A convex optimization method for determining the safety factor of a block and related products

Abstract

The present invention relates to the field of rock engineering, and specifically relates to a convex optimization method for determining the safety factor of a block and related products. The method includes obtaining the structural plane parameters of the block that makes up the block and constructing a normal vector matrix describing the movement direction of the block; by constructing the objective function and constraint conditions of quadratic programming, determining the unit vector of the block movement direction in three-dimensional space, solving the unit vector of the block movement direction, classifying the block into four types, and determining the safety factor of the block according to the block type. The present invention does not involve many cumbersome and abstract mathematical theorems and concepts in the block theory, avoids the cumbersome calculation steps of the vector method in the block theory, reduces the difficulty of the vector method in program implementation, and can analyze the stability problem of the block only by constructing simple optimization objective functions and constraint equations.

Description

Technical Field

[0001] The present invention relates to the field of rock engineering, and particularly to a convex optimization method for determining the safety factor of a block and related products. Background Art

[0002] The block theory is a very important theory in the analysis method of rock mass stability. Its essence is to use geometric methods (topology and set theory) to study the types and mobility of blocks composed of structural planes and free faces. According to the static equilibrium conditions of the blocks, the safety factor is obtained to evaluate its stability.

[0003] The core of the block theory is to find the key blocks on the free face so as to take engineering treatment measures for them to maintain the stability of the rock mass. Regarding the determination of key blocks, there is a set of strict procedures (Shi Genhua. Stereographic Projection Method for Rock Mass Stability Analysis. Chinese Science, 1977; Shi Genhua. Geometric Method for Rock Mass Stability Analysis. Chinese Science, 1981): (1) First, the finiteness theorem is used to judge whether the block is a finite block or an infinite block according to whether the block cone of the block is an empty set. (2) Secondly, the mobility theorem is used to judge whether the finite block is a movable block or an immovable block according to whether the joint cone of the block is an empty set. (3) Then, the kinematic theorem is used to judge whether the movable block is a block that can move and maintain stability without friction, a free-falling block, a single-sided sliding block or a double-sided sliding block according to the sufficient and necessary conditions of free fall, single-sided sliding and double-sided sliding. (4) Finally, the limit equilibrium method is used to calculate the safety factor of the block, and the unstable block with a safety factor less than 1 is the key block.

[0004] The above steps are implemented by the vector method of the block theory in a computer. However, there are the following two difficulties: (1) The block theory involves a large number of abstract mathematical concepts and theorems, making it difficult for engineering practitioners to understand. (2) The above steps are computationally cumbersome, and it is difficult to implement them in a computer program. The above difficulties limit the wide use of the block theory, and it is necessary to propose an alternative method with simple concepts and easy programming. Summary of the Invention

[0005] The purpose of the present invention is to overcome the deficiencies of the prior art, and provide a convex optimization method for determining the safety factor of a block and related products, which does not involve many cumbersome and abstract mathematical theorems and concepts in the block theory, avoids the cumbersome calculation steps of the vector method of the block theory, reduces the difficulty of implementing the vector method in a program, and can analyze the stability problem of the block only by constructing a simple optimization objective function and constraint equation.

[0006] The present invention is realized by the following technical solutions:

[0007] A convex optimization method for determining the safety factor of a block, comprising:

[0008] Obtain the structural plane parameters that make up the block, where the structural plane parameters include direction parameters, dip angles, dip direction angles, and internal friction angles;

[0009] Construct a normal vector matrix that describes the movement direction of the block according to the structural plane parameters;

[0010] By constructing the objective function and constraint conditions of quadratic programming, determine the unit vector of the movement direction of the block in three-dimensional space, where the objective function is to maximize the inner product of the unit vector of the resultant external force on the block and the unit vector of the movement direction of the block in three-dimensional space, and the constraint conditions include the quadratic norm constraint of the unit vector of the movement direction, the geometric relationship constraint between the movement direction of the block and the normal vector of the structural plane, the component constraints of the unit vector of the movement direction of the block, and the modulus length constraint of the unit vector of the movement direction of the block;

[0011] Solve the unit vector of the movement direction of the block, and classify the block into four types: immovable block, movable and stable block without friction, free-falling block, and sliding block;

[0012] Determine the safety factor of the block according to the block type.

