CN111444649B - Slope system reliability analysis method based on intensity reduction method - Google Patents

Abstract

The application provides a slope system reliability analysis method based on intensity reduction, wherein an intensity reduction SRM is adopted to evaluate a stability coefficient, an initial sampling strategy and an active learning function are adopted, active learning agent models ASVM, ARBF and AK of an original limit state function LSF are constructed, monte Carlo simulation MCS and the active learning agent models are combined to evaluate the failure probability of a slope system, the influence of random variables and related parameters on the slope stability can be quantized, 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

Reliability analysis method of slope system based on strength reduction method

technical field

The present application relates to the field of soil slope stability analysis, in particular to a slope system reliability analysis method based on the strength reduction method.

Background technique

Slope stability evaluation is a complex geotechnical engineering problem, and its input parameters are uncertain. The traditional deterministic analysis method using the stability factor (FS) may not truly reflect the safety of the slope. To quantify the effects of uncertainty, probabilistic methods are widely used in slope reliability analysis.

A slope may be damaged along different sliding surfaces, and the failure of any sliding surface will cause the failure of the slope, thus forming a series of system problems. Accurate and effective reliability analysis of such complex problems is the main problem faced by the application of probabilistic methods in geotechnical engineering practice.

The direct simulation method is one of the probability methods, such as Monte Carlo simulation (MCS) and importance sampling (IS), which can make unbiased estimation of the failure probability P _f,s of the slope system, but at present, most scholars use limit equilibrium. The reliability analysis is carried out using the LEM method. When this method is combined with the MCS, it needs to search for the critical slip surface with the minimum FS in each simulation, so the amount of calculation is very large. A more critical issue is that LEM mainly uses randomly generated slip surfaces, which may miss critical slip surfaces and thus provide a biased estimate of P _f,s .

To effectively combine LEM with MCS, another common approach is to identify some typical sliding surfaces (RSSs) that contribute the most to P _f,s , and then, considering the correlation between different RSSs, P _f,s can be easily calculated. In the prior art, RSSs are identified by randomly generating a large number of potential sliding surfaces. However, a challenge for this RSSs-based method is how to choose a reasonable threshold for the correlation coefficient between RSSs to achieve computational efficiency and accuracy.

To improve computational efficiency, surrogate models are widely used together with MCS. Many general and advanced surrogate models are used for slope reliability analysis, such as Gaussian process regression method, swarm intelligence support vector machine method and multivariate adaptive regression spline method. LEM is often chosen as a deterministic analysis method to evaluate the FS of slopes. The advantages of LEM are its simplicity and low computational cost, but its main disadvantage is that it is difficult to locate when the critical slip surface is not known in advance; The test sliding surface is usually assumed to be circular, which may not be suitable for complex slope systems, especially when the slope has weak interlayers.

Therefore, it is still necessary to develop a reliable and efficient method for reliability analysis of slope system.

SUMMARY OF THE INVENTION

The present application provides a method for analyzing the reliability of a slope system based on a strength reduction method to overcome the above technical problems.

In order to solve the above-mentioned problems, the present application discloses a slope system reliability analysis method based on the strength reduction method, including:

Step S1: in the standard normal space, use the initial sampling point strategy to generate the training sample set of the slope system;

Step S2: Convert the sample points of the undetermined functional response G(u) in the training sample set from the standard normal space to the physical space, and use the intensity reduction method to calculate the corresponding G(u) of the sample points converted to the physical space. u);

Step S3: In the standard normal space, use the training sample set and G(u) to train a surrogate model;

Step S4: using the trained surrogate model to predict the functional response of all sample points in the Monte Carlo simulation MCS pool, and 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 coefficient of variation of the failure probability calculated by the last five iterations is less than a preset convergence threshold;

Step S6: when the coefficient of variation of the failure probability calculated in the last five iterations is not less than the preset convergence threshold, use the active learning function in combination with the trained surrogate model, and select from the MCS pool to be located in the standard normal space. The optimal sample point is added to the training sample set, and steps S2 to S6 are repeated;

Step S7: When the coefficient of variation of the failure probability calculated by the last five iterations is less than the preset convergence threshold, the failure probability calculated by the last iteration in the preset matrix is used as the result of the reliability analysis of the slope system.

Further, in step S1, in the standard normal space, the step of using the initial sampling point strategy to generate the training sample set of the slope system includes:

In the standard normal space, the 3-σ rule is used to construct the initial training sample set of the slope system; the initial training sample set includes a plurality of sample points u, where u represents the random variable u in the standard normal space the vector;

For each u in the initial training sample set, determine whether the u satisfies any of the following conditions:

The u has n-1 u equal to -3, another u equal to 0 or 3, the n represents the number of u in u; or the n elements of the u are all the same, all equal to -3, 0 or 3;

If the u is satisfied, then keep the u in the initial training sample set;

If the u is not satisfied, remove the u from the initial training sample set;

When the initial training sample set is judged, the training sample set S is obtained.

Further, in step S2, the sample points of the undetermined functional response G(u) in the training sample set are converted from the standard normal space to the physical space, and the intensity reduction method is used to calculate the samples converted to the physical space. The steps of the point corresponding to G(u) include:

Let the standard normal space be U and the physical space be X;

After the sample points of the undetermined G(u) in the S are converted from the U to X, the sample points are converted from u to x;

Using a given linear function g(x), compute the functional response of said x:

g(x)=FS(x)-1(1);

where FS is the stability factor calculated using the strength reduction method embedded in FLAC ^3D ;

The corresponding G(u) can be obtained by formula (1), which satisfies:

g(x)=G(u) (2).

