Slope reliability design method and device based on conversion performance function gradient

Abstract

The invention provides a slope reliability design method and device based on a conversion performance function gradient, and relates to the field of reliability design of slope engineering.The method comprises the steps that a performance function expression is determined according to a design variable and a random variable, an original performance function is mapped to a U space, and a mapped performance function is obtained; the design variables are separated, a separated design variable expression is obtained, and a target performance function represented by the separated design variable expression is defined; determining a performance function gradient according to the target performance function, processing the iterative checking point by using the performance function gradient, and determining the coordinate of the next checking point; and determining the iterative checking point as a target design point until the distance between the coordinate of the iterative checking point and the coordinate of the next checking point is smaller than or equal to a preset tolerance, and determining a target design variable of the target reliability index according to the target design point. According to the method, a traditional trial and error method is abandoned, target design variables are rapidly obtained, and powerful support is provided for slope reliability design.

Description

Technical Field

[0001] The present invention relates to the field of reliability design of slope engineering, and particularly to a slope reliability design method and device based on the gradient of a transformed performance function. Background Art

[0002] The reliability analysis of slope engineering is to determine the failure probability or reliability index of slope engineering with known design parameters, which can be determined by traditional analysis methods such as the first-order second-moment method and the Monte Carlo method. Conversely, the reliability design of slope engineering is to determine design parameters based on the target failure probability or reliability index, which belongs to a typical inverse problem and is difficult to solve directly.

[0003] The traditional reliability design uses a trial-and-error method, that is, first assume a design parameter, and then use the traditional method to calculate the reliability to see if the corresponding reliability index is the same as the target value. Generally, multiple iterations are required to approach the target reliability index, and the calculation process is complex and the calculation efficiency is not high. In addition, it can also be solved by the inverse reliability method. This method requires setting the initial checking point and the trial values of the design parameters, calculating the gradients of the performance function with respect to the checking point and the design parameters, using them as the iteration directions, and performing iterative updates on the checking point and the design parameters through the set iteration supplement. When both the checking point and the design parameters no longer change, the design point and the design parameters can be obtained. The existing inverse reliability iteration method requires setting the iteration step size, and also requires iterating on both the checking point and the design parameters at the same time, and judging the convergence conditions of the two iterations, and the process is relatively complex.

[0004] In summary, there is currently no technical solution that can solve the above technical problems, and there is no slope reliability design method and device based on the gradient of a transformed performance function. Summary of the Invention

[0005] The present invention provides a slope reliability design method and device based on the gradient of a transformed performance function. Through the transformation and reconstruction of the performance function, the calculation formula of the performance function gradient is simplified, the iteration of design parameters can be avoided, and the working efficiency of slope reliability design is improved.

[0006] In a first aspect, the present invention provides a slope reliability design method based on the gradient of a transformed performance function, including:

[0007] Determine the expression of the performance function in the original parameter X space according to the design variables of the slope engineering and the random variables affecting the structural function. According to the relationship between the original parameter X space and the standard normal U space random vector, map the original performance function to the U space to obtain the mapped performance function;

[0008] Separate the design variables from the mapped functional function to obtain the separated design variable expression, which is used to characterize the design variable failure threshold that makes the functional function equal to 0, and define the target functional function represented by the separated design variable expression in the U space;

[0009] Select the mean value of the random variables as the initial checking point in the U space for iteration, and repeat the following steps:

[0010] For any iteration checking point, determine the functional function gradient of the iteration checking point according to the target functional function, and process the iteration checking point using the functional function gradient to determine the coordinates of the next checking point corresponding to the iteration checking point;

[0011] Until the distance between the coordinates of the iteration checking point and the coordinates of the next checking point is less than or equal to the preset tolerance, determine the iteration checking point as the target design point, determine the target design variables of the target reliability index according to the target design point, and generate an indication instruction according to the target design variables, where the indication instruction is used to guide the slope engineering design according to the target design variables to meet the requirements of slope reliability design.

[0012] According to the slope reliability design method based on the transformed functional function gradient provided by the present invention, the determination of the functional function expression in the original parameter X space according to the design variables of the slope engineering and the random variables affecting the structural function includes:

[0013] Determine the random vector θ of the random variables x 1 ,x 2 ,…,x m as θ = [x 1 ,x 2 ,…,x m ;

[0014] According to the design variables ξ of the slope engineering and the random vector θ, determine the functional function expression Z X = g(ξ, θ).

[0015] According to the slope reliability design method based on the transformed functional function gradient provided by the present invention, the standardized normal vector of the random vector θ is u, and the relationship between the random vector θ and the standardized normal vector u is θ = f(u);

[0016] The mapping of the original functional function to the U space according to the relationship between the original parameter X space and the random vector in the standard normal U space to obtain the mapped functional function includes:

[0017] Z U = g(ξ, f(u)) = g ′ (ξ, u)

[0018] wherein, Z U is the mapped functional function.