[0013] Optionally, the number of structural planes that make up the block is not less than 2 groups; the direction parameter D i is determined by the relative position of the block and the structural plane. When the block is above the structural plane, take D i = 1, otherwise take D i = -1, i = 1, 2,..., n, where n is the number of structural planes of the block, and n ≥ 2.

[0014] Specifically, the construction method of the normal vector matrix includes:

[0015] Determine the normal vector v i pointing to the inside of the block for the i-th structural plane, v i = D i ·(sinα i ·sinβ i , sinα i ·cosβ i , cosα i ), where D i is the direction parameter of the i-th structural plane, α i is the dip angle of the i-th structural plane, and β i is the dip direction angle of the i-th structural plane;

[0016] Arrange the normal vectors of all structural planes in rows to form the structural plane normal vector matrix A,

[0017] Optionally, the objective function is: maximizer T x, and the constraint conditions are: where x = [x 1 x 2 x 3 T is the unit vector of the block movement direction in three-dimensional space, Q = diag(2, 2, 2) is a diagonal matrix, r is the unit vector of the resultant external force acting on the block, and inf is infinity.

[0018] Specifically, the interior point method is used to solve the objective function and the constraint conditions to obtain the unit vector x of the block movement direction, and the obtained vector x is judged;

[0019] If ‖x‖ < ε, then let r = -r, where ε is an integer and ε → 0 + , and recalculate the unit vector x of the new block movement direction through the objective function and the constraint conditions, and then judge the failure mode;

[0020] If |‖x‖ - 1| < ε, then directly obtain the unit vector x of the block movement direction and judge the failure mode.

[0021] Specifically, according to the unit vector x of the block movement direction, x = [x 1 x 2 x 3 T , and the method for judging the block type includes:

[0022] If ‖x‖ < ε, where ε is an integer and ε → 0 + , then the block type is an immovable block;

[0023] If |‖x‖ - 1| < ε and x 3 > ε, then the block type is a movable block that can maintain stability without friction;

[0024] If |‖x‖ - 1| < ε and x 3 < - ε, then calculate the discriminant vector T of the failure mode, T =

[0025]

[0026] If all elements in the discriminant vector T are 1, then the block type is a free-falling block, otherwise the block type is a sliding block.

[0027] Furthermore, the sliding block includes a single-sided sliding block and a double-sided sliding block;

[0028] If the discriminant vector T only includes 1 zero element, then the block type is a single-sided sliding block; if the discriminant vector T only includes 2 zero elements, then the block type is a double-sided sliding block.​​

[0029] Specifically, if the block is an immovable block or a movable block that can maintain stability without friction, the block safety factor FoS = +∞;

[0030] If the block is a free-falling block, the block safety factor FoS = 0;

[0031] If the block is a single-plane sliding block, the block safety factor is the internal friction angle of the i-th structural plane;

[0032] If the block is a double-plane sliding block, the block safety factor where F as is the anti-sliding force per unit volume of the block, and F sl is the sliding-down force per unit volume of the block.

[0033] where N i and N j are the normal forces on the structural planes i and j respectively, and are the internal friction angles of the structural planes i and j respectively;

[0034] F sl = |r·(v i ×v j )| / |v i ×v j |, where v i and v j are the unit normal vectors pointing into the block on the i-th and j-th structural planes respectively;

[0035]

[0036] A convex optimization terminal for determining the block safety factor includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the convex optimization method for determining the block safety factor as described above is implemented.

[0037] A computer program product includes a computer program / instructions. When the computer program / instructions are executed by a processor, the convex optimization method for determining the block safety factor as described above is implemented.

[0038] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0039] The present invention mainly determines the safety factor of a block and evaluates its stability through geometric analysis and optimization calculations. This method first determines whether the block is a finite block or an infinite block according to the finiteness theorem; then, through the mobility theorem, it determines whether the block is a movable block; then, using the kinematic theorem, it further analyzes the specific motion mode of the movable block, such as free fall, single-sided sliding or double-sided sliding; finally, the limit equilibrium method is used to calculate the safety factor of the block. The whole process is based on geometric calculations and vector analysis, and can be automatically realized by a computer program, reducing human judgment errors.

