Structural static response global sensitivity solving method based on homotopy proxy model

Abstract

A structural static response global sensitivity solving method based on a homotopy agent model is characterized by comprising the steps that firstly, a free vibration control equation is reconstructed based on a homotopy analysis method, then a random residual error expression of the control equation is established, and parameters in the random residual error expression are determined by minimizing the random residual error; a displacement function expression can be obtained by substituting the optimal h value into a sample formula, global sensitivity analysis can be carried out on the displacement function expression, and the technical problem that modal analysis cannot be carried out through a traditional PC method and a Kriging method is solved. The algorithm is reasonable, the displacement field of the whole structure can be obtained, the problem of a large number of random input can be efficiently solved, samples are not needed, the calculation result is not affected by the samples, good stability and calculation efficiency are embodied in sensitivity analysis of high-dimensional random output, and the method has good application prospects. And efficient solution of the structural static response global sensitivity is realized.

Description

Technical Field

[0001] The invention relates to a method for solving the global sensitivity of a structural static response, in particular to a method for solving the global sensitivity of a structural static response based on a homotopy proxy model, and belongs to the technical field of structural engineering. Background Art

[0002] With the advancement of technology and economic development, people pay more and more attention to the safety of static displacement of buildings, bridges and other engineering structures. In the production and construction process of actual structures, there are inherent uncertainties in the structural parameters, which leads to differences between their static displacement and the design value. Therefore, considering the randomness of structural parameters, in-depth analysis of the static displacement response of the structure is of great significance to the design and optimization of the structure.

[0003] Real structures always inevitably have randomness. The uncertainty quantification and parameter sensitivity analysis of random structures have always been a hot topic. Parameter sensitivity analysis helps to find out the main factors that affect the uncertainty of structural response, and can be used to simplify the number of random variables in the system and structural optimization analysis. Sensitivity analysis includes local sensitivity and global sensitivity. Local sensitivity can only reflect the impact of random parameters on structural response when they fluctuate around the mean, and the results are only reliable when the relationship between input and output is weakly nonlinear; global sensitivity has been widely studied and applied because it can reflect the impact of random parameters changing in the entire sample space on the uncertainty of structural response.

[0004] At present, a variety of global sensitivity indices have been proposed, among which the global sensitivity based on variance, namely the Sobol index, has been widely used in various projects because of its simple calculation and ability to reflect the contribution of input variance to output variance. The traditional global sensitivity index calculation uses the Monte Carlo simulation method (MCS), but it is very time-consuming because it involves a double-layer cycle. Later, some people proposed proxy models such as Kriging and PC to solve the Sobol index. These methods can only obtain the sensitivity of a certain point displacement in the structural displacement field, so they have been widely developed and applied in single output sensitivity analysis. However, for multi-output problems, it is generally necessary to perform response decomposition or principal component analysis on the high-dimensional output, and then establish proxy models for the response amount after dimensionality reduction. These methods, whether in the principal component analysis of the response or in the process of establishing the proxy model, rely on a sufficient number of samples, especially when the random input or output dimension of the system is high, the number of samples required is still not small. Therefore, the accuracy, efficiency and robustness of the global sensitivity analysis based on these methods are closely related to the selection of sample points. Based on the above problems, the present invention is proposed. Summary of the invention

[0005] The purpose of the present invention is to address the defects and deficiencies that traditional methods are difficult to obtain the displacement field of the entire structure, require the use of samples, and the calculation results are affected by the samples. There are also drawbacks of large amount of calculation and inaccurate calculation, which are not conducive to the design and optimization of the structure. Now an algorithm is provided that is reasonable and can obtain the displacement field of the entire structure, and can efficiently solve problems with a large number of random inputs, does not require the use of samples, and the calculation results are not affected by the samples. It embodies good stability and calculation efficiency in the sensitivity analysis of high-dimensional random outputs, and realizes an efficient solution to the global sensitivity of the static response of the structure. A method for solving the global sensitivity of the static response of the structure based on a homotopy surrogate model is provided.

[0006] To achieve the purpose of the above invention, the technical solution of the present invention is: a method for solving the global sensitivity of structural static response based on homotopy surrogate model, characterized by comprising the following steps:

[0007] Step 1: Consider the uncertainty of the elastic modulus of the material, describe it as a random field or express it with independent random variables, and establish the static equilibrium equation of the random parameter structure;

[0008] Step 2: Based on the idea of ​​homotopy analysis method, the control equation of the structural displacement response under static load is reconstructed to obtain the zero-order deformation equation;

[0009] Step 3: Express the displacement as a homotopy series form with respect to the random variable, solve the zero-order deformation equation, obtain the coefficients of each term in the displacement expression, and the expansion order of the series is determined by a preset error threshold;

[0010] Step 4: Introduce the L2 norm of the vector to define the random residual error and minimize it to obtain the optimal h value and the optimal homotopy series solution. After obtaining the optimal h value, insert the sample into the formula to obtain the displacement function expression, and then perform a global sensitivity analysis on it;

[0011] Step 5: The i-th random variable ξ i Corresponding displacement global sensitivity index S i Calculate S i By sorting the random variables from large to small, the main structural parameters that affect the static response of the structure can be obtained.

