CN111444649A - Slope system reliability analysis method based on intensity reduction method - Google Patents

Slope system reliability analysis method based on intensity reduction method Download PDF

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CN111444649A
CN111444649A CN202010214034.6A CN202010214034A CN111444649A CN 111444649 A CN111444649 A CN 111444649A CN 202010214034 A CN202010214034 A CN 202010214034A CN 111444649 A CN111444649 A CN 111444649A
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CN111444649B (en
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张天龙
曾鹏
李天斌
孙小平
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Chengdu Univeristy of Technology
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Abstract

The application provides a slope system reliability analysis method based on intensity reduction, wherein an intensity reduction SRM is adopted to evaluate a stability coefficient, an initial sampling strategy and an active learning function are adopted, active learning agent models ASVM, ARBF and AK of an original extreme state function L SF are constructed, Monte Carlo simulation MCS and the active learning agent models are combined to evaluate the failure probability of a slope system, the influence of random variables and related parameters on the slope stability can be quantified, the number of initial sample points is greatly reduced, the calculation efficiency is effectively improved, sliding surfaces of any shape in a soil slope can be automatically identified, and the method is more convenient for reliability analysis of a layered slope with a complex geometric shape.

Description

Slope system reliability analysis method based on intensity reduction method
Technical Field
The application relates to the field of soil slope stability analysis, in particular to a slope system reliability analysis method based on a strength reduction method.
Background
Slope stability evaluation is a complex geotechnical engineering problem, and input parameters of the slope stability evaluation are uncertain. Traditional deterministic analysis methods using stability coefficients (FS) may not truly reflect the safety of the slope. To quantify the effect of uncertainty, probabilistic methods are widely used in slope reliability analysis.
A slope may break along different sliding surfaces, and the breaking of any one sliding surface causes the slope to break, creating a series of system problems. Accurate and effective reliability analysis of such complex problems is a major problem faced by the application of probabilistic methods in geotechnical engineering practice.
The direct simulation method IS one of probability methods, such as Monte Carlo Simulation (MCS) and Importance Sampling (IS) to determine the failure probability P of the slope systemf,sUnbiased estimation was performed, but currently most scholars performed reliability analysis using the limit balance method (L EM), which, when combined with MCS, required searching for the critical slip plane with the smallest FS in each simulation and therefore was computationally intensive a more critical problem was that L EM primarily used randomly generated slip planes, which may miss critical slip planes, thereby providing P with larger deviationsf,sAnd (6) estimating the value.
To effectively combine L EM with MCS, another common approach is to identify some pairs Pf,sThe most contributing representative slip planes (RSSs), then, taking into account the correlation between the different RSSs, P can be easily calculatedf,s. It is known in the art to identify RSSs by randomly generating a large number of potential slip planes. However, one challenge faced by such RSSs-based approaches is how to select a reasonable threshold for the correlation coefficient between RSSs to achieve computational efficiency and accuracy.
L EM is often selected as a deterministic analysis method to evaluate the FS of a side slope, and &lttttranslation = L "&tttL &ltt/T &gttEM has advantages of its simplicity and low computational cost but has major disadvantages of being difficult to locate when the critical sliding surface is not known in advance, and furthermore, the test sliding surface is generally assumed to be circular, which may not be suitable for a complex side slope system, especially when a soft interlayer exists in the side slope.
Therefore, it is still necessary to develop a reliable and efficient slope system reliability and efficiency analysis method.
Disclosure of Invention
The application provides a slope system reliability analysis method based on a strength reduction method, and aims to overcome the technical problems.
In order to solve the above problems, the present application discloses a slope system reliability analysis method based on a strength reduction method, including:
step S1: generating a training sample set of the slope system by utilizing an initial sampling point strategy in a standard normal space;
step S2: converting the sample points of undetermined functional response G (u) in the training sample set from the standard normal space to a physical space, and calculating G (u) corresponding to the sample points converted to the physical space by using an intensity reduction method;
step S3: training a proxy model in the standard normal space using the training sample set and G (u);
step S4: predicting the functional response of all sample points in the Monte Carlo simulation MCS pool by using the trained agent model, calculating the failure probability of the current iteration according to the predicted functional response, and recording the failure probability of the current iteration in a preset matrix;
step S5: judging whether the variation coefficient of the failure probability calculated by the last five iterations is smaller than a preset convergence threshold value or not;
step S6: when the variation coefficient of the failure probability calculated by the last five iterations is not smaller than a preset convergence threshold, selecting an optimal sample point located in a standard normal space from the MCS pool by using an active learning function in combination with the trained agent model, adding the optimal sample point into the training sample set, and repeating the steps S2-S6;
step S7: and when the variation coefficient of the failure probability calculated by the last five times of iteration is smaller than a preset convergence threshold value, taking the failure probability calculated by the last iteration in the preset matrix as a result of the reliability analysis of the slope system.
Further, in step S1, in the standard normal space, the step of generating the training sample set of the slope system by using the initial sampling point strategy includes:
in a standard normal space, constructing an initial training sample set of the slope system by using a 3-sigma rule; the initial training sample set comprises a plurality of sample points u, wherein u represents a vector of random variables u in the standard normal space;
for each u in the initial training sample set, judging whether the u meets any one of the following conditions:
n-1 of the u is equal to-3, the other u is equal to 0 or 3, and n represents the number of u in the u; or n elements of said u are all the same, all equal to-3, 0 or 3;
if the u is satisfied, keeping the u in the initial training sample set;
if the u is not satisfied, removing the u from the initial training sample set;
and when the initial training sample set is judged, obtaining the training sample set S.
