A method for solving global sensitivity of structural elastic buckling based on random homotopy analysis

Abstract

A structure elastic buckling global sensitivity solution method based on random homotopy analysis is characterized by: using finite element method and minimum potential energy principle, the control equation of the deterministic structure elastic stability analysis is established, the control equation is restructured based on the idea of homotopy analysis method, the buckling load and the mode are expressed as the homotopy series form about the random variable, the restructured control equation is solved, the coefficients in the series expression of the buckling load and the mode are obtained, the random residual error expression of the control equation is established, the parameter sensitivity of the yield load is calculated based on the sobol index, the parameter sensitivity of the buckling mode is calculated based on the covariance decomposition index, the final expression is obtained, the algorithm is reasonable, the buckling load and the mode can be efficiently solved, and the method does not need samples, the calculation efficiency is improved while the stability of the results is guaranteed, and the parameter global sensitivity analysis can be carried out.

Description

TECHNICAL FIELD

[0001] The application relates to a structural elastic buckling global sensitivity solving method, in particular to a structural elastic buckling global sensitivity solving method based on random homotopy analysis, and belongs to the technical field of structural engineering. BACKGROUND

[0002] With the rapid development of economy and civil engineering infrastructure, structural safety is also paid more and more attention by people, and structural stability is one of the main problems related to structural safety. The traditional structural stability analysis is carried out in the framework of a deterministic structure system. However, in real engineering, there are many uncertainties in material properties, geometric sizes and the like. Therefore, fully considering these uncertainties in structural modeling will make the analysis result more reasonable, and is of great significance to the safety evaluation of the structure.

[0003] In actual engineering, many problems involve multiple uncertain parameters, and the influence degree of these parameters on the result may be different. Sensitivity analysis can help us determine which parameters are more sensitive to the result, so as to find important influence parameters and optimize the structure design. Sensitivity analysis includes local sensitivity analysis and global sensitivity analysis, the former is suitable for a narrow range of application, and is generally used to calculate the sensitivity of parameters near the mean value or near the design value. The latter is more widely applicable, and can be used for parameter sensitivity analysis in the entire sample space. Among them, the global sensitivity analysis based on variance can reflect the contribution of input variable variance to output response variance, and is widely used in actual engineering.

[0004] Among the existing methods for solving global sensitivity, the MCS method (Monte Carlo simulation method) needs to select a large number of samples and uses double-loop calculation, which will result in a huge amount of calculation. The Kriging method (Kriging method) and other proxy model methods can only calculate the global sensitivity of a single output, and for a multi-output model, a proxy model needs to be established for each single output, and in the process of establishing the proxy model, samples are needed, and the accuracy and convergence of the calculation result are affected by the selected sample points. In the elastic buckling analysis of the random parameter structure, the buckling load is a single scalar output, and the buckling mode is a vector type multi-output. Therefore, although the MCS method can calculate the global sensitivity of the buckling load and the mode, the amount of calculation is huge. The calculation efficiency of the Kriging method in solving the buckling load is significantly improved compared with the MCS method, but these methods are difficult to solve the buckling mode and perform global sensitivity analysis, which is not conducive to the design and optimization of the structure. Based on the above problems, the present application is proposed. SUMMARY

[0005] The purpose of the present application is to solve the random buckling mode which is difficult to solve by the traditional method, the calculation workload is huge, and the parameter global sensitivity analysis of the buckling mode cannot be carried out, which is not conducive to the design and optimization of the structure, and the present application provides a kind of structure elastic buckling global sensitivity solving method based on random homotopy analysis, which has reasonable algorithm, small calculation amount, can efficiently solve the buckling load and mode, and the method does not need sample in the process of establishing the proxy model between the buckling load or buckling mode and the random parameter, thereby avoiding the influence of sample point selection on the convergence of the calculation result, improving the calculation efficiency while ensuring the stability of the result, and being able to carry out parameter global sensitivity analysis.