[0040] The method for analyzing the stability of a block provided by the present invention simplifies the complex mathematical concepts and theorems in the traditional block theory, reducing the understanding difficulty for engineers; through the vectorized analysis method, the calculation process is significantly simplified, making the implementation of the computer program more intuitive and easier, and greatly improving the operability and popularity of the application. Brief Description of the Drawings

[0041] The drawings illustrate exemplary embodiments of the present invention and are used together with the description to explain the principles of the present invention. These drawings are included to provide a further understanding of the present invention, and the drawings are included in this specification and form a part of this specification, and do not constitute a limitation on the embodiments of the present invention.

[0042] Figure 1 It is a schematic flow chart of a convex optimization method for determining the safety factor of a block according to the present invention.

[0043] Figure 2 It is the morphology and stereographic projection diagram of the block in the second embodiment according to the present invention.

[0044] Figure 3 It is the morphology and stereographic projection diagram of the block in the third embodiment according to the present invention.

[0045] Figure 4 It is the morphology and stereographic projection diagram of the block in the fourth embodiment according to the present invention.

[0046] Figure 5 It is the morphology and stereographic projection diagram of the block in the fifth embodiment according to the present invention.

[0047] Figure 6 It is the morphology and stereographic projection diagram of the block in the sixth embodiment according to the present invention. Detailed Embodiments

[0048] To make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the drawings and embodiments. It can be understood that the specific embodiments described herein are only used to explain the relevant content and do not limit the present invention.

[0049] In addition, it should be noted that for the convenience of description, only the parts related to the present invention are shown in the drawings.

[0050] Without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the drawings and in combination with the embodiments.

[0051] Embodiment 1

[0052] As Figure 1 shown, a convex optimization method for determining the safety factor of a block is provided, including:

[0053] S1. Obtain the structural plane parameters of the block, where the structural plane parameters include direction parameters, dip angles, dip directions, and internal friction angles.

[0054] The number of structural planes forming the block is not less than 2 groups, generally 3 groups or 4 groups. The direction parameter D i is determined by the relative position of the block and the structural plane. When the block is above the structural plane, D i = 1, otherwise D i = -1, i = 1, 2,..., n, where n is the number of structural planes of the block and n ≥ 2. The attitude parameters α i and β i of the structural plane are obtained by on-site compass testing, and the internal friction angle of the structural plane is determined by laboratory testing or engineering analogy method.

[0055] S2. According to the structural plane parameters, construct a normal vector matrix describing the movement direction of the block. The construction method of the normal vector matrix includes:

[0056] Determine the normal vector v i pointing to the inside of the block for the i-th structural plane, v i = D i ·(sinα i ·sinβ i , sinα i ·cosβ i , cosα i ), where D i is the direction parameter of the i-th structural plane, α i is the dip angle of the i-th structural plane, and β i is the dip direction of the i-th structural plane;

[0057] Arrange the normal vectors of all structural planes in rows to form the structural plane normal vector matrix A,

[0058] S3. By constructing the objective function and constraint conditions of quadratic programming, determine the unit vector of the block movement direction in three-dimensional space. The objective function is to maximize the inner product of the unit vector of the resultant external force acting on the block and the unit vector of the block movement direction in three-dimensional space. The constraint conditions include the quadratic norm constraint of the movement direction unit vector, the geometric relationship constraint between the block movement direction and the normal vector of the structural plane, the component constraints of the block movement direction unit vector, and the modulus length constraint of the block movement direction unit vector.

[0059] The objective function is: maximizer T x, and the constraint conditions are: where x = [x 1 x 2 x 3 T is the unit vector of the block movement direction in three-dimensional space, Q = diag(2, 2, 2) is a diagonal matrix, r is the unit vector of the resultant external force acting on the block, and inf is infinity.

[0060] Generally, the components x i of x take values between -1 and 1. In addition, x also needs to satisfy the constraint condition of ‖x‖ = 1, that is However, in order to construct the constraint equation of the standard form of quadratic programming with quadratic constraints, the constraint condition is relaxed to ‖x‖ ≤ 1, that is

[0061] r represents the unit vector of the resultant external force acting on the block. When the resultant external force acting on the block is only gravity, the unit vector r = [0 0 -1] T ; when the resultant external force acting on the block includes water pressure or seismic inertia force, its components are recalculated under the condition of ‖r‖ = 1.