Further, in step S2, the step of converting the sample points in the training sample set for which the functional response g (u) is not determined from the standard normal space to the physical space, and calculating g (u) corresponding to the sample points converted to the physical space by using an intensity reduction method includes:
let the standard normal space be U and the physical space be X;
converting the sample point of undetermined G (U) in S from U to X, and then converting the sample point from U to X;
calculating the functional response of x using a given linear function g (x):
g(x)＝FS(x)-1 (1)；
wherein FS is F L AC3DThe stability coefficient calculated by the embedded strength reduction method;
the corresponding G (u) can be obtained by the formula (1), and satisfies the following conditions:
g(x)＝G(u) (2)。
further, in step S3, when the surrogate model is a Support Vector Machine (SVM) surrogate model, the step of training the surrogate model includes, in the standard normal space, using the training sample set and g (u):
training the SVM proxy model by using the training sample set in the standard normal space; wherein, one sample point of the ith simulation in the current S
Figure BDA0002423785850000041
Satisfy the requirement of
Figure BDA0002423785850000042
The vector of the sample points of (1) is located at one side, satisfies
Figure BDA0002423785850000043
Is located on the other side;
searching for an optimal classification hyperplane h (u) using the SVM proxy model for the current S:
Figure BDA0002423785850000044
Figure BDA0002423785850000045
in the above formula, w and e represent unknown parameters, wTRepresenting the transpose of the w matrix, yiIs that
Figure BDA0002423785850000046
A classification symbol of (a), represents positive or negative;
calculating a distance vector V (u) from all sample points in the current S to the H (u):
Figure BDA0002423785850000047
(5) in the formula (I), the compound is shown in the specification,
Figure BDA0002423785850000048
the sample point with the minimum distance H (u) in the current S is represented as a support vector; n is a radical ofSVIs composed of
Figure BDA0002423785850000049
The number of (2); omegaiObtaining a weight coefficient representing the ith sample point by optimizing and solving the formula (4);
Figure BDA00024237858500000410
to represent
Figure BDA00024237858500000411
Transposing the matrix;
and determining the classification condition of each sample point in the current S according to the positive or negative of the classification sign of V (u).
Further, in step S3, when the agent model is a Kriging agent model, the step of training the agent model includes, in the standard normal space, using the training sample set and g (u):
in the standard normal space, training a Kriging agent model by using the training sample set to obtain an agent expression corresponding to G (u)
Figure BDA00024237858500000412
Figure BDA00024237858500000413
(6) Wherein L (u) represents a function representing the trend of G (u) obtained from regression analysis, z (u) is an assumed smooth Gaussian process, and the ith simulated sample point u is determined when L (u) is 0(i)And sample point u of the jth simulation(j)The covariance between is:
Figure BDA00024237858500000414
(7) in the formula (I), the compound is shown in the specification,
Figure BDA00024237858500000415
is shown byThe path variance, R (-) is a Gaussian kernel function, expressed as:
Figure BDA00024237858500000416
(8) in the formula, θ represents a vector of unknown coefficients θ; where, theta and sigmazBy all points in the current sample set S
Figure BDA00024237858500000417
And corresponding thereto
Figure BDA00024237858500000418
Obtained in combination with the maximum likelihood estimate.
Further, in step S3, when the proxy model is a radial basis function RBF proxy model, the step of training the proxy model includes, in the standard normal space, using the training sample set and g (u):
in the standard normal space, the RBF proxy model is trained by utilizing the training sample set to obtain a proxy expression corresponding to G (u)
Figure BDA0002423785850000051
Figure BDA0002423785850000052
(9) In the formula (I), the compound is shown in the specification,
Figure BDA0002423785850000053
representing a sample point of the ith simulation in the current S, N representing the number of the sample points in the current S, rho and b representing vectors of unknown coefficients rho and b in the RBF proxy model respectively, N representing the number of random variables in u, and u representing the number of the random variables in ujRepresents the jth random variable in u, and Ψ (-) represents a kernel function;
using a linear kernel function psi (a) ═ a, and dividing each sample point
Figure BDA0002423785850000054
Substituting into the formula (9) to solveThe unknown coefficients;
Figure BDA0002423785850000055
in the above equation, the unknown coefficient of the n +1 th term is determined by the orthogonality condition, ΨijRepresents the distance value between the i, j modeled sample points calculated using the Ψ (·).
Further, in step S4, the step of predicting the functional responses of all sample points in the monte carlo simulation MCS pool by using the trained proxy model, and calculating the failure probability of the current iteration according to the predicted functional responses includes:
Figure BDA0002423785850000056
Figure BDA0002423785850000057
in the above formula, NSPRepresenting a number of sample points in the MCS pool;
when the agent model is an SVM agent model, V (u) obtained by the trained SVM agent model(i)) Instead of G (u)(i)) Substituting the formula (11) and the formula (12) to obtain the failure probability of the current iteration;
when the agent model is RBF agent model or Kriging agent model, the obtained agent model after training is used
Figure BDA0002423785850000061
Instead of G (u)(i)) And substituting the formula (11) and the formula (12) to obtain the failure probability of the current iteration.
Further, in step S5, the step of determining whether the coefficient of variation of the failure probability calculated in the last five iterations is smaller than the preset convergence threshold includes:
standard deviation of failure probability calculated from the last five iterations
Figure BDA0002423785850000062
And average value
Figure BDA0002423785850000063
Calculating the coefficient of variation
Figure BDA0002423785850000064
Figure BDA0002423785850000065
And judging whether the variation coefficient is smaller than a preset convergence threshold value or not.
