A dimensionality reduction search method for slope sliding surface based on Laplace equation

Abstract

The present invention discloses a method for reducing the dimension of the slope sliding surface search based on the Laplace equation, which includes the following steps: solving a seepage velocity field according to the seepage tensor and the Laplace equation contained in the stress; integrating the streamline according to the velocity field; calculating the safety factor of each streamline according to the stress state on the streamline, and the streamline with the minimum safety factor is the dangerous sliding surface. By solving the anisotropic Laplace equation, the pure optimization problem is transformed into a boundary value problem, and the two-dimensional slope stability optimization problem is transformed into a one-dimensional problem, improving the efficiency of the critical slip line search. The present invention is based on the finite element elastic stress field and the closed virtual seepage field, neither requiring iterative work nor making assumptions about the shape of the slip line and the inter-slice force, capable of reducing the dimension for optimization, greatly reducing the workload of searching for the dangerous slip arc; abandoning the assumption of the sliding surface shape and being based on the finite element elastic stress field, and abandoning the assumption of the inter-slice force.

Description

Technical Field

[0001] The present invention belongs to the technical field of geotechnical engineering slopes, and particularly relates to a method for dimensionality reduction search of slope slip surfaces based on the Laplace equation. Background Art

[0002] A core task in slope stability analysis is the search for dangerous slip surfaces. Various mathematical optimization methods and evolutionary algorithms have shown their respective capabilities in the search for dangerous slip arcs. However, regardless of the optimization algorithm, its efficiency highly depends on the dimension of the optimization problem, and the computational time complexity increases non-linearly and sharply with the increase in the problem dimension. In two-dimensional problems, when the slip surface is assumed to be a circular arc, the optimization dimension is 3; while when the slip surface is non-circular, the dimension rises to dozens or hundreds.

[0003] The limit equilibrium slice method (LEM) is the most commonly used method for slope analysis. Duncan summarized the characteristics of some classical limit equilibrium analysis methods (Duncan J M. State of the Art: Limit Equilibrium and Finite-Element Analysis of Slopes[J]. Journal of Geotechnical Engineering, 1996, 122(7): 577-596.), including the Swedish slice method, Bishop's slice method, unbalanced thrust method, Janbu's slice method, Morgenstern and Price method, Spencer method, etc. Since the equilibrium equations obtained by slicing are indeterminate, in order to obtain the solutions of the equations (the stresses on the slip surface), all the above methods invariably adopt various assumptions of inter-slice forces. For the stability evaluation method of composite foundation filled slope based on the simplified Bishop method (CN202011497841.X), for the slope reinforced by piles arranged in a square pattern, by dividing the sliding mass into several blocks with and without piles, the areas and weights of each slice are calculated separately, and the stability coefficient of the composite foundation filled slope is calculated based on the simplified Bishop method. In fact, the essential difference among the above methods lies in the different assumptions of inter-slice forces. The assumption of inter-slice forces becomes both the greatest feature and the greatest drawback of the limit slice method because the assumption leads to inaccurate stress solutions on the slip surface, thus affecting the judgment of the slope safety state. Summary of the Invention

[0004] The search for dangerous slip surfaces is a core task in slope stability analysis. The efficiency of existing various optimization search algorithms severely depends on the dimension of the optimization problem, and their computational time complexity increases non-linearly and sharply with the increase of the problem dimension, seriously affecting the search efficiency. The purpose of the present invention is to provide an efficient method for dimensionality reduction optimization analysis of slope stability problems: the position and trend of the slip surface inside the slope body are determined by numerically solving the Laplace equation, and dimensionality reduction is achieved by using streamlines. Because when the solution conditions are certain, the streamline passing through any point in space is unique.

[0005] The present invention is implemented at least by one of the following technical solutions.

