CN111339487A - Slope system failure probability calculation method based on radial basis function RBF - Google Patents

Abstract

The application provides a slope system failure probability calculation method based on a Radial Basis Function (RBF), which is characterized in that a Strength Reduction (SRM) method is provided for evaluating a stability coefficient, an initial sampling strategy and an active learning function are adopted, an active learning radial basis function (ARBF) proxy model of an original Limit State Function (LSF) is constructed, Monte Carlo Simulation (MCS) and the ARBF proxy model are combined to evaluate the failure probability of a slope system, the influence of random variables and related parameters thereof 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 failure probability calculation method based on radial basis function RBF

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 Radial Basis Function (RBF).

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 perform reliability analysis using the limit balance method (LEM), which, when combined with MCS, requires searching for the critical sliding surface with the smallest FS in each simulation, and is therefore computationally expensive. A more critical issue is that LEMs primarily use randomly generated slip planes that may miss critical slip planes, thereby providing a more highly biased P_f,sAnd (6) estimating the value.

To effectively combine LEM with MCS, another common method 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.

To improve computational efficiency, the proxy model is widely applied along with the MCS. Many generic and advanced surrogate models are used for slope reliability analysis, such as gaussian process regression, group intelligent support vector machine, and multivariate adaptive regression splines. LEM is often chosen as a deterministic analysis method to evaluate FS of a slope, the advantage of LEM is its simplicity and low computational cost, but its main disadvantages are: difficult to locate when the critical sliding surface is not known in advance; furthermore, the test sliding surface is usually assumed to be circular, which may not be suitable for complex slope systems, especially when there is a weak interlayer in the 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 Radial Basis Function (RBF) to overcome the technical problems.

In order to solve the above problems, the present application discloses a slope system failure probability calculation method based on a radial basis function RBF, 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: in the standard normal space, training a Radial Basis Function (RBF) proxy model by 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 RBF agent model, calculating the failure probability of the current iteration according to the predicted functional response, and recording the failure probability 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 RBF 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, the step S2 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 FLAC^3DThe stability coefficient calculated by the embedded strength reduction method;

g (x) the corresponding G (u) can be obtained by the formula (1).

Further, in step S3, the step of training the radial basis function RBF proxy model in the standard normal space by using the training sample set g (u) includes:

in the standard normal space, the RBF agent model is trained by utilizing the training sample set to obtain G (u) corresponding toProxy expression of

(2) In the formula (I), the compound is shown in the specification,

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 u_jRepresents 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

Substituting the formula (2) into the formula (2) to solve the unknown coefficient;

(3) in which the unknown coefficient of the n +1 th term is determined by the orthogonality condition, Ψ_ijRepresents 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 RBF proxy model, and calculating the failure probability of the current iteration according to the predicted functional responses includes:

obtained by using the trained RBF agent model

Instead of G (u)⁽ⁱ⁾) Substituting the formula 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 epsilon 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 RBF agent model_cComprises the following steps:

wherein u is_TRepresents one sample point in the MCS pool, d (u)_TS) represents said u_TMinimum distance from sample point in current S, d (S) is targetThe limit value of the minimum distance, wherein 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 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 LEM, and is more convenient to perform reliability analysis on the layered slope with complex geometric shape;

according to the method, an initial sampling point strategy is adopted, an ARBF (autoregressive moving field) proxy model replacing an original limit state function LSF (local start function) is developed by combining an active learning function, the failure probability of a slope system is evaluated by combining an MCS (modulation and coding scheme) with the ARBF proxy model, 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 quantified.

Drawings

FIG. 1 is a flowchart illustrating steps of a slope system failure probability calculation method based on a radial basis function RBF 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 LSS;

FIG. 3(b) is a diagram of the fitting performance of the ARBF proxy model;

FIG. 4 is a flowchart of a slope system reliability analysis performed by using an ARBF 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 diagram of the fitting performance of the LHS sample to the ARBF agent 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 LHS sample and the ARBF 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 radial basis function RBF according to the present application is shown, and specifically 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;

proper selection of the initial sample points can speed up the convergence of the training process. The initial training sample set may be constructed using Latin Hypercube Sampling (LHS), but this may not be suitable for 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 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.