Further, in step S6, an optimal sample point u in a standard normal space is selected from the MCS pool by using an active learning function in combination with the trained proxy modelcComprises the following steps:
when the agent model is Kriging agent model, the u iscThe calculation formula (2) includes:
Figure BDA0002423785850000066
wherein u isTRepresents the sample point in the MCS pool T in the U space,
Figure BDA0002423785850000067
representing passing the Kriging agent model
Figure BDA0002423785850000068
A predicted standard deviation;
when the agent model is an RBF agent model, the ucThe calculation formula (2) includes:
Figure BDA0002423785850000069
Figure BDA00024237858500000610
wherein u isTIndicating the MCSOne sample point in the pool, d (u)TS) represents said uTThe minimum distance between the target and the sample point in the current S, d (S) is the limit value of the target minimum distance, lambda is a scale factor, and lambda is more than or equal to 0.1 and less than or equal to 0.5;
when the proxy model is an SVM proxy model, adopting
Figure BDA00024237858500000611
Instead of the former
Figure BDA00024237858500000612
Substituting into (15) and (16) to calculate uc。
Further, the method further comprises:
introducing an explicit highly nonlinear function g (x)' as a test, and verifying the steps S2-S6, wherein:
Figure BDA0002423785850000071
compared with the prior art, the method has the following advantages:
the application provides a slope system reliability analysis method based on SRM, which can automatically identify sliding surfaces in any shape in soil slopes, does not need to identify critical sliding surfaces like L EM, and is more convenient to perform reliability analysis on layered slopes with complex geometric shapes;
according to the method, an initial sampling point strategy is adopted, three active learning agent models of ASVM, ARBF and AK are developed by combining an active learning function, an agent model of an original extreme state function L SF is constructed, MCS and the active learning agent model are combined to evaluate the failure probability of the slope system, the number of initial sample points is greatly reduced, the calculation efficiency is effectively improved, and the influence of random variables and related parameters thereof on the slope stability can be quantized.
Drawings
FIG. 1 is a flowchart illustrating steps of a slope system reliability analysis method based on intensity reduction method according to the present application;
FIG. 2(a) is a schematic diagram of sample point locations generated by a conventional 3-sigma rule;
FIG. 2(b) is a schematic diagram of sample point locations generated by the improved 3-sigma rule of the present application;
FIG. 3(a) is a schematic diagram of a fit of 5000 MCS samples and true L SS;
FIG. 3(b) is a diagram showing the fitting performance of the AK model;
FIG. 3(c) is a graphical representation of the fit performance of ARBF;
FIG. 3(d) is a schematic diagram of the fit performance of an ASVM;
FIG. 4 is a flowchart of a slope system reliability analysis using an active learning agent model and a strength reduction method according to the present application;
FIG. 5 is a schematic diagram of the slope geometry of case one;
FIG. 6 is a graphical illustration of the calculated FS and grid density relationships for three cases of the present application;
FIG. 7 is a graph of the probability of failure prediction for case one;
FIG. 8 is a graph of the fitting performance of the L HS samples with different agent models in case one;
FIG. 9 is a schematic diagram of the slope geometry of case two;
FIG. 10 is a failure probability prediction graph for case two;
FIG. 11 is a graph showing the fitting performance of L HS samples with different agent models in case two;
fig. 12 is a schematic diagram of the slope geometry of case three;
FIG. 13 is a failure probability prediction graph for case three.
Detailed Description
In order to make the aforementioned objects, features and advantages of the present application more comprehensible, the present application is described in further detail with reference to the accompanying drawings and the detailed description.
Referring to fig. 1, a flowchart illustrating steps of a slope system reliability analysis method based on intensity reduction method according to the present application is shown, and specifically, the method may include the following steps:
step S1: generating a training sample set of the slope system by utilizing an initial sampling point strategy in a standard normal space;
the initial training sample set may be constructed using Latin hypercube sampling (L HS), but this may not be applicable to some models with lower probability of failure because it must contain two types of points (e.g., G (u))>0 and G (u)<0). Conventional 3-sigma may achieve this well because it may roughly reflect the general trend of g (u) in the whole sampling space and contains two types of points. However, this method requires about 3nInitial sample points, where n is the number of random variables; therefore, it may not be suitable for a problem that contains many random variables (e.g., a 10 random variable problem requires 59049 initial sample points, which is clearly unacceptable in practice).
Therefore, the application proposes an improved 3-sigma rule, whose basic idea is to balance the number of two region points, destabilized and non-destabilized, and speed up the training process of the separation of the safe region and the ineffective region. To this end, the sampling range of each random variable is treated as [ -3, 3] in an uncorrelated standard normal space (also referred to as U-space), and step S1 may include the following sub-steps:
substep 1-1: in a standard normal space, constructing an initial training sample set of the slope system by using a 3-sigma rule; the initial training sample set comprises a plurality of sample points u, where u represents a vector of random variables u;
substeps 1-2: for each u in the initial training sample set, judging whether the u meets any one of the following conditions:
n-1 of the u is equal to-3, the other u is equal to 0 or 3, and n represents the number of u in the u; or n of said u are all the same and equal to-3, 0 or 3;
if the u is satisfied, keeping the u in the initial training sample set;
substeps 1-3: if the u is satisfied, keeping the u in the initial training sample set; and if the u is not satisfied, removing the u from the initial training sample set, and obtaining the training sample set S when the initial training sample set is judged to be finished.
In the present applicationOne sample point u includes a plurality of random variables, e.g. u1,u2,…,un. Wherein u is1＝u2＝…＝unWhen-3, the stability factor FS is the smallest, i.e. the point is the Most Dangerous Point (MDP) in the training sample set S.
The training sample set S obtained through the initial sampling point strategy finally generates 2n +3 initial sample points, and compared with the traditional 3-sigma, the number of the initial sample points can be greatly reduced when a slope system has a large number of random variables. Fig. 2 shows the sample point cases generated in U-space by the conventional 3-sigma rule and the improved 3-sigma rule of the present application when 3 random variables are considered. Wherein fig. 2(a) shows a schematic diagram of sample point locations generated by a conventional 3-sigma rule; fig. 2(b) shows a schematic diagram of sample point locations generated by the improved 3-sigma rule of the present application.
Step S2: converting the sample points of undetermined functional response G (u) in the training sample set from the standard normal space to a physical space, and calculating G (u) corresponding to the sample points converted to the physical space by using an intensity reduction method;
the reliability analysis method can quantify the influence of random variables and related parameters thereof on the slope stability. Let the standard normal space be U and the physical space be X;
converting the sample points of undetermined G (U) in the S from the U space to an X space, wherein the sample points are converted from U to X, and the X represents a vector of random variables in the X space;
calculating the functional response of x using a given linear function g (x):
g(x)＝FS(x)-1 (1)；
wherein FS is F L AC3DThe stability factor calculated by the intensity reduction method (hereinafter, both expressed as SRM) of the mid-insert;
g (x) the corresponding G (u) can be obtained by the formula (1).
In the prior art, if formula (1) is used for directly calculating the failure probability of the slope system, the failure probability P is calculatedf,sCan be expressed as:
Pf,s＝P[g(x)＜0]＝∫g(x)＜0f(x)dx；
where f (x) represents the joint Probability Density Function (PDF) of the random variables involved. But since g (x) is implicit, directly calculating the integral in the equation is difficult to achieve. Thus, the present application transforms the vector x into U of sample points in the uncorrelated standard normal space, such that the extreme state surface can be rewritten as g (U) 0, g (U) is a mapping of g (x) in the uncorrelated standard normal space U.
After the above transformation, if the failure probability P can be provided according to the conventional MCSf,sUnbiased estimation of (d). However, in this method, when P isf,s＝10-2And the coefficient of variation of MCS
Figure BDA0002423785850000102
Then, a model needs about 104Sub-simulation for time-consuming reliability analysis (e.g. using F L AC)3DAnd analysis by SRM) are unacceptable, there are several variations of MCS in the prior art, such as L HS, IS, and subset modeling (SS), which may reduce the calculated P to some extentf,sThereby reducing the number of simulations required. But at present, the target P of many civil engineering projectsf,sAt 10-3To 10-5This further increases the required computational effort, making it difficult for even these MCS variants to meet the computational demands.
Therefore, the present application proposes a Monte Carlo simulation based on a proxy model, which can use a small number of sample points to construct an explicit prediction model of G (u)
Figure BDA0002423785850000101
Replacement of original F L AC with proxy model3DAnd SRM analysis, thereby greatly improving the efficiency and reducing the calculation cost. Thus, the failure probability calculation result can be quickly given by using the MCS or the variant thereof.
Step S3: training a proxy model in the standard normal space using the training sample set and G (u);
in the application, three methods of a Support Vector Machine (SVM), Kriging (Kriging) and a Radial Basis Function (RBF) are adopted to establish 3 side slope system failure probability agent models, namely an SVM agent model, a Kriging agent model and an RBF agent model, and the conditions are as follows:
firstly, SVM proxy model:
when the surrogate model is a Support Vector Machine (SVM) surrogate model, in the standard normal space, the training sample set and G (u) are utilized, and the step of training the surrogate model comprises the following steps:
training the SVM proxy model by using the training sample set in the standard normal space; wherein, one sample point of the ith simulation in the current S
Figure BDA0002423785850000111
Satisfy the requirement of
Figure BDA0002423785850000112
The vector of the sample points of (1) is located at one side, satisfies
Figure BDA0002423785850000113
Is located on the other side;
searching for an optimal classification hyperplane h (u) using the SVM proxy model for the current S:
Figure BDA0002423785850000114
Figure BDA0002423785850000115
in the above formula, w and e represent unknown parameters, wTRepresenting the transpose of the w matrix, yiIs that
Figure BDA0002423785850000116
A classification symbol of (a), represents positive or negative;
calculating a distance vector V (u) from all sample points in the current S to the H (u):
Figure BDA0002423785850000117
(5) in the formula (I), the compound is shown in the specification,
Figure BDA0002423785850000118
the sample point with the minimum distance H (u) in the current S is represented as a support vector; n is a radical ofSVIs composed of
Figure BDA0002423785850000119
The number of (2); omegaiObtaining a weight coefficient representing the ith sample point by optimizing and solving the formula (4);
Figure BDA00024237858500001110
to represent
Figure BDA00024237858500001111
Transposing the matrix;
and determining the classification condition of each sample point in the current S according to the positive or negative of the classification sign of V (u).
It should be noted that, the above solution is linear classification hyperplane, and to obtain nonlinear classification hyperplane, kernel function may be used
Figure BDA00024237858500001112
Substitution to Gaussian Kernel function
Figure BDA00024237858500001113
Second, Kriging agent model:
kriging is another powerful tool for constructing surrogate models, and in particular, is an interpolation method based on statistical assumptions. In step S3, when the surrogate model is a Support Vector Machine (SVM) surrogate model, the step of training the surrogate model includes, in the standard normal space, using the training sample set and g (u):
in the standard normal space, training a Kriging agent model by using the training sample set to obtain an agent expression corresponding to G (u)
Figure BDA00024237858500001114
Figure BDA00024237858500001115
(6) Wherein L (u) represents a function representing the trend of G (u) obtained from regression analysis, z (u) is an assumed smooth Gaussian process, and the ith simulated sample point u is determined when L (u) is 0(i)And sample point u of the jth simulation(j)The covariance between is:
Figure BDA0002423785850000121
(7) in the formula (I), the compound is shown in the specification,
Figure BDA0002423785850000122
representing the process variance, R (-) is a gaussian kernel function, expressed as:
Figure BDA0002423785850000123
(8) in the formula, θ represents a vector of unknown coefficients θ; where, theta and sigmazBy all points in the current sample set S
Figure BDA0002423785850000124
And corresponding thereto
Figure BDA0002423785850000125
Obtained in combination with the maximum likelihood estimate.
Thirdly, RBF agent model:
RBF is another accurate interpolation method. Its advantages are simple structure and easy implementation. When the proxy model is a Radial Basis Function (RBF) proxy model, in the standard normal space, using the training sample set and G (u), the step of training the proxy model includes:
in the standard normal space, the RBF proxy model is trained by utilizing the training sample set to obtain a proxy expression corresponding to G (u)
Figure BDA0002423785850000051
Figure DEST_PATH_FDA00024237858400000314
(9) In the formula (I), the compound is shown in the specification,
Figure BDA0002423785850000053
representing a sample point of the ith simulation in the current S, N representing the number of the sample points in the current S, rho and b representing vectors of unknown coefficients rho and b in the RBF proxy model respectively, N representing the number of random variables in u, and u representing the number of the random variables in ujRepresents the jth random variable in u, and Ψ (-) represents a kernel function;
using a linear kernel function psi (a) ═ a, and dividing each sample point
Figure BDA0002423785850000127
Substituting the formula (9) to solve the unknown coefficient;
Figure BDA0002423785850000128
in the above equation, the unknown coefficient of the n +1 th term is determined by the orthogonality condition, ΨijRepresents the distance value between the i, j modeled sample points calculated using the Ψ (·).
The three proxy models described above are all alternatives to the present application, but are not limited thereto. In particular, any agent model can be trained to perform calculations.
Step S4: predicting the functional response of all sample points in the Monte Carlo simulation MCS pool by using the trained agent model, calculating the failure probability of the current iteration according to the predicted functional response, and recording the failure probability of the current iteration in a preset matrix;
in the application, all the sample points in the MCS pool T are substituted into the proxy model for calculation, and the functional response of the corresponding sample points can be predicted.
In the present application, the failure probability calculation formula is as follows:
Figure BDA0002423785850000131
Figure BDA0002423785850000132
in the above formula, NSPRepresenting a number of sample points in the MCS pool;
when the agent model is an SVM agent model, V (u) obtained by the trained SVM agent model(i)) Instead of G (u)(i)) Substituting the formula (11) and the formula (12) to obtain the failure probability of the current iteration;
when the agent model is RBF agent model or Kriging agent model, the obtained agent model after training is used
Figure BDA0002423785850000133
Instead of G (u)(i)) And substituting the formula (11) and the formula (12) to obtain the failure probability of the current iteration. The predetermined matrix may be set during a preparation phase, and a matrix is initialized to record the failure probability of each iteration of the system. Iteration means that a deterministic model is calculated by continuously using different sample points in the process of constructing the proxy model.
Step S5: judging whether the variation coefficient of the failure probability calculated by the last five iterations is smaller than a preset convergence threshold value or not;
there are typically two convergence criteria (i) sample points near the hyperplane limit state plane (L SS) are sufficiently dense, or (ii) prediction P isf,sIs sufficiently small. The same criteria apply to the present application. Therefore, step S5 may specifically include, when implemented:
standard deviation of failure probability calculated from the last five iterations
Figure BDA0002423785850000134
And average value
Figure BDA0002423785850000135
Calculating the coefficient of variation
Figure BDA0002423785850000136
Figure BDA0002423785850000141
Judging whether the variation coefficient is smaller than a preset convergence threshold value or not; in the present application, 0.001 is preferred.
Step S6: when the variation coefficient of the failure probability calculated by the last five iterations is not smaller than a preset convergence threshold, selecting an optimal sample point located in a standard normal space from the MCS pool by using an active learning function in combination with the trained agent model, adding the optimal sample point into the training sample set, and repeating the steps S2-S6;
in the present application, a large sample pool T (e.g., 200000 points) is first generated using MCS, and the setting of the learning function is related to the actual analysis objective and aims to rank a set of candidate points in the MCS pool T. in the slope reliability analysis of the present application, the learning function needs to give an optimal sample point during each iteration to update the current proxy model, and this optimal sample point should satisfy both conditions, (i) it is located near the extreme state surface (L SS), (ii) redundant information is avoided (i.e., away from the sample points in the existing S).
Therefore, in step S6, the optimal sample point u in the normal space is selected from the MCS pool by using the active learning function in combination with the trained proxy modelcComprises the following steps:
when the agent model is Kriging agent model, the u iscThe calculation formula (2) includes:
Figure BDA0002423785850000142
wherein u isTRepresents the sample point in the MCS pool T in the U space,
Figure BDA0002423785850000143
representing passing the Kriging agent model
Figure BDA0002423785850000144
A predicted standard deviation;
when the agent model is an RBF agent model, the ucThe calculation formula (2) includes:
Figure BDA0002423785850000145
Figure BDA0002423785850000146
wherein u isTRepresents one sample point in the MCS pool, d (u)TS) represents said uTThe minimum distance between the target and the sample point in the current S, d (S) is the limit value of the target minimum distance, lambda is a scale factor, and lambda is more than or equal to 0.1 and less than or equal to 0.5;
when the proxy model is an SVM proxy model, adopting
Figure BDA0002423785850000151
Instead of the former
Figure BDA0002423785850000152
Substituting into (15) and (16) to calculate ucIn the present application, λ is further preferably 0.2.
The active learning agent model, namely the ASMs agent model, is established through the formula. Specifically, three ASMs proxy models, namely active learning kriging AK, active learning support vector machine ASVM and active learning radial basis function ARBF, are respectively established for the listed three proxy models. It is emphasized that the active learning technique of the present application does not randomly select new sample points to increase the training sample set S, but starts with a small number of sample points in S, in order to enrich S by adding targeted candidate samples (optimal sample points) one by one during the training process. And selecting the optimal sample point from the MCS pool T according to the active learning function. Once a candidate sample is selected from the MCS pool T, it is added to the training sample set S to update the corresponding ASMs proxy model, whereby a continuous training process can be started using the initial sampling strategy and active learning function to improve the accuracy of the prediction.
In this application, FIG. 3 shows the fitting performance of different ASMs for highly non-linear functions, where FIG. 3(a) shows a schematic of a 5000 MCS samples and a true L SS fit, and FIGS. 3(b) through 3(d) show sample points and corresponding fitting results provided by different ASMs (i.e., AK, ARBF, and ASVM). the results show that three ASMs have many sample points around L SS that provide most of the information for constructing a proxy model, and thus enable good interpolation or fitting of an entire finite sample space with a small number of sample points.
Step S7: and when the variation coefficient of the failure probability calculated by the last five times of iteration is smaller than a preset convergence threshold value, taking the failure probability calculated by the last iteration in the preset matrix as a result of the reliability analysis of the slope system.
The method is mainly divided into a preparation stage, an iteration stage and an output stage by integrating the steps S1 to S7.
A preparation stage: (i) and (3) generating an initial sample point in the U space by using an initial sampling point strategy, transferring the initial sample point from the U space to a physical space X, and determining the real response of g (X) by using the SRM. (ii) A reasonable convergence threshold is selected for the active learning process. (iii) Generating a reusable MCS pool to calculate P for each iterationf,sAnd provides the optimal sample points to enrich the training sample set S to update the ASMs proxy model.