[0006] A method for dimensionality reduction search of slope slip surfaces based on the Laplace equation, comprising the following steps:

[0007] First step, select the Laplace equation as the control equation for the guiding potential field of the slip surface inside the slope;

[0008] Second step, based on the elastic stress field and the Mohr-Coulomb failure criterion, construct a quantity for representing the local stability of a certain point in the material: the maximum stress failure ratio;

[0009] Third step, determine the permeability tensor that can reflect the stress failure state of the slope according to the magnitude and direction of the maximum stress failure ratio;

[0010] Fourth step, use the finite element method to solve the Laplace equation to obtain the guiding potential field of the slip surface inside the slope, and construct a cluster of reasonable slip surfaces according to this field;

[0011] Fifth step, for each slip surface, calculate its safety factor according to the stress state on the slip surface, and obtain the dangerous slip surface with the minimum safety factor.

[0012] Furthermore, the specific process of constructing the maximum stress failure ratio is as follows:

[0013] The stress failure ratio is defined as:

[0014]

[0015] In the formula: SFR(θ) is the stress failure ratio at θ, τ is the shear stress on the slip surface, τ f is the shear strength of the soil on the slip surface, θ is twice the angle between the SFR plane and the major principal stress plane relative to the angle between the major principal stress plane and the maximum principal stress plane, and when the principal stress acts at a certain point, let

[0016]

[0017] In the formula: σ′ c and σ′ qThey are all intermediate variables, σ′1 is the maximum normal stress in the Mohr circle, and σ′3 is the minimum normal stress in the Mohr circle; there is

[0018]

[0019] In the formula: c is the cohesion of the slope soil mass, is the internal friction angle of the slope soil mass, and a and b are both intermediate variables;

[0020] Equation (3) has a maximum value. Taking the derivative of it and setting dSFR / dθ = 0, the maximum value of SFR(θ) is obtained, that is, the maximum stress failure ratio SFR max and its direction, where SFR max The angle of the plane relative to the major principal stress acting plane:

[0021]

[0022] Furthermore, for a given point, there are two failure planes, and the sliding direction of the slope points to the activated plane. Considering the slope sliding direction, the maximum stress failure ratio SFR max The angle of the plane relative to the x-axis is:

[0023]

[0024] In the formula: β max is the angle of the SFR max plane relative to the x-axis, α is the angle between the major principal stress acting plane and the x-axis direction, and θ max is the angle of the SFR max plane relative to the major principal stress acting plane, σ′ x and σ′ y are the horizontal and vertical stress values of the soil element respectively, and SD is the parameter for adjusting the sliding direction.

[0025] Furthermore, the specific process of determining the permeability tensor that can reflect the stress failure state of the slope is as follows:

[0026] Define the angle of the principal axis of the permeability tensor relative to the x-axis as β max ;

[0027] For all material points, assume there is a constant anisotropy ratio R k = k max / k min , where k max is the maximum permeability coefficient in the element, and k min is the minimum permeability coefficient in the element. The role of anisotropy is to make the streamline flow in the direction of the maximum value SFR of the stress failure ratio SFR(θ) max ;

[0028] There is a scaling factor κ for the permeability of each point in the material, and the scaling factor is based on the SFR of each integration point max Scale the permeability so that more streamlines flow within the area of SFR max to increase the chance of finding the critical slip surface:

[0029]

[0030] Where: K g is the permeability tensor of the governing equation of the guiding potential field, κ is the scaling factor, R is the rotation tensor, K l is the original permeability tensor, K min is the minimum value of the permeability coefficient within the element, R k is the constant anisotropy ratio; SFR K0 is the stress failure ratio in the elastic state, k b is the background flow factor, and the maximum value of (SFR max - SFR K0 ) for all integration points is M SFR , p is the weight for adjusting the scale; β max is the angle of the SFR max plane relative to the x-axis.

[0031] Furthermore, the maximum principal direction of the permeability tensor is exactly the same as that of SFR max and they are similar in shape. SFR(θ) cannot be represented by a symmetric second-order tensor, but K can. It is impossible to obtain a K that is the same as SFR.