Further, in step S3, when the surrogate model is a support vector machine SVM surrogate model, in the standard normal space, using the training sample set and G(u), the step of training the surrogate model includes:

满足

的样本点的向量位于一侧，满足

的样本点位于另一侧；In the standard normal space, the SVM surrogate model is trained by using the training sample set; wherein, a sample point of the ith simulation in the current S is

Satisfy

The vector of sample points of is on one side, satisfying

The sample point of is on the other side;

For the current S, use the SVM surrogate model to search for the optimal classification hyperplane H(u):

的分类符号，表示正或负；In the above formula, w and e represent unknown parameters, w ^T represents the transpose of the w matrix, and y _i is

The classification symbol of , indicating positive or negative;

Calculate the distance vector V(u) from all sample points in the current S to the H(u):

为支持向量，表示当前S中距离H(u)最小的样本点；N_SV为

的数目；ω_i通过公式(4)优化求解获得，表示第i个样本点的权重系数；

表示

矩阵的转置；In formula (5),

is the support vector, representing the sample point with the smallest distance H(u) in the current S; N _SV is

The number of ; ω _i is obtained by the optimization solution of formula (4), which represents the weight coefficient of the ith sample point;

express

transpose of a matrix;

According to whether the classification sign of V(u) is positive or negative, the classification status of each sample point in the current S is determined.

Further, in step S3, when the surrogate model is a Kriging surrogate model, in the standard normal space, using the training sample set and G(u), the step of training the surrogate model includes:

In the standard normal space, use the training sample set to train the Kriging surrogate model to obtain the surrogate expression corresponding to G(u)

In formula (6), L(u) represents the function representing the trend of G(u) obtained from regression analysis, and z(u) is an assumed stationary Gaussian process; when L(u)=0, the ith simulation The covariance between the sample point u ⁽ⁱ⁾ of and the sample point u ^(j) of the jth simulation is:

表示过程方差，R(·)为高斯核函数，表示为：In formula (7),

represents the process variance, R( ) is a Gaussian kernel function, expressed as:

及其对应的

结合最大似然估计来获得。In formula (8), θ represents the vector of unknown coefficients θ; among them, θ and σ _z pass through all points in the current sample set S

and its corresponding

Combined with maximum likelihood estimation to obtain.

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

In the standard normal space, use the training sample set to train the RBF surrogate model, and obtain the surrogate expression corresponding to G(u)

表示当前S中第i次模拟的一个样本点，N表示当前S 中的样本点数目，ρ、b分别表示所述RBF代理模型中未知系数ρ、b的向量， n表示u中随机变量的个数，u_j表示u中第j个随机变量，Ψ(·)表示核函数；In formula (9),

represents a sample point of the ith simulation in the current S, N represents the number of sample points in the current S, ρ and b represent the vectors of unknown coefficients ρ and b in the RBF surrogate model, respectively, and n represents the number of random variables in u number, u _j represents the jth random variable in u, Ψ( ) represents the kernel function;

代入(9)式，求解所述未知系数；Using the linear kernel function Ψ(a)=a, each sample point is

Substitute into formula (9) to solve the unknown coefficient;

In the above formula, the unknown coefficient of the n+1th term is determined by the orthogonal condition, and Ψ _ij represents the distance value between the sample points of i and j simulated twice by using the Ψ(·).

Further, in step S4, the steps of using the trained surrogate model to predict the functional responses of all sample points in the Monte Carlo simulation MCS pool, and calculating the failure probability of the current iteration according to the predicted functional responses include:

In the above formula, N _SP represents the number of sample points in the MCS pool;

When the surrogate model is an SVM surrogate model, use V(u ⁽ⁱ⁾ ) obtained from the trained SVM surrogate model to replace G(u ⁽ⁱ⁾ ), and substitute it into equations (11) and (12) for calculation to obtain the current iteration of probability of failure;

代替G(u⁽ⁱ⁾)，代入(11)(12)式计算，获得当前迭代的失效概率。When the surrogate model is an RBF surrogate model or a Kriging surrogate model, use the

Substitute G(u ⁽ⁱ⁾ ) into equations (11) and (12) for calculation to obtain the failure probability of the current iteration.

Further, in step S5, the step of judging whether the coefficient of variation of the failure probability calculated by the last five iterations is less than the preset convergence threshold includes:

和平均值

计算变异系数

Standard deviation of failure probabilities calculated from the last five iterations

and average

Calculate the coefficient of variation

It is judged whether the coefficient of variation is smaller than a preset convergence threshold ε.

Further, in the step S6, the step of selecting the optimal sample point _uc located in the standard normal space from the MCS pool by using the active learning function in combination with the trained surrogate model includes:

When the surrogate model is the Kriging surrogate model, the calculation formula of the _uc includes:

表示通过所述Kriging 代理模型

预测的标准差；where u _T represents the sample points in the MCS pool T in U space,

represented by the Kriging proxy model

the standard deviation of the forecast;

When the surrogate model is the RBF surrogate model, the calculation formula of the _uc includes:

Among them, u _T represents a sample point in the MCS pool, d(u _T , S) represents the minimum distance between the u _T and the sample point in the current S, d(S) is the limit of the minimum distance of the target, λ is the scale factor, 0.1≤λ≤0.5;

代替

代入(15)、(16)式计算u_c。When the proxy model is the SVM proxy model, use

replace

Substitute into equations (15) and (16) to calculate u _c .