An iteration stage: the three modules are mainly used for executing an iterative task and comprise a numerical analysis module, a Monte Carlo module and an active learning and convergence judging module, and the three modules are interactively operated at the stage to generate a sequential process. The agent model needs to iteratively update candidate samples selected from the MCS pool T according to the corresponding active learning function unless a convergence criterion is satisfied. The detailed interaction of these three modules is shown in fig. 4.
An output stage: and stopping iteration after the convergence condition is met, and selecting the failure probability of the last iteration calculation as the final estimation of the slope system reliability.
In addition, in a preferred embodiment of the present application, in order to verify the convergence criterion and the proxy model proposed in the present application, the method further includes the following steps:
introducing an explicit highly nonlinear function g (x)' as a test, and verifying the steps S2-S6, wherein:
Figure BDA0002423785850000161
the result shows that the three agent models listed in the application can establish the agent model of the actual functional response G (U) in the U space based on a small number of training samples.
To further illustrate the reliability analysis method of the present application, a verification analysis is next performed using three typical reference slopes as cases.
In addition, the application also applies a plurality of widely used methods based on polynomial expansion (PCE), such as a Quadratic Response Surface Method (QRSM) and a sparse PCE minimum angle regression method (SPCE-L AR), to the following three cases and carries out comparative research with the method of the application.
In order to verify the calculation accuracy of the reliability analysis method in the slope system reliability analysis, in three cases, 10000 times of Latin hypercube sampling (L HS) simulation are carried out on an original limit state function (L SF) directly based on SRM, and the system failure probability P provided by L HSf,sTo measure computational efficiency, the number of sample points required for each analysis (and also the number of times a numerical analysis is performed) is used because of the large number of calculations involved when introducing numerical methods (e.g., F L AC as used in this application)3D) The computational effort required by the rest of the algorithm is usually negligibleAnd (6) counting. Therefore, the number of sample points can be used as a general index of the calculation efficiency of the practical problem: the larger the sample size, the lower the efficiency.
Case one: single layer slope
FIG. 5 shows the slope geometry for case one, statistical information of soil parameters is shown in Table 1, and failure probability is shown in Table 2. Table 2 shows the number of sample points and P for 10000L HS simulations using different reliability methodsf,sAnd (6) obtaining the result.
Table 1: soil parameter statistics for case one
Figure BDA0002423785850000171
Table 2: failure probability of case-system obtained by different methods
Figure BDA0002423785850000172
In the above table:
a indicates that the model convergence condition-dependent certainty factor is set to 0.99.
b represents NE ═ the number of numerical analyses.
c represents the mean and 99.76% confidence interval.
d represents the relative error from the L HS mean.
It should be noted that, in order to obtain a reasonable finite difference grid to ensure efficiency and accuracy, fig. 6 shows the relationship between FS and the number of cells in the grid, which is calculated under the condition that parameter variables are averaged based on the reduced intensity method, and it is observed that FS is a monotonically decreasing function of grid density. As shown in fig. 6, the optimal density point may be selected to determine the grid density after which point FS does not have a tendency to decrease significantly as the grid density increases. The optimal mesh density and final finite difference mesh of case one of the present application is shown in fig. 5, which has a FS of 1.34.
FIG. 7 shows the probability of failure P for a system of cases using different ASMs, at different sample pointsf,sAnd with L HS results and 99.76% confidence bandIn case one, 10000 times L HS sample point positions and classification situations in a two-dimensional U space, and prediction situations of different agent models on an actual L SS are shown in FIG. 8.
Case two: two-layer slope
Fig. 9 shows the slope geometry of case two, with statistical information on soil parameters as shown in table 3 and failure probability as shown in table 4.
Table 3: soil parameter statistics for case two
Figure BDA0002423785850000181
In the above table:
a represents the undrained shear strength;
b represents a coefficient of variation.
Table 4: failure probability of case two system obtained by different methods
Figure BDA0002423785850000182
Figure BDA0002423785850000191
In the above table:
a represents NE ═ the number of numerical analyses;
b represents the mean and 99.76% confidence interval;
c represents the relative error with respect to the L HS mean.
In table 3, the undrained shear strength of the two clay layers was considered as a random variable. Using the mean values of random variables for analysis, the case-two-best density finite difference grid of the present application is shown in fig. 9 with 3625 cells corresponding to an FS of 1.926, and the best density can be determined from the fitted curve shown in fig. 6. FIG. 10 shows the prediction of failure probability P for case two system using different ASMs under different training sample point numbersf,sAnd with L HS results and 99.76% confidence interval line as a reference FIG. 10 reflects three ASMs as training samplesFigure 11 shows the sample point locations and classifications of L HS in 10000 times in two-dimensional U space and the predictions of actual L SS for different surrogate models in case two.
Case three: three-layer slope
Fig. 12 shows the slope geometry for case three, with statistical information on soil parameters as shown in table 5 and failure probability as shown in table 6.
Table 5: soil parameter statistics for case three
Figure BDA0002423785850000192
In the above table:
a represents a coefficient of variation.
Table 6: failure probability of case three systems obtained by different methods
Figure BDA0002423785850000193
Figure BDA0002423785850000201
In the above table:
a represents NE ═ the number of numerical analyses;
b represents the mean and 99.76% confidence interval;
c represents the relative error from the L HS mean.
Using the mean values of the random variables for analysis, the final finite difference grid for case three of the present application with the optimal density, which can be determined by fitting the curve shown in fig. 6, is shown in fig. 12 with a FS of 1.36. FIG. 13 shows the probability of failure P for the three-system case prediction using different ASMs at different sample pointsf,sAnd the L HS results and 99.76% confidence interval line were used as a reference.
From the three cases, in the slope reliability analysis based on the SRM, the grid density has a significant influence on the FS result: the FS value is gradually reduced along with the increase of the grid density, and the FS value tends to be stable when the grid density is larger. Therefore, before performing an SRM-based reliability analysis, a sensitivity analysis should be performed to obtain the optimal grid density for a given slope. Furthermore, in slope reliability analysis, the use of non-uniform grids to improve computational efficiency may not be a judicious choice, since the difference in random variables may result in (deterministic) critical sliding planes of different shapes and locations.
