CN111428363A

CN111428363A - Slope system failure probability calculation method based on Support Vector Machine (SVM)

Info

Publication number: CN111428363A
Application number: CN202010214048.8A
Authority: CN
Inventors: 曾鹏; 张天龙; 李天斌; 孙小平; 王宇豪
Original assignee: Chengdu Univeristy of Technology
Current assignee: Chengdu Univeristy of Technology
Priority date: 2020-03-24
Filing date: 2020-03-24
Publication date: 2020-07-17
Anticipated expiration: 2040-03-24
Also published as: CN111428363B

Abstract

The utility model provides a slope system failure probability calculation method based on support vector machine SVM, put forward intensity reduction method SRM and assess the stability coefficient, and adopt initial sampling point strategy and initiative learning function, the initiative learning support vector machine ASVM proxy model of original extreme state function L SF has been constructed, combine Monte Carlo simulation MCS and ASVM proxy model to assess the failure probability of slope system, can quantify the influence of random variable and relevant parameter to the slope stability, greatly reduced initial sample point, effectively improved computational efficiency, can automatic identification soil property side slope in the glide plane of arbitrary shape, it is more convenient when carrying out the reliability analysis to the stratiform side slope that has complicated geometry.

Description

Slope system failure probability calculation method based on Support Vector Machine (SVM)

Technical Field

The application relates to the field of soil slope stability analysis, in particular to a slope system failure probability calculation method based on a Support Vector Machine (SVM).

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 system_f,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 deviations_f,sAnd (6) estimating the value.

To effectively combine L EM with MCS, another common approach is to identify some pairs P_f,sThe most contributing representative slip planes (RSSs), then, taking into account the correlation between the different RSSs, P can be easily calculated_f,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 very urgent to develop a reliable and efficient failure probability calculation method for the reliability of the slope system.

Disclosure of Invention

The application provides a slope system failure probability calculation method based on a Support Vector Machine (SVM) so as to overcome the technical problems.

In order to solve the technical problem, the application discloses a slope system failure probability calculation method based on a Support Vector Machine (SVM), which comprises 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;

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 Support Vector Machine (SVM) proxy model by using the training sample set G (u) in the standard normal space;

step S4: predicting the functional response of all sample points in the Monte Carlo simulation MCS pool by using the trained SVM proxy 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 in a standard normal space from the MCS pool by using an active learning function in combination with the trained SVM surrogate 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 AC^3DThe 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, the training of the SVM proxy model in the normal space using the training sample set g (u) includes:

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

Satisfy the requirement of

The vector of the sample points of (1) is located at one side, satisfies

Is located on the other side;

searching for an optimal classification hyperplane h (u) using the SVM proxy model for the current S:

in the above formula, w and e represent unknown parameters, w^TRepresenting the transpose of the w matrix, y_iIs that

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):

(5) in the formula (I), the compound is shown in the specification,

the sample point with the minimum distance H (u) in the current S is represented as a support vector; n is a radical of_SVIs composed of

The number of (2); omega_iObtaining a weight coefficient representing the ith sample point by optimizing and solving the formula (4);

to represent

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 S4, the step of predicting the functional responses of all sample points in the monte carlo simulation MCS pool by using the trained SVM proxy model, and calculating the failure probability of the current iteration according to the predicted functional responses includes:

v (u) obtained by using trained SVM proxy model⁽ⁱ⁾) Instead of G (u)⁽ⁱ⁾) Substituting the formula (6) and the formula (7) for calculation to obtain the failure probability of the current iteration;

in the above formula, N_SPRepresents the number of sample points in the MCS pool.

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

And average value

Calculating the coefficient of variation

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 the standard normal space is selected from the MCS pool by using an active learning function in combination with the trained SVM surrogate model_cThe calculation formula (2) includes:

wherein u is_TRepresents one sample point in the MCS pool, d (u)_TS) represents said u_TThe minimum distance from 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.