[0032] Furthermore, when there is a weak soil layer in a multi-layer slope, the permeability of the weak soil layer will be increased by 10 - 1000 times according to its thickness to ensure that more streamlines pass through it.

[0033] Furthermore, for the boundary sensitivity problem involved, the sensitivity of the boundary is suppressed by strengthening the anisotropy of the permeability tensor.

[0034] Furthermore,: when there is groundwater action, a real seepage field needs to be applied simultaneously.

[0035] Furthermore, there are two methods to generate a virtual seepage field according to the boundary conditions: one is to specify the inlet and outlet areas and apply a constant water head on them; the other is to apply a linearly varying water head on all ground surfaces according to the ground elevation.

[0036] Furthermore, during the streamline integration process, if the horizontal component V x of the seepage velocity at the integration point is opposite to the direction of slope sliding, or the mobilized shear stress is opposite to the direction of slope movement, then it is modified to the velocity direction V parallel to the direction of the passive failure surface. For a right landslide, it is equal to For the left landslide, it is equal to the internal friction angle of the soil mass.

[0037] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0038] By solving the anisotropic Laplace equation, the pure optimization problem is transformed into a boundary value problem, reducing the optimization dimension. This method converts the two-dimensional slope stability optimization problem into a one-dimensional problem, greatly improving the efficiency of searching for the critical slip line. This method is based on the finite element elastic stress field and the closed virtual seepage field, without the need for iterative work or assumptions about the slip line shape and inter-slice forces. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 is a flowchart of a method for reducing the dimension of the slope sliding surface based on the Laplace equation according to an embodiment of the present invention;

[0040] Figure 2 is a schematic diagram of the stress failure ratio structure provided in an embodiment of the present invention;

[0041] Figure 3 is a mapping relationship diagram of the permeability tensor and the stress failure ratio provided in an embodiment of the present invention;

[0042] Figure 4 is a diagram of the method for applying the water head boundary of the virtual seepage model provided in an embodiment of the present invention;

[0043] Figure 5 is a diagram of the variation of the safety factor of the slip line along the water head boundary provided in an embodiment of the present invention;

[0044] Figure 6 is an analysis diagram for defining the safety factor provided in an embodiment of the present invention;

[0045] Figure 7 is an analysis diagram of the results of the safety factor and the critical sliding surface in Embodiment 1 of the present invention;

[0046] Figure 8 is an analysis diagram of the results of the safety factor and the critical sliding surface in Embodiment 2 of the present invention;

[0047] Figure 9 is an analysis diagram of the results of the safety factor and the critical sliding surface in Embodiment 3 of the present invention;

[0048] Figure 10 is an analysis diagram of the results of the safety factor and the critical sliding surface in Embodiment 4 of the present invention;

[0049] Figure 11 is an analysis diagram of the results of the safety factor and the critical sliding surface in Embodiment 5 of the present invention. DETAILED IMPLEMENTATION METHOD

[0050] The specific implementation method of the present invention will be described in detail below with reference to the accompanying drawings. It should be understood that the specific implementation method described herein is only used to illustrate and explain the present invention, and is not used to limit the present invention. Embodiment

[0051] A slope sliding surface dimensionality reduction search method based on the Laplace equation includes the following steps:

[0052] (1) Select the Laplace equation as the control equation for the guiding potential field of the internal sliding surface of the slope.

[0053] (2) Based on the elastic stress field and the Mohr-Coulomb failure criterion, construct a quantity representing the local stability of a certain point in the material: the maximum stress failure ratio SFR max . Among them, the construction steps are as follows:

[0054] As Figure 2 shown, the stress failure ratio is defined as:

[0055]

[0056] In the formula: SFR(θ) is the stress failure ratio at θ, τ is the shear stress on the sliding surface, τ f is the shear strength of the soil on the sliding surface.