[0012] Furthermore, the static equilibrium equation of the random parameter structure established in step 1 is recorded as formula 1, as follows:

[0013]

[0014] Where K0 represents the stiffness matrix when the structural parameters take the mean value.

[0015] Furthermore, in step 2, based on the idea of ​​homotopy analysis method, the control equation of the structural displacement response under static load is reconstructed into formula 2:

[0016]

[0017] Where K0 represents the stiffness matrix when the structural parameters take the mean value, U0 is the displacement field of the structure when the structural parameters take the mean value, ξ={ξ1,ξ2,···,ξ n} are n random variables describing the randomness of structural materials, F0 represents the mean load vector, F i is the coefficient of certainty;

[0018] Compared with the original control equation, this equation introduces two new parameters p and h. The parameter p can establish the relationship between the mean solution U0 of the structural system and the true solution U(ξ) of the equation. This relationship is called homology, and then the homology series solution of U(ξ) is established. h is an important parameter that controls the convergence properties of the homology series.

[0019] Furthermore, the derivation method and steps in step three are as follows:

[0020] For formula 2, Ψ(ξ, h, 0) = U0, Ψ(ξ, h, 1) = U(ξ, h). Expanding Ψ(ξ, h, p) into a Maclaurin series about p gives formula 3:

[0021]

[0022] For Formula 3, when parameter p=1, Formula 4 can be obtained:

[0023]

[0024] Take the r-times partial derivative of both sides of formula 2 with respect to the parameter p and set p = 0 to obtain the recursive relationship formula 5:

[0025]

[0026] Where I represents the N×N dimensional identity matrix,

[0027] Substituting formula 5 into formula 4, we can get formula 6:

[0028]

[0029] In the formula is a function of the random variable h.

[0030] Furthermore, the specific method of step 4 is as follows:

[0031] By introducing the L2 norm of the vector to define the random residual error and minimizing it, we can obtain the optimal h value and the optimal homotopy series solution, which is recorded as Formula 7:

[0032]

[0033] After obtaining the optimal h value, we can input the sample into Formula 6 to get the displacement function expression U(ξ), and then we can perform a global sensitivity analysis on it.

[0034] Furthermore, the specific method of step five is as follows:

[0035] The calculation method of the displacement global sensitivity index Si corresponding to the i-th random variable ξi is as follows: First, a sample matrix A with a number of G is generated for n random variables (G×n) and B (G×n) , B (G×n) The i-th column of (G×n) The sample matrix is ​​generated by replacing the i-th column Calculate A using Formula 6 (G×n) and C (G×n) The displacement sample U corresponding to the sample matrix [A] and Then S i The calculation formula is as follows, recorded as Formula 10:

[0036]

[0037] The beneficial effects of the present invention are:

[0038] 1. The present invention constructs a proxy model to quickly and accurately calculate the statistical characteristics of the structural displacement field under the condition of considering the randomness of parameters. Compared with the existing calculation method, it does not need to use samples, and the calculation results are not affected by the samples. The entire displacement field proxy model can be conveniently established in the form of vectors, which solves the technical problem that the traditional PC method and Kriging method can only calculate the sensitivity of a certain point in the displacement field, and can also efficiently solve problems with a large number of random inputs.

[0039] 2. The present invention first reconstructs the structural static response control equation based on the homotopy analysis method, then establishes the random residual error expression of the control equation, determines the parameters therein by minimizing the random residual error, and finally substitutes the parameters into the homotopy series solution to obtain the displacement field and perform statistical characteristic analysis on it.

[0040] 3. The algorithm of the present invention is reasonable, which overcomes the drawbacks of high computational cost, large amount of computation and inaccurate calculation of traditional methods, and embodies good stability and computational efficiency in the sensitivity analysis of high-dimensional random output, and realizes the efficient solution of the global sensitivity of structural static response, which can provide scientific design basis and technical support for structural design, optimization and long-term operation and maintenance. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 It is a cross-sectional view of the cantilever beam of Example 1 of the present invention.