Further, the method further comprises:

introducing an explicit highly nonlinear function g (x)' as a test, and verifying the steps S2-S6, wherein:

compared with the prior art, the method has the following advantages:

the application provides a slope system reliability analysis method based on the emphasis reduction method SRM, which can automatically identify the sliding surface with any shape in the soil slope, does not need to identify the critical sliding surface like L EM, and is more convenient for the reliability analysis of the layered slope with complex geometric shape;

according to the method, an initial sampling point strategy is adopted, an ASVM proxy model replacing an original extreme state function L SF is developed by combining an active learning function, and a Monte Carlo simulation MCS and the ASVM proxy model are combined to evaluate the failure probability of the slope system, so that 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 failure probability calculation method based on a support vector machine SVM 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 of the fitting performance of the ASVM proxy model;

FIG. 4 is a flowchart of a slope system reliability analysis performed by the ASVM proxy model and the 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 representation 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 diagram of the fitting performance of the L HS samples to the ASVM proxy model 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 of the fitting performance of the L HS samples with the ASVM proxy model 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 failure probability calculation method based on a support vector machine SVM according to the present application is shown, and specifically, the method may include the following steps:

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).

The application provides an improved 3-sigma rule, and the basic idea is to balance the numbers of unstable and unstable area points and accelerate the training process of separating a safe area from a failure area. 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 this application, a sample point u comprises a plurality of random variables u, such as u₁,u₂,…,u_n. Wherein u is₁＝u₂＝…＝u_nWhen-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.

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 AC^3DThe stability factor calculated by the intensity reduction method (hereinafter, both expressed as SRM) of the mid-insert;

g(x)＝G(u) (2)。

in the prior art, if formula (1) is used for directly calculating the failure probability of the slope system, the failure probability P is calculated_f,sCan be expressed as:

P_f,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 MCS_f,sUnbiased estimation of (d). However, in this method, when P is_f,s＝10^-2And the coefficient of variation of MCS

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 extent_f,sThereby reducing the number of simulations required. But at present, the target P of many civil engineering projects_f,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 application proposes Monte Carlo simulation based on the SVM (support vector machine) surrogate model, and can construct the explicit prediction model of G (u) by using a small number of sample points

Replacement of original F L AC by SVM proxy model^3DAnd 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.

Satisfy the requirement of

The vector of the sample points of (1) is located at one side, satisfies

Is located on the other side;

A classification symbol of (a), represents positive or negative;

(5) in the formula (I), the compound is shown in the specification,

to represent

Transposing the matrix;

It should be noted that, the above solution is linear classification hyperplane, and to obtain nonlinear classification hyperplane, kernel function may be used

Substitution to Gaussian Kernel function

in the application, all the sample points in the MCS pool T are substituted into the SVM proxy model for calculation, and the functional response V (u) of the corresponding sample point can be predicted.

V (u) obtained by using trained SVM proxy model⁽ⁱ⁾) Instead of G (u)⁽ⁱ⁾) Substituting the formula for calculation 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.

there are typically two convergence criteria (i) sample points near the extreme state plane (L SS) are sufficiently dense, or (ii) prediction P is predicted_f,sIs sufficiently small. The same criteria apply to the present application. Therefore, 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:

And average value

Calculating the coefficient of variation

And judging whether the variation coefficient is smaller than a preset convergence threshold value or not. In the present application, 0.001 is preferred.

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, an optimal sample point u in a standard normal space is selected from the MCS pool by combining an active learning function with the trained SVM surrogate model_cThe calculation formula (2) includes:

wherein u is_TRepresents one sample point in the MCS pool, d (u)_TS) represents said u_TThe minimum distance from 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. λ is further preferably 0.2 in the present application.

In step S6, when the variation coefficient of the failure probability calculated in the last five iterations is not less than the preset convergence threshold, the optimal sample point in the MCS pool T is further screened by the active learning function of equations (9) and (10), and after being added to the training sample set S, the current agent model is updated, so that an active learning support vector machine ASVM agent model is constructed, and updating is implemented in subsequent iterations. 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. Therefore, a continuous training process can be started by utilizing the initial sampling point strategy and the active learning function, and the prediction precision is improved.