[0057] Among them, θ is twice the angle between plane C and plane D, and the principal stress acts on Figure 2 point D in, let

[0058]

[0059] Then there is

[0060]

[0061] There is a maximum value in formula (3). Take the derivative of it and let dSFR / dθ = 0 to obtain the maximum value SFR of SFR(θ) max and its direction, where θ max is the direction relative to the action surface of the major principal stress.

[0062]

[0063] For a given point, there are two failure planes. Which plane is activated depends on the sliding direction (left or right) of the slope.

[0064] Finally, considering the slope sliding direction, the angle of the SFR max plane (pM or pM') relative to the x-axis is:

[0065]

[0066] In the formula: β max is the angle of the SFR max plane with respect to the x-axis, α is the angle between the principal stress acting plane and the x-axis direction, and θ max is the angle of the SFR max plane with respect to the major principal stress acting plane, σ′ x and σ′ y are the horizontal and vertical stress values of the soil element respectively, and SD is the parameter for adjusting the sliding direction.

[0067] (3) Determine an infiltration tensor that can reflect the stress failure state of the slope according to the magnitude and direction of the SFR max . The construction process is as follows:

[0068] Define that the angle of the main axis of the infiltration tensor with respect to the x-axis is set to β max .

[0069] For all material points, assume a constant anisotropy ratio, R k = k max / k min , where k max is the maximum value of the infiltration coefficient in the element, and k min is the minimum value of the infiltration coefficient in the element. Here, R k is usually set to 10. The role of anisotropy is to make the streamline flow in the direction of the SFR max .

[0070] Each point in the material has a scaling factor α for the permeability. The role of this factor is to scale the permeability according to the SFR max of each integration point, so that more streamlines can flow in the area where the SFR max is larger, increasing the chance of finding the critical slip surface.

[0071]

[0072] In the formula: K g is the infiltration tensor of the control equation for guiding the potential field, α is the scaling factor, R is the rotation tensor, K l is the original infiltration tensor, K min is the minimum value of the infiltration coefficient in the element, and R k is the constant anisotropy ratio.

[0073] In the expression of the scaling factor α, the SFR K0 is the stress failure ratio in the elastic state, and k b is a background flow factor, which is set to 0.2M SFR , where M SFRis the maximum value of (SFR max -SFR K0 ) for all integration points of the model. Additionally, p is the weight for adjusting the scale and is usually set to be between 1 and 2.

[0074] As an example, when there is a weak soil layer in a multi-layer slope, the permeability of the weak soil layer will increase by 10 - 1000 times according to its thickness to ensure that more streamlines pass through it.

[0075] Regarding the boundary sensitivity problem involved, the sensitivity of the boundary can be suppressed by strengthening the anisotropy of the permeability tensor.

[0076] Furthermore, to generate a virtual seepage field, appropriate boundary conditions should be given, such as Figure 4 , and there are usually two methods: one is to specify the inlet and outlet areas and apply a constant water head on them; the other is to apply a linearly varying water head on all ground surfaces according to the ground elevation.

[0077] (4) Use the finite element method to solve the Laplace equation to obtain the potential field of the slip surface inside the slope, and based on this field, construct a reasonable cluster of slip surfaces. During the streamline integration process, if the horizontal component V x of the seepage velocity at the integration point is opposite to the slope sliding direction, or the mobilized shear stress is opposite to the slope movement direction, then it is modified to the velocity direction V parallel to the direction of the passive failure surface. For a right landslide, it is equal to For a left landslide, it is equal to is the internal friction angle of the soil mass.