[0019] According to the slope reliability design method based on the gradient of the transformed functional function provided by the present invention, separating the design variables from the mapped functional function to obtain an expression of the separated design variables, including:

[0020] Let Z U be 0, separating the design variable ξ from the mapped functional function to obtain an expression of the separated design variable ξ = h(u), which is used to characterize the failure threshold of the design variable that makes the functional function equal to 0.

[0021] According to the slope reliability design method based on the gradient of the transformed functional function provided by the present invention, defining the objective functional function represented by the expression of the separated design variables in the U space, including:

[0022] When the design variable ξ is positively correlated with the original functional function, the objective functional function g ″ (ξ, u) = ξ - h(u);

[0023] When the design variable ξ is negatively correlated with the original functional function, the objective functional function g ″ (ξ, u) = h(u) - ξ.

[0024] According to the slope reliability design method based on the gradient of the transformed functional function provided by the present invention, determining the functional function gradient of the iterative check point according to the objective functional function, including:

[0025] When the design variable ξ is positively correlated with the original functional function, the functional function gradient of the iterative check point is

[0026] When the design variable ξ is negatively correlated with the original functional function, the functional function gradient of the iterative check point is

[0027] According to the slope reliability design method based on the gradient of the transformed functional function provided by the present invention, processing the iterative check point by using the functional function gradient to determine the coordinates of the next check point corresponding to the iterative check point, including:

[0028] For any iterative check point, the gradient of the objective functional function g″(ξ, u) in the U space at the iterative check point is expressed as: or Then the coordinates of the next check point corresponding to the iterative check point are expressed as:

[0029]

[0030] Or,

[0031] where β T is the target reliability index, is the gradient with respect to the standardized normal vector u in the U - space, and u k is the coordinate of the iterative checking point.

[0032] According to the slope reliability design method based on the gradient of the transformation performance function provided by the present invention, after determining the coordinates of the next checking point corresponding to the iterative checking point, the method further includes:

[0033] When the distance between the coordinates of the iterative checking point and the coordinates of the next checking point is greater than the preset tolerance, for the next checking point, determine the performance function gradient of the next checking point according to the target performance function, process the next checking point using the performance function gradient, and determine the coordinates of the next - again checking point corresponding to the next checking point;

[0034] Until the distance between the coordinates of the next checking point and the coordinates of the next - again checking point is less than or equal to the preset tolerance.

[0035] According to the slope reliability design method based on the gradient of the transformation performance function provided by the present invention, the slope reliability design includes the reliability design of the slope height and the slope reliability design without an explicit performance function expression.

[0036] In a second aspect, a slope reliability design device based on the gradient of the transformation performance function is provided, including:

[0037] A determination unit, which is used to determine the performance function expression in the original parameter X - space according to the design variables of the slope engineering and the random variables affecting the structural function, and map the original performance function to the U - space according to the relationship between the original parameter X - space and the standardized normal U - space random vector to obtain the mapped performance function;

[0038] A separation unit, which is used to separate the design variables from the mapped performance function to obtain a separated design variable expression, which is used to characterize the failure threshold of the design variables that makes the performance function equal to 0, and define a target performance function represented by the separated design variable expression in the U - space;

[0039] A repetition unit, which is used to select the mean value of the random variables as the initial checking point in the U - space for iteration, and repeat the following steps:

[0040] For any iterative checking point, determine the gradient of the performance function of the iterative checking point according to the target performance function, process the iterative checking point by using the gradient of the performance function, and determine the coordinates of the next checking point corresponding to the iterative checking point;

[0041] Until the distance between the coordinates of the iterative checking point and the coordinates of the next checking point is less than or equal to a preset tolerance, determine the iterative checking point as the target design point, determine the target design variables of the target reliability index according to the target design point, and generate an indication instruction according to the target design variables. The indication instruction is used to guide the design of the slope engineering according to the target design variables to meet the requirements of slope reliability design.

[0042] By mapping the original performance function from the original parameter X space to the standard normal U space and performing iterative calculations in this space, the present invention can more accurately capture the variation laws and interactions of various random variables affecting slope reliability. Using the gradient of the performance function to update the iterative checking point can gradually approach the target design point, thereby obtaining a more accurate target reliability index and target design variables, providing strong support for slope reliability design;

[0043] By accurately calculating the failure threshold of the design variables, it can ensure that the slope engineering has sufficient stability during the designed service life, reduce the safety risks caused by slope instability. Through the inverse reliability approximation method, the design variables can be further optimized to improve the overall stability and durability of the slope engineering; Using an iterative algorithm to update the checking point can gradually narrow the search range, reduce the calculation amount, and improve the design efficiency; By mapping the original performance function to the U space through standardization processing, the calculation process can be simplified, the calculation difficulty of slope reliability design can be reduced, the possibility of disasters such as landslides and collapses can be reduced through scientific design, over-conservative design can be avoided, resource allocation can be optimized, and data support can be provided for engineering code revision and insurance risk assessment. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] In order to more clearly illustrate the technical solutions in the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0045] Figure 1 is one of the flow diagrams of the slope reliability design method based on the transformed performance function gradient provided by the present invention;