Further, the method also includes:

The explicit highly nonlinear function g(x)' is introduced as a test, and steps S2 to S6 are verified, wherein:

Compared with the prior art, the present application includes the following advantages:

This application proposes a slope system reliability analysis method based on SRM, which can automatically identify the sliding surface of any shape in the soil slope. It is no longer necessary to identify the critical sliding surface like LEM. For layered slopes with complex geometric shapes It is more convenient to perform reliability analysis;

This application adopts the initial sampling point strategy, combined with the active learning function, develops three active learning surrogate models of ASVM, ARBF and AK, constructs the surrogate model of the original limit state function LSF, and combines the MCS and the active learning surrogate model. Evaluating 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.

Description of drawings

Fig. 1 is the step flow chart of a kind of slope system reliability analysis method based on strength reduction method of the present application;

Figure 2(a) is a schematic diagram of the sample point positions generated by the traditional 3-σ rule;

Figure 2(b) is a schematic diagram of the sample point positions generated by the improved 3-σ rule of the present application;

Figure 3(a) is the fitting diagram of 5000 MCS samples and the real LSS;

Figure 3(b) is a schematic diagram of the fitting performance of the AK model;

Figure 3(c) is a schematic diagram of the fitting performance of ARBF;

Figure 3(d) is a schematic diagram of the fitting performance of ASVM;

Fig. 4 is the flow chart that the application adopts active learning proxy model and strength reduction method to carry out the reliability analysis of slope system;

Figure 5 is a schematic diagram of the slope geometry of Case 1;

6 is a schematic diagram of the relationship between FS and grid density calculated for the three cases of the present application;

Figure 7 is the failure probability prediction diagram of Case 1;

Figure 8 is a schematic diagram of the fitting performance between the LHS samples and different surrogate models in Case 1;

Figure 9 is a schematic diagram of the slope geometry of Case 2;

Figure 10 is the failure probability prediction diagram of Case 2;

Figure 11 is a schematic diagram of the fitting performance between the LHS samples and different surrogate models in Case 2;

Figure 12 is a schematic diagram of the slope geometry of Case 3;

Fig. 13 is the failure probability prediction graph of Case 3.

Detailed ways

In order to make the above objects, features and advantages of the present application more clearly understood, the present application will be described in further detail below with reference to the accompanying drawings and specific embodiments.

Referring to FIG. 1 , a flow chart of the steps of a method for analyzing the reliability of a slope system based on the strength reduction method of the present application is shown, which may specifically include the following steps:

Step S1: in the standard normal space, use the initial sampling point strategy to generate the training sample set of the slope system;

Reasonable selection of initial sample points can speed up the convergence of the training process. The initial training sample set can be constructed with Latin Hypercube Sampling (LHS), but this may not work for some models with low failure probability, since it must contain two classes of points (eg, G(u)>0 and G(u) <0). The traditional 3-σ can serve this purpose well, because it can roughly reflect the general trend of G(u) in the whole sampling space, and contains two kinds of points. However, this method requires about 3 ⁿ initial sample points, where n is the number of random variables; therefore, it may not be suitable for problems with many random variables (e.g., a problem with 10 random variables requires 59049 initial sample points , which is clearly unacceptable in practice).

Therefore, this application proposes an improved 3-σ rule, the basic idea of which is to balance the number of points in the two regions of instability and non-instability, and to speed up the training process for the separation of the safety domain and the failure domain. To this end, the sampling range of each random variable is regarded as [-3, 3] in an uncorrelated standard normal space (also called U-space), and step S1 can include the following sub-steps:

Sub-step 1-1: In the standard normal space, use the 3-σ rule to construct the initial training sample set of the slope system; the initial training sample set includes a plurality of sample points u, where u represents the random variable u. vector;

Sub-step 1-2: For each u in the initial training sample set, determine whether the u satisfies any of the following conditions:

The u has n-1 u equal to -3, another u equal to 0 or 3, the n represents the number of u in u; or the n u of the u are all the same, all equal to -3,0 or 3;

If the u is satisfied, then keep the u in the initial training sample set;

Sub-step 1-3: if the u is satisfied, keep the u in the initial training sample set; if the u is not satisfied, remove the u from the initial training sample set, when the After the initial training sample set is judged, the training sample set S is obtained.

In this application, a sample point u includes multiple random variables, such as u ₁ , u ₂ , . . . , u _n . Among them, when u ₁ =u ₂ =...= _un =-3, the stability coefficient FS is the smallest, that is, the point is the most dangerous point (MDP) in the training sample set S.

The training sample set S obtained by the initial sampling point strategy in this application finally generates 2n+3 initial sample points. Compared with the traditional 3-σ, when the slope system has a large number of random variables, the number of initial sample points can be greatly reduced. Fig. 2 shows the situation of sample points generated in U-space by the traditional 3-σ rule and the improved 3-σ rule of the present application when three random variables are considered. Among them, Fig. 2(a) shows a schematic diagram of the sample point position generated by the traditional 3-σ rule; Fig. 2(b) shows a schematic diagram of the sample point position generated by the improved 3-σ rule of the present application.

Step S2: Convert the sample points of the undetermined functional response G(u) in the training sample set from the standard normal space to the physical space, and use the intensity reduction method to calculate the corresponding G(u) of the sample points converted to the physical space. u);

The reliability analysis method can quantify the influence of random variables and their related parameters on slope stability. Let the standard normal space be U and the physical space be X;

After converting the sample points of the undetermined G(u) in the S from the U space to the X space, the sample points are converted from u to x, and x represents a vector of random variables in the X space;

Using a given linear function g(x), compute the functional response of said x:

g(x)=FS(x)-1(1);

where FS is the stability coefficient calculated using the intensity reduction method embedded in FLAC ^3D (referred to as SRM hereinafter);

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

In the prior art, if formula (1) is used to directly calculate the failure probability of the 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. But since g(x) is implicit, it is difficult to directly calculate the integral in the above formula. Therefore, this application transforms the vector x into the u of the sample points in the uncorrelated standard normal space, so that the limit state surface can be rewritten as G(u)=0, G(u) is g(x) in the uncorrelated A mapping in the standard normal space U.

时，一个模型需要大约10⁴次模拟，对于耗时的可靠性分析(如使用FLAC^3D和SRM进行的分析)来说，计算量难以接受。已有技术中，MCS也有几种变体，例如LHS、 IS和子集模拟(SS)，它们可以一定程度上减少计算的P_f,s的变异系数，从而减少所需的模拟次数。但目前许多土木工程项目的目标P_f,s在10^-3到10^-5之间，这进一步增加了所需的计算工作量，使得即使是这些MCS变体难以满足计算量的要求。After the above transformation, according to the traditional MCS can provide an unbiased estimate of the failure probability P _f,s . But in this method, when P _f,s =10 ^-2 and the coefficient of variation of MCS

, a model requires about ¹⁰⁴ simulations, which is computationally unacceptable for time-consuming reliability analyses such as those performed with FLAC ^3D and SRM. In the prior art, there are also several variants of MCS, such as LHS, IS, and subset simulation (SS), which can reduce the coefficient of variation of the calculated P _f,s to a certain extent, thereby reducing the number of simulations required. But many civil engineering projects currently target P _f,s between 10 ^-3 and 10 ^-5 , which further increases the required computational effort, making it difficult for even these MCS variants to meet the computational demands.

用代理模型来代替原来的FLAC^3D和 SRM分析，从而大大提高了效率，降低计算成本。这样，利用MCS或其变体可以快速给出失效概率计算结果。Therefore, this application proposes a Monte Carlo simulation based on a surrogate model, which can use a small number of sample points to construct an explicit prediction model for G(u).

Replacing the original FLAC ^3D and SRM analysis with surrogate models greatly improves efficiency and reduces computational costs. In this way, the use of MCS or its variants can quickly give failure probability calculations.

Step S3: In the standard normal space, use the training sample set and G(u) to train a surrogate model;

In this application, three methods of support vector machine (SVM), Kriging and radial basis function (RBF) are used to establish three surrogate models of failure probability of slope system, which are SVM surrogate model respectively, The Kriging proxy model and the RBF proxy model are as follows:

1. SVM proxy model:

When the surrogate model is a SVM surrogate model, in the standard normal space, using the training sample set and G(u), the steps of training the surrogate model include:

满足

的样本点的向量位于一侧，满足

的样本点位于另一侧；In the standard normal space, the SVM surrogate model is trained by using the training sample set; wherein, a sample point of the ith simulation in the current S is

Satisfy

The vector of sample points of is on one side, satisfying

The sample point of is on the other side;

For the current S, use the SVM surrogate model to search for the optimal classification hyperplane H(u):

的分类符号，表示正或负；In the above formula, w and e represent unknown parameters, w ^T represents the transpose of the w matrix, and y _i is

The classification symbol of , indicating positive or negative;

Calculate the distance vector V(u) from all sample points in the current S to the H(u):

为支持向量，表示当前S中距离H(u)最小的样本点；N_SV为

的数目；ω_i通过公式(4)优化求解获得，表示第i个样本点的权重系数；

表示

矩阵的转置；In formula (5),

is the support vector, representing the sample point with the smallest distance H(u) in the current S; N _SV is

The number of ; ω _i is obtained by the optimization solution of formula (4), which represents the weight coefficient of the ith sample point;

express

transpose of a matrix;

According to whether the classification sign of V(u) is positive or negative, the classification status of each sample point in the current S is determined.

替换为高斯核函数

It should be noted that the above solution is a linear classification hyperplane. To obtain a nonlinear classification hyperplane, the kernel function can be

Replace with Gaussian kernel function

2. Kriging proxy model:

Kriging is another powerful tool for building surrogate models, in particular, an interpolation method based on statistical assumptions. In step S3, when the surrogate model is a SVM surrogate model, in the standard normal space, using the training sample set and G(u), the steps of training the surrogate model include:

In the standard normal space, use the training sample set to train the Kriging surrogate model to obtain the surrogate expression corresponding to G(u)

In formula (6), L(u) represents the function representing the trend of G(u) obtained from regression analysis, and z(u) is an assumed stationary Gaussian process; when L(u)=0, the ith simulation The covariance between the sample point u ⁽ⁱ⁾ of and the sample point u ^(j) of the jth simulation is:

表示过程方差，R(·)为高斯核函数，表示为：In formula (7),

represents the process variance, R( ) is a Gaussian kernel function, expressed as:

及其对应的

结合最大似然估计来获得。In formula (8), θ represents the vector of unknown coefficients θ; among them, θ and σ _z pass through all points in the current sample set S

and its corresponding

Combined with maximum likelihood estimation to obtain.

3. RBF proxy model:

RBF is another accurate interpolation method. Its advantage is that it is easy to implement due to its simple construction process. , when the surrogate model is a radial basis function RBF surrogate model, in the standard normal space, using the training sample set and G(u), the steps of training the surrogate model include:

In the standard normal space, use the training sample set to train the RBF surrogate model, and obtain the surrogate expression corresponding to G(u)

表示当前S中第i次模拟的一个样本点，N表示当前S中的样本点数目，ρ、b分别表示所述RBF代理模型中未知系数ρ、b的向量， n表示u中随机变量的个数，u_j表示u中第j个随机变量，Ψ(·)表示核函数；In formula (9),

Represents a sample point of the ith simulation in the current S, N represents the number of sample points in the current S, ρ and b represent the vectors of the unknown coefficients ρ and b in the RBF surrogate model, respectively, and n represents the number of random variables in u. number, u _j represents the jth random variable in u, Ψ( ) represents the kernel function;

代入(9)式，求解所述未知系数；Using the linear kernel function Ψ(a)=a, each sample point is

Substitute into formula (9) to solve the unknown coefficient;

In the above formula, the unknown coefficient of the n+1th term is determined by the orthogonal condition, and Ψ _ij represents the distance value between the sample points of i and j simulated twice by using the Ψ(·).

The above three proxy models are all optional solutions of the present application, but are not limited thereto. In specific implementation, any surrogate model can be trained for calculation.

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

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

In this application, the failure probability calculation formula is as follows:

In the above formula, N _SP represents the number of sample points in the MCS pool;

When the surrogate model is an SVM surrogate model, use V(u ⁽ⁱ⁾ ) obtained from the trained SVM surrogate model to replace G(u ⁽ⁱ⁾ ), and substitute it into equations (11) and (12) for calculation to obtain the current iteration of probability of failure;

代替G(u⁽ⁱ⁾)，代入(11)(12)式计算，获得当前迭代的失效概率。上述预设矩阵可在准备阶段设置，初始化一个矩阵来记录系统每一次迭代的失效概率。迭代是指代理模型构建过程中连续使用不同的样本点计算确定性模型。When the surrogate model is an RBF surrogate model or a Kriging surrogate model, use the

Substitute G(u ⁽ⁱ⁾ ) into equations (11) and (12) for calculation to obtain the failure probability of the current iteration. The above preset matrix can be set in the preparation stage, and a matrix is initialized to record the failure probability of each iteration of the system. Iteration refers to the continuous use of different sample points to calculate the deterministic model during the surrogate model building process.

Step S5: judging whether the coefficient of variation of the failure probability calculated by the last five iterations is less than a preset convergence threshold;

A reasonable convergence criterion should stop the training process in time, thereby reducing the required training sample set when the current surrogate model is stable. There are usually two convergence criteria: (i) the sample points near the hyperplane limit state surface (LSS) are sufficiently dense, or (ii) the fluctuation of the predicted P _f,s is sufficiently small. This application applies the same guidelines. Therefore, step S5 may specifically include:

和平均值

计算变异系数

Standard deviation of failure probabilities calculated from the last five iterations

and average

Calculate the coefficient of variation

It is judged whether the coefficient of variation is smaller than a preset convergence threshold ε; in this application, preferably ε=0.001.

Step S6: when the coefficient of variation of the failure probability calculated in the last five iterations is not less than the preset convergence threshold, use the active learning function in combination with the trained surrogate model, and select from the MCS pool to be located in the standard normal space. The optimal sample point is added to the training sample set, and steps S2 to S6 are repeated;

In this application, first, a large sample pool T (eg: 200,000 points) is generated using MCS, and the setting of the learning function is closely related to the actual analysis purpose, aiming to rank a set of candidate points in the MCS pool T. In the slope reliability analysis of this application, the learning function needs to give an optimal sample point in each iteration process to update the current surrogate model. This optimal sample point should satisfy two conditions at the same time: (i ) which is located near the limit state surface (LSS) and (ii) avoids redundant information (i.e., far from the existing sample points in S).

Therefore, in the step S6, using the active learning function in combination with the trained surrogate model, the step of selecting the optimal sample point _uc located in the standard normal space from the MCS pool includes:

When the surrogate model is the Kriging surrogate model, the calculation formula of the _uc includes:

表示通过所述Kriging 代理模型

预测的标准差；where u _T represents the sample points in the MCS pool T in U space,

represented by the Kriging proxy model

the standard deviation of the forecast;

When the surrogate model is the RBF surrogate model, the calculation formula of the _uc includes:

Among them, u _T represents a sample point in the MCS pool, d(u _T , S) represents the minimum distance between the u _T and the sample point in the current S, d(S) is the limit of the minimum distance of the target, λ is the scale factor, 0.1≤λ≤0.5;

代替

代入(15)、 (16)式计算u_c，本申请将λ进一步优选为0.2。When the proxy model is the SVM proxy model, use

replace

Substitute into equations (15) and (16) to calculate u _c , and in the present application, λ is more preferably 0.2.

The present application establishes an active learning surrogate model, namely the ASMs surrogate model, through the above formula. Specifically, for the three surrogate models listed, three ASMs surrogate models, namely active learning kriging AK, active learning support vector machine ASVM and active learning radial basis function ARBF, are established respectively. It should be emphasized that the active learning technology of this application does not randomly select new sample points to increase the training sample set S, but starts from a small number of sample points in S, the purpose is to add targeted candidates one by one during the training process. samples (optimal sample points) to enrich S. The optimal sample point selected from the MCS pool T according to the active learning function. Once a candidate sample is selected from the MCS pool T, it is added to the training sample set S to update the corresponding ASMs surrogate model, thereby utilizing the initial sampling strategy and active learning function, a continuous training process can be started, improving the Prediction accuracy.

In this application, Fig. 3 shows the fitting performance of different ASMs for highly nonlinear functions, in which Fig. 3(a) shows the fitting diagram of 5000 MCS samples and the real LSS, Fig. 3(b) to Figure 3(d) shows the sample points provided by different ASMs (i.e., AK, ARBF, and ASVM) and the corresponding fitting results. The results show that the three ASMs have many sample points near the LSS, which provide most of the information for constructing the surrogate model and thus enable good interpolation or fitting to the entire limited sample space with a small number of sample points .

Step S7: When the coefficient of variation of the failure probability calculated by the last five iterations is less than the preset convergence threshold, the failure probability calculated by the last iteration in the preset matrix is used as the result of the reliability analysis of the slope system.

Combining steps S1 to S7, the present application is mainly divided into a preparation stage, an iterative stage and an output stage.

Preliminary stage: (i) Use the initial sampling point strategy to generate initial sample points in U space, transfer them from U space to physical space X, and use SRM to determine the true response of g(x). (ii) Choose a reasonable convergence threshold ε for the active learning process. (iii) Generate a reusable MCS pool to compute P _f,s for each iteration and provide optimal sample points to enrich the training sample set S to update the ASMs surrogate model.

Iterative stage: Iterative tasks are mainly performed by three modules, including numerical analysis module, Monte Carlo module, active learning and convergence discrimination module, which work interactively in this stage to generate sequential processes. The surrogate model needs to be iteratively updated with candidate samples picked from the MCS pool T according to the corresponding active learning function, unless the convergence criteria are met. The detailed interaction of these three modules is shown in Figure 4.

Output stage: Stop the iteration after meeting the convergence conditions, and select the failure probability calculated by the last iteration as the final estimate of the reliability of the slope system.

In addition, in a preferred embodiment of the present application, in order to verify the above-mentioned convergence criterion and the surrogate model proposed by the present application, the following steps are also included:

The explicit highly nonlinear function g(x)' is introduced as a test, and steps S2 to S6 are verified, wherein:

The results show that the three surrogate models listed in this application can all establish surrogate models of actual functional response G(u) in U space based on a small number of training samples.

In order to further illustrate the reliability analysis method of the present application, three typical reference slopes are used as cases for verification analysis.

Since the shear modulus and bulk modulus of the soil have little effect on the FS of the slope, their values are assumed to be 30 MPa and 100 MPa, respectively, in the three cases; the random variables involved are considered to be independent and uncorrelated. In addition, this application also applies some widely used polynomial expansion (PCE) based methods, such as quadratic response surface method (QRSM) and sparse PCE least angle regression (SPCE-LAR) in the following three cases, and A comparative study was carried out with the method of the present application.

In order to verify the calculation accuracy of the reliability analysis method of the present application in the reliability analysis of the slope system, in three cases, directly based on the SRM, 10,000 Latin hypercube sampling (LHS) samples were performed on the original limit state function (LSF). )simulation. The probability of system failure, P _f,s, provided by the LHS will be considered as a "reference" or "exact" value. To measure computational efficiency, the number of sample points required for each analysis (also the number of numerical analyses performed) is used. This is because when computationally expensive numerical methods (such as FLAC ^3D used in this application) are introduced, the computational effort required by other parts of the algorithm is often negligible. Therefore, the number of sample points can be used as a general indicator of computational efficiency for practical problems: the larger the sample size, the lower the efficiency.

Case 1: Single-story slope

Figure 5 shows the slope geometry of case 1, the soil parameter statistics are shown in Table 1, and the failure probability is shown in Table 2. Table 2 shows the number of sample points and P _f,s results using different reliability methods under 10,000 LHS simulations.

Table 1: Soil parameter statistics for Case 1

Table 2: Failure probabilities of case 1 systems obtained by different methods

In the table above:

a indicates that the coefficient of certainty associated with the model convergence condition is set to 0.99.

b represents NE = number of numerical analyses.

c represents mean and 99.76% confidence interval.

d represents the relative error from the LHS mean.

It should be noted that, in order to obtain a reasonable finite-difference mesh to ensure efficiency and accuracy, Figure 6 shows the difference between the calculated FS and the number of cells in the mesh based on the reduced strength method under the condition that the parameter variables are averaged. It is observed that FS is a monotonically decreasing function of grid density. As shown in Fig. 6, the optimal density point can be selected to determine the grid density, after this point, the FS does not tend to decrease significantly as the grid density increases. The optimal mesh density and final finite difference mesh of Case 1 of this application are shown in Figure 5, and its FS is 1.34.

Figure 7 shows the predicted failure probability P _f,s of the case-1 system with different ASMs under different sample points, and uses the LHS results and the 99.76% confidence interval line as a reference. Figure 8 shows the location and classification of sample points of 10,000 LHSs in the two-dimensional U space in Case 1, and the predictions of actual LSSs by different surrogate models.

Case 2: Two-story slope

Figure 9 shows the slope geometry of Case 2, 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 2

In the table above:

a represents the undrained shear strength;

b represents the coefficient of variation.

Table 4: Failure probabilities of the case 2 system obtained by different methods

In the table above:

a represents NE = the number of numerical analyses;

B stands for mean and 99.76% confidence interval;

C represents the relative error with respect to the LHS mean.

In Table 3, the undrained shear strengths of the two clay layers are considered as random variables. Using the average value of random variables for analysis, the finite difference grid of the optimal density of Case 2 of this application is shown in Figure 9. There are 3625 cells, and the corresponding FS is 1.926. The optimal density can be obtained through the simulation shown in Figure 6. The fitting curve is determined. Figure 10 shows the prediction of failure probability P _f,s of the case 2 system using different ASMs under the condition of different training sample points, and takes the LHS results and the 99.76% confidence interval line as a reference. Figure 10 reflects the convergence of the three ASMs as the number of training samples increases. Figure 11 shows the sample point location and classification of 10,000 LHSs in the two-dimensional U space in Case 2, and the predictions of actual LSSs by different surrogate models.

Case 3: Three-story slope

Figure 12 shows the slope geometry of Case 3, the soil parameter statistics are shown in Table 5, and the failure probability is shown in Table 6.

Table 5: Statistics of soil parameters for Case 3

In the table above:

a represents the coefficient of variation.

Table 6: Failure probabilities of case three systems obtained by different methods

In the table above:

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 mean.

Using the average value of random variables for analysis, the final finite difference grid with optimal density in case three of this application is shown in Figure 12, and its FS is 1.36. The optimal density can be determined by the fitting curve shown in Figure 6. Fig. 13 shows the prediction of failure probability P _f,s of the case three system with different ASMs under different sample points, and takes the LHS results and the 99.76% confidence interval line as a reference.

It can be seen from the above three cases that in the SRM-based slope reliability analysis, the grid density has a significant impact on the FS results: as the grid density increases, the FS value gradually decreases, and the greater the grid density, the higher the FS value. more stable. Therefore, a sensitivity analysis should be performed to obtain the optimal mesh density for a given slope before performing an SRM-based reliability analysis. Furthermore, in slope reliability analysis, using a non-uniform grid to improve computational efficiency may not be a wise choice, since differences in random variables may produce (deterministic) critical slip surfaces of different shapes and locations.

For a single-layer slope, such as case 1, the limit state surface G(u)=0, the LSS has a certain linearity, because the slope system is mainly controlled by a failure mode. However, for slopes containing multiple layers of soil, the LSS is highly nonlinear because the failure probability of the slope system may be controlled by multiple failure modes at the same time. For example, as shown in Figure 11, the LSS of case two can be approximated as a combination of two linear problems.

The results of the above three cases show that the ASVM, AK, and ARBF methods proposed in this application are always able to estimate P _f,s well for slope systems with multiple layers of soil and random variables. In cases one and three, their relative errors relative to the LHS results were within 5%; in case two, the value was about 10%. The main reason for the relatively large error in case two may be: when P _f,s is small (0.91%), the LHS sample size (10000) is too small, resulting in a relatively wide 99.76% confidence interval for the P _f,s estimate (0.64%～1.18%). Nevertheless, the accuracy of the method proposed in this application is generally better than other methods, especially for slopes with multiple layers of soil; for example, in case two, the QRMS results in a large relative error (-50.55%) compared to the LHS results ; P _f,s predicted by SPCE-LAR is even far from the LHS reference value (relative error 178.02%).

In terms of computational cost, for the above three cases, the number of sample points required by the ASMs proposed in this application is usually less than 100, which is generally considered to be computationally feasible in engineering practice. Compared with the existing methods, it greatly reduces The amount of calculation is improved, and the calculation efficiency is improved. Among them, the ARBF surrogate model with a linear kernel function (without unknown parameters in the kernel function) outperforms the ASVM and AK surrogate models in terms of model stability (see Figures 7, 10, and 13), and requires fewer sample points. On the one hand, the introduction of an active learning function greatly speeds up the model training; on the other hand, a concise surrogate model (using fewer additional parameters, such as unknown parameters in the kernel function) may be more stable during the iterative analysis process. The ASVM surrogate model, although using the same learning function as ARBF, can converge to a sufficient value of P _f,s , but requires the most sample points and has a larger fluctuation range. The performance of the widely used AK model is between ARBF and ASVM.

The various embodiments in this specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments, and the same and similar parts between the various embodiments may be referred to each other.

The reliability analysis method of the slope system based on the strength reduction method provided by the present application has been introduced in detail above. In this article, specific examples are used to illustrate the principle and implementation of the present application. The descriptions of the above embodiments are only for the purpose of Help to understand the method of the present application and its core idea; meanwhile, for those of ordinary skill in the art, according to the idea of the present application, there will be changes in the specific implementation and application scope. In summary, the content of this specification It should not be construed as a limitation of this application.

Claims (7)

1. the slope system reliability analysis method based on strength reduction method, is characterized in that, comprises: Step S1: in the standard normal space, use the initial sampling point strategy to generate the training sample set of the slope system; Step S2: Convert the sample points of the undetermined functional response G(u) in the training sample set from the standard normal space to the physical space, and use the intensity reduction method to calculate the corresponding G(u) of the sample points converted to the physical space. u); Step S3: In the standard normal space, use the training sample set and G(u) to train a surrogate model, where the surrogate model is a support vector machine SVM surrogate model, a Kriging surrogate model or a radial basis function RBF surrogate Model; Step S4: using the trained surrogate model to predict the functional response of all sample points in the Monte Carlo simulation MCS pool, and 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 coefficient of variation of the failure probability calculated by the last five iterations is less than a preset convergence threshold; Step S6: when the coefficient of variation of the failure probability calculated in the last five iterations is not less than the preset convergence threshold, use the active learning function in combination with the trained surrogate model, and select from the MCS pool to be located in the standard normal space. The optimal sample point is added to the training sample set, and steps S2 to S6 are repeated; Step S7: when the coefficient of variation of the failure probability calculated by the last five iterations is less than a preset convergence threshold, the failure probability calculated by the last iteration in the preset matrix is used as the result of the reliability analysis of the slope system; In step S1, in the standard normal space, the steps of using the initial sampling point strategy to generate the training sample set of the slope system include: In the standard normal space, the 3-σ rule is used to construct the initial training sample set of the slope system; the initial training sample set includes a plurality of sample points u, where u represents the random variable u in the standard normal space the vector; For each u in the initial training sample set, determine whether the u satisfies any of the following conditions: The u has n-1 u equal to -3, another u equal to 0 or 3, the n represents the number of u in u; or the n elements of the u are all the same, all equal to -3, 0 or 3; If the u is satisfied, then keep the u in the initial training sample set; If the u is not satisfied, remove the u from the initial training sample set; When the initial training sample set is judged, obtain the training sample set S; In step S4, the steps of using the trained surrogate model to predict the functional responses of all sample points in the Monte Carlo simulation MCS pool, and calculating the failure probability of the current iteration according to the predicted functional responses include:

In the above formula, N _SP represents the number of sample points in the MCS pool; When the surrogate model is an SVM surrogate model, use V(u ⁽ⁱ⁾ ) obtained from the trained SVM surrogate model to replace G(u ⁽ⁱ⁾ ), and substitute it into equations (11) and (12) for calculation to obtain the current iteration of probability of failure; When the surrogate model is an RBF surrogate model or a Kriging surrogate model, use the

Substitute G(u ⁽ⁱ⁾ ) into (11) (12) for calculation to obtain the failure probability of the current iteration; In the step S6, using the active learning function in combination with the trained surrogate model, the step of selecting the optimal sample point _uc located in the standard normal space from the MCS pool includes: When the surrogate model is the Kriging surrogate model, the calculation formula of the _uc includes:

where u _T represents the sample points in the MCS pool T in U space,

represented by the Kriging proxy model

the standard deviation of the forecast; When the surrogate model is the RBF surrogate model, the calculation formula of the _uc includes:

Among them, u _T represents a sample point in the MCS pool, d(u _T , S) represents the minimum distance between the u _T and the sample point in the current S, d(S) is the limit of the minimum distance of the target, λ is the scale factor, 0.1≤λ≤0.5; When the proxy model is the SVM proxy model, use

replace

Substitute into equations (15) and (16) to calculate u _c . 2. The method according to claim 1, wherein in step S2, the sample points of the undetermined functional response G(u) in the training sample set are converted from the standard normal space to the physical space, and The steps of calculating G(u) corresponding to the sample point converted to physical space by using the intensity reduction method include: Let the standard normal space be U and the physical space be X; After converting the sample points of the undetermined G(u) in the S from the U to X, the sample points are converted from u to x; Using a given linear function g(x), compute the functional response of said x: g(x)=FS(x)-1(1); where FS is the stability factor calculated using the strength reduction method embedded in FLAC ^3D ; The corresponding G(u) can be obtained by formula (1), which satisfies: g(x)=G(u) (2). 3. The method according to claim 2, wherein in step S3, when the surrogate model is a support vector machine (SVM) surrogate model, in the standard normal space, the training sample set and G(u), the steps of training the surrogate model include: In the standard normal space, use the training sample set to train the SVM surrogate model; wherein, a sample point u _(i)S of the i-th simulation in the current S satisfies G(u _(i)S ) The vector of sample points > 0 is on one side, and the sample points satisfying G(u _(j)S )<0 are on the other side; For the current S, use the SVM surrogate model to search for the optimal classification hyperplane H(u):

In the above formula, w and e represent unknown parameters, w ^T represents the transpose of the w matrix, y _i is the classification symbol of G(u _(i)S ), indicating positive or negative; Calculate the distance vector V(u) from all sample points in the current S to the H(u):

In formula (5),

is the support vector, representing the sample point with the smallest distance H(u) in the current S; N _SV is

The number of ; ω _i is obtained by the optimization solution of formula (4), which represents the weight coefficient of the ith sample point;

express

transpose of a matrix; According to whether the classification sign of V(u) is positive or negative, the classification status of each sample point in the current S is determined. 4. The method according to claim 2, wherein in step S3, when the surrogate model is a Kriging surrogate model, in the standard normal space, the training sample set and G(u ), the steps to train the surrogate model include: In the standard normal space, use the training sample set to train the Kriging surrogate model to obtain the surrogate expression corresponding to G(u)

In formula (6), L(u) represents the function representing the trend of G(u) obtained from regression analysis, and z(u) is an assumed stationary Gaussian process; when L(u)=0, the ith simulation The covariance between the sample point u ⁽ⁱ⁾ of and the sample point u ^(j) of the jth simulation is:

In formula (7), σ _2z represents the process variance, and R( ) is the Gaussian kernel function, which is expressed as:

In formula (8), θ represents the vector of unknown coefficients θ; among them, θ and σ _z pass through all points u _(i)S in the current sample set S and their corresponding G(u _(i)S ), combined with the maximum likelihood Estimated to obtain. 5. The method according to claim 2, wherein, in step S3, when the surrogate model is a radial basis function (RBF) surrogate model, in the standard normal space, the training sample set is used and G(u), the steps to train the surrogate model include: In the standard normal space, use the training sample set to train the RBF surrogate model, and obtain the surrogate expression corresponding to G(u)

In formula (9), u _(i)S represents a sample point of the i-th simulation in the current S, N represents the number of sample points in the current S, and ρ and b respectively represent the unknown coefficients ρ and b in the RBF surrogate model. The vector of , n represents the number of random variables in u, u _j represents the jth random variable in u, and Ψ( ) represents the kernel function; Adopt linear kernel function Ψ(a)=a, substitute each sample point u _(i)S into (9) formula, solve described unknown coefficient;

In the above formula, the unknown coefficient of the n+1th term is determined by the orthogonal condition, and Ψ _ij represents the distance value between the sample points of i and j simulated twice by using the Ψ(·). 6. The method according to claim 1, wherein, in step S5, the step of judging whether the coefficient of variation of the failure probability calculated by the last five iterations is less than a preset convergence threshold comprises: Standard deviation of failure probabilities calculated from the last five iterations

and mean

Calculate the coefficient of variation

It is judged whether the coefficient of variation is smaller than a preset convergence threshold ε. 7. The method according to claim 2, wherein the method further comprises: The explicit highly nonlinear function g(x)' is introduced as a test, and steps S2 to S6 are verified, wherein:

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