[0078] (5) For each streamline (slip surface), calculate its safety factor according to the stress state on its slip surface, and obtain the dangerous slip surface with the minimum safety factor. The safety factor calculation formula is:

[0079]

[0080] Combined with Figure 6 , A and B are two endpoints of any slip surface, i is a segment on AB, τ is the shear stress on the slip surface, τ f is the shear strength of the soil mass on the slip surface, ΔL is the length of the unit segment, F s is the safety factor, S a is the total anti-sliding force of the ΔL unit segment, S m is the total sliding force of the ΔL unit segment, ΔL i is the length of the i-th unit segment, τ i is the shear strength of the soil mass on the i-th unit segment. Here, if the normal stress σ i acting on the ΔL segment is less than 0, then τ fiTake 0 because the soil mass cannot bear tensile force.

[0081] Example 1 is a simple homogeneous slope; assume the dimensions of a homogeneous slope are as Figure 7 , the unit weight of the soil is 17.64 kN / m 3 , the cohesion is 9.8 kPa, the internal friction angle is 10°, the elastic modulus is 100 MPa, and the Poisson's ratio is 0.45. As Figure 7 shown, for stress analysis, the left and right sides are fixed in the x-direction, and the bottom is completely fixed. For seepage analysis, both the left and right sides and the bottom of the slope are impermeable, and the second method is used to apply the closed model head boundary: where h ABCD is the head value of the ABCD surface, y A is the ground elevation at point A, y is the applied head value, y D is the ground elevation at point D. Among them, k min = 0.1, R k = 10, p = 2, k b = 0.2. Using the above method, the safety factor and the critical slip surface are obtained as Figure 7 , which is consistent with the calculation results of other methods.

[0082] Example 2 is a two-layer slope with groundwater; assume the dimensions of this two-layer slope are as Figure 8 , the unit weight of the upper layer of soil is 15 kN / m 3 , the cohesion is 5 kPa, the internal friction angle is 20°, the elastic modulus is 15 MPa, and the Poisson's ratio is 0.4. The unit weight of the lower layer of soil is 18 kN / m 3 , the cohesion is 10 kPa, the internal friction angle is 25°, the elastic modulus is 12 MPa, and the Poisson's ratio is 0.37. As Figure 8 shown, for stress analysis, the left and right sides are fixed in the x-direction, and the bottom is completely fixed. For seepage analysis, both the left and right sides and the bottom of the slope are impermeable, and the second method is used to apply the closed model head boundary: where k min = 0.1, R k = 10, p = 2, k b = 0.2. Using the above method, the safety factor and the critical slip surface are obtained as Figure 8 , which is consistent with the calculation results of other methods.

[0083] Example 3 is a heterogeneous three-layer slope; assume the slope dimensions are as Figure 9 , the unit weight of the upper layer of soil is 19.5 kN / m 3 , the cohesion is 0, the internal friction angle is 38°, the elastic modulus is 100 MPa, and the Poisson's ratio is 0.4. The unit weight of the middle layer of soil is 19.5 kN / m 3, the cohesion is 5.3 kPa, the internal friction angle is 23°, the elastic modulus is 100 MPa, and the Poisson's ratio is 0.4. The unit weight of the lower layer soil is 19.5 kN / m 3 , the cohesion is 7.2 kPa, the internal friction angle is 20°, the elastic modulus is 100 MPa, and the Poisson's ratio is 0.4. As Figure 9 shown, for stress analysis, the left and right sides are fixed in the x - direction, and the bottom is completely fixed. For seepage analysis, both the left and right sides and the bottom of the slope are impermeable, and the second method is used to apply the closed - model head boundary: where k min = 0.1, R k = 10, p = 2, k b = 0.2. Using the above method, the safety factor and the critical slip surface are obtained as Figure 9 , which is consistent with the calculation results of other methods.