[0042] Figure 2 It is a box diagram of the global sensitivity S1 of the displacement of point C in Example 1 of the present invention.

[0043] Figure 3 This is a diagram of the sensitivity field SF1 of the cantilever beam in Example 1 of the present invention.

[0044] Figure 4 It is a finite element model diagram of the cable-stayed bridge according to Example 1 of the present invention.

[0045] Figure 5 It is a deck load distribution diagram and a bridge tower load distribution diagram of the cable-stayed bridge according to Example 2 of the present invention.

[0046] Figure 6 It is the global sensitivity in the x, y, and z directions of point A on the top of the tower in Example 2 of the present invention.

[0047] Figure 7 It is the global sensitivity field in the y direction of the bridge deck centerline of Example 2 of the present invention.

[0048] Figure 8 It is a global sensitivity diagram of the bending moment at point C at the bottom of the bridge tower pier in Example 2 of the present invention.

[0049] Fig. 9 This is a table of vertical load values ​​for the cable-stayed bridge according to Example 2 of the present invention.

[0050] Fig.10 This is a table of wind load values ​​for the cable-stayed bridge according to Example 2 of the present invention. DETAILED DESCRIPTION

[0051] The present invention is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0052] See also Figures 1 to 10 The present invention constructs a proxy model to quickly and accurately calculate the statistical characteristics of the structural displacement field under the condition of considering the randomness of parameters, thereby realizing the efficient solution of the statistical characteristics of the structural displacement field. The specific structure and principle are as follows:

[0053] First, considering the uncertainty of the elastic modulus of the material, describe it as a random field or express it with independent random variables, and establish the static equilibrium equation of the random parameter structure, which is recorded as Formula 1, as follows:

[0054]

[0055] Where K0 represents the stiffness matrix when the structural parameters take the mean value.

[0056] Subsequently, based on the idea of ​​homotopy analysis method, the control equation of the structural displacement response under static load was reconstructed to obtain the zero-order deformation equation. The control equation of the structural displacement response under static load was reconstructed into Formula 2:

[0057]

[0058] Where K0 represents the stiffness matrix when the structural parameters take the average value, U0 is the displacement field of the structure when the structural parameters take the average value, ξ={ξ1,ξ2,···,ξ n} are n random variables describing the randomness of structural materials, F0 represents the mean load vector, F i is the coefficient of certainty;

[0059] Compared with the original control equation, this equation introduces two new parameters p and h. The parameter p can establish the relationship between the mean solution U0 of the structural system and the true solution U(ξ) of the equation. This relationship is called homology, and then the homology series solution of U(ξ) is established. h is an important parameter that controls the convergence properties of the homology series.

[0060] Then, the displacement is expressed as a homotopy series form about random variables, and the zero-order deformation equation is solved to obtain the coefficients of each term in the displacement expression. The expansion order of the series is determined by a preset error threshold. The derivation method and steps are as follows:

[0061] For formula 2, Ψ(ξ, h, 0) = U0, Ψ(ξ, h, 1) = U(ξ, h). Expanding Ψ(ξ, h, p) into a Maclaurin series about p gives formula 3:

[0062]

[0063] For Formula 3, when parameter p=1, Formula 4 can be obtained:

[0064]

[0065] Take r partial derivatives of both sides of formula 2 with respect to parameter p and set p = 0 to obtain the recursive relationship formula 5:

[0066]

[0067] Where I represents the N×N dimensional identity matrix,

[0068] Substituting formula 5 into formula 4, we can get formula 6:

[0069]

[0070] In the formula is a function of the random variable h.

[0071] Then, the L2 norm of the vector is introduced to define the random residual error and minimize it to obtain the optimal h value and the optimal homotopy series solution. After obtaining the optimal h value, the sample is inserted into the formula to obtain the displacement function expression, and then a global sensitivity analysis can be performed on it.

[0072] The specific method is as follows:

[0073] By introducing the L2 norm of the vector to define the random residual error and minimizing it, we can obtain the optimal h value and the optimal homotopy series solution, which is recorded as Formula 7:

[0074]

[0075] After obtaining the optimal h value, we can input the sample into Formula 6 to get the displacement function expression U(ξ), and then we can perform a global sensitivity analysis on it.

[0076] Finally, the i-th random variable ξ i Corresponding displacement global sensitivity index S i The calculation method of the displacement global sensitivity index Si corresponding to the i-th random variable ξi is as follows: First, generate a sample matrix A with a number of G for n random variables (G×n) and B (G×n) , B (G×n) The i-th column of (G×n) The sample matrix is ​​generated by replacing the i-th column Calculate A using Formula 6 (G×n) and C (G×n) The displacement sample U corresponding to the sample matrix [ A ] and Then S i The calculation formula is as follows, recorded as Formula 10:

[0077]

[0078] S i By sorting the random variables from large to small, the main structural parameters that affect the static response of the structure can be obtained.