[0046] Figure 2 is the second flow diagram of the slope reliability design method based on the transformed performance function gradient provided by the present invention;

[0047] Figure 3 It is the iterative schematic diagram provided by the present invention from the iterative check point to the next check point;

[0048] Figure 4 It is the schematic diagram provided by the present invention for determining the target design variables;

[0049] Figure 5 It is the slope schematic diagram provided by the present invention;

[0050] Figure 6 It is the slope numerical model schematic diagram provided by the present invention;

[0051] Figure 7 It is the structural schematic diagram of the slope reliability design device based on the gradient of the transformed performance function provided by the present invention. Detailed implementation manners

[0052] To make the objectives, technical solutions and advantages of the present invention clearer, the technical solutions in the present invention will be clearly and completely described below with reference to the accompanying drawings in the present invention. Obviously, the described embodiments are some but not all of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present invention without making creative efforts shall fall within the protection scope of the present invention.

[0053] The present invention discloses a reverse reliability approximation method for slope U - space design points based on the gradient of the transformed performance function. Its innovative points and core ideas include: proposing an equivalent transformation method for the performance function of separable design variables and proposing a single - loop design point approximation method for the U - space transformed performance function. Figure 1 It is one of the flow schematic diagrams of the slope reliability design method based on the gradient of the transformed performance function provided by the present invention. The slope reliability design method based on the gradient of the transformed performance function includes:

[0054] Step 101: Determine the expression of the performance function in the original parameter X - space according to the design variables of the slope project and the random variables affecting the structural function. According to the relationship between the original parameter X - space and the standard normal U - space random vector, map the original performance function to the U - space to obtain the mapped performance function;

[0055] Step 102: Separate the design variables from the mapped performance function to obtain the separated design variable expression, which is used to characterize the design variable failure threshold when the performance function is equal to 0, and define the target performance function expressed by the separated design variable expression in the U - space;

[0056] Step 103: Select the mean value of the random variables as the initial check point in the U - space for iteration, and repeat the following steps:

[0057] For any iterative checking point, determine the functional function gradient of the iterative checking point according to the target functional function, process the iterative checking point by using the functional function gradient, and determine the coordinates of the next checking point corresponding to the iterative checking point;

[0058] Until the distance between the coordinates of the iterative checking point and the coordinates of the next checking point is less than or equal to a preset tolerance, determine the iterative checking point as the target design point, determine the target design variables of the target reliability index according to the target design point, and generate an indication instruction according to the target design variables, where the indication instruction is used to guide the slope engineering design according to the target design variables to meet the requirements of slope reliability design.

[0059] In step 101, determining the functional function expression in the original parameter X space according to the design variables of the slope project and the random variables affecting the structural function includes:

[0060] Determine the random variables x 1 , x 2 , …, x m of the random vector θ = [x 1 , x 2 , …, x m ;

[0061] According to the design variables ξ of the slope project and the random vector θ, determine the functional function expression Z X = g(ξ, θ).

[0062] Optionally, the design variables of a slope engineering problem are ξ, and the random variables affecting the structural function are x 1 , x 2 , …, x m , where m is the number of random variables, and its vector form is θ = [x 1 , x 2 , …, x m , assuming that the functional function expression in the original parameter (X) space is: Z X = g(ξ, θ), and the standardized normal vector of the random vector θ is u, and the relationship between the two is θ = f(u).

[0063] Optionally, map the original functional function to the U space, define the original functional function, and establish the original functional function Z according to the design variables of the slope project (such as slope height H, inclination angle α) and random variables (such as soil unit weight γ, cohesion c, and internal friction angle X = g(ξ, θ), where: ξ is the design variable vector; θ is the random variable vector (subject to a specific probability distribution).

[0064] Then, perform the standardization of the random variable: convert the random variable θ into a standard normal variable u through a probability integral transformation (such as where is the cumulative distribution function (CDF) of θ, and Φ is the CDF of the standard normal distribution. i

[0065] Optionally, the standard normal vector of the random vector θ is u, and the relationship between the random vector θ and the standard normal vector u is θ = f(u);

[0066] According to the relationship between the original parameter X space and the random vector in the standard normal U space, map the original performance function to the U space to obtain the mapped performance function, including:

[0067] Z U = g(ξ, f(u)) = g ′ (ξ, u)

[0068] where Z U is the mapped performance function.

[0069] Finally, perform the mapping of the performance function, convert the original performance function Z X = g(ξ, θ) into the U - space form Z U = g ′ (ξ, u), ensuring that Z U = 0 corresponds to the slope failure boundary.

[0070] The present invention standardizes the random variable with a non - standard normal distribution into a standard normal distribution, which is convenient for subsequent calculation of the reliability index β T based on the standard normal distribution. By standardization, the complex distribution dependence of the random variable is reduced, providing a consistent basis for subsequent gradient solution.

