Structural elastic buckling global sensitivity solving method based on random homotopy analysis

Abstract

A structural elastic buckling global sensitivity solving method based on random homotopy analysis is characterized in that a finite element method and a minimum potential energy principle are utilized to establish a control equation of deterministic structural elastic stability analysis, the control equation is reconstructed based on the thought of a homotopy analysis method, and the structural elastic buckling global sensitivity is obtained. The method comprises the following steps of: reconstructing a control equation, expressing a buckling load and a modal as a homotopy series form about a random variable, solving the reconstructed control equation to obtain various coefficients in a buckling load and modal series expression, establishing a random residual error expression of the control equation, calculating parameter sensitivity of a yield load based on a sobol index, and calculating the parameter sensitivity of the buckling load. And calculating the parameter sensitivity of the buckling mode based on the covariance decomposition index to obtain a final expression. The method is reasonable, can efficiently solve the buckling load and the mode, does not need a sample, can ensure the stability of the result while improving the calculation efficiency, and can perform parameter global sensitivity analysis.

Description

Technical Field

[0001] The invention relates to a method for solving global sensitivity of structural elastic buckling, in particular to a method for solving global sensitivity of structural elastic buckling based on stochastic homotopy analysis, and belongs to the technical field of structural engineering. Background Art

[0002] With the rapid development of economy and civil engineering infrastructure, people pay more and more attention to structural safety, and structural stability is one of the main issues related to structural safety. Traditional structural stability analysis is carried out under the framework of deterministic structural system. However, in real engineering, there are always many uncertainties in material properties, geometric dimensions, etc. Therefore, fully considering these uncertainties in structural modeling will make the analysis results more reasonable, which is of great significance to the safety assessment of the structure.

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

[0004] Among the various existing methods for solving global sensitivity, the MCS method (Monte Carlo simulation method) requires the selection of a large number of samples and the use of double-layer loop calculations, which will result in a huge amount of calculations. The proxy model methods such as the Kriging method can only calculate the global sensitivity under a single output. For multi-output models, it is necessary to establish a proxy model for each single output, and in the process of establishing the proxy model, it is necessary to use samples. The accuracy and convergence of the calculation results are affected by the selected sample points. In the elastic buckling analysis of random parameter structures, 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 mode, its calculation workload is huge. The Kriging method has significantly improved computational efficiency in solving the buckling load compared to 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 invention is proposed. Summary of the invention

[0005] The purpose of the present invention is to address the defects and shortcomings of traditional methods, such as difficulty in solving random buckling modes, huge computational workload, inability to perform global sensitivity analysis of parameters of buckling modes, and being unfavorable to structural design and optimization. The present invention provides a reasonable algorithm with small computational workload, which can efficiently solve buckling loads and modes. In addition, this method does not require samples in the process of establishing a proxy model between buckling loads or buckling modes and random parameters, thereby avoiding the influence of sample point selection on the convergence of calculation results. While improving calculation efficiency, it can ensure the stability of the results and perform global sensitivity analysis of parameters. A method for solving global sensitivity of structural elastic buckling based on random homotopy analysis 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 elastic buckling based on stochastic homotopy analysis, characterized by comprising the following steps:

[0007] Step 1: Using the finite element method and the principle of minimum potential energy, considering the uncertainty of the material elastic modulus, establish the control equation for deterministic structural elastic stability analysis:

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

[0009] Where 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 matrices, respectively. 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 2: Reconstruct the control equation in step 1 based on the idea of ​​homotopy analysis method;

[0011] Step 3: Express the buckling load and mode in the form of homotopy series with respect to random variables, solve the reconstructed control equation, and obtain the coefficients of each term in the buckling load and mode series expressions;

[0012] Step 4: Establish a random residual error expression for the control equation, and determine the value of the parameter h in the buckling load and modal expression by minimizing the random residual error;

[0013] Step 5: Calculate the parameter sensitivity of the yield load based on the Sobol index, and calculate the parameter sensitivity of the buckling mode based on the covariance decomposition index to obtain the final expression.

