CN117744444B - BSPLS-RSM-based structural seismic vulnerability curved surface prediction method - Google Patents
BSPLS-RSM-based structural seismic vulnerability curved surface prediction method Download PDFInfo
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Abstract
The invention discloses a structural seismic vulnerability curved surface prediction method based on BSPLS-RSM. Firstly, building a structural finite element model, determining input parameters of a total data set, and carrying out nonlinear power analysis to obtain output parameters; uniformly designing and selecting a training set and a testing set, and training by the training set to obtain a spline function partial least square response surface substitution model; inputting the input parameters of the test set into a substitution model to obtain a fitting value of the output parameters of the test set, and respectively obtaining a vulnerability curved surface of the nonlinear dynamic analysis and a vulnerability curved surface of the spline function partial least square response surface substitution model by the fitting values of the test set and the test set by using a linear fitting method; comparing the two vulnerable curved surface fitting effects, and judging the decision coefficient R until the decision coefficient R is more than or equal to 0.9; and obtaining a structure and component vulnerability model and failure probability considering all random variables by using the determined response surface substitution model, so that weak links of the structure and the component are analyzed, and necessary basis is provided for earthquake resistance reinforcement and risk assessment.
Description
Technical Field
The invention relates to a structural seismic vulnerability curved surface prediction method based on BSPLS-RSM (English Basic SPLINE PARTIAL LEAST Squares Response Surface Model, namely a Basic spline partial least square response surface substitution model), which analyzes structural seismic response and structural damage from the aspect of probability, utilizes a response surface substitution model to perform fitting and prediction, and belongs to the technical field of predicting a ancient tower structural vulnerability curved surface by the response surface substitution model.
Background
Structural seismic response analysis and structural damage analysis, collectively referred to as seismic vulnerability analysis, refer to the probability of a structure or component reaching or exceeding a certain limit state. The earthquake-resistant performance of the structure and the component is quantitatively described from the perspective of probability, the relation between the earthquake intensity and the structural damage degree is macroscopically reflected, and necessary basis is provided for researches such as determining weak links, earthquake resistance reinforcement and risk assessment of the structure and the component. The factors influencing the vulnerability of the structure and the components are numerous and can influence each other, and for the structural vulnerability analysis with high nonlinearity and multiple dimensions, the traditional nonlinear power analysis has high calculation cost and low efficiency, so that the response surface model is proposed to replace the nonlinear power analysis. Partial least squares processes multiple collinearity and high-dimensional data by reducing the correlation between independent variables, taking into account the weights between independent and dependent variables, to extract the principal components, thereby building a regression model. The partial least squares method combines principal component analysis and least squares methods and can be used to address data problems with non-linearities, multi-dimensions and strong correlations of independent variables. The spline function interpolation fitting is to divide the whole definition domain into a plurality of spline spaces, fit the spline functions by adopting cubic spline functions in each spline space, find the spline functions with proper width by adjusting the width parameters, and finally achieve small fitting error and smooth curve.
Disclosure of Invention
Aiming at the problems, the invention provides a structural seismic vulnerability curved surface prediction method based on BSPLS-RSM, and the experimental design adopts a uniform design to select a small number of sample points to cover the total sample points so as to reduce the calculation cost of the sample points. The response surface substitution model adopts a basic spline function and a partial least squares fitting regression model so as to ensure that the substitution model can solve the problem of high nonlinearity and ensure the accuracy of the substitution model. In theoretical vulnerability analysis, input parameters and output parameters are determined, a linear fitting method based on a probability earthquake demand model is used for obtaining a vulnerability curved surface, the fitting effect of the vulnerability curved surface obtained by nonlinear dynamic analysis and a response surface substitution model is compared, and the response surface substitution model is used for obtaining a structure and component vulnerability model and failure probability considering random variables, so that weak links of the structure and component can be analyzed, and necessary basis is provided for researches such as earthquake resistance reinforcement, risk assessment and the like. Compared with the vulnerability analysis of the nonlinear dynamic analysis, the response surface substitution model provided by the invention greatly reduces the time cost. Compared with a least square method, the partial least square method combines principal component analysis and the least square method, and can be used for processing data problems with nonlinearity, multidimensional and strong correlation of independent variables. Compared with a polynomial function, the basic spline function disclosed by the invention is beneficial to reducing the risk of overfitting by using a low-order polynomial to interpolate on each subinterval.