[0062] S4. Solve the unit vector of the block movement direction.

[0063] Use the interior point method to solve the objective function and constraint conditions to obtain the unit vector x of the block movement direction, and judge the obtained vector x;

[0064] If ‖x‖ < ε, then let r = -r, where ε is an integer and ε → 0 + , and recalculate the unit vector x of the new block movement direction through the objective function and constraint conditions, and then judge the failure mode, that is, after updating the unit vector x, jump to S5;

[0065] If |‖x‖ - 1| < ε, directly obtain the unit vector x of the block movement direction and judge the failure mode, that is, directly jump to S5.

[0066] S5. Determine the failure mode of the block and the corresponding sliding surface according to the unit vector of the movement direction. There are four types of blocks in block theory: immovable blocks, movable blocks that can maintain stability without friction, free-falling blocks, and sliding blocks. Among them, sliding blocks are further divided into single-sided sliding blocks and double-sided sliding blocks.

[0067] The methods for determining the block type include:

[0068] If ‖x‖ < ε, where ε is an integer and ε → 0 + , that is, ε is a very small value close to 0, then the block type is an immovable block;

[0069] If |‖x‖ - 1| < ε and x 3 > ε, then the block type is a movable block that can maintain stability without friction;

[0070] If |‖x‖ - 1| < ε and x 3 < - ε, then calculate the discriminant vector T of the failure mode, sign(·) is the sign function.

[0071] If all elements in the discriminant vector T are 1, then the block type is a free-falling block; otherwise, the block type is a sliding block.

[0072] And if the discriminant vector T only includes 1 zero element and the rest are all 1, then the block type is a single-sided sliding block, and the structural plane corresponding to the row number where the zero element is located is the sliding surface of the single-sided sliding.

[0073] If the discriminant vector T only includes 2 zero elements and the rest are all 1, then the block type is a double-sided sliding block, and the structural planes corresponding to the row numbers of the two zero elements are the sliding surfaces of the double-sided sliding.

[0074] S6. Determine the safety factor of the block according to the block type.

[0075] If the block is an immovable block or a movable block that can maintain stability without friction, the block safety factor FoS = +∞; that is, the block is absolutely stable under this resultant external force.

[0076] If the block is a free-falling block, then the block safety factor FoS = 0.

[0077] If the block is a sliding block, then the block safety factor is related to the structural plane parameters (dip angle α i , dip direction angle β i and internal friction angle ), and the limit equilibrium formula for single-sided sliding or double-sided sliding is used to calculate the safety factor.

[0078] If the block is a single-sided sliding block, the safety factor of the block is the internal friction angle of the i-th structural plane;

[0079] If the block is a double-sided sliding block, the safety factor of the block where F as is the anti-sliding force per unit volume of the block, and F sl is the down-sliding force per unit volume of the block.

[0080] where N i and N j are the normal forces on the structural planes i and j respectively, and are the internal friction angles of the structural planes i and j respectively;

[0081] F sl = |r·(v i × v j )| / |v i × v j |, where v i and v j are the unit normal vectors pointing into the block on the i-th and j-th structural planes respectively;

[0082]

[0083] Example 2

[0084] This example provides a single-sided sliding block with 5 planes.

[0085] As Figure 2 (a) shows, the block contains 5 planes, specifically 4 structural planes J1 - J4 and 1 free face EF. The dip angle and dip direction angle parameters of the structural planes and the free face are shown in Table 1. Note that the method proposed in the present invention does not require the parameters of the free face. From Figure 2 (a) the block morphology diagram and perspective view shown, it can be seen that the block is respectively above the structural plane J1, above the structural plane J2, below the structural plane J3, and above the structural plane J4. Therefore, the direction parameters D 1 = 1, D 2 = 1, D 3 = -1, D 4 = 1. The resultant external force only considers gravity, that is, r = [0 0 -1] T . All structural planes have the same internal friction angle

[0086] Table 1 Parameters of the structural planes and free face of the block in Example 1

[0087]

[0088] Figure 2 (b) is the stereographic projection diagram of the block, which shows the projections of the block cone (the empty set does not exist) and the joint cone (the shaded area). The failure mode and movement direction of the block can be identified from the stereographic projection diagram through the relevant theorems of the block theory. Figure 2 (b) shows (The projection of the resultant external force r on the structural plane J4) is the movement direction of the block, and the failure mode is single-plane sliding, with the sliding plane being the structural plane J4. The results obtained from this stereographic projection diagram are used to verify the accuracy of the results of the method proposed in this paper.