[0071] In step 102, separate the design variables and define the target performance function. First, decouple the design variables, and separate the design variable ξ from the mapped performance function Z U = g ′ (ξ, u) so that it is only related to the random variable u through the failure threshold expression ξ = h(u).

[0072] Optionally, separate the design variable from the mapped performance function to obtain the expression of the separated design variable, including:

[0073] Let Z U be 0, and separate the design variable ξ from the mapped performance function to obtain the expression of the separated design variable ξ = h(u), which is used to characterize the failure threshold of the design variable when the performance function is equal to 0.

[0074] ​Optionally, the objective function defined in the U space and expressed by the separated design variable expression includes:

[0075] When the design variable ξ is positively correlated with the original objective function, the objective function g is adopted ″ (ξ, u) = ξ - h(u);

[0076] When the design variable ξ is negatively correlated with the original objective function, the objective function g is adopted ″ (ξ, u) = h(u) - ξ, and the original objective function and the transformed objective function are equivalent.

[0077] Optionally, in the U space, the objective function is h(u), and u that satisfies β needs to be found through iteration T such that the corresponding ξ = h(u d ) is the final design parameter. The design variables of the present invention no longer directly participate in the gradient calculation and are only indirectly associated through h(u), avoiding the coupled iteration of design variables and random variables in the traditional method. Only the random variables need to be adjusted in the iteration process, significantly reducing the computational complexity. d

[0078] Optionally, the function gradient of the iterative check point determined according to the objective function includes:

[0079] When the design variable ξ is positively correlated with the original objective function, the function gradient of the iterative check point is

[0080] When the design variable ξ is negatively correlated with the original objective function, the function gradient of the iterative check point is

[0081] Optionally, according to the definition of reliability, in the U space, the design point is at the intersection of the circle (for two random variables), sphere (for three random variables), or hypersphere (for four or more random variables) with the target reliability index as the radius and the limit state plane g ′ (ξ, u) = 0. Let the target reliability index of a certain engineering problem be β T , then the design point in the U space can be expressed as:

[0082]

[0083] In the formula, is the gradient of the objective function, is the modulus of the gradient vector. Since the performance function contains the design parameter ξ, when calculating the gradient of the performance function, the trial value of the design parameter ξ needs to be input. Therefore, in the iterative process of traditional inverse reliability analysis, it is necessary to adjust both the design parameter and the checking point simultaneously during the iterative process, and the design parameter can be obtained only when both converge. The process is complex and inefficient.

[0084] Therefore, the present invention maps the original performance function in the X space to the U space and redefines it according to the failure threshold of the design parameter. When the design variable ξ is positively correlated with the performance function, the gradient of the original performance function can be converted into the gradient of the newly defined performance function, that is: Similarly, when the design variable ξ is negatively correlated with the original performance function, the gradient of the original performance function can be converted into: Since the design parameter failure threshold expression h(u) does not contain the design variable ξ, the trial value of the design variable does not need to be input during the iterative process, nor is it necessary to solve the gradient of the performance function with respect to the design variable, which will greatly simplify the iterative process.

[0085] In step 103, the mean value of the random variable in the U space (i.e., the zero vector [0, 0,..., 0]) is taken as the initial checking point u 0 , because it corresponds to the highest probability density region and is usually close to the optimal solution. Take the derivative of the objective function h(u) to obtain the gradient The using the gradient of the performance function to process the iterative checking point and determining the coordinates of the next checking point corresponding to the iterative checking point includes:

[0086] For any iterative checking point, the gradient of the objective performance function g″(ξ, u) in the U space at the iterative checking point is expressed as: or Then the coordinates of the next checking point corresponding to the iterative checking point are expressed as:

[0087]

[0088] Or,

[0089] where β T is the target reliability index, is the gradient with respect to the standardized normal vector u in the U space, and u k are the coordinates of the iterative checking point.

[0090] By mapping the original performance function from the original parameter X space to the standard normal U space and performing iterative calculations within this space, the present invention can more accurately capture the variation laws and interactions of various random variables affecting the slope reliability. By using the gradient of the performance function to update the iterative checking points, the target design point can be gradually approximated, thereby obtaining a more accurate target reliability index and target design variables, providing strong support for slope reliability design;

[0091] By accurately calculating the failure thresholds of the design variables, it is possible to ensure that the slope project has sufficient stability within the designed service life, reducing the safety risks caused by slope instability. Through the inverse reliability approximation method, the design variables can be further optimized to improve the overall stability and durability of the slope project; Using an iterative algorithm to update the checking points can gradually narrow the search range, reduce the computational amount, and improve the design efficiency; By mapping the original performance function to the U space through standardization processing, the calculation process can be simplified and the computational difficulty of slope reliability design can be reduced.