The above purpose is achieved by the following technical scheme:
the structural earthquake vulnerability curved surface prediction method based on BSPLS-RSM is characterized by comprising the following steps of:
s1, building a structure finite element model, determining an input parameter X of a total data set, and performing nonlinear power analysis to obtain an output parameter Y;
S2, selecting a training set (train_X, train_Y) and a Test set (test_X, test_Y) by using a uniform design of the total data set, and training the training set to obtain a spline function partial least square response surface substitution model;
S3, inputting the Test set input parameters test_X into the spline function partial least square response surface substitution model obtained in the step S2 to obtain a fitting value Fit_test_Y of the Test set output parameters, and respectively obtaining a nonlinear dynamic analysis vulnerability curved surface and a spline function partial least square response surface substitution model vulnerability curved surface by using a linear fitting method based on a probability earthquake demand model in theoretical vulnerability analysis of the Test set (test_X, test_Y) and the Test set fitting value (test_X, fit_test_Y);
S4, comparing the fitting effect of the two vulnerable curved surfaces in the step S3, judging through a decision coefficient R, if R is more than or equal to 0.9, determining that the response surface replaces the vulnerable curved surface of the model, and if R is less than or equal to 0.9, adjusting the width parameter of the basic spline response surface replacement model until R is more than or equal to 0.9;
s5, obtaining a structure and component vulnerability model and failure probability considering all random variables by using the determined response surface substitution model, so that weak links of the structure and the component are analyzed, and necessary basis is provided for research of earthquake resistance reinforcement and risk assessment.
Further, the total data set in step S2 is a training set and a testing set selected by using a uniform design, and the specific steps are as follows:
S211, a distribution point P n (k), i.e., for a i e h, i=1, 2,..s,
Pn(k)＝(ka1,ka2,...,kas)(mod q)；k＝1,2,...,q
Wherein a s is a natural number; s is the number of variables; q is the number of sample points of each variable; k represents a kth sample point in each variable sample point; h is a positive integer less than q, and the greatest common divisor satisfying h and q is 1;
(ka 1,ka2,...,kas) (mod q) calculating a divisor of the known remainder q for each term in brackets;
s212, obtaining uniform distribution parameters xi of each distribution point P n (k), namely,
a＝(a1,a2,...,as)
avk＝kav(mod q)
Wherein v is the v-th variable of the variables s; ka (mod q) is the calculation of the divisor for the known remainder q and the dividend ka;
S213, finding a distribution point P n (k) which enables the uniformly distributed parameter xi to be minimum, wherein the distribution point is a training sample point;
The step S2 is that a spline function partial least square response surface substitution model is obtained by training a training set, and the specific steps are as follows:
S221, transferring the training set input parameter Train_X to a basic spline space, defining a segment number M i, a segment length h i and a node xi i,l-1, dividing each independent variable into (M i +3) variables of the basic spline space, namely,
ξi,l-1＝min(xi)+max(l-1)hi,(l＝0,1,...,Mi+2)
Wherein X i is the ith variable of the training set input parameter train_x; m i is the number of basic spline space segments of the ith variable; h i is the base spline space segment length of the ith variable; ζ i,l-1 is the first-1 node of the i-th variable; l is the first node of the base spline space;
S222, converting the ith variable X i of the training set input parameter train_x into a variable z i of the base spline space, in which the variable z i is, that is,
Wherein the method comprises the steps ofThe value of the jth sample point of x i on the ith node of the base spline space corresponding to x i; omega b is a B-order B-spline basis function; /(I)Inputting the jth sample point of the ith variable of the parameter train_x for the training set;
S223, normalizing the training set output parameter Train Y and the base spline space variable to eliminate errors caused by different orders of magnitude, that is,
Wherein Y c is the c-th sample point of the training set output parameter Train _ Y,Is the mean of y c, s y is the variance of y c,Normalization to y c; /(I)The value of the jth sample point of x i on the ith node of the base spline space corresponding to x i; For/> S i,l is/>Variance of/>For/>Is normalized by (2); n is the number of sample points of the training set;
S224, generating a multiple linear function, obtaining regression coefficients by using a partial least square method, namely,
Wherein the method comprises the steps ofIs a multiple linear function; a i,l is the regression coefficient of the multiple linear function; p is the variable number of the training set input parameter train_X; /(I)Normalizing the value of the first node of the basic spline space corresponding to x i; epsilon is the fitting error of a multi-element linear function;