For a single layer slope, as in case one, the extreme state plane g (u) ═ 0, L SS has some linearity, since the slope system is primarily controlled by one failure mode, however, for a slope containing multiple layers of earth, L SS has a high degree of nonlinearity since the failure probability of the slope system may be controlled by multiple failure modes simultaneously.
The results of the three cases show that the ASVM, AK and ARBF method provided by the application can always well estimate the P of the slope system with multilayer soil and random variablesf,sIn case one and case three, their relative error with respect to the L HS results is within 5%, in case two, the value is about 10%, the major cause of the relatively large error in case two may be when P isf,sVery small (0.91%), L HS sample size (10000) is too small, resulting in Pf,sEstimated to have a relatively wide 99.76% confidence interval (0.64% to 1.18%). although so, the accuracy of the method proposed by the present application is generally better than other methods, particularly for multi-layer soil slopes, e.g., in case two, QRMS produces a large relative error (-50.55%) compared to L HS results, SPCE-L AR predicted Pf,sEven away from the L HS reference (relative error 178.02%).
In terms of calculation cost, for the three cases, the number of sample points required by the ASMs provided by the application is generally less than 100, which is generally considered to be computationally feasible in engineering practice, and compared with the existing method, the calculation amount is greatly reduced, and the calculation efficiency is improved. Among them, the ARBF proxy model with linear kernel function (no unknown parameters in kernel function) is superior to ASVM and AK proxy in model stabilityThe model (see fig. 7, 10 and 13) and fewer sample points are required. On one hand, the introduction of the active learning function greatly accelerates the model training speed; on the other hand, a compact proxy model (using fewer additional parameters, such as unknown parameters in the kernel function) may be more stable in the iterative analysis process. The ASVM proxy model, though using the same learning function as the ARBF, can converge to a sufficient value of Pf,sBut requires the most sample points and has a large fluctuation range. While the widely used AK model performs between ARBF and ASVM.
The embodiments in the present specification are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other.
The slope system reliability analysis method based on the intensity reduction method provided by the application is described in detail, specific examples are applied in the method to explain the principle and the implementation mode of the application, and the description of the examples is only used for helping to understand the method and the core idea of the application; meanwhile, for a person skilled in the art, according to the idea of the present application, there may be variations in the specific embodiments and the application scope, and in summary, the content of the present specification should not be construed as a limitation to the present application.

Claims (10)

1. The method for analyzing the reliability of the slope system based on the intensity reduction method is characterized by comprising the following steps of:
step S1: generating a training sample set of the slope system by utilizing an initial sampling point strategy in a standard normal space;
step S2: converting the sample points of undetermined functional response G (u) in the training sample set from the standard normal space to a physical space, and calculating G (u) corresponding to the sample points converted to the physical space by using an intensity reduction method;
step S3: training a proxy model in the standard normal space using the training sample set and G (u);
step S4: predicting the functional response of all sample points in the Monte Carlo simulation MCS pool by using the trained agent model, calculating the failure probability of the current iteration according to the predicted functional response, and recording the failure probability of the current iteration in a preset matrix;
step S5: judging whether the variation coefficient of the failure probability calculated by the last five iterations is smaller than a preset convergence threshold value or not;
step S6: when the variation coefficient of the failure probability calculated by the last five iterations is not smaller than a preset convergence threshold, selecting an optimal sample point located in a standard normal space from the MCS pool by using an active learning function in combination with the trained agent model, adding the optimal sample point into the training sample set, and repeating the steps S2-S6;
step S7: and when the variation coefficient of the failure probability calculated by the last five times of iteration is smaller than a preset convergence threshold value, taking the failure probability calculated by the last iteration in the preset matrix as a result of the reliability analysis of the slope system.
2. The method of claim 1, wherein in step S1, the step of generating the training sample set of the slope system by using an initial sampling point strategy in a standard normal space comprises:
in a standard normal space, constructing an initial training sample set of the slope system by using a 3-sigma rule; the initial training sample set comprises a plurality of sample points u, wherein u represents a vector of random variables u in the standard normal space;
for each u in the initial training sample set, judging whether the u meets any one of the following conditions:
n-1 of the u is equal to-3, the other u is equal to 0 or 3, and n represents the number of u in the u; or n elements of said u are all the same, all equal to-3, 0 or 3;
if the u is satisfied, keeping the u in the initial training sample set;
if the u is not satisfied, removing the u from the initial training sample set;
and when the initial training sample set is judged, obtaining the training sample set S.
3. The method according to claim 2, wherein in step S2, the step of transforming the sample points in the training sample set for which the functional response g (u) is not determined from the standard normal space to the physical space, and calculating g (u) corresponding to the sample points transformed to the physical space by using an intensity reduction method comprises:
let the standard normal space be U and the physical space be X;
converting the sample point of undetermined G (U) in S from U to X, and then converting the sample point from U to X;
calculating the functional response of x using a given linear function g (x):
g(x)＝FS(x)-1 (1)；
wherein FS is F L AC3DThe stability coefficient calculated by the embedded strength reduction method;
the corresponding G (u) can be obtained by the formula (1), and satisfies the following conditions:
g(x)＝G(u) (2)。
4. the method according to claim 3, wherein in step S3, when the proxy model is a SVM (support vector machine) proxy model, the step of training the proxy model comprises using the training sample set and G (u) in the standard normal space:
training the SVM proxy model by using the training sample set in the standard normal space; wherein, one sample point of the ith simulation in the current S
Figure FDA0002423785840000021
Satisfy the requirement of
Figure FDA0002423785840000022
The vector of the sample points of (1) is located at one side, satisfies
Figure FDA0002423785840000023
Sample point ofIs positioned at the other side;
searching for an optimal classification hyperplane h (u) using the SVM proxy model for the current S:
Figure FDA0002423785840000024
Figure FDA0002423785840000025
in the above formula, w and e represent unknown parameters, wTRepresenting the transpose of the w matrix, yiIs that
Figure FDA0002423785840000026
A classification symbol of (a), represents positive or negative;
calculating a distance vector V (u) from all sample points in the current S to the H (u):
Figure FDA0002423785840000031
(5) in the formula (I), the compound is shown in the specification,
Figure FDA0002423785840000032
the sample point with the minimum distance H (u) in the current S is represented as a support vector; n is a radical ofSVIs composed of
Figure FDA0002423785840000033
The number of (2); omegaiObtaining a weight coefficient representing the ith sample point by optimizing and solving the formula (4);
Figure FDA0002423785840000034
to represent
Figure FDA0002423785840000035
Transposing the matrix;
and determining the classification condition of each sample point in the current S according to the positive or negative of the classification sign of V (u).