[0092] Optionally, Figure 2 is the second schematic flow chart of the slope reliability design method based on the gradient of the transformed performance function provided by the present invention. After determining the coordinates of the next checking point corresponding to the iterative checking point, the method further includes:

[0093] When the distance between the coordinates of the iterative checking point and the coordinates of the next checking point is greater than the preset tolerance, for the next checking point, determine the gradient of the performance function of the next checking point according to the target performance function, and use the gradient of the performance function to process the next checking point to determine the coordinates of the next checking point corresponding to another checking point;

[0094] Until the distance between the coordinates of the next checking point and the coordinates of another checking point is less than or equal to the preset tolerance. For example, when the distance between adjacent checking points ||u k+1 -u k || ≤ ∈ (such as ∈ = 0.0005), stop the iteration. After the iteration converges, substitute ξ = h(u d ) to obtain the target design variables (such as H = h(u d ).

[0095] As Figure 2 shown, the reliability design process based on the gradient of the transformed performance function is as follows:

[0096] Step 1: Transform the performance function, map the original performance function g(ξ, θ) in the X space to the performance function g ′ (ξ, u) in the U space, and according to the design parameter failure threshold expression ξ = h(u), define the new performance function as g ″ (ξ, u) = ξ - h(u), or g″ (ξ, u) = h(u) - ξ。

[0097] Step 2: Select the initial checking point u 0 , and select the mean value of the random variables as the initial checking point in the U space, i.e., u 0 = [u 1 , u 2 …, u m = [0, 0, …, 0], where m is the number of random samples.

[0098] Step 3: Calculate the gradient of the transformation performance function at each checking point When taking the partial derivative with respect to the random variables, the design variable ξ in the transformation performance function g ″ (ξ, u) = ξ - h(u) or g ″ (ξ, u) = h(u) - ξ can be regarded as a constant. At this time, the gradient of the transformation performance function is equal to or

[0099] Step 4: Determine the new checking point u k+1 , and the i-th component of the new checking point can be calculated as u i = β T cosω i .

[0100] Step 5: Judge whether the convergence condition is satisfied. Judge whether the convergence condition ||u k+1 - u k ‖ ≤ ∈ is satisfied, where ∈ is a preset tolerance, such as 0.0005. Otherwise, repeat Steps 3 - 5 until the convergence condition is satisfied.

[0101] Step 6: Determine the design parameters at the design point. When the convergence condition is satisfied, the checking point u k (or u k+1 ) is approximately the design point u d , and at this time, the design parameters that satisfy the target reliability index can be determined as ξ = h(u d ).

[0102] Figure 3 is the iteration schematic diagram from the iterative checking point to the next checking point provided by the present invention, Figure 4 is the schematic diagram for determining the target design variables provided by the present invention. The iteration schematic diagram from the checking point u k to the checking point u k+1 is as shown in Figure 3 . For the checking point u k , its gradient vector (L1) is in the opposite direction to the radius vector (L2). Therefore, there is:

[0103]

[0104] In the formula, the subscript i represents the relevant variable of the i-th component, and the new checking point u k+1 has coordinates:

[0105]

[0106] When the distance between u k+1 and u k is less than a small tolerance (such as 0.001), it indicates that the checking point on the circle is tangent to the performance function. At this time, the checking point is the design point u d , and the design parameter can be calculated as ξ = h(u d ), as shown in Figure 4 .

[0107] The present invention iterates the U-space variables in a single loop, avoiding the computational redundancy caused by traditional multi-variable coupled iteration. By adjusting the gradient direction through the target reliability index, it ensures that the iteration path directly faces the optimal solution. The present invention realizes the high efficiency and automation of slope inverse reliability design through standardized transformation, variable decoupling, and gradient-driven iteration, solves the problems of complex iteration, strong coupling, and insufficient accuracy in traditional methods, and has significant engineering practical value.

[0108] Optionally, the slope reliability design includes the reliability design of the slope height and the slope reliability design without an explicit performance function expression.

[0109] As the first preferred embodiment of the present invention, Figure 5 is the slope schematic diagram provided by the present invention, which discloses the reliability design of a slope height H. The unit weight γ, cohesion c, and internal friction angle of the soil mass are random variables, and their mean values and standard deviations are as shown in Figure 5 . In the figure, θ and ψ are the dip angle of the slip surface and the slope angle respectively. The performance function of the slope is:

[0110]

[0111] Existing research shows that when the slope height is H = 10m, its reliability index is β = 3.4249. In this embodiment, β T is set to 3.4249, and the convergence condition is set as ||u k+1 -u k || ≤ 0.0005, and inverse reliability iteration is performed by the method of the present invention. According to the method of the present invention, the failure threshold expression of the design parameter H in the U-space is obtained as:

[0112]

[0113] The gradient of the transformed performance function can be expressed as It can be directly calculated through the built-in difference function "diff()" of the Sympy module in the python program. The data generated during the iteration process is shown in Table 1. For example, for the first checking point, the first component of the gradient is The first coordinate of the next checking point can be updated to u 1 = β T cosω i = 3.4249×0.402 = 1.377.

[0114] When iterating to the 6th step, the convergence condition ||u k+1 - u k || ≤ 0.0005 is satisfied. The checking point at this time is the design point that meets the target reliability index of β T = 3.4249, that is, u d = [1.451, -2.105, -2.279]. The design value of the slope height can be calculated as H * = h(u d ) = 9.968m. This design value is in good agreement with H = 10m, proving that the method of the present invention has high precision.