[0079] The following is further described in conjunction with two specific embodiments:

[0080] Example 1: Global sensitivity analysis of the displacement field of a cantilever beam structure under static load

[0081] like Figure 1 The cantilever beam shown in the figure has the following parameters for each material: the cantilever beam is divided into three sections: reinforced concrete, the overlapped part of concrete and steel bars, and steel bars. Each section is 2.0 m long and the total length is 6.0 m. The average elastic modulus of the three sections is 40×10 6 kN / m 2 , 60×10 6 kN / m 2 and 210×10 6 kN / m 2 , density is 2360kg / m 3 、3000kg / m 3 and 7850kg / m 3 The moments of inertia of reinforced concrete and steel bars around the y-axis are 2.1×10 -3 m 4 and 2.4×10 -5 m 4 The finite element model consists of 60 beam elements, and only the deflection and rotation are considered, with a total of 120 degrees of freedom. Assume that the elastic moduli of these three segments obey lognormal, Beta and Weibull distributions, respectively, and generate 4 variables corresponding to each, and a total of 12 random variables.

[0082] Four methods, namely the method of the present invention (HSM), PC, Kriging and MCS (10,000 samples), were used for sensitivity analysis. HSM takes a fourth-order expansion (i.e., r=4) and MCS can obtain sensitivity indicators of all node displacements. PC (takes a second-order expansion) and Kriging can only perform sensitivity analysis on one displacement degree of freedom at a time. Figure 2 It can be seen from the box plot that the method of the present invention is not affected by the sample and is a horizontal straight line, which is closest to the MCS method. The results of the PC method and the Kriging method are greatly affected by the sample. When the sample is small, the stability of the calculation result is poor, which means that when there are many random variables, a sufficient number of samples are needed to ensure the stability of the result. Figure 3 The sensitivity field SF1 of the method of the present invention and the MCS method is given. PC and Kriging can only target one degree of freedom of displacement, so the deflections of points A, B, and C are taken out and their error bars are drawn. It can be seen from the figure that the method of the present invention is closer to the MCS result. Figure 3It can also be seen that the sensitivity index of the same parameter at different coordinate positions is different, and may even change from the most important to the least important. SF1 is the sensitivity field corresponding to the variable ξ1. In the figure, the influence of the sensitivity result of the variable ξ1 gradually decreases. It can be seen that it is very necessary to make sensitivity analysis of the responses of different points of interest.

[0083] Example 2: Global sensitivity analysis of the displacement field of a long-span cable-stayed bridge under static load

[0084] like Figure 4 The figure shows a finite element model of a long-span cable-stayed bridge. The cable-stayed bridge deck is 696 meters long, 28 meters wide, and the two bridge towers are 162 meters high. The finite element model contains 236 cable elements and 370 beam elements for the bridge towers, beams, and bridge deck main beams. Each node has three displacement and three rotation degrees of freedom, and the displacement field dimension of the entire bridge is 2214. Considering the bridge's own weight and the vehicle load equivalent to a uniformly distributed load, and applying an equivalent static wind load along the y direction, the load distribution is as follows Figure 5 The numerical values ​​are shown in the attached Fig. 9 and attached Fig.10 .

[0085] Considering the uncertainty of the elastic modulus of bridge deck beams, bridge towers, and cables, the mean values ​​of their elastic moduli are 3.5×10 10 kN / m 2 , 3.5×10 10 kN / m 2 and 2.1×10 15 kN / m 2 The elastic modulus of the main beam and bridge tower obeys the translational log-normal distribution, and the elastic modulus of the cable obeys the beta distribution. The randomness of the elastic modulus of the main beam, bridge tower and cable is described by four random variables respectively, and there are 12 independent random variables in total for the whole bridge.

[0086] like Figure 6 As shown in the figure, the Sobol index of the displacement sensitivity of point A along the x, y, and z directions is calculated by the method in this paper. It can be seen from the figure that the sensitive parameters of the displacement of the tower top in different directions are not the same. Figure 7 As shown in the figure, this is the sensitivity field of the bridge deck centerline in the y direction. The figure gives the sensitivity indicators of all points on the bridge deck centerline. It can be seen that variables ξ2 and ξ3 have a greater impact on the y-direction displacement of the bridge deck midspan. Figure 8 As shown, the Sobol sensitivity index of the tower bottom C point bending moment is given, and it can be seen that the tower bottom bending moment is mainly affected by ξ8. Due to the large model, the calculated MCS time is thousands of times the calculation time of the method of the present invention. Therefore, MCS is too time-consuming and costly to solve large model examples, which reflects the superiority of the method of the present invention.