[0006] To achieve the purpose of the above-mentioned application, the technical solution of the present application is: a kind of structure elastic buckling global sensitivity solving method based on random homotopy analysis, characterized by comprising the following steps:

[0007] Step one, using finite element method and minimum potential energy principle, considering the uncertainty of material elastic modulus, the control equation of deterministic structure elastic stability analysis is established:

[0008] (K-FK g )D=0 (1)

[0009] Wherein K and K g are the elastic stiffness matrix and geometric stiffness matrix of the structure respectively, F and D are the eigenvalue and eigenvector matrix respectively, and the minimum eigenvalue and the corresponding eigenvector obtained by solving the equation are the buckling load and buckling mode of the deterministic structure respectively;

[0010] Step two, the control equation in step one is reconstructed based on the idea of homotopy analysis method;

[0011] Step three, the buckling load and mode are expressed as homotopy series form about random variables, the control equation after reconstruction is solved, and each coefficient in the series expression of buckling load and mode is obtained;

[0012] Step four, the random residual error expression of the control equation is established, and the value of the parameter h in the buckling load and mode expression is determined by making the random residual error minimum;

[0013] Step five, the parameter sensitivity of yield load is calculated based on sobol index, and the parameter sensitivity of buckling mode is calculated based on covariance decomposition index, and the final expression is obtained.

[0014] Further, the specific method of the step two is as follows:

[0015] The uncertainty of material parameters (such as elastic modulus) is described by random field or by mutually independent random variables, then the elastic stiffness matrix K can be expressed by formula (2):

[0016]

[0017] where K0is the elastic stiffness matrix when the structure parameters take the mean values, K i is the deterministic coefficient matrix, ξ i is a random variable, and ΔKis the random part of the elastic stiffness matrix. The random buckling eigenvalue equation of the structure is as follows:

[0018]

[0019] Based on the random homotopy analysis, the equation (3) is restructured as

[0020] (1-p)[k0Ω(ξ,h,p)-Φ(ξ,h,p)k g Ω(ξ,h,p)]

[0021]

[0022] where p∈[0,1] and h≠0 are auxiliary parameters; Φ(ξ,h,0) and Ω(ξ,h,0) correspond to the buckling load and buckling mode when the structure parameters take the design values, respectively, and Φ(ξ,h,1) and Ω(ξ,h,1) correspond to the random buckling load and buckling mode of the structure after considering the randomness of the parameters, respectively; it can be seen that when the parameter p increases from 0 to 1, Φ(ξ,h,p) and Ω(ξ,h,p) change from the buckling load and mode of the original deterministic structure system to the buckling load and mode of the random structure system, respectively.

[0023] Further, the specific method of the step three is as follows:

[0024] First, Taylor series expansion of Φ(ξ,h,p) and Ω(ξ,h,p) is carried out at p=0:

[0025]

[0026] Since Φ(ξ,h,0)=F0 and Ω(ξ,h,0)=D0, the accurate solutions of the random structure buckling load and buckling mode can be obtained at p=1:

[0027]

[0028] Subsequently, F m and D m (m≥1) can be obtained by taking the m-th order partial derivative of the parameter p in the equation (4) and setting p=0, and they have the following recursive expression:

[0029]

[0030] ​

[0031] where,

[0032] Finally, F m and D m are substituted into equation (7) and rearranged to obtain the expressions for the random buckling load and buckling mode as follows:

[0033]

[0034] where F(ξ, h) and D(ξ, h) are the buckling eigenvalue or buckling mode, respectively, and F0and D0are deterministic coefficients; β m，l (h)(l = 1, 2,..., m) are functions of h only, where h = (-2, 0);

[0035]

[0036] Further, the specific method of step four is as follows:

[0037] The p-order central moment of the random residual error of equation (3) is defined as:

[0038]

[0039] where, a i and b i (i = 1, 2,..., n) are the upper and lower bounds of the ith random variable ξ i , respectively, and f ξi is the probability density function of ξ i ;

[0040] The value of h is determined by minimizing the p-order central moment of the random residual error by introducing the L 2 norm of vector:

[0041]

[0042] Further, the specific method of step five is as follows:

[0043] The ith random variable ξ i corresponds to the global sensitivity index S i of the buckling load and the global sensitivity index