[0115] Table 1 Iteration data

[0116] Iterative step <![CDATA[u 1 > <![CDATA[u 2 > <![CDATA[u 3 > <![CDATA[cosω 1 > <![CDATA[cosω 2 > <![CDATA[cosω 3 > <![CDATA[||u k+1 -u k ||]]> H / m 1 0 0 0 0.402 -0.402 -0.823 / 2 1.377 -1.377 -2.817 0.455 -0.600 -0.658 0.8997 3 1.558 -2.055 -2.254 0.421 -0.612 -0.669 0.1295 4 1.441 -2.097 -2.293 0.425 -0.615 -0.665 0.0042 5 1.454 -2.105 -2.277 0.424 -0.615 -0.665 0.0008 6 1.451 -2.105 -2.279 0.0002 9.968

[0117] As the second embodiment of the present invention, Figure 6 is a schematic diagram of the slope numerical model provided by the present invention, which discloses a slope reliability design without a display function expression. When the slope height has been determined, according to the target reliability index, the inclination angle α of the slope is designed. As Figure 6 shown, where the distance from the left side of the model to the slope toe is 20m, the distance from the right side of the model to the slope toe is 5m, the height of the slope is 5m, and the unit weight of the soil is ρ = 1764kg·m -3 , the cohesion and internal friction angle of the soil follow a lognormal distribution, where the mean value is μ c = 9.8kPa, the standard deviation is σ v = 9.8kPa and Assume that the distance from the left side of the model to the slope top is x. Then the horizontal distance from the slope top to the slope toe is 20 - x. When the slope height is known, the slope toe α and x are in one-to-one correspondence and positively correlated. Therefore, the size of the inclination angle α can be designed by designing the distance x. Existing research shows that when x = 10, the probability of slope instability is 7.17%, that is, the reliability index is β = 1.4632.

[0118] Assume that x is unknown, and the target reliability index is set to βT = 1.4632, and the method of the present invention is used to determine the magnitude of x. Since there is no display function in this embodiment, the numerical difference method is used to solve the gradient of the function, and its expression is:

[0119]

[0120] where Δu i represents a small increment of the i-th component u i , for example, -0.1, ξ and respectively represent the failure thresholds of the design parameters at the checking points u = [u 1 ,…u i ,…] and u Δi = [u 1 ,…(u i + Δu i ),…]. These failure thresholds are determined by numerical iteration method.

[0121] The steps for reliability design using the method of the present invention are as follows: First, the initial checking point in the U space is selected as the mean point, that is, u = [0, 0]. At this point, the failure threshold of x is determined to be x = 15.12 m by numerical iteration method. Let the increments of the two random variables be both Δu i = -0.1, then the initial checking points can be changed to u Δ1 = [-0.1, 0] and u Δ2 = [0, -0.1] respectively. At these two points, the failure thresholds of x are respectively and 15.00 m. According to these three failure thresholds, the gradient of the function in the U space can be calculated, that is, the partial derivatives in two directions are:

[0122]

[0123] and

[0124]

[0125] Then, the coefficients cosω 1 and cosω 2 are -0.913 and -0.406 respectively. Thus, the checking point is updated to obtain the checking point of the second iteration step. Repeat the above steps. When the distance between two adjacent checking points is less than the preset tolerance, the iteration can be stopped. The data during the iteration process is shown in Table 2. It can be seen that the final design value of x converges to x = 10 m, which is consistent with the expected result, indicating that the present invention is accurate.

[0126] Table 2 Iteration data

[0127]

[0128]

[0129] Through the transformation and reconstruction of the functional function, the present invention simplifies the calculation formula of the gradient of the functional function, avoids the iteration of design parameters, is simpler and more efficient compared with the traditional inverse reliability iteration method, and uses the built-in difference function "diff()" of the Sympy module in the python program to calculate the gradient of the transformed functional function, greatly simplifying the calculation process of the gradient of complex functional functions in the standard normal space.

[0130] Traditional inverse reliability design requires synchronous iteration of the design variable ξ and the random variable θ, resulting in high computational complexity and being prone to falling into local optima. Through variable decoupling, the present invention only needs to perform a single-loop iteration of the random variable u in the U space, without repeatedly adjusting the design variable ξ, and the iteration efficiency is significantly improved. After standardizing the random variables, the reliability index can be directly calculated using the standard normal distribution theory, avoiding the complex integral operations of non-normal distributions. Traditional trial-and-error methods or numerical iteration methods rely on manual experience to select step sizes, with slow convergence speeds and uncontrollable accuracies. The gradient direction of the present invention directly points to the design point in the U space corresponding to the target reliability index, ensuring the shortest iteration path and fast convergence speed.