[0089] Next, the method proposed in this paper is used to analyze the movement method and failure mode of the block. Since the solution obtained by the convex optimization method is an approximate solution rather than an exact solution, a minimum value ε needs to be introduced as the criterion for determining whether the solution is 0. Here, ε = 1e-4 is taken.

[0090] (1) The required structural plane parameters have been given in Table 1.

[0091] (2) Calculate the structural plane normal vector v pointing to the inside of the block from the structural plane parameters i ;

[0092] v 1 = D 1 ·(sinα 1 ·sinβ 1 , sinα 1 ·cosβ 1 , cosα 1 ) = (0.7500, -0.4330, 0.5000)

[0093] v 2 = D 2 ·(sinα 2 ·sinβ 2 , sinα 2 ·cosβ 2 , cosα 2 ) = (-0.6124, -0.3536, 0.7071)

[0094] v 3 = D 3 ·(sinα 3 ·sinβ 3 , sinα 3 ·cosβ 3 , cosα 3 ) = (0.0000, 0.1736, -0.9848)

[0095] v4 = D 4 ·(sinα 4 @ sinβ 4 , sinα 4 @ cosβ 4 , cosα 4 ) = (-0.0594, -0.3368, 0.9397)

[0096] Further obtain the matrix

[0097] (3) Construct the objective function and constraint equations in the standard form of quadratic programming with quadratic constraints:

[0098] maximize r T x

[0099]

[0100] 0 ≤ Ax ≤ inf

[0101] -1 ≤ x i ≤ 1, i = 1, 2, 3,

[0102] (4) Use the interior point method to calculate and obtain the global optimal solution of the movement direction x = [-0.1632, -0.9254, -0.3420]. |‖x‖ - 1| = 3.9135e-09 < ε, directly proceed to the next step.

[0103] (5) |‖x‖ - 1| < ε and x 3 = -0.3420 < -ε, continue to calculate the failure mode discrimination vector T = sign(Ax) = sign([0.1073, 0.1853, 0.1761, -2.388e-10]) = [1 1 1 0]

[0104] There is only 1 zero element in T and the remaining elements are all 1. The failure mode of the block is single-sided sliding, and the row number of the zero element corresponds to the structural plane J4 as the sliding plane of single-sided sliding. It can be seen that the result of this method is consistent with the stereographic projection result.

[0105] (6) Select the formula for single-sided sliding according to the discrimination result of the previous step to calculate the safety factor

[0106] Example 3

[0107] Similarly, it is a convex block containing 5 planes. The morphology and stereographic projection of the block are shown in Figure 3 . The sliding direction of the block obtained according to the stereographic projection is (Projection of the resultant external force r on the intersection edge of structural planes J1 and J2), the failure mode is double-sided sliding, and the sliding planes are structural planes J1 and J2.

[0108] (1) The parameters of the three-block structural planes J1-J3 and the free face in Example 3 are the same as those in the two-block example, except for the parameters of structural plane J4, specifically α 4 = 50°, β 4 = 150° and D 4 = 1.

[0109] (2) The corresponding calculated v 4 = (0.3830, -0.6634, 0.6428) and matrix

[0110] (3) Construct the objective function and constraint equations in the standard form of quadratic programming with quadratic constraints.

[0111] (4) Using the interior point method, the global optimal solution of the movement direction is calculated as x = [-0.1296, -0.8375, -0.5309]. |‖x‖ - 1| = 5.2219e-10 < ε, and directly proceed to the next step.

[0112] (5) |‖x‖ - 1| < ε and x 3 = -0.5309 < -ε, continue to calculate the failure mode discrimination vector T = sign(Ax) = sign([2.2452e-11, 6.845e-10, 0.37744, 0.1647]) = [0 0 1 1].

[0113] T contains 2 zero elements and the remaining elements are all 1. The failure mode of the block is double-sided sliding, and the row numbers of the zero elements correspond to the sliding planes of structural planes J1 and J2 for double-sided sliding. It can be seen that the results of this method are consistent with the stereographic projection results.