S225, after obtaining a regression model in the spline space, changing the regression model into an original space to obtain a nonlinear regression model of y versus x, namely obtaining a response surface substitution model trained by a training set, namely,
Wherein ε is the error of a multiple linear function; Predicted values for the response surface substitution model; beta 0 is a custom parameter that changes spline space to original space; beta i,l is a custom parameter that changes spline space to original space; /(I) Inputting the jth sample point of the ith variable of the parameter train_x for the training set; /(I)Is the mean value of y c; /(I)For/>Is the average value of (2); a i,l is the regression coefficient of the multiple linear function;
The B-spline basis function described in step S222 has the following specific formula:
Wherein Ω d,b is the d-th B-order B-spline basis function; f is an input variable; b is the order of the B spline basis function; d is the number of B spline basis functions; delta is each node of the segmented interval;
the partial least square method in step S224 obtains regression coefficients, which specifically includes the steps of:
s2241, the independent variable group and the dependent variable group are standardized, the standardized independent variable group and the standardized dependent variable group are A and B respectively, namely,
Wherein A and B are independent variable groups and dependent variable groups after standardization; a 1 and a 1 are independent and dependent variable groups before normalization; and/> Is the mean of variables A 1 and B 1; var (A 1) and var (B 1) are the variances of variables A 1 and B 1;
S2242, the weight W 1 is determined according to the correlation of a and B, that is,
Wherein the superscript T denotes a matrix transpose; II 2 is the square operation of the European norm;
S2242, weights a to get a principal component T 1, that is,
T1＝AW1
S2243, the regression coefficients C 1 of B and T 1, that is,
S2244, the regression coefficients P 1 of T 1 and a, that is,
S2245, residual matrices E 1 and F 1 of the normalized independent variables and dependent variables, respectively, that is,
F1＝B-T1C1
S2246, replacing A and B with residual matrixes E 1 and F 1, repeating the above process for iteration until the residual matrixes meet the precision requirement or the number of main components is used up,
S2247, a partial least squares regression equation of the independent variable A 1 and the dependent variable B 1 is obtained.
Further, the linear fitting method based on the probabilistic earthquake demand model in the theoretical vulnerability analysis in the step S3 is used for obtaining the vulnerability curved surface, and the specific steps are as follows:
s311, obtaining structural response through nonlinear dynamic analysis and a response surface substitution model;
s312, drawing a probability demand model and fitting, taking logarithms for input parameters and output parameters respectively, and then carrying out logarithmic linear fitting, namely,
ln(Sd)＝G1ln(x1)+G2ln(x2)+...+Gnln(xn)+ε
S d is a structural requirement parameter; x n is a structural parameter, a material parameter or a seismic intensity parameter; g n is a regression coefficient; epsilon is the fitting error;
s313, substituting the vulnerability function formula to obtain the vulnerability curved surface, namely,
Wherein P is failure probability; s d is a structure demand parameter; x n is a structural parameter, a material parameter or a seismic intensity parameter; LS e is the structural limit state for the e-th performance level; s c is structural resistance, namely a value corresponding to a structural limit state LS; beta c and beta d are the logarithmic standard deviations of the structural demand parameters and structural resistance, respectively; Φ is a standard normal distribution function.
Further, the fitting effect of the two vulnerable curved surfaces in the step S3 is compared in the step S4, and the fitting effect is judged by determining the coefficient R, and the specific formula is as follows:
wherein the method comprises the steps of In order to replace the failure probability of the vulnerable curved surface of the model by the response surface, P f is the failure probability of the vulnerable curved surface of the nonlinear dynamic analysis,/>Is the average value of P f.
Further, step S5 uses the response surface substitution model to obtain a structure and component vulnerability model and failure probability considering random variables, so that weak links of the structure and the component can be analyzed, necessary basis is provided for researches such as earthquake resistance reinforcement, risk assessment and the like, and the specific steps are as follows:
s511, the structure can select random variables from the aspects of material parameters, structure parameters and earthquake motion intensity parameters;
s512, determining the value range and interval of the random variable to obtain sample points;
s513, inputting the sample points into a response surface substitution model of the vulnerable curved surface to obtain failure probability;
S514, analyzing weak links of the structure and performing earthquake-resistant reinforcement.