[0044] MS i The calculation method is as follows: first, generate two sample matrices A (G×n) and B (G×n) of G number of n random variables, replace the ith column of B (G×n) with the ith column of A (G×n) to generate sample matrix Calculate A(G×n) and The sample matrix corresponds to the buckling load and modal sample, F A (1×G) 、 D A (N×G) and Then S i and MS i The calculation expression is as follows:

[0045]

[0046] The beneficial effects of the present application are:

[0047] 1. The present application restructures the free vibration control equation by using the idea of homotopy analysis method, obtains the zero-order deformation equation, and establishes the random residual error expression of the control equation, and the statistical characteristics of the two can be obtained by using the series expression of the natural frequency and the natural mode.

[0048] 2. The present application does not need samples when establishing the proxy model between the buckling load and the random variable, thereby fundamentally avoiding the influence of the selection of sample points on the convergence of the calculation results, improving the calculation efficiency while ensuring the stability of the results.

[0049] 3. The algorithm of the present application is reasonable and has small calculation amount, and overcomes the disadvantages of high calculation cost, large calculation amount and inaccurate calculation of the traditional method, realizes efficient solution of the buckling load and the mode, and can provide scientific design basis and technical support for structural design, optimization and long-term operation and maintenance. BRIEF DESCRIPTION OF DRAWINGS

[0050] Figure 1 is the calculation flowchart of the present application.

[0051] Figure 2 is the probability density contrast chart of the buckling characteristic value obtained by using the method of the present application and the traditional MCS method and Kriging method.

[0052] Figure 3 is the mean contrast chart of the buckling mode of the present application.

[0053] Figure 4 is the mean square deviation contrast chart of the buckling mode of the present application.

[0054] Figure 5 is the sensitivity contrast chart of the buckling load by using the method of the present application and the traditional MCS method.

[0055] Figure 6 is the sensitivity contrast chart of the buckling mode by using the method of the present application and the traditional MCS method. DETAILED DESCRIPTION

[0056] The application is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0057] Referring to Figures 1 to 6 A structure elastic buckling global sensitivity solving method based on random homotopy analysis of the application, characterized in that comprising the following steps:

[0058] Step one, using the finite element method and the principle of minimum potential energy, considering the uncertainty of material elastic modulus, the control equation of the deterministic structure elastic stability analysis is established:

[0059] (K-FK g )D=0 (1)

[0060] Wherein K and K s are the elastic stiffness matrix and the geometric stiffness matrix of the structure respectively, F and D are the eigenvalue and eigenvector matrix respectively, the minimum eigenvalue and the corresponding eigenvector obtained by solving the equation are the buckling load and the buckling mode of the deterministic structure respectively;

[0061] Step two, the control equation in step one is reconstructed based on the idea of homotopy analysis method;

[0062] Step three, the buckling load and mode are expressed as homotopy series form about random variables, the control equation after reconstruction is solved, and each coefficient in the series expression of buckling load and mode is obtained;

[0063] Step four, the random residual error expression of the control equation is established, and the value of the parameter h in the expression of buckling load and mode is determined by making the random residual error minimum;

[0064] Step five, the parameter sensitivity of yield load is calculated based on sobol index, and the parameter sensitivity of buckling mode is calculated based on covariance decomposition index, and the final expression is obtained.

[0065] The specific method of step two is as follows:

[0066] The uncertainty of material parameters (such as elastic modulus) is described by random field or by mutually independent random variables, then the elastic stiffness matrix K can be expressed by formula (2):

[0067]

[0068] Wherein K0 is the elastic stiffness matrix when the structure parameters take the mean value, K i is the deterministic coefficient matrix, ξ i is the random variable, and ΔK is the random part of the elastic stiffness matrix. At this time, the random buckling eigenvalue equation of the structure is as follows:

[0069]

[0070] Based on the random homotopy analysis, the equation (3) is restructured as

[0071]

[0072] where p∈[0, 1] and h≠0 are auxiliary parameters; Φ(ξ, h, 0) and Ω(ξ, h, 0) are the buckling load and buckling mode shape corresponding to the design value of the structure parameter, respectively; Φ(ξ, h, 1) and Ω(ξ, h, 1) are the random buckling load and buckling mode shape of the structure considering the randomness of the parameter, respectively; it can be seen that when the parameter p increases from 0 to 1, Φ(ξ, h, p) and Ω(ξ, h, p) change from the buckling load and mode shape of the original deterministic structure system to the buckling load and mode shape of the random structure system, respectively.

