrow><mo>|</mo><msubsup><mi>&amp;Phi;</mi><mi>a</mi><mi>T</mi></msubsup><msub><mi>&amp;Phi;</mi><mi>e</mi></msub><msup><mo>|</mo><mn>2</mn></msup></mrow><mrow><mo>(</mo><msubsup><mi>&amp;Phi;</mi><mi>a</mi><mi>T</mi></msubsup><msub><mi>&amp;Phi;</mi><mi>a</mi></msub><mo>)</mo><mo>(</mo><msubsup><mi>&amp;Phi;</mi><mi>e</mi><mi>T</mi></msubsup><msub><mi>&amp;Phi;</mi><mi>e</mi></msub><mo>)</mo></mrow></mfrac><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow> In the formula, M represents the modal confidence matrix, Φ _a and Φ _e represent the analytical and experimental mode shapes respectively, and finally determine the following objective optimization function: <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mi>min</mi><mi></mi><mi>J</mi><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>=</mo><mi>&amp;epsiv;</mi><msup><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mi>T</mi></msup><mi>W</mi><mi>&amp;epsiv;</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>s</mi><mo>.</mi>mo><mi>t</mi><mo>.</mo><msub><mi>p</mi><mi>l</mi></msub><mo>&amp;le;</mo><mi>p</mi><mo>&amp;le;</mo><msub><mi>p</mi><mi>u</mi></msub></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow> In the formula, p is the elastic parameter vector to be corrected, J(p) represents the objective function, ε(p) is the residual vector constructed by modal frequency f _a and modal frequency f _e and ε(p)=f _e -f _a (p), Wε=round(max(f _e ) diag(f _e )), represents the weighting matrix obtained according to the experimental modal frequency values, round(·), max(·) and diag(·) represent Rounding, maximum value operation and diagonal matrix operation; the physical meaning of the objective function is: within the variation range of the parameters [p _l , p _u ], find the optimal parameters so that the experimental modal frequency and the analytical modal frequency vector The two-norm of the difference is the smallest; 4.3) Construct an optimization inverse problem based on the constructed objective function (6) to modify the elastic parameters of the composite finite element model to obtain effective elastic parameters.