[0131] Combining the first embodiment and the second embodiment, the present invention can realize engineering application value. The high precision (error < 0.3%) of the present invention method under explicit functional functions is verified by using the slope height H design, meeting the actual engineering requirements. Using the slope inclination angle α design demonstrates the support ability for implicit functional functions and can be seamlessly connected to numerical simulation tools.

[0132] Traditional methods need to rely on numerical simulations (such as limit equilibrium analysis) for repeated trial calculations, which are time-consuming and prone to missing the global optimal solution. Through gradient-driven iterative optimization, it directly approximates the failure boundary in the probability space (U space), significantly reducing the computational amount. For example, in the optimization of slope support parameters, the design cycle is effectively shortened; although Monte Carlo simulation can evaluate the failure probability, the computational cost for high-dimensional problems is extremely high. The present invention combines gradient information and response surface technology to quickly construct an approximate model, balancing accuracy and efficiency. For example, in the risk analysis of rainfall-induced landslides, sensitive parameters (such as permeability coefficient, groundwater level) can be accurately identified. In the design of mountain highway slopes, the spacing of retaining piles can be automatically adjusted to meet specific reliability requirements. In slope support design, parameters such as the length of anchor cables and the density of steel mesh can be determined, reducing the landslide failure probability and finding the optimal balance among cost, safety, and construction feasibility, assisting engineers in formulating economic and reasonable design strategies. The present invention reduces the possibility of disasters such as landslides and collapses through scientific design, avoids overly conservative designs, optimizes resource allocation, and provides data support for engineering code revisions and insurance risk assessments.

[0133] Figure 7 It is a structural schematic diagram of a slope reliability design device based on the gradient of the transformation function. The slope reliability design device based on the gradient of the transformation function includes:

[0134] A determination unit 1, which is used to determine the functional function expression in the original parameter X space according to the design variables of the slope project and the random variables affecting the structural function, and map the original functional function to the U space according to the relationship between the original parameter X space and the standard normal U space random vector to obtain the mapped functional function;

[0135] A separation unit 2, which is used to separate the design variables from the mapped functional function to obtain the separated design variable expression, which is used to characterize the failure threshold of the design variables that make the functional function equal to 0, and define the target functional function represented by the separated design variable expression in the U space;

[0136] A repetition unit 3, which is used to select the mean value of the random variables as the initial checking point for iteration in the U space, and repeat the following steps:

[0137] For any iteration checking point, determine the functional function gradient of the iteration checking point according to the target functional function, process the iteration checking point by using the functional function gradient, and determine the coordinates of the next checking point corresponding to the iteration checking point;

[0138] Until the distance between the coordinates of the iteration checking point and the coordinates of the next checking point is less than or equal to the preset tolerance, determine the iteration checking point as the target design point, determine the target design variables of the target reliability index according to the target design point, and generate an instruction according to the target design variables. The instruction is used to guide the slope project design according to the target design variables to meet the requirements of slope reliability design.

[0139] By mapping the original functional function from the original parameter X space to the standard normal U space and performing iterative calculations in this space, the present invention can more accurately capture the variation laws and interactions of various random variables affecting the slope reliability. Using the gradient of the functional function to update the iteration checking point can gradually approach the target design point, thereby obtaining a more accurate target reliability index and target design variables, providing strong support for slope reliability design;

[0140] By accurately calculating the failure thresholds of design variables, the slope engineering can be ensured to have sufficient stability within the designed service life, reducing the safety risks caused by slope instability. Through the inverse reliability approximation method, the design variables can be further optimized to improve the overall stability and durability of the slope engineering. The iterative algorithm is used to update the checking points, which can gradually narrow the search range, reduce the computational amount, and improve the design efficiency. By mapping the original performance function to the U space through standardization processing, the calculation process can be simplified and the computational difficulty of slope reliability design can be reduced.

[0141] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed to multiple network units. Some or all of the modules can be selected according to actual needs to achieve the purpose of the solution of this embodiment. Those of ordinary skill in the art can understand and implement it without creative efforts.

[0142] Through the description of the above embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus a necessary general hardware platform, and of course also by hardware. Based on this understanding, the above technical solutions, in essence, or the part that contributes to the prior art can be embodied in the form of a software product, which can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., including several instructions to enable a computer device (which can be a personal computer, server, or network device, etc.) to execute the methods described in each embodiment or some parts of the embodiments.

[0143] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A slope reliability design method based on the gradient of the transfer function, characterized in that: include: According to the design variables of the slope engineering and the random variables that affect the structural function, the functional function expression of the original parameter X space is determined, and according to the relationship between the original parameter X space and the standard normal U space random vector, the original functional function is mapped to the U space to obtain the mapped functional function; Separating the design variable from the mapped performance function to obtain a separated design variable expression for characterizing a design variable failure threshold that makes the performance function equal to 0, and defining a target performance function represented by the separated design variable expression in U space; Select the mean of the random variable in the U space as the initial verification point for iteration, and repeat the following steps: For any iterative calculation point, determine the performance function gradient of the iterative calculation point according to the target performance function, process the iterative calculation point using the performance function gradient, and determine the coordinates of the next calculation point corresponding to the iterative calculation point; Until the distance between the coordinates of the iterative verification point and the coordinates of the next verification point is less than or equal to the preset tolerance, the iterative verification point is determined as the target design point, the target design variable of the target reliability index is determined according to the target design point, and an indication instruction is generated according to the target design variable. The indication instruction is used to guide the slope engineering design according to the target design variable to meet the requirements of slope reliability design.