[0073] The specific method of the third step is as follows:

[0074] First, Taylor series expansion of Φ(ξ, h, p) and Ω(ξ, h, p) is carried out at p=0:

[0075]

[0076] Since Φ(ξ, h, 0)=F0 and Ω(ξ, h, 0)=D0, the exact solution of the random structure buckling load and buckling mode shape can be obtained at p=1:

[0077]

[0078] Subsequently, F m and D m (m≥1) can be obtained by taking the m-th partial derivative of the parameter p through equation (4) and setting p=0, and they have the following recursive expression:

[0079]

[0080]

[0081]

[0082] Finally, F m and D m are brought into equation (7), and after combining like terms, the expression of the random buckling load and buckling mode shape can be obtained as follows:

[0083]

[0084] where F(ξ, h) and D(ξ, h) are the buckling eigenvalue or buckling mode shape, respectively, and F0 and D0 are the deterministic coefficients; β m，l (h)(l=1, 2,..., m) is a function containing only the parameter h, where h= (-2, 0).​​

[0085]

[0086] The specific method of step four is as follows:

[0087] The p-order origin moment of the random residual error of equation (3) is defined as:

[0088]

[0089] In the formula, a i and b i (i=1, 2,..., n) are the upper limit and lower limit of the ith random variable ξ i , and f ξi is the probability density function of ξ i ;

[0090] The value of h is determined by introducing the L 2 norm of the vector, and minimizing the p-order origin moment of the above random residual error:

[0091]

[0092] The specific method of step five is as follows:

[0093] The ith random variable ξ i corresponds to the global sensitivity index S i of the buckling load and the global sensitivity index MS i of the buckling mode, and the calculation method is as follows: first, generate two sample matrices A (G×n) and B (G×n) with G numbers of n random variables, replace the ith column of B (G×n) with the ith column of A (G×n) to generate a sample matrix Calculate the buckling load and modal sample corresponding to the sample matrix A (G×n) and respectively, F A (1×G) , D A (N×G) and Then, the calculation expressions of S i and MS i are as follows:

[0094]

[0095] Step four, Taylor series expansion of Φ(ξ, h, p) and Ω(ξ, h, p) at p=0:

[0096]

[0097] Since Φ(ξ, h, 0) = F0and Ω(ξ, h, 0) = D0, the exact solution of the buckling load and buckling mode of the random structure can be obtained at p = 1:

[0098]

[0099] Step five, F m and D m (m≥1) can be obtained by taking the m-th order partial derivative of equation (4) with respect to p and setting p = 0:

[0100] There is the following recursive expression:

[0101]

[0102]

[0103] where,

[0104] Step six, F m and D m are brought into equation (7) and after combining like terms, the expression of the random buckling load and buckling mode can be obtained as follows:

[0105]

[0106] where F(ξ, h) and D(ξ, h) are the buckling eigenvalues or buckling modes, F0and D0are deterministic coefficients; β m，l (h) (l = 1, 2,..., m) are functions containing only the parameter h, where h = (-2, 0);

[0107]

[0108] The specific method of step four is as follows:

[0109] Define the p-th order origin moment of the random residual error of equation (3) as:

[0110]

[0111] where, a i and b i (i = 1, 2,..., n) are the upper and lower bounds of the i-th random variable ξ i , and f ξi is the probability density function of ξ i ;

[0112] By introducing the L 2 norm of the vector, the value of h is determined by minimizing the p-th order origin moment of the above random residual error:

[0113]

[0114] The specific method of the step five is as follows:

[0115] The i-th random variable ξ i The global sensitivity index S of the buckling load i And the global sensitivity index MS of the buckling mode i The calculation method is as follows: first, two groups of sample matrices A (G×n) And B (G×n) of G numbers are generated for n random variables (G×n) The i-th column of B (G×n) is replaced with the i-th column of A The sample matrix (G×n) And The buckling load and mode sample corresponding to the sample matrix are calculated respectively, F A (1×G) , D A (N×G) And Then S i And MS i The calculation expression is as follows:

[0116]

[0117] In order to clearly understand the present application, the present application will be further described in detail below in combination with examples, but this description will not constitute a limitation on the present application. In order to facilitate description, the method of the present application is named as SHA method.