[0014] Furthermore, the specific method of step 2 is as follows:

[0015] Using random fields or independent random variables to describe the uncertainty of material parameters (such as elastic modulus), the elastic stiffness moment K can be expressed by formula (2):

[0016]

[0017] Among them, K0 is the elastic stiffness matrix when the structural parameters take the average value, K i is the deterministic coefficient matrix, ξ i is a 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:

[0018]

[0019] Reconstructing equation (3) based on stochastic homotopy analysis,

[0020]

[0021] In the formula: p∈[0,1], h≠0 is an auxiliary parameter; Φ(ξ,h,0) and Ω(ξ,h,0) correspond to the buckling load and buckling mode when the structural 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 structural system to the buckling load and mode of the random structural system, respectively.

[0022] Furthermore, the specific method of step three is as follows:

[0023] First, expand Φ(ξ,h,p) and Ω(ξ,h,p) in Taylor series at p=0:

[0024]

[0025]

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

[0027]

[0028] Then F m and D m (m≥1) By taking the mth partial derivative of the parameter p through equation (4) and setting p=0, we can get them as follows

[0029] The following recursive expression:

[0030]

[0031]

[0032] in,

[0033] Finally, F m and D m Substituting into equation (7) and merging similar terms, we can obtain the expressions of random buckling load and buckling mode as follows:

[0034]

[0035]

[0036] where F(ξ,h) and D(ξ,h) are the buckling eigenvalues ​​or buckling modes, respectively, and F . and D . is the coefficient of certainty; β m，l (h)(l=1, 2, ..., m) is a function containing only parameter h, where h=(-2, 0);

[0037]

[0038] Furthermore, the specific method of step 4 is as follows:

[0039] The p-order origin moment of the random residual error of equation (3) is defined as:

[0040]

[0041] In the formula, a i and b i (i=1,2,...,n) are the i-th random variables ξ i The upper and lower bounds of f ξi For i The probability density function of

[0042] By introducing the vector L 2 norm, let the p-order origin moment of the above random residual be the minimum to determine the value of h:

[0043]

[0044] Furthermore, the specific method of step five is as follows:

[0045] The i-th random variable ξ i The global sensitivity index S corresponding to the buckling load i and buckling mode global sensitivity index

[0046] MS i The calculation method is as follows: First, generate two sets of sample matrices 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 (G×n) and Buckling load and modal sample corresponding to the sample matrix, F A (1×G) , D A (N×G) and Then S i and MS i The calculation expression is as follows:

[0047]

[0048]

[0049] The beneficial effects of the present invention are:

[0050] 1. The present invention utilizes the idea of ​​homotopy analysis method to reconstruct the free vibration control equation, obtains the zero-order deformation equation, and establishes the random residual error expression of the control equation. The statistical characteristics of the two can be obtained by using the series expression of the natural frequency and the natural mode.

[0051] 2. The present invention does not require samples when establishing a proxy model between buckling loads and random variables, thereby fundamentally avoiding the influence of sample point selection on the convergence of calculation results, and ensuring the stability of results while improving calculation efficiency.

[0052] 3. The algorithm of the present invention is reasonable and the calculation amount is small. It overcomes the disadvantages of high calculation cost, large calculation amount and inaccurate calculation of traditional methods, and realizes the efficient solution of buckling load and mode, which can provide scientific design basis and technical support for structural design, optimization and long-term operation and maintenance. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a calculation flow chart of the present invention.

[0054] Figure 2 It is a probability density comparison diagram of the buckling eigenvalues ​​solved by the method of the present invention and the traditional MCS method and Kriging method.

[0055] Figure 3 It is a comparison diagram of the mean values ​​of the buckling modes of the present invention.

[0056] Figure 4 It is a comparison diagram of the mean square error of the buckling mode of the present invention.

[0057] Figure 5It is a comparison diagram of the sensitivity of buckling load using the method of the present invention and the traditional MCS method.

[0058] Figure 6 It is a comparison diagram of the sensitivity of buckling modes using the method of the present invention and the traditional MCS method. DETAILED DESCRIPTION

[0059] The present invention is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0060] See also Figures 1 to 6 The present invention provides a method for solving the global sensitivity of structural elastic buckling based on stochastic homotopy analysis, which is characterized by comprising the following steps:

[0061] Step 1: Using the finite element method and the principle of minimum potential energy, considering the uncertainty of the material elastic modulus, establish the control equation for deterministic structural elastic stability analysis:

[0062] (K-FK g )D=0 (1)

[0063] Where 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 matrices, respectively. The minimum eigenvalue and the corresponding eigenvector obtained by solving the equation are the buckling load and buckling mode of the deterministic structure, respectively.

[0064] Step 2: Reconstruct the control equation in step 1 based on the idea of ​​homotopy analysis method;

[0065] Step 3: Express the buckling load and mode in the form of homotopy series with respect to random variables, solve the reconstructed control equation, and obtain the coefficients of each term in the buckling load and mode series expressions;

[0066] Step 4: Establish a random residual error expression for the control equation, and determine the value of the parameter h in the buckling load and modal expression by minimizing the random residual error;

[0067] Step 5: Calculate the parameter sensitivity of the yield load based on the Sobol index, and calculate the parameter sensitivity of the buckling mode based on the covariance decomposition index to obtain the final expression.

[0068] The specific method of step 2 is as follows:

[0069] Using random fields or independent random variables to describe the uncertainty of material parameters (such as elastic modulus), the elastic stiffness moment K can be expressed by formula (2):

[0070]

[0071] Among them, K0 is the elastic stiffness matrix when the structural parameters take the average value, K i is the deterministic coefficient matrix, ξ i is a 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:

[0072]

[0073] Reconstructing equation (3) based on stochastic homotopy analysis,

[0074]

[0075] In the formula: p∈[0,1], h≠0 is an auxiliary parameter; Φ(ξ,h,0) and Ω(ξ,h,0) correspond to the buckling load and buckling mode when the structural 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 structural system to the buckling load and mode of the random structural system, respectively.

[0076] The specific method of step three is as follows:

[0077] First, expand Φ(ξ,h,p) and Ω(ξ,h,p) in Taylor series at p=0:

[0078]

[0079]

[0080] Since Φ(ξ,h,0)=F0 and Ω(ξ,h,0)=D0, then the exact solution of the buckling load and buckling mode of the random structure can be obtained when p=1:

[0081]

[0082] Then F m and D m (m≥1) By taking the mth partial derivative of the parameter p through equation (4) and setting p=0, we can get them as follows

[0083] The following recursive expression:

[0084]

[0085]

[0086] in,

[0087] Finally, F mand D m Substituting into equation (7) and merging similar terms, we can obtain the expressions of random buckling load and buckling mode as follows:

[0088]

[0089]

[0090] where F(ξ,h) and D(ξ,h) are the buckling eigenvalues ​​or buckling modes, respectively, and F . and D . is the coefficient of certainty; β m，l (h)(l=1, 2, ..., m) is a function containing only parameter h, where h=(-2, 0);

[0091]

[0092] The specific method of step 4 is as follows:

[0093] The p-order origin moment of the random residual error of equation (3) is defined as:

[0094]

[0095] In the formula, a i and b i (i=1,2,...,n) are the i-th random variables ξ i The upper and lower bounds of f ξi For i The probability density function of

[0096] By introducing the vector L 2 norm, let the p-order origin moment of the above random residual be the minimum to determine the value of h:

[0097]

[0098] The specific method of step five is as follows:

[0099] The i-th random variable ξ i The global sensitivity index S corresponding to the buckling load i and buckling mode global sensitivity index

[0100] MS i The calculation method is as follows: First, generate two sets of sample matrices 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 (G×n) and Buckling load and modal sample corresponding to the sample matrix, F A (1×G) , D A (N×G) and Then S i and MS i The calculation expression is as follows:

[0101]

[0102]

[0103] Step 4: Expand Φ(ξ,h,p) and Ω(ξ,h,p) into Taylor series at p=0:

[0104]

[0105]

[0106] Since Φ(ξ,h,0)=F0 and Ω(ξ,h,0)=D0, then the exact solution of the buckling load and buckling mode of the random structure can be obtained when p=1:

[0107]

[0108] Step 5. F m and D m (m≥1) By taking the mth partial derivative of the parameter p through equation (4) and setting p=0, we can get

[0109] There is the following recursive expression:

[0110]

[0111]

[0112] in,

[0113] Step 6: F m and D m Substituting into equation (7) and merging similar terms, we can obtain the expressions of random buckling load and buckling mode as follows:

[0114]

[0115]

[0116] where F(ξ,h) and D(ξ,h) are the buckling eigenvalues ​​or buckling modes, respectively, and F . and D . is the coefficient of certainty; βm，l (h)(l=1, 2, ..., m) is a function containing only parameter h, where h=(-2, 0);

[0117]

[0118] The specific method of step 4 is as follows:

[0119] The p-order origin moment of the random residual error of equation (3) is defined as:

[0120]

[0121] In the formula, a i and b i (i=1,2,...,n) are the i-th random variables ξ i The upper and lower bounds of f ξi For i The probability density function of

[0122] By introducing the vector L 2 norm, let the p-order origin moment of the above random residual be the minimum to determine the value of h:

[0123]

[0124] The specific method of step five is as follows:

[0125] The i-th random variable ξ i The global sensitivity index S corresponding to the buckling load i and buckling mode global sensitivity index MS i The calculation method is as follows: First, generate two sets of sample matrices 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 (G×n) and Buckling load and modal sample corresponding to the sample matrix, F A (1×G) , D A (N×G) and Then S i and MS i The calculation expression is as follows:

[0126]

[0127]

[0128] In order to facilitate a clear understanding of the present invention, the present invention is further described in detail below in conjunction with the embodiments, but this description does not constitute a limitation of the present invention. For ease of description, the method of the present invention is named SHA method.

[0129] Example 1: This paper 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 conducts a parameter global sensitivity analysis on the buckling load and mode. The total length of the column is 6m, which is divided into three sections: upper, middle and lower, 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 with an elastic modulus of 40×10 6 kN / m 2 , the cross-sectional area is 0.16m 2 , the section moment of inertia is 0.0021m 4 ; The middle part is a reinforced concrete structure with an elastic modulus of 60×10 6 kN / m 2 The cross-sectional area is 0.16m 2 , the section moment of inertia is 0.0021m 4 ; The upper part is a steel structure with an elastic modulus of 210×10 6 kN / m 2 , the cross-sectional area is 35.5×10 -4 m 2 , the section moment of inertia is 2.4×10 -5 m 4 The elastic moduli of the three parts obey normal distribution, beta distribution and Weibull distribution respectively, 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 with a unit length of 0.1m.

[0130] The method proposed in this invention (SHA method) is applied to calculate the first-order eigenvalue and eigenvector, and the average value of the buckling eigenvalue is 24.69. In order to verify the accuracy and calculation efficiency of the method of the present invention, the probability density distribution of the buckling eigenvalue, the mean and mean square error 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 proxy model). The buckling eigenvalue probability density diagrams of the three methods are shown in the attached figure. Figure 2 shown.

[0131] By the attached Figure 2 It can be seen that the probability density plots of the buckling eigenvalues ​​of the SHA method and the MCS method are quite consistent, while the probability density plot of the Kriging method is less consistent in the tail.

[0132] The Kriging method cannot calculate the buckling mode. The mean and mean square error diagrams of the buckling modes of the SHA method and the MCS method are shown in the attached figure. Figure 3 , Attachment Figure 4 shown.

[0133] By the attached Figure 3 , Attachment Figure 4 It can be seen that the mean and mean square error of the buckling modes of the SHA method and the MCS method are quite consistent.

[0134] The buckling load sensitivity analysis diagrams of the three methods are shown in the attached figure. Figure 5 shown.

[0135] By the attached Figure 5 It can be seen that the mean and mean square error of the buckling modes of the SHA method and the MCS method are quite consistent. However, the Kriging method is almost the same as the MCS method except for point X9, but has large errors at other points.

[0136] Buckling modal sensitivity analysis diagrams of SAH method and MCS method are shown in the attached figure. Figure 6 It can be seen that the buckling mode sensitivities of the two methods are almost the same, both relatively low, and the buckling mode sensitivity of the method of the present invention is higher.