[0114] (6) Select the formula for double-sided sliding according to the discrimination result of the previous step to calculate the safety factor

[0115]

[0116] FoS = F as / F s＝ 1.2533.

[0117] Example 4

[0118] It is also a convex block containing 5 planes. The morphology and stereographic projection of the block in Example 4 are shown in Figure 4 . According to the stereographic projection, the joint cone of the block is an empty set, so the block is an immovable block.

[0119] (1) The parameters of the four-block structural planes J1 - J3 and the free face in Example 4 are the same as those in Example 2, with the difference being the parameters of structural plane J4. Specifically, α 4 = 20°, β 4 = 20° and D 4 = 1.

[0120] (2) The correspondingly calculated v 4 = (-0.1170, -0.3214, 0.9397) and the matrix

[0121] (3) Construct the objective function and constraint equations in the standard form of quadratic programming with quadratic constraints.

[0122] (4) Using the interior point method, the global optimal solution for the movement direction is obtained as x = [-3.8368e-11, 3.4362e-11, 3.7597e-11].

[0123] ‖x‖ = 6.3769e-11 < ε, let r = -r, go back to (3) and recalculate x = [-1.9721e-16, -1.842e-17, 1.6705e-16], then proceed to the next step.

[0124] (5) ‖x‖ = 2.5911e-16 < ε, the block is an immovable block.

[0125] (6) The safety factor of the immovable block is +∞.

[0126] Example 5

[0127] Provide a movable convex block containing 4 planes that can maintain stability without friction.

[0128] Example 5 is a convex block containing 4 planes. The morphology and stereographic projection of the block are shown in Figure 5 . According to the stereographic projection, the joint cone of the block exists, so the block is movable. However, it does not satisfy the necessary and sufficient conditions for free fall, single-sided sliding, and double-sided sliding. Therefore, it is movable and can maintain stability without friction.

[0129] (1) The parameters of the structural planes and free face of the block in Example 5 are shown in Table 2.

[0130] Table 2 Parameters of the structural planes and free face of the block in Example 4

[0131]

[0132] (2) Calculate the structural plane normal vector v pointing into the block from the structural plane parameters i

[0133] v 1 = D 1 ·(sinα 1 ·sinβ 1 , sinα 1 @cosβ 1 , cosα 1 ) = (0.7500, 0.4330, 0.5000)

[0134] v 2 = D 2 @(sinα 2 @sinβ 2 , sinα 2 @cosβ 2 , cosα 2 ) = (-0.6124, 0.3536, 0.7071)

[0135] v 3 = D 3 ·(sinα 3 ·sinβ 3 , sinα 3 ·cosβ 3 , cosα 3 ) = (0.0000, -0.9397, -0.3420)

[0136] Further obtain the matrix

[0137] (3) Construct the objective function and constraint equations in the standard form of quadratic programming with quadratic constraints.

[0138] (4) Use the interior point method to calculate and obtain the global optimal solution of the movement direction x = [-6.1503e-10, -9.3540e-10, 1.7729e-09]. ‖x‖ = 2.0968e-09 < ε, let r = -r, go back to (3) and recalculate x = [-8.5216e-06, -0.3420, 0.9397], and then enter the next step.

[0139] (5) |‖x‖ - 1| = 5.1237e-10 < ε and x 3 = 0.939 > ε, the block is a movable block that can maintain stability without friction.

[0140] (6) The safety factor of the movable block that can maintain stability without friction is +∞.

[0141] Example Six

[0142] Provide a convex block with 6 planes. The morphology and stereographic projection of the block are shown inFigure 6 . According to the stereographic projection, the failure mode of the block is free fall, and the movement direction is the same as the direction of the resultant external force.

[0143] (1) The parameters of the block structural plane and free face in Example 6 are shown in Table 3.