The method has the following beneficial effects: the experimental design of the invention adopts a uniform design to select a small number of sample points to cover the total sample points so as to reduce the calculation cost of the sample points. The response surface substitution model adopts a basic spline function and a partial least squares fitting regression model so as to ensure that the substitution model can solve the problem of high nonlinearity and ensure the accuracy of the substitution model. In theoretical vulnerability analysis, input parameters and output parameters are determined, a linear fitting method based on a probability earthquake demand model is used for obtaining a vulnerability curved surface, the fitting effect of the vulnerability curved surface obtained by nonlinear dynamic analysis and a response surface substitution model is compared, and the response surface substitution model is used for obtaining a structure and component vulnerability model and failure probability considering random variables, so that weak links of the structure and component can be analyzed, and necessary basis is provided for researches such as earthquake resistance reinforcement, risk assessment and the like.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a first performance level vulnerability surface comparison graph of a nonlinear dynamics analysis and response surface substitution model;
FIG. 3 is a second performance level vulnerability surface comparison graph of a nonlinear dynamics analysis and response surface substitution model;
FIG. 4 is a third performance level vulnerability surface comparison graph of a nonlinear dynamics analysis and response surface substitution model;
FIG. 5 is a fourth performance level vulnerability surface comparison graph of a nonlinear dynamics analysis and response surface substitution model.
Detailed Description
The present invention will be described in further detail with reference to examples and embodiments. It should not be construed that the scope of the present disclosure is limited to the following embodiments, and all techniques implemented based on the present disclosure are within the scope of the present disclosure.
Examples
Taking ancient tower architecture as an example, the structural earthquake vulnerability curved surface prediction method based on BSPLS-RSM of the embodiment comprises the following steps:
s1, building a structure finite element model, determining an input parameter X of a total data set, and performing nonlinear power analysis to obtain an output parameter Y;
S2, selecting a training set (train_X, train_Y) and a Test set (test_X, test_Y) by using a uniform design of the total data set, and training the training set to obtain a spline function partial least square response surface substitution model;
S3, inputting the Test set input parameters test_X into the spline function partial least square response surface substitution model obtained in the step S2 to obtain a fitting value Fit_test_Y of the Test set output parameters, and respectively obtaining a nonlinear dynamic analysis vulnerability curved surface and a spline function partial least square response surface substitution model vulnerability curved surface by using a linear fitting method based on a probability earthquake demand model in theoretical vulnerability analysis of the Test set (test_X, test_Y) and the Test set fitting value (test_X, fit_test_Y);
S4, comparing the fitting effect of the two vulnerable curved surfaces in the step S3, judging through a decision coefficient R, if R is more than or equal to 0.9, determining that the response surface replaces the vulnerable curved surface of the model, and if R is less than or equal to 0.9, adjusting the width parameter of the basic spline response surface replacement model until R is more than or equal to 0.9;
S5, obtaining a model of vulnerability of the ancient tower structure and failure probability considering each random variable by using the determined response surface substitution model, so that weak links of the ancient tower structure are analyzed, and necessary basis is provided for researches such as seismic reinforcement, risk assessment and the like.
The building structure finite element model in the step S1 takes an ancient tower as an example, and specific information of the ancient tower is as follows:
The height of the column is 34.2m, the cross section of the bottom layer is annular, the diameter of the bottom of the column is 9.72m, the column is made of masonry, two spiral stairs are arranged in the column, and the double stairs spiral around the center of the column. The elastic modulus of the masonry is E=818.7 MPa, the Poisson ratio lambda=0.15 and the density rho=1707 kg/m 3.