[0084] Example 4 is a layered slope with a weak layer between two relatively strong layers; assume the slope dimensions are as Figure 10 , the unit weight of the upper layer soil is 18.82 kN / m 3 , the cohesion is 29.4 kPa, the internal friction angle is 12°, the elastic modulus is 100 MPa, and the Poisson's ratio is 0.4. The unit weight of the middle layer soil is 18.82 kN / m 3 , the cohesion is 9.8 kPa, the internal friction angle is 5°, the elastic modulus is 100 MPa, and the Poisson's ratio is 0.4. The unit weight of the lower layer soil is 18.82 kN / m 3 , the cohesion is 294 kPa, the internal friction angle is 40°, the elastic modulus is 100 MPa, and the Poisson's ratio is 0.4. As Figure 10 shown, for stress analysis, the left and right sides are fixed in the x - direction, and the bottom is completely fixed. For seepage analysis, both the left and right sides and the bottom of the slope are impermeable, and the second method is used to apply the closed - model head boundary: where k min = 0.1, R k = 10, p = 2, k b = 0.2. Using the above method, the safety factor and the critical slip surface are obtained as Figure 10 , which is consistent with the calculation results of other methods.

[0085] Example 5 is a three - layer slope with the downhill area flooded and a weak layer sandwiched between two relatively hard layers; assume the slope dimensions are as Figure 11 , the unit weight of the upper layer soil is 15 kN / m 3 , the cohesion is 20 kPa, the internal friction angle is 30°, the elastic modulus is 15 MPa, and the Poisson's ratio is 0.33. The unit weight of the middle layer soil is 18 kN / m 3, the cohesion is 0, the internal friction angle is 10°, the elastic modulus is 2 MPa, and the Poisson's ratio is 0.45. The unit weight of the lower soil layer is 20 kN / m 3 , the cohesion is 100 kPa, the internal friction angle is 30°, the elastic modulus is 100 MPa, and the Poisson's ratio is 0.35. As Figure 11 shown, for stress analysis, the left and right sides are fixed in the x direction, and the bottom is completely fixed. For seepage analysis, both the left and right sides and the bottom of the slope are impermeable, and the second method is used to apply the closed model head boundary: where the permeability coefficient k of the middle soil layer min = 1, the permeability coefficients k of the upper and lower soil layers min = 0.1, R k = 10, p = 2, k b = 0.2. Using the above method, the safety factor and the critical slip surface are obtained as Figure 11 , which is consistent with the calculation results of other methods.

[0086] The preferred embodiments of the present invention disclosed above are only used to help illustrate the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to the specific embodiments described. Obviously, many modifications and variations can be made according to the content of this specification. These embodiments are selected and specifically described in this specification to better explain the principles and practical applications of the present invention, so that those skilled in the art can understand and utilize the present invention well. The present invention is only limited by the claims and their full scope and equivalents.

Claims

1. A slope sliding surface dimensionality reduction search method based on the Laplace equation, characterized in that: It includes the following steps: First step: Select the Laplace equation as the control equation for the guiding potential field of the internal sliding surface of the slope; Second step: Based on the elastic stress field and the Mohr-Coulomb failure criterion, construct a quantity used to represent the local stability of a certain point in the material: the maximum stress failure ratio; The specific process of constructing the maximum stress failure ratio is as follows: The stress failure ratio is defined as: Where: SFR(θ) is the stress failure ratio at θ, τ is the shear stress on the slip surface, τ f is the shear strength of the soil on the slip surface, θ is twice the angle between the SFR plane and the major principal stress plane relative to the major principal stress plane, and the principal stress acts at a certain point. Let where: σ c ′ and σ q ′ are both intermediate variables, σ1′ is the maximum normal stress in the Mohr circle, and σ3′ is the minimum normal stress in the Mohr circle; there is where: c is the cohesion of the slope soil mass, is the internal friction angle of the slope soil mass, and a and b are both intermediate variables; Equation (3) has a maximum value. By taking its derivative and setting dSFR / dθ = 0, the maximum value of SFR(θ), i.e., the maximum stress failure ratio SFR, is obtained max and its direction, where SFR max is the angle between the plane and the major principal stress plane: Third step: Determine the permeability tensor that can reflect the stress failure state of the slope according to the magnitude and direction of the maximum stress failure ratio; Fourth step: Use the finite element method to solve the Laplace equation to obtain the guiding potential field of the internal sliding surface of the slope, and construct a cluster of sliding surfaces according to this field; Fifth step: For each sliding surface, calculate its safety factor according to the stress state on its sliding surface, and obtain the dangerous sliding surface with the minimum safety factor.