[0144] Table 3 Parameters of the structural plane and free face of the block in Example 6

[0145]

[0146] (2) Calculate the normal vector v of the structural plane pointing into the block from the structural plane parameters i

[0147] v 1 = D 1 @(sinα 1 @sinβ 1 , sinα 1 ·cosβ 1 , cosα 1 ) = (-0.5868, -0.4924, -0.6428)

[0148] v 2 = D 2 ·(sinα 2 ·sinβ 2 , sinα 2 ·cosβ 2 , cosα 2 ) = (-0.1330, 0.7544, -0.6428)

[0149] v 3 = D 3 ·(sinα 3 ·sinβ 3 , sinα 3 ·cosβ 3 , cosα 3 ) = (0.7660, 0.0000, -0.6428)

[0150] v 4 = D 4 ·(sinα 4 ·sinβ 4 , sinα 4 ·cosβ 4 , cosα 4 ) = (0.1710, 0.4698, -0.8660)

[0151] v 5 = D 5 ·(sinα 5·sinβ 5 ,sinα 5 ·cosβ 5 ,cosα 5 )=(0.3214, -0.8830, -0.3420)

[0152] Further obtain the matrix

[0153] (3) Construct the objective function and constraint equations in the standard form of quadratic programming with quadratic constraints.

[0154] (4) Use the interior point method to calculate and obtain the global optimal solution of the motion direction x = [-6.4677e-6, 1.557e-5, -1.0000]. |‖x‖ - 1| = 8.8307e-8 < ε, directly proceed to the next step.

[0155] (5) |‖x‖ - 1| < ε and x 3 = -1.0000 < -ε, continue to calculate the failure mode discrimination vector T = sign(Ax) = sign([0.6428, 0.6428, 0.6428, 0.8660, 0.3420]) = [1 1 1 1 1], all elements of T are 1, and the failure mode of the block is free fall.

[0156] (6) The safety factor of the free-falling block is 0.

[0157] Example Seven

[0158] A convex optimization terminal for determining the safety factor of a block, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the convex optimization method for determining the safety factor of a block as described above.

[0159] The memory can be used to store software programs and modules. The processor runs the software programs and modules stored in the memory to execute various functional applications and data processing of the terminal. The memory mainly includes a program storage area and a data storage area. Among them, the program storage area can store an operating system, execution programs required for at least one function, etc.

[0160] The data storage area can store data created according to the use of the terminal, etc. In addition, the memory can include high-speed random access memory, and can also include non-volatile memory, such as at least one magnetic disk storage device, a flash memory device, or other non-volatile solid-state storage devices.

[0161] A computer program product includes a computer program / instructions. When the computer program / instructions are executed by the processor, it implements the convex optimization method for determining the safety factor of a block as described above.

[0162] A computer program product includes a computer program or set of instructions for performing a specific task or implementing a specific function. These programs or instructions are designed to be executable by a processor to achieve a series of predefined steps or operations. The program product may be stored in various forms of computer storage media, such as memory, hard disk, solid state drive, optical disc, or other forms of digital storage devices. It may exist in the form of compiled binary code or in the form of scripts or bytecode executable by an interpreter. Through carefully designed algorithms and logical instructions, the program product enables the processor to process data in a specific order and manner to complete various functions such as data analysis, user interaction, device control, etc.

[0163] In the description of this specification, the description with reference to terms such as "one embodiment / way", "some embodiments / ways", "example", "specific example", or "some examples", etc. means that the specific features, structures, materials, or characteristics described in connection with the embodiment / way or example are included in at least one embodiment / way or example of this application. In this specification, the schematic expressions of the above terms do not necessarily refer to the same embodiment / way or example. Moreover, the specific features, structures, materials, or characteristics described can be combined in a suitable manner in any one or more embodiments / ways or examples. In addition, without contradiction, those skilled in the art can combine and combine the different embodiments / ways or examples described in this specification and the features of different embodiments / ways or examples.

[0164] In addition, the terms "first" and "second" are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include at least one of the features. In the description of this application, "a plurality" means at least two, such as two, three, etc., unless otherwise clearly and specifically defined.

[0165] Those skilled in the art should understand that the above embodiments are only for clearly explaining the present invention and are not intended to limit the scope of the present invention. For those skilled in the art, other changes or modifications can be made on the basis of the above invention, and these changes or modifications are still within the scope of the present invention.

Claims

1. A convex optimization method for determining the safety factor of a block, characterized in that: include: Obtaining structural surface parameters constituting the block, the structural surface parameters including direction parameters, dip angle, inclination angle and internal friction angle; According to the structural surface parameters, a normal vector matrix describing the direction of block movement is constructed; By constructing the objective function and constraint conditions of quadratic programming, the unit vector of the block's motion direction in three-dimensional space is determined, wherein the objective function is to maximize the inner product of the unit vector of the resultant external force on the block and the unit vector of the block's motion direction in three-dimensional space, and the constraint conditions include the quadratic norm constraint of the unit vector of the motion direction, the geometric relationship constraint between the block's motion direction and the structural surface normal vector, the component constraints of the unit vector of the block's motion direction, and the modulus length constraint of the unit vector of the block's motion direction; Solve for the unit vector of the block's motion direction and classify the blocks into four types: immovable blocks, movable blocks that can remain stable without friction, free-falling blocks, and sliding blocks; Determine the block safety factor according to the block type; Among them, according to the unit vector x of the block motion direction, x=[x1x2x3] T , the methods for determining the block type include: If ‖x‖<ε, ε is an integer and ε→0 + , then the block type is an immovable block; If |‖x‖-1|<ε and x3>ε, the block type is movable and friction is not required to maintain a stable block; If |‖x‖-1|<ε and x3<-ε, then calculate the discriminant vector T of the failure mode, T= If all elements in the discriminant vector T are 1, the block type is a free-fall block, otherwise the block type is a sliding block; If the block is an immovable block or a movable block that can remain stable without friction, the block safety factor FoS = +∞; If the block is a free-falling block, the block safety factor FoS = 0; If the block is a single-sided sliding block, the block safety factor is is the internal friction angle of the i-th structural surface; If the block is a double-sided sliding block, the block safety factor is Among them, F as F is the anti-sliding force per unit volume of the block, sl is the sliding force per unit volume of the block; in, N i and N j are the normal forces on the structural surfaces i and j, and are the internal friction angles of structural surfaces i and j respectively; F sl =|r·(v i ×v j )| / |v i ×v j |, where v i and v j are the unit normal vectors of the i-th and j-th structural surfaces pointing to the inside of the block respectively; 2. A convex optimization method for determining a block safety factor according to claim 1, characterized in that: The number of structural faces constituting the block is not less than 2 groups; the direction parameter D i Determined by the relative position of the block and the structural surface, when the block is located above the structural surface, D i =1, otherwise take D i =-1, i = 1, 2,…, n, n is the number of structural faces of the block, n ≥ 2.

3. A convex optimization method for determining a block safety factor according to claim 1, characterized in that: The construction methods of the normal vector matrix include: Determine the normal vector v of the i-th structural surface pointing to the inside of the block i , v i =D i ·(sinα i ·sinβ i ,sinα i ·cosβ i ,cosα i ), where D i is the direction parameter of the i-th structural surface, α i is the inclination angle of the i-th structural surface, β i is the inclination angle of the i-th structural surface; Arrange the normal vectors of all structural surfaces in rows to form the structural surface normal vector matrix A.

4. A convex optimization method for determining a block safety factor according to claim 1, characterized in that: The objective function is: maximizer T x, the constraints are: Where x = [x1 x2 x3] T is the unit vector of the block's motion direction in three-dimensional space, Q=diag(2,2,2) is a diagonal matrix, r is the unit vector of the resultant external force acting on the block, and inf is infinity.

5. A convex optimization method for determining a block safety factor according to claim 4, characterized in that: The interior point method is used to solve the objective function and constraint conditions to obtain the unit vector x of the block motion direction, and the vector x obtained by the solution is judged; If ‖x‖<ε, let r=-r, ε is an integer and ε→0 + , and recalculate the new unit vector x of the block movement direction through the objective function and constraint conditions, and then judge the failure mode; If |‖x‖-1|<ε, the unit vector x of the block motion direction can be directly obtained to determine the failure mode.

6. A convex optimization method for determining a block safety factor according to claim 1, characterized in that: The sliding block body includes a single-sided sliding block body and a double-sided sliding block body; If the discriminant vector T includes only one zero element, the type of the block is a single-sided sliding block; if the discriminant vector T includes only two zero elements, the type of the block is a double-sided sliding block.

7. A convex optimization terminal for determining a block safety factor, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the computer program, the convex optimization method for determining the safety factor of a block as described in any one of claims 1-6 is implemented.

8. A computer program product comprising a computer program / instructions, characterized in that When the computer program / instructions are executed by a processor, the convex optimization method for determining the safety factor of a block as described in any one of claims 1 to 6 is implemented.

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