5. The method of claim 3, wherein in step S3, when the agent model is Kriging agent model, the step of training agent model comprises the steps of, in the standard normal space, using the training sample set and G (u):
in the standard normal space, training a Kriging agent model by using the training sample set to obtain an agent expression corresponding to G (u)
Figure FDA0002423785840000036
Figure FDA0002423785840000037
(6) Wherein L (u) represents a function representing the trend of G (u) obtained from regression analysis, z (u) is an assumed smooth Gaussian process, and the ith simulated sample point u is determined when L (u) is 0(i)And sample point u of the jth simulation(j)The covariance between is:
Figure FDA0002423785840000038
(7) in the formula (I), the compound is shown in the specification,
Figure FDA0002423785840000039
representing the process variance, R (-) is a gaussian kernel function, expressed as:
Figure FDA00024237858400000310
(8) in the formula, θ represents a vector of unknown coefficients θ; where, theta and sigmazBy all points in the current sample set S
Figure FDA00024237858400000311
And corresponding thereto
Figure FDA00024237858400000312
Obtained in combination with the maximum likelihood estimate.
6. The method according to claim 3, wherein in step S3, when the proxy model is a Radial Basis Function (RBF) proxy model, the step of training the proxy model comprises using the training sample set and G (u) in the standard normal space:
in the standard normal space, the RBF proxy model is trained by utilizing the training sample set to obtain a proxy expression corresponding to G (u)
Figure FDA00024237858400000313
Figure FDA00024237858400000314
(9) In the formula (I), the compound is shown in the specification,
Figure FDA00024237858400000315
representing a sample point of the ith simulation in the current S, N representing the number of the sample points in the current S, rho and b representing vectors of unknown coefficients rho and b in the RBF proxy model respectively, N representing the number of random variables in u, and u representing the number of the random variables in ujRepresents the jth random variable in u, and Ψ (-) represents a kernel function;
using a linear kernel function psi (a) ═ a, and dividing each sample point
Figure FDA0002423785840000041
Substituting the formula (9) to solve the unknown coefficient;
Figure FDA0002423785840000042
in the above equation, the unknown coefficient of the n +1 th term is determined by the orthogonality condition, ΨijRepresents the distance value between the i, j modeled sample points calculated using the Ψ (·).
7. The method according to any one of claims 4 to 6, wherein in step S4, the step of predicting the functional response of all sample points in the Monte Carlo simulation MCS pool by using the trained proxy model, and calculating the failure probability of the current iteration according to the predicted functional response comprises:
Figure FDA0002423785840000043
Figure FDA0002423785840000044
in the above formula, NSPRepresenting a number of sample points in the MCS pool;
when the agent model is an SVM agent model, V (u) obtained by the trained SVM agent model(i)) Instead of G (u)(i)) Substituting the formula (11) and the formula (12) to obtain the failure probability of the current iteration;
when the agent model is RBF agent model or Kriging agent model, the obtained agent model after training is used
Figure FDA0002423785840000045
Instead of G (u)(i)) And substituting the formula (11) and the formula (12) to obtain the failure probability of the current iteration.
8. The method of claim 7, wherein the step of determining whether the coefficient of variation of the failure probability calculated in the last five iterations is smaller than a preset convergence threshold in step S5 comprises:
standard deviation of failure probability calculated from the last five iterations
Figure FDA0002423785840000046
And average value
Figure FDA0002423785840000047
Calculating the coefficient of variation
Figure FDA0002423785840000048
Figure FDA0002423785840000051
And judging whether the variation coefficient is smaller than a preset convergence threshold value or not.
9. The method according to any one of claims 4 to 6, wherein in step S6, an optimal sample point u in a normal space is selected from the MCS pool by using an active learning function in combination with the trained proxy modelcComprises the following steps:
when the agent model is Kriging agent model, the u iscThe calculation formula (2) includes:
Figure FDA0002423785840000052
wherein u isTRepresents the sample point in the MCS pool T in the U space,
Figure FDA0002423785840000053
representing passing the Kriging agent model
Figure FDA0002423785840000054
A predicted standard deviation;
when the agent model is an RBF agent model, the ucThe calculation formula (2) includes:
Figure FDA0002423785840000055
Figure FDA0002423785840000056
wherein u isTRepresents one sample point in the MCS pool, d (u)TS) represents said uTThe minimum distance between the target and the sample point in the current S, d (S) is the limit value of the target minimum distance, lambda is a scale factor, and lambda is more than or equal to 0.1 and less than or equal to 0.5;
when the proxy model is an SVM proxy model, adopting
Figure FDA0002423785840000057
Instead of the former
Figure FDA0002423785840000058
Substituting into (15) and (16) to calculate uc。
10. The method of claim 3, further comprising:
introducing an explicit highly nonlinear function g (x)' as a test, and verifying the steps S2-S6, wherein:
Figure FDA0002423785840000059
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CN112733392B (en) * 2020-10-30 2022-02-15 成都理工大学 Slope system reliability analysis method based on two classifications
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CN113722920A (en) * 2021-08-30 2021-11-30 河海大学 Rapid slope earthquake reliability calculation method based on FLAC3D-Python secondary development
CN113722920B (en) * 2021-08-30 2024-03-19 河海大学 Rapid calculation method for reliability of side slope earthquake based on FLAC3D-Python secondary development

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