2. The slope sliding surface dimensionality reduction search method based on the Laplace equation according to claim 1, characterized in that: For a given point, there are two failure planes and the sliding direction of the slope points to the activated plane. Considering the slope sliding direction, the maximum stress failure ratio SFR max The angle of the plane with respect to the x-axis is: Where: β max is the angle of the SFR max plane relative to the x-axis, α is the angle between the principal stress acting plane and the x-axis direction, θ max is the angle of the SFR max plane relative to the major principal stress acting plane, σ x ′ and σ y ′ are the horizontal and vertical stress values of the soil element respectively, and SD is the parameter for adjusting the sliding direction.

3. A method for reducing the dimension of the slope sliding surface search based on the Laplace equation according to claim 1, characterized in that: The specific process of determining the permeability tensor that can reflect the stress failure state of the slope is as follows: The angle that defines the principal axis of the permeability tensor relative to the x-axis is set to β max ; For all material points, a constant anisotropy ratio R is assumed k = k max / k min , where k max is the maximum value of the permeability coefficient in the element, and k min is the minimum value of the permeability coefficient in the element. The effect of anisotropy is to make the streamline flow in the direction of the maximum value SFR of the stress failure ratio SFR(θ) max ; There is a scaling coefficient κ for the permeability of each point in the material, and the scaling coefficient is based on the SFR of each integration point max Scale the permeability so that more flow lines flow within the area of the SFR max and increase the chance of finding the critical slip surface: Where: K g is the permeability tensor of the governing equation of the guiding potential field, κ is the scaling factor, R is the rotation tensor, K l is the original permeability tensor, K min is the minimum value of the permeability coefficient within the element, R k is the constant anisotropy ratio; SFR K0 is the stress failure ratio in the elastic state, k b is the background flow factor, and the maximum value of (SFR max - SFR K0 ) at all integration points is M SFR , and p is the weight for adjusting the scale; β max is an SFR max The angle of the plane relative to the x-axis.

4. A method for reducing the dimension of the slope sliding surface search based on the Laplace equation according to claim 3, characterized in that: Seepage tensor and SFR max are completely consistent in the maximum principal direction and similar in shape.

5. A method for dimensionality reduction search of slope sliding surface based on Laplace equation according to claim 3, characterized in that: When there are weak soil layers in a multi-layer slope, the permeability of the weak soil layer will be increased by 10 - 1000 times according to its thickness to ensure that more streamlines pass through it.

6. A method for dimensionality reduction search of slope sliding surface based on Laplace equation according to claim 3, characterized in that: For the boundary sensitivity problem involved, the sensitivity of the boundary is suppressed by strengthening the anisotropy of the permeability tensor.

7. A method for dimensionality reduction search of slope slip surface based on Laplace equation according to claim 3, characterized in that: When there is groundwater action, it is necessary to apply the real seepage field simultaneously.

8. A method for reducing the dimension of the slope sliding surface search based on the Laplace equation according to claim 7, characterized in that: There are two methods to generate a virtual seepage field according to the boundary conditions: one is to specify the inlet and outlet areas and apply a constant water head on them; the other is to apply a linearly varying water head on all the ground surfaces according to the ground elevation.

9. A method for dimensionality reduction search of slope sliding surface based on Laplace equation according to claim 1, characterized in that: During the streamline integration process, if the horizontal component V of the seepage velocity at the integration point x , is opposite to the slope sliding direction, or the mobilized shear stress is opposite to the slope movement direction, then it is modified to the velocity direction V parallel to the passive failure surface direction, which for a right landslide is equal to For a left landslide, it is equal to where is the angle of internal friction of the soil mass.

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