[0137] From the above examples, it can be seen that the method proposed in the present invention is almost the same as the MCS method in terms of calculation accuracy. In terms of calculation time, the SHA method is 37.4s, the MCS method is 237.8s, and the Kriging method is 52.6s. Therefore, this method has the significant advantage of less calculation time. However, the calculation accuracy of the Kriging method is obviously inferior to the other two methods, and it cannot calculate the buckling mode.

[0138] 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 elastic buckling based on stochastic homotopy analysis, characterized by The following steps are involved: Step 1: Using the finite element method and the principle of minimum potential energy, considering the uncertainty of the material elastic modulus, establish the control equation for deterministic structural elastic stability analysis: (K-FK g )D=0 (1) Where 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 matrices, respectively. The minimum eigenvalue and the corresponding eigenvector obtained by solving the equation are the buckling load and buckling mode of the deterministic structure, respectively. Step 2: Reconstruct the control equation in step 1 based on the idea of ​​homotopy analysis method; Step 3: Express the buckling load and mode in the form of homotopy series with respect to random variables, solve the reconstructed control equation, and obtain the coefficients of each term in the buckling load and mode series expressions; Step 4: Establish a random residual error expression for the control equation, and determine the value of the parameter h in the buckling load and modal expression by minimizing the random residual error; Step 5: Calculate the parameter sensitivity of the yield load based on the Sobol index, and calculate the parameter sensitivity of the buckling mode based on the covariance decomposition index to obtain the final expression.

2. The method for solving global sensitivity of structural elastic buckling based on stochastic homotopy analysis according to claim 1 is characterized by: The specific method of step 2 is as follows: Using random fields or independent random variables to describe the uncertainty of material parameters (such as elastic modulus), the elastic stiffness moment K can be expressed by formula (2): Among them, K0 is the elastic stiffness matrix when the structural parameters take the average value, K i is the deterministic coefficient matrix, ξ i is a 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: Reconstructing equation (3) based on stochastic homotopy analysis, In the formula: p∈[0,1], h≠0 is an auxiliary parameter; Φ(ξ,h,0) and Ω(ξ,h,0) correspond to the buckling load and buckling mode when the structural 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 structural system to the buckling load and mode of the random structural system, respectively.

3. The method for solving global sensitivity of structural elastic buckling based on stochastic homotopy analysis according to claim 1 is characterized by: The specific method of step three is as follows: First, expand Φ(ξ,h,p) and Ω(ξ,h,p) in Taylor series at p=0: Since Φ(ξ,h,0)=F0 and Ω(ξ,h,0)=D0, then the exact solution of the buckling load and buckling mode of the random structure can be obtained when p=1: Then F m and D m (m≥1) By taking the m-th partial derivative of the parameter p through equation (4) and setting p=0, we can get the following recursive expression: in, Finally, F m and D m Substituting into equation (7) and merging similar terms, we can obtain the expressions of random buckling load and buckling mode as follows: where F(ξ,h) and D(ξ,h) are the buckling eigenvalues ​​or buckling modes, respectively, and F . and D . is the coefficient of certainty; β m，l (h)(l=1, 2, ..., m) is a function containing only parameter h, where h=(-2, 0); 4. The method for solving global sensitivity of structural elastic buckling based on stochastic homotopy analysis according to claim 1 is characterized by: The specific method of step 4 is as follows: The p-order origin moment of the random residual error of equation (3) is defined as: In the formula, a i and b i (i=1,2,...,n) are the i-th random variables ξ i The upper and lower bounds of For i The probability density function of By introducing the vector L 2 norm, let the p-order origin moment of the above random residual be the minimum to determine the value of h:

5. The method for solving global sensitivity of structural elastic buckling based on stochastic homotopy analysis according to claim 1 is characterized by: The specific method of step five is as follows: The i-th random variable ξ i The global sensitivity index S corresponding to the buckling load i and the buckling mode global sensitivity index MS i The calculation method is as follows: First, generate two sets of sample matrices 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 (G×n) and Buckling load and modal sample corresponding to the sample matrix, F A (1×G) , D A (N×G) and Then S i and MS i The calculation expression is as follows:

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