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 FLAC^3DThe 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 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

When, a model needs to be largeAbout 10⁴Sub-simulation for time-consuming reliability analysis (e.g., using FLAC)^3DAnd analysis by SRM), the amount of computation is unacceptable. In the prior art, there are several variations of MCS, such as LHS, IS and Subset Simulation (SS), which can 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 RBF proxy model, and can use a small number of sample points to construct an explicit prediction model of G (u)

Replacing original FLAC by RBF 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.

Step S3: in the standard normal space, training a Radial Basis Function (RBF) proxy model by using the training sample set and G (u);

RBF is another accurate interpolation method. Its advantages are simple structure and easy implementation. In step S3, the step of training the radial basis function RBF proxy model in the standard normal space by using the training sample set and g (u) 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)

(2) In the formula (I), the compound is shown in the specification,

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 u_jRepresents 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

Substituting the formula (2) into the formula (2) to solve the unknown coefficient;

in the above equation, the unknown coefficient of the n +1 th term is determined by the orthogonality condition, Ψ_ijRepresents the distance value between the i, j modeled sample points calculated using the Ψ (·).

Step S4: predicting the functional response of all sample points in the Monte Carlo simulation MCS pool by using the trained RBF agent model, calculating the failure probability of the current iteration according to the predicted functional response, and recording the failure probability in a preset matrix;

in the application, all sample points in the MCS pool T are substituted into the RBF proxy model for calculation, and the functional response of the corresponding sample points can be predicted

The method comprises the following steps of predicting the functional response of all sample points in a Monte Carlo simulation MCS pool by using a trained RBF agent model, and calculating the failure probability of the current iteration according to the predicted functional response:

obtained by using the trained RBF agent model

Instead of G (u)⁽ⁱ⁾) Substituting the formula 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.

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;

a reasonable convergence criterion should stop the training process in time, thereby reducing the required training sample set when the current proxy model is stable. There are generally two convergence criteria: (i) sufficiently dense sample points near the hyperplane limit state plane (LSS), or (ii) predicting P_f,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

And average value

Calculating the coefficient of variation

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

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 RBF surrogate 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 by using MCS, and the setting of the learning function is related to the actual analysis purpose, aiming to order a set of candidate points in the MCS pool T. In the slope reliability analysis of the present application, the learning function needs to provide an optimal sample point in each iteration process for updating the current agent model, and the optimal sample point should satisfy two conditions at the same time: (i) it is located near the extreme state surface (LSS), (ii) redundant information is avoided (i.e., away from sample points in the existing S).

Therefore, in step S6, the optimal sample point u in the standard normal space is selected from the MCS pool by using the active learning function in combination with the trained RBF proxy model_cComprises the following steps:

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 points in the MCST pool are further screened by the active learning functions of formulas (7) and (8), and added to the training sample set S to update the current proxy model, thereby constructing an active learning radial basis function ARBF proxy model and implementing updating 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 an initial sampling strategy and an active learning function, and the prediction precision is improved.

In this application, fig. 3(a) shows a fitting schematic diagram of 5000 MCS samples and true LSSs, and fig. 3(b) shows sample points and corresponding fitting results provided by the ARBF proxy model. The results show that the ARBF proxy model has many sample points near the LSS, which provide most of the information for constructing the proxy model, and thus can interpolate or fit well to the entire limited 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 epsilon is chosen 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 ARBF agent 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 perform iterative update of the agent model according to candidate samples selected from the MCS pool T by the corresponding active learning function unless the 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:

the result shows that the ARBF agent model of the application can establish the agent expression of the actual functional response G (U) in the U space based on a small amount of training samples

In order to further explain the reliability analysis method of the application, the application uses three typical reference slopes as cases for verification analysis.

Because the shear modulus and the bulk modulus of the soil body have small influence on the FS of the side slope, the shear modulus and the bulk modulus are respectively assumed to be 30MPa and 100MPa in three cases; the random variables involved are considered independently uncorrelated. In addition, the present application also applies some widely used methods based on polynomial expansion (PCE), such as the Quadratic Response Surface Method (QRSM) and sparse PCE minimum angle regression method (SPCE-LAR), to the following three cases, and performs comparative studies with the method of the present 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 (LHS) simulation are carried out on the original LSF directly based on the SRM. LHS provided System failure probability P_f,sWill be considered to be a "reference" or "exact" value. To measure the computational efficiency, the number of sample points required for each analysis (and also the number of times the numerical analysis is performed) is used. This is because when introducingInto a computationally intensive numerical method (FLAC 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, with statistical information on soil parameters as shown in table 1 and failure probability as shown in table 2. Table 2 shows sample point numbers and P using different reliability methods under 10000 LHS simulations_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 LHS average.

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 an ARBF agent model at different sample points_f,sAnd the LHS results and 99.76% confidence interval line were used as references. Fig. 8 shows sample point positions and classification of LHS in 10000 times in two-dimensional U space and prediction of actual LSS by ARBF agent 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 LHS average.

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 utilization of ARBF surrogate model for different training sample point numbersFailure probability P of case two system_f,sAnd the LHS results and 99.76% confidence interval line were used as references. FIG. 10 reflects that the ARBF proxy model converges as the number of training samples increases. Fig. 11 shows the sample point position and classification of LHS in 10000 times in two-dimensional U space and the prediction of actual LSS by ARBF agent model 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

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 LHS average.

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 of the three-system case prediction using the ARBF agent model with different sample points_f,sAnd the LHS results and 99.76% confidence interval line were 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, with the limit state plane g (u) equal to 0, LSS has some linearity, since the slope system is mainly controlled by a failure mode. However, for a slope containing multiple layers of earth, its LSS is highly non-linear, since the failure probability of the slope system may be controlled by multiple failure modes simultaneously. For example, as shown in fig. 11, case two's LSS may be approximated as a combination of two linear problems.

The results of the three cases show that the ARBF agent model provided by the application can well estimate P of the slope system with multilayer soil and random variables_f,s. In case one and case three, their relative errors with respect to LHS results were 3.35% and-1.64%, respectively; in case two, this value is about 10%. The main reasons for the relatively large error of case two may be: when P is present_f,sVery small (0.91%) LHS sample size (10000) is too small, resulting in P_f,sIt is estimated to have a relatively wide 99.76% confidence interval (0.64% to 1.18%). Nevertheless, the accuracy of the method proposed by the present application is generally superior to other methods, particularly for slopes of multi-layer soils; for example, in case two, QRMS produces a large relative error (-50.55%) compared to LHS results; SPCE-LAR predicted P_f,sEven away from the LHS reference (relative error 178.02%).

In terms of calculation cost, for the three cases, the number of sample points required by the ARBF 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 radial basis function RBF 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 (8)

1. The slope system failure probability calculation method based on the radial basis function RBF is characterized by comprising 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: in the standard normal space, training a Radial Basis Function (RBF) proxy model by 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 RBF agent model, calculating the failure probability of the current iteration according to the predicted functional response, and recording the failure probability 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 RBF 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.

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 the step S2 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 FLAC^3DThe stability coefficient calculated by the embedded strength reduction method;

g (x) the corresponding G (u) can be obtained by the formula (1).

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

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)

(2) In the formula (I), the compound is shown in the specification,

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 u_jRepresents 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

Substituting the formula (2) into the formula (2) to solve the unknown coefficient;

(3) in which the unknown coefficient of the n +1 th term is determined by the orthogonality condition, Ψ_ijRepresents the distance value between the i, j modeled sample points calculated using the Ψ (·).

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 RBF proxy model, and calculating the failure probability of the current iteration according to the predicted functional response comprises:

obtained by using the trained RBF agent model

Instead of G (u)⁽ⁱ⁾) Substituting the formula 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.

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:

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 epsilon or not.

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 RBF proxy model_cComprises the following steps:

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.

8. 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:

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