[0118] Example one: the present application takes a reinforced concrete-steel composite column with a fixed lower end and a free upper end as an example to solve the buckling load and mode, and performs parameter global sensitivity analysis on the buckling load and mode. The column is 6m long, which is divided into upper, middle and lower three sections, each section is 2m long. The lower part is fixed to the ground, and the free end is subjected to an axial load of 100kN vertically downward. The lower part is a reinforced concrete structure, the elastic modulus is 40×10 6 kN / m 2 , the cross-sectional area is 0.16m 2 , and the cross-sectional moment of inertia is 0.0021m 4 ; the middle part is a reinforced concrete structure, the elastic modulus is 60×10 6 kN / m 2 , the cross-sectional area is 0.16m 2 , and the cross-sectional moment of inertia is 0.0021m 4 ; the upper part is a steel structure, the elastic modulus is 210×10 6 kN / m 2, the cross-sectional area is 35.5*10 -4 m 2 , the cross-sectional moment of inertia is 2.4*10 -5 m 4 The elastic modulus of the three parts respectively obeys normal distribution, beta distribution and weibull distribution, each part corresponds to 4 random variables, and the structure contains 12 random variables in total. The finite element model of the column is divided into 60 units, and the unit length is 0.1m.

[0119] The first order eigenvalue and eigenvector are calculated by the method (SHA method) provided by the application, and the mean of the buckling eigenvalue obtained is 24.69. In order to verify the accuracy and calculation efficiency of the method of the application, the probability density distribution of the buckling eigenvalue, the mean and mean square deviation of the buckling mode, the global sensitivity index of the buckling load and the buckling mode are compared with the MCS method and the Kriging method (using 200 samples to establish a surrogate model), and the buckling eigenvalue probability density graphs of the three methods are compared as shown in the accompanying Figure 2 .

[0120] As shown in the accompanying Figure 2 , the buckling eigenvalue probability density graphs of the SHA method and the MCS method are quite consistent, and the probability density graph of the Kriging method is poor in tail consistency.

[0121] The Kriging method cannot calculate the buckling mode, and the mean and mean square deviation graphs of the buckling mode of the SHA method and the MCS method are compared as shown in the accompanying Figure 3 , the accompanying Figure 4 .

[0122] As shown in the accompanying Figure 3 , the accompanying Figure 4 , the mean and mean square deviation of the buckling mode of the SHA method and the MCS method are quite consistent.

[0123] The buckling load sensitivity analysis graphs of the three methods are shown in the accompanying Figure 5 .

[0124] As shown in the accompanying Figure 5 , the mean and mean square deviation of the buckling mode of the SHA method and the MCS method are quite consistent. And the Kriging method is quite different from the MCS method at point X9, and has a large error at other points.

[0125] The buckling mode sensitivity analysis graphs of the SHA method and the MCS method are shown in the accompanying Figure 6 . It can be seen that the buckling mode sensitivity of the two methods is almost the same, and is relatively low, and the buckling mode sensitivity of the method of the application is relatively high.

[0126] From the above examples, it can be seen that the method proposed in the application is similar to the MCS method in terms of calculation accuracy. In terms of calculation time, the SHA method is 37.4 s, the MCS method is 237.8 s, and the Kriging method is 52.6 s, so the method has a significant advantage of less calculation time. The calculation accuracy of the Kriging method is obviously not as good as the other two methods, and it cannot calculate the buckling mode.

[0127] The above is a further detailed description of the application in combination with the specific embodiments, and it cannot be considered that the specific embodiments of the application are limited to these descriptions. For ordinary skilled persons in the technical field to which the application belongs, various simple replacements, improvements and changes can be made without departing from the concept of the application, and various simple replacements, improvements and changes made should be considered to belong to the protection scope of the application.

Claims

1. A structure elastic buckling global sensitivity solution method based on random homotopy analysis, characterized in that The method comprises the following steps: Step one, considering the uncertainty of material elastic modulus, the control equation of elastic stability analysis of deterministic structure is established by using finite element method and minimum potential energy principle: (K-FK g )D = 0 (1) where K and K g are the structural and geometric stiffness matrices, respectively, and F and D are the eigenvalue and eigenvector matrices, respectively. The smallest eigenvalue and corresponding eigenvector obtained by solving the equation are the buckling load and buckling mode of the deterministic structure, respectively. Step two, the control equation in step one is reconstructed based on the idea of homotopy analysis method, and the specific method is as follows: The uncertainty of material parameters is described by using random field or using independent random variables, and then the elastic stiffness matrix K can be expressed as formula (2): where K0is the elastic stiffness matrix with the structure parameters taking mean values, K i is the deterministic coefficient matrix, ξ i is a random variable, and ΔKis the random part of the elastic stiffness matrix. In this case, the random buckling eigenvalue equation of the structure is as follows: Based on random homotopy analysis, the equation (3) is reconstructed, In the formula, p is in [0, 1], h≠0 is an auxiliary parameter; Φ(ξ, h, 0) and Ω(ξ, h, 0) respectively correspond to the buckling load and buckling mode when the structure parameter takes the design value, Φ(ξ, h, 1) and Ω(ξ, h, 1) respectively correspond to the random buckling load and buckling mode of the structure after considering the randomness of the parameter; it can be seen that when the parameter p increases from 0 to 1, Φ(ξ, h, p) and Ω(ξ, h, p) respectively change from the buckling load and mode of the original deterministic structure system to the buckling load and mode of the random structure system; Step three, the buckling load and mode are expressed in the form of homotopy series about random variables, the reconstructed control equation is solved, and the coefficients in the series expression of the buckling load and mode are obtained; Step four, the random residual error expression of the control equation is established, and the value of the parameter h in the expression of the buckling load and mode is determined by minimizing the random residual error; Step five, the parameter sensitivity of the yield load is calculated based on the sobol index, and the parameter sensitivity of the buckling mode is calculated based on the covariance decomposition index, and the final expression is obtained.

2. The method of claim 1, wherein the method is based on a random homotopy analysis. The specific method of step three is as follows: Firstly, Taylor series expansion is carried out on Φ(ξ, h, p) and Ω(ξ, h, p) at p=0: Since Φ(ξ, h, 0)=F0 and Ω(ξ, h, 0)=D0, the exact solution of the random structure buckling load and buckling mode can be obtained at p=1: Subsequently F m and D m where m > 1, have the following recursive expressions, obtained by taking the m-th derivative of p with respect to equation (4) and setting p = 0: wherein Finally, F m and D m After sorting by combining like terms, the expressions for the random buckling load and buckling mode are as follows: where F(ξ, h) and D(ξ, h) are the buckling eigenvalue or buckling mode, respectively, and F0and D0are deterministic coefficients; β m,l (h), where l = 1, 2,..., m, is a function containing only the parameter h, where h = (-2, 0); 3. The method of claim 1, wherein: The specific method of step four is as follows: The p-order origin moment of the random residual error of equation (3) is defined as: Where a i and b i , where i=1,2,...,n, are the i-th random variable ξ i The upper and lower bounds of for ξ i The probability density function of By introducing the L 2 norm of the vector, the value of h is determined by minimizing the p-th order central moment of the random residual error.

4. The method of claim 1, wherein: The specific method of step five is as follows: The ith random variable ξ i The global sensitivity index of buckling load S i And the global sensitivity index of buckling mode MS i The calculation method is as follows: first, generate two groups of sample matrices A (G×n) And B (G×n) Replace the ith column of B (G×n) With the ith column of A (G×n) To generate sample matrix Calculate the buckling load and modal sample corresponding to sample matrix A (G×n) And Sample matrix A (1×G) , D A (N×G) And Then S i And MS i The calculation expression is as follows:

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