The input parameters for determining the total data set in the step S1 specifically include the following steps:
s111, determining an input parameter X 1 as ground peak acceleration PGA, wherein the value range is 0.01g to 2.469g;
S112, determining the input parameter X 2 as the magnitude M, wherein the value range is 5.09 to 7.62;
the total data set input parameters described in step S1 specifically include:
S121, determining an output parameter Y as a maximum interlayer displacement angle;
s122 divides the performance of the masonry tower building under the action of an earthquake into 5 damage states and 4 limit state limits according to a building structure damage level division table in the example, wherein the first performance level limit state limit value LS 1 =1/565, the second performance level limit state limit value LS 2 =1/376, the third performance level limit state limit value LS 3 =1/188 and the fourth performance level limit state limit value LS 4 =1/150;
the training sample points are selected by using uniform design in the step S2, and the specific steps are as follows:
S211, a distribution point P n (k), i.e., for a i e h, i=1, 2,..s,
Pn(k)＝(ka1,ka2,...,kas)(mod q)；k＝1,2,...,q
Wherein a s is a natural number; s is the number of variables; q is the number of sample points of each variable; k represents a kth sample point in each variable sample point; h is a positive integer less than q, and the greatest common divisor satisfying h and q is 1;
(ka 1,ka2,...,kas) (mod q) the calculation of the divisor of the known remainder q is performed for each term in brackets.
S212, obtaining uniform distribution parameters xi of each distribution point P n (k), namely,
a＝(a1,a2,...,as)
avk＝kav(mod q)
Wherein v is the v-th variable of the variables s; ka (mod q) is the calculation of the divisor for the known remainder q and the dividend ka.
S213, finding a distribution point P n (k) which enables the uniformly distributed parameter xi to be minimum, wherein the distribution point is a training sample point;
The training in the step S2 is carried out to obtain a cubic spline function partial least square response surface substitution model, and the specific steps are as follows:
S221, transferring the training set input parameter Train_X to a basic spline space, defining a segment number M i, a segment length h i and a node xi i,l-1, dividing each independent variable into (M i +3) variables of the basic spline space, namely,
ξi,l-1＝min(xi)+max(l-1)hi,(l＝0,1,...,Mi+2)
Wherein X i is the ith variable of the training set input parameter train_x; m i is the number of basic spline space segments of the ith variable; h i is the base spline space segment length of the ith variable; ζ i,l-1 is the first-1 node of the i-th variable; l is the first node of the base spline space.
S222, converting the ith variable X i of the training set input parameter train_x into a variable z i of the base spline space, in which the variable z i is, that is,
Wherein the method comprises the steps ofThe value of the jth sample point of x i on the ith node of the base spline space corresponding to x i; omega b is a B-order B-spline basis function; /(I)The jth sample point of the ith variable of the parameter train_x is input for the training set.
S223, normalizing the training set output parameter Train Y and the base spline space variable to eliminate errors caused by different orders of magnitude, that is,
Wherein Y c is the c-th sample point of the training set output parameter Train _ Y,Is the mean of y c, s y is the variance of y c,Normalization to y c; /(I)The value of the jth sample point of x i on the ith node of the base spline space corresponding to x i; For/> S i,l is/>Variance of/>For/>Is normalized by (2); n is the number of sample points of the training set.
S224, generating a multiple linear function, obtaining regression coefficients by using a partial least square method, namely,
Wherein the method comprises the steps ofIs a multiple linear function; a i,l is the regression coefficient of the multiple linear function; p is the variable number of the training set input parameter train_X; /(I)Normalizing the value of the first node of the basic spline space corresponding to x i; epsilon is the fitting error of the multiple linear function.
S225, after obtaining a regression model in the spline space, changing the regression model into an original space to obtain a nonlinear regression model of y versus x, namely obtaining a response surface substitution model trained by a training set, namely,
Wherein ε is the error of a multiple linear function; Predicted values for the response surface substitution model; beta 0 is a custom parameter that changes spline space to original space; beta i,l is a custom parameter that changes spline space to original space; /(I) Inputting the jth sample point of the ith variable of the parameter train_x for the training set; /(I)Is the mean value of y c; /(I)For/>Is the average value of (2); a i,l is the regression coefficient of the multiple linear function.
The cubic B-spline basis function described in step S222 has the following specific formula:
Wherein Ω 3 is a cubic base spline function.
The partial least square method in step S224 obtains regression coefficients, which specifically includes the steps of:
s2241, the independent variable group and the dependent variable group are standardized, the standardized independent variable group and the standardized dependent variable group are A and B respectively, namely,
Wherein A and B are independent variable groups and dependent variable groups after standardization; a 1 and a 1 are independent and dependent variable groups before normalization; and/> Is the mean of variables A 1 and B 1; var (A 1) and var (B 1) are the variances of variables A 1 and B 1;
S2242, the weight W 1 is determined according to the correlation of a and B, that is,
Wherein the superscript T denotes a matrix transpose; II 2 is the square operation of the European norm;
S2242, weights a to get a principal component T 1, that is,
T1＝AW1
S2243, the regression coefficients C 1 of B and T 1, that is,
S2244, the regression coefficients P 1 of T 1 and a, that is,
S2245, residual matrices E 1 and F 1 of the normalized independent variables and dependent variables, respectively, that is,
F1＝B-T1C1
S2246, replacing A and B with residual matrixes E 1 and F 1, repeating the above process for iteration until the residual matrixes meet the precision requirement or the number of main components is used up,
S2247, a partial least squares regression equation of the independent variable A 1 and the dependent variable B 1 is obtained.
The linear fitting method based on the probability earthquake demand model in the application theory vulnerability analysis in the step S3 obtains a vulnerability curved surface, and the specific steps are as follows:
s311, obtaining structural response through nonlinear dynamic analysis and a response surface substitution model;
s312, drawing a probability demand model and fitting, taking logarithms for input parameters and output parameters respectively, and then carrying out logarithmic linear fitting, namely,
ln(Sd)＝G1ln(x1)+G2ln(x2)+...+Gnln(xn)+ε
S d is a structural requirement parameter; x n is a structural parameter, a material parameter or a seismic intensity parameter; g n is a regression coefficient; epsilon is the fitting error;
s313, substituting the vulnerability function formula to obtain the vulnerability curved surface, namely,
Wherein P is failure probability; s d is a structure demand parameter; x n is a structural parameter, a material parameter or a seismic intensity parameter; LS e is the structural limit state for the e-th performance level; s c is structural resistance, namely a value corresponding to a structural limit state LS; beta c and beta d are the logarithmic standard deviations of the structural demand parameters and structural resistance, respectively; Φ is a standard normal distribution function.
The fitting effect of the vulnerable curved surfaces obtained by comparing the nonlinear power analysis and the response surface substitution model in the step S4 is good, the determining coefficients R of the vulnerable curved surfaces in four limit states are 0.9759, 0.9846, 0.9905 and 0.9933 respectively, the fitting effect is good, the vulnerable curved surfaces in four limit states are compared with those in fig. 2 to 5, the black wire frame curved surface is the vulnerable curved surface obtained by the nonlinear power analysis, and the black round dot is the vulnerable curved surface obtained by the response surface substitution model:
The vulnerability model and the failure probability of the ancient tower structure under the ground peak acceleration PGA and the earthquake magnitude M are obtained by using the response surface substitution model in the step S5, so that necessary basis is provided for researches such as risk assessment and the like of the ancient tower structure, and the specific steps are as follows:
S511, determining the value range of the ground peak acceleration PGA to be 0.01g to 1.0g, and the interval to be 0.01g. The value range of the vibration level M is determined to be 5-9, and the interval is 0.1. There are 100x40 = 4000 sets of sample points;
S512, inputting the sample points into a response surface substitution model of the vulnerable curved surface to obtain failure probability;
As can be seen from fig. 2 to 5, S513 shows that the probability of failure of the ancient tower structure decreases with an increase in the limit value of the performance level limit state, and increases with an increase in the ground peak acceleration PGA, and slowly increases with an increase in the seismic level M. The probability of 5 damage states of the temporary ancient tower structure when an earthquake occurs can be analyzed from the graph, weak links of the ancient tower structure are analyzed, earthquake resistance reinforcement is carried out, and necessary basis is provided for researches such as risk assessment of the ancient tower structure.
Claims (4)
1. The structural earthquake vulnerability curved surface prediction method based on BSPLS-RSM is characterized by comprising the following steps of:
s1, building a structure finite element model, determining an input parameter X of a total data set, and performing nonlinear power analysis to obtain an output parameter Y;
S2, selecting a training set (train_X, train_Y) and a Test set (test_X, test_Y) by using a uniform design of the total data set, and training the training set to obtain a spline function partial least square response surface substitution model;
S3, inputting the Test set input parameters test_X into the spline function partial least square response surface substitution model obtained in the step S2 to obtain a fitting value Fit_test_Y of the Test set output parameters, and respectively obtaining a nonlinear dynamic analysis vulnerability curved surface and a spline function partial least square response surface substitution model vulnerability curved surface by using a linear fitting method based on a probability earthquake demand model in theoretical vulnerability analysis of the Test set (test_X, test_Y) and the Test set fitting value (test_X, fit_test_Y);
S4, comparing the fitting effect of the two vulnerable curved surfaces in the step S3, judging through a decision coefficient R, if R is more than or equal to 0.9, determining that the response surface replaces the vulnerable curved surface of the model, and if R is less than or equal to 0.9, adjusting the width parameter of the basic spline response surface replacement model until R is more than or equal to 0.9;
S5, obtaining a structure and component vulnerability model and failure probability considering all random variables by using the determined response surface substitution model, so as to analyze weak links of the structure and the component and provide necessary basis for research of earthquake resistance reinforcement and risk assessment;
the total data set in step S2 is a training set and a testing set selected by using a uniform design, and the specific steps are as follows:
S211, a distribution point P n (k), i.e., for a i e h, i=1, 2,..s,
Pn(k)＝(ka1,ka2,...,kas)(mod q)；k＝1,2,...,q
Wherein a s is a natural number; s is the number of variables; q is the number of sample points of each variable; k represents a kth sample point in each variable sample point; h is a positive integer less than q, and the greatest common divisor satisfying h and q is 1; (ka 1,ka2,...,kas) (mod q) calculating a divisor of the known remainder q for each term in brackets;
s212, obtaining uniform distribution parameters xi of each distribution point P n (k), namely,
a＝(a1,a2,...,as)
avk＝kav(mod q)
Wherein v is the v-th variable of the variables s; ka (mod q) is the calculation of the divisor for the known remainder q and the dividend ka;
S213, finding a distribution point P n (k) which enables the uniformly distributed parameter xi to be minimum, wherein the distribution point is a training sample point;
The step S2 is that a spline function partial least square response surface substitution model is obtained by training a training set, and the specific steps are as follows:
S221, transferring the training set input parameter Train_X to a basic spline space, defining a segment number M i, a segment length h i and a node xi i,l-1, dividing each independent variable into (M i +3) variables of the basic spline space, namely,
ξi,l-1＝min(xi)+max(l-1)hi,(l＝0,1,...,Mi+2)
Wherein X i is the ith variable of the training set input parameter train_x; m i is the number of basic spline space segments of the ith variable; h i is the base spline space segment length of the ith variable; ζ i,l-1 is the first-1 node of the i-th variable; l is the first node of the base spline space;
S222, converting the ith variable X i of the training set input parameter train_x into a variable z i of the base spline space, in which the variable z i is, that is,
Wherein the method comprises the steps ofThe value of the jth sample point of x i on the ith node of the base spline space corresponding to x i; omega b is a B-order B-spline basis function; /(I)Inputting the jth sample point of the ith variable of the parameter train_x for the training set;
S223, normalizing the training set output parameter Train Y and the base spline space variable to eliminate errors caused by different orders of magnitude, that is,
Wherein Y c is the c-th sample point of the training set output parameter Train _ Y,Is the mean of y c, s y is the variance of y c,/>Normalization to y c; /(I)The value of the jth sample point of x i on the ith node of the base spline space corresponding to x i; /(I)For/>S i,l is/>Variance of/>For/>Is normalized by (2); n is the number of sample points of the training set;
S224, generating a multiple linear function, obtaining regression coefficients by using a partial least square method, namely,
Wherein the method comprises the steps ofIs a multiple linear function; a i,l is the regression coefficient of the multiple linear function; p is the variable number of the training set input parameter train_X; /(I)Normalizing the value of the first node of the basic spline space corresponding to x i; epsilon is the fitting error of a multi-element linear function;
S225, after obtaining a regression model in the spline space, changing the regression model into an original space to obtain a nonlinear regression model of y versus x, namely obtaining a response surface substitution model trained by a training set, namely,
Wherein ε is the error of a multiple linear function; Predicted values for the response surface substitution model; beta 0 is a custom parameter that changes spline space to original space; beta i,l is a custom parameter that changes spline space to original space; /(I) Inputting the jth sample point of the ith variable of the parameter train_x for the training set; /(I)Is the mean value of y c; /(I)For/>Is the average value of (2); a i,l is the regression coefficient of the multiple linear function;
The B-spline basis function described in step S222 has the following specific formula:
Wherein Ω d,b is the d-th B-order B-spline basis function; f is an input variable; b is the order of the B spline basis function; d is the number of B spline basis functions; delta is each node of the segmented interval;
the partial least square method in step S224 obtains regression coefficients, which specifically includes the steps of:
s2241, the independent variable group and the dependent variable group are standardized, the standardized independent variable group and the standardized dependent variable group are A and B respectively, namely,
Wherein A and B are independent variable groups and dependent variable groups after standardization; a 1 and a 1 are independent and dependent variable groups before normalization; and/> Is the mean of variables A 1 and B 1; var (A 1) and var (B 1) are the variances of variables A 1 and B 1;
S2242, the weight W 1 is determined according to the correlation of a and B, that is,
Wherein the superscript T denotes a matrix transpose; II 2 is the square operation of the European norm;
S2242, weights a to get a principal component T 1, that is,
T1＝AW1
S2243, the regression coefficients C 1 of B and T 1, that is,
S2244, the regression coefficients P 1 of T 1 and a, that is,
S2245, residual matrices E 1 and F 1 of the normalized independent variables and dependent variables, respectively, that is,
F1＝B-T1C1
S2246, replacing A and B with residual matrixes E 1 and F 1, repeating the above process for iteration until the residual matrixes meet the precision requirement or the number of main components is used up,
S2247, a partial least squares regression equation of the independent variable A 1 and the dependent variable B 1 is obtained.
2. The method for predicting the structural seismic vulnerability curved surface based on BSPLS-RSM according to claim 1, wherein the method for obtaining the vulnerability curved surface by using the linear fitting method based on the probabilistic seismic demand model in the theoretical vulnerability analysis in step S3 comprises the following specific steps:
s311, obtaining structural response through nonlinear dynamic analysis and a response surface substitution model;
s312, drawing a probability demand model and fitting, taking logarithms for input parameters and output parameters respectively, and then carrying out logarithmic linear fitting, namely,
ln(Sd)＝G1ln(x1)+G2ln(x2)+...+Gnln(xn)+ε
S d is a structural requirement parameter; x n is a structural parameter, a material parameter or a seismic intensity parameter; g n is a regression coefficient; epsilon is the fitting error;
s313, substituting the vulnerability function formula to obtain the vulnerability curved surface, namely,
Wherein P is failure probability; s d is a structure demand parameter; x n is a structural parameter, a material parameter or a seismic intensity parameter; LS e is the structural limit state for the e-th performance level; s c is structural resistance, namely a value corresponding to a structural limit state LS; beta c and beta d are the logarithmic standard deviations of the structural demand parameters and structural resistance, respectively; Φ is a standard normal distribution function.
3. The method for predicting the structural seismic vulnerability curved surface based on BSPLS-RSM according to claim 1, wherein the fitting effect of the two vulnerability curved surfaces in the step S3 is compared in the step S4, and the fitting effect is judged by determining a coefficient R, and the specific formula is as follows:
wherein the method comprises the steps of In order to replace the failure probability of the vulnerable curved surface of the model by the response surface, P f is the failure probability of the vulnerable curved surface of the nonlinear dynamic analysis,/>Is the average value of P f.
4. The method for predicting the structural seismic vulnerability curved surface based on BSPLS-RSM as set forth in claim 1, wherein the step S5 uses the response surface substitution model to obtain a structural and component vulnerability model and failure probability considering random variables, thereby analyzing weak links of the structural and component, providing necessary basis for seismic reinforcement and risk assessment research, and comprises the following specific steps:
S511, selecting random variables from the aspects of material parameters, structural parameters and earthquake motion intensity parameters by the structure;
s512, determining the value range and interval of the random variable to obtain sample points;
s513, inputting the sample points into a response surface substitution model of the vulnerable curved surface to obtain failure probability;
S514, analyzing weak links of the structure and performing earthquake-resistant reinforcement.
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