2. The slope reliability design method based on the transfer function gradient according to claim 1 is characterized in that: The functional function expression of the original parameter X space is determined according to the design variables of the slope engineering and the random variables affecting the structural function, including: Determine the random variables x1, x2, …, x that affect the structure function m The random vector θ=[x1,x2,…,x m ]; According to the design variable ξ of the slope engineering and the random vector θ, the functional expression Z of the original parameter X space is determined: X =g(ξ,θ).

3. The slope reliability design method based on the transfer function gradient according to claim 2 is characterized in that: The standard normalized vector of the random vector θ is u, and the relationship between the random vector θ and the standard normalized vector u is θ=f(u); The method maps the original function to the U space according to the relationship between the original parameter X space and the standard normal U space random vector to obtain the mapped function, including: Z U =g(ξ,f(u))=g ′ (ξ,u) Among them, z U It is the function after mapping.

4. The slope reliability design method based on the transfer function gradient according to claim 3 is characterized in that: The design variables are separated from the mapped functional function to obtain separated design variable expressions, including: Order Z U is 0, the design variable ξ is separated from the mapped performance function, and a separated design variable expression ξ=h(u) is obtained, which is used to characterize the design variable failure threshold that makes the performance function equal to 0.

5. The slope reliability design method based on the transfer function gradient according to claim 4 is characterized in that: The objective performance function represented by the separated design variable expression is defined in the U space, including: When the design variable ξ is positively correlated with the original performance function, the objective performance function g″(ξ,u)=ξ-h(u) is adopted; When the design variable ξ is negatively correlated with the original performance function, the objective performance function g″(ξ,u)=h(u)-ξ is adopted.

6. The slope reliability design method based on the transfer function gradient according to claim 5 is characterized in that: Determining the performance function gradient of the iterative verification point according to the objective performance function includes: When the design variable ξ is positively correlated with the original performance function, the performance function gradient of the iterative verification point is When the design variable ξ is negatively correlated with the original performance function, the performance function gradient of the iterative verification point is 7. The slope reliability design method based on the transfer function gradient according to claim 6 is characterized in that: The step of processing the iterative calculation point by using the performance function gradient to determine the coordinates of the next calculation point corresponding to the iterative calculation point includes: For any iterative verification point, the gradient of the objective performance function g″(ξ,u) in U space at the iterative verification point is expressed as: or Then the coordinates of the next verification point corresponding to the iterative verification point are expressed as: or, Among them, β T As a reliable indicator of the target, To find the gradient of the standard normalized vector u in U space, u k are the coordinates of the iterative verification points.

8. The slope reliability design method based on the transfer function gradient according to claim 1 is characterized in that: After determining the coordinates of the next verification point corresponding to the iterative verification point, the method further includes: In the case where the distance between the coordinates of the iterative verification point and the coordinates of the next verification point is greater than the preset tolerance, for the next verification point, the performance function gradient of the next verification point is determined according to the objective performance function, the next verification point is processed using the performance function gradient, and the coordinates of another verification point corresponding to the next verification point are determined; Until the distance between the coordinates of the next verification point and the coordinates of yet another verification point is less than or equal to the preset tolerance.

9. The slope reliability design method based on the transfer function gradient according to claim 1 is characterized in that: The slope reliability design includes the reliability design of the slope height and the reliability design of the slope without display function expression.

10. A slope reliability design device based on the gradient of the transfer function, characterized in that: include: A determination unit, the determination unit is used to determine the functional function expression of the original parameter X space according to the design variables of the slope engineering and the random variables affecting the structural function, and map the original functional function to the U space according to the relationship between the original parameter X space and the standard normal U space random vector to obtain the mapped functional function; A separation unit, the separation unit is used to separate the design variable from the mapped performance function, obtain a separated design variable expression, which is used to characterize a design variable failure threshold that makes the performance function equal to 0, and define a target performance function represented by the separated design variable expression in U space; A repeating unit is used to select the mean of the random variable in the U space as the initial verification point for iteration, and repeat the following steps: For any iterative calculation point, determine the performance function gradient of the iterative calculation point according to the target performance function, process the iterative calculation point using the performance function gradient, and determine the coordinates of the next calculation point corresponding to the iterative calculation point; Until the distance between the coordinates of the iterative verification point and the coordinates of the next verification point is less than or equal to the preset tolerance, the iterative verification point is determined as the target design point, the target design variable of the target reliability index is determined according to the target design point, and an indication instruction is generated according to the target design variable. The indication instruction is used to guide the slope engineering design according to the target design variable to meet the requirements of slope reliability design.

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