[0087] The above content is a further detailed description of the present invention in combination with specific implementation methods. It cannot be considered that the specific implementation of the present invention is limited to these descriptions. For ordinary technicians in the technical field to which the present invention belongs, the present invention will also have various simple substitutions, improvements and changes without departing from the concept of the present invention. Various simple substitutions, improvements and changes made should be deemed to belong to the scope of protection of the present invention.

Claims

1. A method for solving the global sensitivity of structural static response based on homotopy surrogate model, characterized by The following steps are involved: Step 1: Consider the uncertainty of the elastic modulus of the material, describe it as a random field or express it with independent random variables, and establish the static equilibrium equation of the random parameter structure; Step 2: Based on the idea of ​​homotopy analysis method, the control equation of the structural displacement response under static load is reconstructed to obtain the zero-order deformation equation; Step 3: Express the displacement as a homotopy series form with respect to the random variable, solve the zero-order deformation equation, obtain the coefficients of each term in the displacement expression, and the expansion order of the series is determined by a preset error threshold; Step 4: Introduce the L2 norm of the vector to define the random residual error and minimize it to obtain the optimal h value and the optimal homotopy series solution. After obtaining the optimal h value, insert the sample into the formula to obtain the displacement function expression, and then perform a global sensitivity analysis on it; Step 5: The i-th random variable ξ i Corresponding displacement global sensitivity index S i Calculate S i By sorting the random variables from large to small, the main structural parameters that affect the static response of the structure can be obtained.

2. The method for solving the global sensitivity of structural static response based on homotopy surrogate model according to claim 1 is characterized in that: The static equilibrium equation of the random parameter structure established in step 1 is recorded as formula 1, as follows: Where K0 represents the stiffness matrix when the structural parameters take the mean value.

3. The method for solving the global sensitivity of structural static response based on homotopy surrogate model according to claim 1 is characterized by: In the step 2, based on the idea of ​​homotopy analysis method, the control equation of the structural displacement response under static load is reconstructed into Formula 2: Where K0 represents the stiffness matrix when the structural parameters take the average value, U0 is the displacement field of the structure when the structural parameters take the average value, ξ={ξ1,ξ2,···,ξ n } are n random variables describing the randomness of structural materials, F0 represents the mean load vector, F i is the coefficient of certainty; Compared with the original control equation, this equation introduces two new parameters p and h. The parameter p can establish the relationship between the mean solution U0 of the structural system and the true solution U(ξ) of the equation. This relationship is called homology, and then the homology series solution of U(ξ) is established. h is an important parameter that controls the convergence properties of the homology series.

4. The method for solving the global sensitivity of structural static response based on homotopy surrogate model according to claim 1 is characterized in that: The derivation method and steps in step 3 are as follows: For formula 2, Ψ(ξ, h, 0) = U0, Ψ(ξ, h, 0) = Ψ(ξ, h). Expanding Ψ(ξ, h, p) into a Maclaurin series about p gives formula 3: For Formula 3, when parameter p=1, Formula 4 can be obtained: Take r partial derivatives of both sides of formula 2 with respect to parameter p and set p = 0 to obtain the recursive relationship formula 5: Where I represents the N×N dimensional identity matrix, Substituting formula 5 into formula 4, we can get formula 6: In the formula is a function of the random variable h.

5. The method for solving the global sensitivity of structural static response based on homotopy surrogate model according to claim 1 is characterized by: The specific method of step 4 is as follows: By introducing the L2 norm of the vector to define the random residual error and minimizing it, we can obtain the optimal h value and the optimal homotopy series solution, which is recorded as Formula 7: After obtaining the optimal h value, we can input the sample into Formula 6 to get the displacement function expression U(ξ), and then we can perform a global sensitivity analysis on it.

6. The method for solving the global sensitivity of structural static response based on homotopy surrogate model according to claim 1 is characterized by: The specific method of step five is as follows: The calculation method of the displacement global sensitivity index Si corresponding to the i-th random variable ξi is as follows: First, a sample matrix A with a number of G is generated for n random variables (G×n) and B (G×n) , B (G×n) The i-th column of (G×n) The sample matrix is ​​generated by replacing the i-th column Calculate A using Formula 6 (G×n) and C (G×n) The displacement sample U corresponding to the sample matrix [A] and Then S i The calculation formula is as follows, recorded as Formula 10:

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