In this application, FIG. 3(a) shows a schematic of a fit of 5000 MCS samples to a real L SS sample, and FIG. 3(b) shows sample points and corresponding fit results provided by the ASVM proxy model. the results show that the ASVM proxy model has many sample points around L SS that provide most of the information for constructing the proxy model, and thus can interpolate or fit well across a limited sample space with a small number of sample points.

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 iteration_f,sAnd providing the optimal sample points to enrich the training sample set S so as to update the ASVM 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. And carrying out iterative updating of the proxy model by using candidate samples selected from the MCS pool T by using an active learning function unless a convergence criterion is met. 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:

the result shows that the ASVM proxy model of the application establishes the proxy expression of the actual functional response G (U) in the U space based on a small number of training samples

Next, in order to further explain the reliability analysis method of the present application, verification analysis is 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 comparison 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, the original reliability analysis method is directly based on the SRML SF was subjected to 10000 Latin hypercube sampling (L HS) simulations L HS provides a system failure probability P_f,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 negligible. 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 methods_f,sAnd (6) obtaining the result.

Table 1: soil parameter statistics for case one

Table 2: failure probability of case-system obtained by different methods

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 predicting cases using ASVM proxy models at different sample points_f,sAnd L HS results and 99.76% confidence interval line as reference FIG. 8 shows the sample point location and classification for L HS 10000 times in two-dimensional U space and the prediction of actual L SS by ASVM proxy model in case one.

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

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

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 by the fit shown in FIG. 6And (5) determining a curve. FIG. 10 shows the prediction of failure probability P of case two system using ASVM proxy model under different sample point numbers_f,sAnd L HS results and 99.76% confidence interval line are taken as reference to reflect that the ASVM proxy model converges with the increase of the number of training samples, FIG. 11 shows the sample point position and classification of L HS in two-dimensional U space 10000 times in case two and the prediction of actual L SS by the ASVM proxy model.

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

In the above table:

a represents a coefficient of variation.

Table 6: failure probability of case three systems obtained by different methods

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, case three of the present application had a final finite difference grid of optimal density, shown in fig. 12, with a FS of 1.36, which was determined by fitting the curve shown in fig. 6. FIG. 13 shows the probability of failure P for the case three system using ASVM proxy model prediction at different sample points_f,sAnd is connected with L HSThe fruit and 99.76% confidence interval line are used as references.

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, the sensitivity analysis is carried out before the reliability analysis based on the SRM, and the optimal grid density of the given slope can be obtained. 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 proxy model provided by the application can well estimate P of the slope system with multilayer soil and random variables_f,sIn case one and case three, the relative error with respect to the L HS results were 4.32% and-2.05%, respectively, with absolute values within 5%, and in case two, the value was about 10%, the major cause of the relatively large error in case two may be when P is present_f,sVery small (0.91%), L HS sample size (10000) is too small, resulting in P_f,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 P_f,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 ASVM proxy model 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.

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 failure probability calculation method based on the support vector machine SVM provided by the application is introduced in detail, a specific example is applied in the method to explain the principle and the implementation mode of the application, and the description of the embodiment 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

1. A slope system failure probability calculation method based on a Support Vector Machine (SVM) is characterized by comprising the following steps:

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:

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;

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;

calculating the functional response of x using a given linear function g (x):

g(x)＝FS(x)-1 (1)；

g(x)＝G(u) (2)。

4. the method according to claim 3, wherein in step S3, the step of training a SVM proxy model in the standard normal space using the training sample set and G (u) comprises:

Satisfy the requirement of

The vector of the sample points of (1) is located at one side, satisfies

Is located on the other side;

A classification symbol of (a), represents positive or negative;

(5) in the formula (I), the compound is shown in the specification,

to represent

Transposing the matrix;

5. The method of claim 4, 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 SVM surrogate model, and calculating the failure probability of the current iteration according to the predicted functional response comprises:

6. The method according to claim 1 or 5, 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:

And average value

Calculating the coefficient of variation

7. The method of claim 4, wherein in step S6, the optimal sample point u in the normal space is selected from the MCS pool by using an active learning function in combination with the trained SVM surrogate model_cThe calculation formula (2) includes:

8. The method of claim 3, further comprising: