CN111339487B - Slope system failure probability calculation method based on radial basis function RBF - Google Patents

Abstract

The method comprises the steps of providing an intensity folding and subtracting SRM method to evaluate stability coefficients, constructing an active learning radial basis function ARBF proxy model of an original limit state function LSF by adopting an initial sampling strategy and an active learning function, and evaluating the failure probability of a slope system by combining a Monte Carlo simulation MCS and the ARBF proxy model, thereby being capable of quantifying the influence of random variables and related parameters on the slope stability, greatly reducing the number of initial sample points, effectively improving the calculation efficiency, automatically identifying sliding surfaces with any shape in an earth slope, and being more convenient when carrying out reliability analysis on a layered slope with complex geometric shapes.

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 assessment is a complex geotechnical engineering problem, and input parameters of the slope stability assessment have uncertainty. Traditional deterministic analysis methods using stability coefficients (FS) may not truly reflect the safety of the slope. To quantify the impact of uncertainty, probabilistic methods are widely used in slope reliability analysis.

A slope may break along different sliding surfaces, any of which may cause damage to the slope, creating a series of system problems. The accurate and effective reliability analysis of the complex problems is a main difficulty faced by the application of the probability method in geotechnical engineering practice.

The direct simulation method IS one of probability methods, such as Monte Carlo Simulation (MCS) and Importance Sampling (IS) can be used for failure probability P of the side slope system _f,s Unbiased estimation is performed, but most students currently use the limit balance method (LEM) for reliability analysis, which, when combined with MCS, requires searching critical sliding surfaces with minimum FS in each simulation, and thus is computationally intensive. More closedThe key problem is that LEM uses mainly randomly generated sliding surfaces, possibly missing critical sliding surfaces, thus providing P with larger deviations _f,s And (5) estimating a value.

To effectively combine LEM with MCS, another common approach is to identify pairs P _f,s Contributing the largest typical sliding surface (RSSs), then, taking into account the correlation between different RSSs, P can be easily calculated _f,s . There are prior art approaches to identifying RSSs by randomly generating a large number of potential sliding surfaces. However, one challenge faced by such RSSs-based approaches is how to select a reasonable threshold for correlation coefficients between RSSs to achieve computational efficiency and accuracy.

In order to improve the computational efficiency, proxy models are widely used along with MCSs. 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 spline. LEM is often chosen as a deterministic analysis method to evaluate the FS of a slope, and has the advantages of its simplicity and low computational cost, but its main drawbacks are: positioning is difficult when the critical sliding surface is not known in advance; furthermore, the test sliding surface is often assumed to be circular, which may not be suitable for complex slope systems, especially when weak interlayers are present on the slope.

Therefore, a reliable and efficient failure probability calculation method is developed, which is very urgent for the reliability of the slope system.

Disclosure of Invention

The application provides a side slope system failure probability calculation method based on a radial basis function RBF, so as 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 radial basis function RBF, including:

step S1: in a standard normal space, generating a training sample set of the slope system by utilizing an initial sampling point strategy;

step S2: converting sample points of the undetermined functional response G (u) in the training sample set from the standard normal space to a physical space, and calculating the G (u) corresponding to the sample points converted to the physical space by using an intensity reduction method;

step S3: training a radial basis function RBF proxy model in the standard normal space by utilizing 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 proxy 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 in 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 in the last five iterations is not smaller than a preset convergence threshold, selecting an optimal sample point positioned in a standard normal space from the MCS pool by utilizing an active learning function in combination with a trained RBF proxy 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 iterations is smaller than a preset convergence threshold, 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, the step of generating a training sample set of the slope system using the initial sampling point strategy in the standard normal space includes:

constructing an initial training sample set of the side slope system in a standard normal space 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:

the u has n-1 u equal to-3, the other u is equal to 0 or 3, and n represents the number of u in u; or n elements of said u are all the same, equal to-3, 0 or 3;

if the u meets the requirement, the u is reserved in the initial training sample set;

if the u does not meet, 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 a sample point of the undetermined G (U) in the S from the U to X, and converting the sample point from U to X;

using a given linear function g (x), the functional response of said x is calculated:

g(x)＝FS(x)-1 (1)；

wherein FS is FLAC ^3D A stability coefficient calculated by an intensity reduction method embedded in the model;

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

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

in the standard normal space, training an RBF proxy model by using the training sample set to obtain a proxy expression corresponding to G (u)

(2) In the method, in the process of the invention,

representing one sample point of the ith simulation in the current S, N representing the number of sample points in the current S, ρ and b representing vectors of unknown coefficients ρ and b in the RBF proxy model, respectively, N representing the number of random variables in u, u _j Representing the j-th random variable in u, and ψ (·) represents the kernel function;

by linear kernelThe number ψ (a) =a, each sample point

Substituting the unknown coefficients into the formula (2), and solving the unknown coefficients;

(3) Wherein the unknown coefficient of the n+1th term is determined by an orthogonal condition, ψ _ij Representing the distance value between the sample points of the i, j two simulations 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 a trained RBF proxy model

Instead of G (u) ⁽ⁱ⁾ ) Substituting the probability of failure of the current iteration into the following formula to calculate;

in the above, N _SP Representing the number of sample points in the MCS pool.

Further, in step S5, the step of determining whether the variation coefficient of the failure probability calculated in the last five iterations is smaller than a preset convergence threshold includes:

standard deviation of failure probability calculated from last five iterations

Mean value->

Calculating the coefficient of variation->

And judging whether the variation coefficient is smaller than a preset convergence threshold epsilon.

Further, in step S6, an optimal sample point u located 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 _c The method comprises the following steps:

/>

wherein u is _T Represents a sample point, d (u _T S) represents the u _T The minimum distance from the sample point in the current S, d (S) is the limit value of the target minimum distance, lambda is the 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 to S6, wherein:

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

the SRM-based slope system reliability analysis method 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 for reliability analysis of the layered slope with complex geometric shape;

the method adopts an initial sampling point strategy, combines an active learning function, develops an ARBF proxy model for replacing an original limit state function LSF, combines the MCS and the ARBF proxy model to evaluate the failure probability of the slope system, greatly reduces the number of initial sample points, effectively improves the calculation efficiency, and can quantify the influence of random variables and related parameters on the slope stability.

Drawings

FIG. 1 is a flow chart of steps of a method for calculating failure probability of a slope system based on a radial basis function RBF;

FIG. 2 (a) is a schematic diagram of sample point locations generated by a conventional 3-sigma rule;

FIG. 2 (b) is a schematic representation of sample point locations generated by the improved 3-sigma rule of the present application;

fig. 3 (a) is a schematic fit of 5000 MCS samples and real LSS;

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

FIG. 4 is a flow chart of the reliability analysis of the side slope system using the ARBF proxy model and strength reduction method of the present application;

FIG. 5 is a schematic illustration of a slope geometry for case one;

FIG. 6 is a schematic diagram of FS versus grid density calculated for three cases of the present application;

fig. 7 is a case-one failure probability prediction graph;

fig. 8 is a schematic diagram of fitting performance of LHS samples in case one to an ARBF proxy model;

FIG. 9 is a schematic diagram of a side slope geometry for case two;

fig. 10 is a failure probability prediction diagram for case two;

fig. 11 is a schematic diagram of fitting performance of LHS samples in case two to an ARBF proxy model;

FIG. 12 is a schematic diagram of a side slope geometry for case three;

fig. 13 is a failure probability prediction diagram of case three.

Detailed Description

In order that the above-recited objects, features and advantages of the present application will become more readily apparent, a more particular description of the invention briefly described above will be rendered by reference to specific embodiments that are illustrated in the appended drawings.

Referring to fig. 1, a step flowchart of a slope system failure probability calculating method based on a radial basis function RBF according to the present application may specifically include the following steps:

step S1: in a standard normal space, generating a training sample set of the slope system by utilizing an initial sampling point strategy;

proper selection of the initial sample point may speed up convergence of the training process. The initial training sample set may be constructed with Latin Hypercube Sampling (LHS), but this may not be applicable to some models with lower failure probabilities because it must contain two classes of points (e.g., G (u)>0 and G (u)<0). The conventional 3-sigma works well for this purpose because it reflects approximately the general trend of G (u) throughout the sampling space and contains two types of points. However, this method requires about 3 ⁿ A number of initial sample points, where n is the number of random variables; thus, it may not be suitable for problems containing many random variables (e.g., the problem of 10 random variables requires 59049 initial sample points, which is obviously unacceptable in practice).

Thus, the present application proposes an improved 3-sigma rule whose basic idea is to balance the number of two regions of instability and non-instability, speeding up the training process of separation of the security domain and the failure domain. To this end, the sampling range of each random variable is considered as [ -3,3] in an uncorrelated standard normal space (also called U-space), step S1 may comprise the sub-steps of:

substep 1-1: constructing an initial training sample set of the side slope system in a standard normal space 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;

substep 1-2: for each u in the initial training sample set, judging whether the u meets any one of the following conditions:

the u has n-1 u equal to-3, the other u is equal to 0 or 3, and n represents the number of u in u; or n of said u are all the same, equal to-3, 0 or 3;

if the u meets the requirement, the u is reserved in the initial training sample set;

substep 1-3: if the u meets the requirement, the u is reserved in the initial training sample set; and if the u does not meet the requirement, 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 complete.

In the present application, a sample point u includes a plurality of random variables u, e.g., u ₁ ,u ₂ ,…,u _n . Wherein u is ₁ ＝u ₂ ＝…＝u _n At = -3, the stability factor FS is minimal, i.e. the point is the Most Dangerous Point (MDP) in the training sample set S.

According to the training sample set S obtained through the initial sampling point strategy, 2n+3 initial sample points are finally generated, and compared with the traditional 3-sigma, the number of initial sample points can be greatly reduced when a slope system has a large number of random variables. FIG. 2 illustrates sample point cases generated in U-space by a conventional 3-sigma rule and the 3-sigma rule modified in this application when 3 random variables are considered. Wherein FIG. 2 (a) shows a sample point location diagram generated by a conventional 3-sigma rule; fig. 2 (b) shows a schematic diagram of sample point locations generated by the 3-sigma rule modified in the present application.

Step S2: converting sample points of the undetermined functional response G (u) in the training sample set from the standard normal space to a physical space, and calculating the 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 on the slope stability. Let the standard normal space be U and the physical space be X;

after a sample point of the undetermined G (U) in the S is converted from the U space to an X space, the sample point is converted from U to X, and X represents a vector of a random variable in the X space;

using a given linear function g (x), the functional response of said x is calculated:

g(x)＝FS(x)-1 (1)；

wherein FS is FLAC ^3D Stability coefficients calculated by the embedded intensity folding method (hereinafter, all are denoted by SRM);

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

In the prior art, if (1) is used for directly calculating the failure probability of a slope system, the failure probability P _f，s Can be expressed as:

P _f，s ＝P[g(x)＜0]＝∫ _g(x)＜0 f(x)dx；

where f (x) represents the joint Probability Density Function (PDF) of the random variables involved. However, since g (x) is implicit, it is difficult to directly calculate the integral in the above equation. Thus, the present application transforms vector x into U for sample points in the uncorrelated normal space, such that the limit state plane can be rewritten as G (U) = 0,G (U) is a mapping of G (x) in the uncorrelated normal space U.

After the transformation, if the failure probability P can be provided according to the traditional MCS _f,s Is an unbiased estimate of (1). But in this method, when P _f,s ＝10 ^-2 And the coefficient of variation of MCS

When a model is needed about 10 ⁴ Sub-simulations, for time-consuming reliability analysis (e.g., using FLAC ^3D And analysis by SRM) is unacceptable. There are also several variants of MCS in the prior art, such as LHS, IS and Subset Simulation (SS), which can reduce the calculated P to some extent _f,s Thereby reducing the number of simulations required. But now object P of many civil engineering projects _f,s At 10 ^-3 To 10 ^-5 Between these, this further increases the computational effort required, making it difficult for even these MCS variants to meet the computational demands.

Accordingly, the present application proposes Monte Carlo simulation based on RBF proxy modelExplicit prediction model of G (u) can be constructed with a small number of sample points

Replacing the original FLAC with the RBF proxy model ^3D And SRM analysis, thereby greatly improving the efficiency and reducing the calculation cost. In this way, failure probability calculation results can be quickly given by using the MCS or a variant thereof.

Step S3: training a radial basis function RBF proxy model in the standard normal space by utilizing the training sample set and G (u);

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

in the standard normal space, training an RBF proxy model by using the training sample set to obtain a proxy expression corresponding to G (u)

(2) In the method, in the process of the invention,

representing one sample point of the ith simulation in the current S, N representing the number of sample points in the current S, ρ and b representing vectors of unknown coefficients ρ and b in the RBF proxy model, respectively, N representing the number of random variables in u, u _j Representing the j-th random variable in u, and ψ (·) represents the kernel function;

each sample point is processed by a linear kernel function ψ (a) =a

Substituting the unknown coefficients into the formula (2), and solving the unknown coefficients; />

In the above formula, the unknown coefficient of the n+1th term is determined by the orthogonal condition, ψ _ij Representing the distance value between the sample points of the i, j two simulations 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 proxy 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 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 the following steps:

obtained by using a trained RBF proxy model

Instead of G (u) ⁽ⁱ⁾ ) Substituting the probability of failure of the current iteration into the following formula to calculate;

in the above, N _SP Representing the number of sample points in the MCS pool.

The preset matrix can be set in a preparation stage, and a matrix is initialized to record the failure probability of each iteration of the system. Iteration refers to the calculation of a deterministic model using different sample points in succession in the proxy model construction process.

Step S5: judging whether the variation coefficient of the failure probability calculated in 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 set of training samples required when the current proxy model is stable. There are generally two convergence criteria: (i) Sample points near the hyperplane Limit State Surface (LSS) are sufficiently dense, or (ii) predict P _f,s Is sufficiently small. The present application applies the same criteria. Thus, step S5 may specifically include, when implemented:

standard deviation of failure probability calculated from last five iterations

Mean value->

Calculating the coefficient of variation->

/>

And judging whether the variation coefficient is smaller than a preset convergence threshold epsilon. In this application, epsilon=0.001 is preferred.

Step S6: when the variation coefficient of the failure probability calculated in the last five iterations is not smaller than a preset convergence threshold, selecting an optimal sample point positioned in a standard normal space from the MCS pool by utilizing an active learning function in combination with a trained RBF proxy model, adding the optimal sample point into the training sample set, and repeating the steps S2-S6;

in the present application, first, a large sample pool T (e.g., 200000 points) is generated by MCS, and the setting of the learning function is closely related to the actual analysis purpose, aiming at ordering 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 in each iteration process, so as to update the current proxy model, and the optimal sample point should simultaneously satisfy two conditions: (i) It is located near the Limit State Surface (LSS), (ii) avoids redundant information (i.e., far from the 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 utilizing the active learning function in combination with the trained RBF proxy model _c The method comprises the following steps:

wherein u is _T Represents a sample point, d (u _T S) represents the u _T The minimum distance from the sample point in the current S, d (S) is the limit value of the target minimum distance, lambda is the scale factor, and lambda is more than or equal to 0.1 and less than or equal to 0.5. The present application further prefers λ to be 0.2.

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, the current proxy model is updated, so that the active learning radial basis function ARBF proxy model is constructed, and the update is realized in the subsequent iterations. It should be 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. With this, a continuous training process can be started by using the initial sampling strategy and the active learning function, and the prediction accuracy is improved.

In this application, fig. 3 (a) shows a schematic fit of 5000 MCS samples and real LSS, and fig. 3 (b) shows sample points and corresponding fit results provided by the ARBF agent model. The results indicate that the ARBF proxy model has many sample points near the LSS that provide most of the information that constructs the proxy model, and thus enables good interpolation or fitting of 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 iterations is smaller than a preset convergence threshold, 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 comprehensive steps S1 to S7 are mainly divided into a preparation stage, an iteration stage and an output stage.

The preparation stage: (i) An initial sample point is generated in the U space by using an initial sample point strategy, and is transferred from the U space to the physical space X, and the real response of g (X) is determined by using the SRM. (ii) A reasonable convergence threshold epsilon is selected for the active learning process. (iii) Generating a reusable MCS pool to calculate P for each iteration _f,s And provides optimal sample points to enrich the training sample set S to update the ARBF agent model.

Iteration stage: the iteration task is mainly executed by three modules, including a numerical analysis module, a Monte Carlo module and an active learning and convergence judging module, and the three modules are interacted at this stage to generate a sequential process. The proxy model needs to perform iterative update of the proxy model according to candidate samples selected from the MCS pool T by the corresponding active learning function unless a convergence criterion is satisfied. The detailed interactions of these three modules are shown in fig. 4.

Output stage: stopping iteration after meeting the convergence condition, and selecting the failure probability calculated in the last iteration as the final estimation of the reliability of the slope system.

In addition, in a preferred embodiment of the present application, to verify the above 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 to S6, wherein:

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

To further illustrate the reliability analysis method of the present application, the present application uses three typical reference slopes as cases for verification analysis.

Since the shear modulus and the bulk modulus of the soil body have little influence on the FS of the slope, the values of the shear modulus and the bulk modulus are respectively assumed to be 30MPa and 100MPa in three cases; the random variables involved are considered independent uncorrelated. In addition, the application also applies a plurality of widely used polynomial expansion (PCE) based methods, such as a Quadratic Response Surface Method (QRSM) and a sparse PCE minimum angle regression method (SPCE-LAR), which are applied in the following three cases, and compared with the method of the application.

In order to verify the calculation accuracy of the reliability analysis method in the reliability analysis of the side slope system, 10000 Latin Hypercube Sampling (LHS) simulations are performed on the original LSF based on SRM directly in three cases. LHS provided system failure probability P _f,s Will be considered a "reference" or "precision" value. To measure the computational efficiency, the number of sample points required for each analysis (also the number of numerical analyses performed) is used. This is because when introducing a computationally intensive numerical method (such as FLAC as used in this application ^3D ) The computational effort required for the rest of the algorithm is typically negligible. Therefore, the number of sample points can be used as a general index of the actual problem calculation efficiency: the larger the sample size, the lower the efficiency.

Case one: single-layer slope

FIG. 5 shows the slope geometry of case one, the statistical information of soil parameters is shown in Table 1, lossThe probability of effectiveness is shown in Table 2. Table 2 shows the number of sample points and P using different reliability methods under 10000 LHS simulations _f,s As a result.

Table 1: case one soil parameter statistics

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

In the table above:

a represents that the certainty factor related to the model convergence condition is set to 0.99.

b represents the number of ne=numerical analyses.

c represents the average and 99.76% confidence interval.

d represents the relative error from the LHS average.

In order to obtain a reasonable finite difference grid to ensure efficiency and accuracy, fig. 6 shows a relationship between FS calculated under the parameter variable averaging condition and the number of cell bodies in the grid based on the reduced intensity method, from which 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 there is no significant trend of decreasing FS as the grid density increases. The optimal grid density and final finite difference grid for case one of the present application is shown in fig. 5 with an FS of 1.34.

FIG. 7 shows the probability of failure P of a case-by-case system using the ARBF proxy model at different sample points _f,s And reference is made to LHS results and a 99.76% confidence interval line. FIG. 8 shows that in case one, inSample point positions and classification conditions of 10000 times LHS in a two-dimensional U space and prediction conditions of actual LSS by an ARBF proxy model.

Case two: two-layer slope

Fig. 9 shows the slope geometry of case two, the soil parameter statistics are shown in table 3, and the failure probability is shown in table 4.

Table 3: soil parameter statistics for case two

In the table above:

a represents non-drainage shear strength;

b represents a coefficient of variation.

Table 4: failure probability of case two system obtained by different methods

In the table above:

a represents the number of ne=numerical analysis;

b represents the average and 99.76% confidence interval;

c represents the relative error with respect to the LHS average.

In table 3, the non-drainage shear strength of the two clay layers was considered as a random variable. The average value of random variables was used for analysis, and a case two optimal density finite difference grid of the present application is shown in fig. 9, with 3625 unit cells, corresponding to FS 1.926, and the optimal density can be determined by the fitted curve shown in fig. 6. FIG. 10 shows the prediction of probability of failure P for case two systems using the ARBF proxy model under different numbers of training samples _f,s And reference is made to LHS results and a 99.76% confidence interval line. FIG. 10 reflects that the ARBF proxy model converges as the number of training samples increases. FIG. 11 shows sample point location and classification of LHS and predicted condition of ARBF proxy model for actual LSS in 10000 times of two-dimensional U space in case twoThe condition is as follows.

Case three: three-layer slope

Fig. 12 shows the slope geometry of case three, the soil parameter statistics are shown in table 5, and the failure probability is shown in table 6.

Table 5: soil parameter statistics for case three

In the table above:

a represents a coefficient of variation.

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

In the table above:

a represents the number of ne=numerical analysis;

b represents the average and 99.76% confidence interval;

c represents the relative error from the LHS average.

Analysis was performed using the average value of the random variables, and case three of the present application had a final finite difference grid with an optimal density, as shown in fig. 12, with an FS of 1.36, which can be determined by the fitted curve shown in fig. 6. FIG. 13 shows the prediction of failure probability P of case three system using ARBF proxy model at different sample points _f,s And reference is made to LHS results and a 99.76% confidence interval line.

From the three cases, in the slope reliability analysis based on the SRM, the grid density has a significant effect on the FS result: as the grid density increases, the FS value gradually decreases, and the greater the grid density, the more stable the FS value tends to be. Therefore, the present application performs a sensitivity analysis to obtain the optimal lattice density for a given slope prior to performing an SRM-based reliability analysis. Furthermore, in slope reliability analysis, the use of non-uniform grids to increase computational efficiency may not be a sensible option, as differences in random variables may result in (deterministic) critical sliding surfaces of different shapes and positions.

For a single-layer slope, as in case one, its limit state surface G (u) =0, lss has a certain linearity, since the slope system is mainly controlled by a failure mode. However, for slopes containing multiple layers of soil, the LSS has a high degree of nonlinearity, since the failure probability of the slope system may be controlled by multiple failure modes simultaneously. For example, as shown in fig. 11, the LSS of case two may be approximated as a combination of two linear problems.

The results of the three cases show that the ARBF proxy model provided by the application can always well estimate the P of the slope system with multiple layers of soil and random variables _f,s . In cases one and three, their relative error with respect to LHS results was 3.35% and-1.64%, respectively; in case two, this value is about 10%. The main reasons for the relatively large errors of case two may be: when P _f,s When small (0.91%), LHS sample size (10000) is too small, resulting in P _f,s It is estimated to have a relatively broad 99.76% confidence interval (0.64% -1.18%). Nevertheless, the accuracy of the method proposed in the present application is generally better than other methods, especially for multi-layer soil slopes; for example, in case two, QRMS produces a larger relative error (-50.55%) compared to LHS results; SPCE-LAR predicted P _f,s Even away from the LHS reference value (relative error 178.02%).

In terms of calculation cost, for the three cases, the number of sample points required by the ARBF proxy model proposed by the application is generally smaller 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.

In this specification, each embodiment is described in a progressive manner, and each embodiment is mainly described by differences from other embodiments, and identical and similar parts between the embodiments are all enough to be referred to each other.

The method for calculating the failure probability of the slope system based on the radial basis function RBF provided by the application is described in detail, and specific examples are applied to the description of the principle and the implementation mode of the application, and the description of the above examples is only used for helping to understand the method and the core idea of the application; meanwhile, as those skilled in the art will have modifications in the specific embodiments and application scope in accordance with the ideas of the present application, the present description should not be construed as limiting the present application in view of the above.

Claims (5)

1. The side slope system failure probability calculation method based on the radial basis function RBF is characterized by comprising the following steps:

step S1: in a standard normal space, generating a training sample set of the slope system by utilizing an initial sampling point strategy;

step S2: converting sample points of the undetermined functional response G (u) in the training sample set from the standard normal space to a physical space, and calculating the G (u) corresponding to the sample points converted to the physical space by using an intensity reduction method;

step S3: training a radial basis function RBF proxy model in the standard normal space by utilizing the training sample set and G (u);

step S4: predicting the functional response of all sample points in a Monte Carlo simulation MCS (modulation and coding scheme) pool by using a trained RBF proxy model, calculating the failure probability of the current iteration according to the predicted functional response, and recording the failure probability in a preset matrix, wherein the preset matrix is an initialization matrix set in a preparation stage and is used for recording the failure probability of each iteration of a system;

step S5: judging whether the variation coefficient of the failure probability calculated in 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 in the last five iterations is not smaller than a preset convergence threshold, selecting an optimal sample point positioned in a standard normal space from the MCS pool by utilizing an active learning function in combination with a trained RBF proxy model, adding the optimal sample point into the training sample set, and repeating the steps S2-S6;

step S7: when the variation coefficient of the failure probability calculated by the last five iterations is smaller than a preset convergence threshold, taking the failure probability calculated by the last iteration in the preset matrix as a result of the reliability analysis of the slope system;

in step S1, the step of generating a training sample set of the slope system using an initial sampling point strategy in a standard normal space includes:

constructing an initial training sample set of the side slope system in a standard normal space 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:

the u has n-1 u equal to-3, the other u is equal to 0 or 3, and n represents the number of u in u; or n elements of said u are all the same, equal to-3, 0 or 3;

if the u meets the requirement, the u is reserved in the initial training sample set;

if the u does not meet, removing the u from the initial training sample set;

when the initial training sample set is judged, obtaining a training sample set S;

the step S2 includes: let the standard normal space be U and the physical space be X;

converting a sample point of the undetermined G (U) in the S from the U to X, and converting the sample point from U to X;

using a given linear function g (x), the functional response of said x is calculated:

g(x)＝FS(x)-1 (1)；

wherein FS is FLAC ^3D A stability coefficient calculated by an intensity reduction method embedded in the model;

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

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 a trained RBF proxy model

Instead of G (u) ⁽ⁱ⁾ ) Substituting the probability of failure of the current iteration into the following formula to calculate;

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in the above, N _SP Representing the number of sample points in the MCS pool.

2. The method according to claim 1, characterized in that in step S3, the step of training a radial basis function RBF proxy model in the standard normal space using the training sample set and G (u) comprises:

in the standard normal space, training an RBF proxy model by using the training sample set to obtain a proxy expression corresponding to G (u)

(2) Wherein u is _(i)S Representing one sample point of the ith simulation in the current S, N representing the number of sample points in the current S, ρ and b representing vectors of unknown coefficients ρ and b in the RBF proxy model, respectively, N representing uNumber of random variables, u _j Representing the j-th random variable in u, and ψ (·) represents the kernel function;

each sample point u is calculated using a linear kernel function ψ (a) =a _(i)S Substituting the unknown coefficients into the formula (2), and solving the unknown coefficients;

(3) Wherein the unknown coefficient of the n+1th term is determined by an orthogonal condition, ψ _ij Representing the distance value between the sample points of the i, j two simulations calculated using the ψ (·).

3. The method according to claim 1, wherein 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 a preset convergence threshold comprises:

standard deviation of failure probability calculated from last five iterations

Mean value->

Calculating the coefficient of variation

And judging whether the variation coefficient is smaller than a preset convergence threshold epsilon.

4. The method of claim 2, wherein in step S6, the most significant in the standard normal space is selected from the MCS pool using an active learning function in combination with the trained RBF proxy modelOptimal sample point u _c The method comprises the following steps:

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wherein u is _T Represents a sample point, d (u _T S) represents the u _T The minimum distance from the sample point in the current S, d (S) is the limit value of the target minimum distance, lambda is the scale factor, and lambda is more than or equal to 0.1 and less than or equal to 0.5.

5. The method according to claim 1, wherein the method further comprises:

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

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