Convex optimization method for determining block safety coefficient and related product

Abstract

The invention relates to the field of rock engineering, in particular to a convex optimization method for determining the safety coefficient of a block and a related product, and the method comprises the steps: obtaining structural plane parameters forming the block, and constructing a normal vector matrix for describing the motion direction of the block; the method comprises the following steps: determining a block motion direction unit vector of a block in a three-dimensional space by constructing a target function and a constraint condition of quadratic programming, solving the block motion direction unit vector, dividing the block into four types, and determining a block safety coefficient according to the block types; according to the method, many tedious and abstract mathematical theorems and concepts in the block theory are not involved, tedious calculation steps of a block theory vector method are avoided, the difficulty of the vector method in program implementation is reduced, and the block stability problem can be analyzed only by constructing a simple optimization objective function and a constraint equation.

Description

Technical Field

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

[0002] Block theory is a very important theory in rock mass stability analysis. Its essence is to apply geometric methods (topology and set theory) to study the types and mobility of blocks composed of structural surfaces and free surfaces, and to calculate the safety factor and evaluate its stability based on the static equilibrium conditions of the blocks.

[0003] The core of block theory is to find the key blocks on the free surface so that engineering treatment measures can be taken to maintain the stability of the rock mass. There is a strict procedure for determining the key blocks (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 determine whether the block is a finite block or an infinite block based on whether the block cone is an empty set. (2) Secondly, the mobility theorem is used to determine whether the finite block is a movable block or an immovable block based on whether the joint cone of the block is an empty set. (3) Then, the kinematic theorem is used to determine whether the movable block is a movable block that can maintain stability without friction, a free fall block, a single-sided sliding block, or a double-sided sliding block based on the necessary and sufficient 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. The unstable block with a safety factor less than 1 is the key block.

[0004] The above steps are implemented in computers through the vector method of block theory. However, there are two difficulties: (1) Block theory involves a large number of abstract mathematical concepts and theorems, which makes it difficult for engineering practitioners to understand; (2) The calculations of the above steps are cumbersome and difficult to implement in computer programs. The above difficulties limit the widespread use of block theory, and it is necessary to propose an alternative method that is simple in concept and easy to program. Summary of the invention

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

[0006] The present invention is achieved through the following technical solutions:

[0007] A convex optimization method for determining a block safety factor comprises:

[0008] Obtaining structural surface parameters constituting the block, the structural surface parameters including direction parameters, dip angle, inclination angle and internal friction angle;

[0009] According to the structural surface parameters, a normal vector matrix describing the direction of block movement is constructed;

[0010] 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;

[0011] 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;

[0012] Determine the block safety factor based on the block type.

[0013] Optionally, 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.

[0014] Specifically, the method for constructing the normal vector matrix includes:

[0015] 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;

[0016] Arrange the normal vectors of all structural surfaces in rows to form the structural surface normal vector matrix A.

[0017] Optionally, the objective function is: maximizerT x, the constraints are: Where x = [x 1 x 2 x 3 ] 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.

[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 motion direction, and the vector x obtained by the solution is judged;

[0019] 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;

[0020] If |‖x‖-1|<ε, the unit vector x of the block motion direction can be directly obtained to determine the failure mode.

[0021] Specifically, according to the unit vector x of the block motion direction, x=[x 1 x 2 x 3 ] T , and the methods for determining the block type include:

[0022] If ‖x‖<ε, 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 movable and does not require friction to remain a stable block;

[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, the block type is a free-fall block, otherwise the block type is a sliding block.

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

[0028] 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.

[0029] Specifically, if the block is an immovable block or a movable block that can remain stable 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-sided sliding block, the block safety factor is is the internal friction angle of the i-th structural surface;

[0032] 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 It is the sliding force per unit volume.

[0033] 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;

[0034] 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;

[0035]

[0036] A convex optimization terminal for determining a block safety factor comprises 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 a block safety factor as described above is implemented.

[0037] A computer program product comprises a computer program / instructions which, when executed by a processor, implement the convex optimization method for determining the block safety factor as described above.

[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 calculation. The method first determines whether the block is a finite block or an infinite block based on the finiteness theorem; then determines whether the block is a movable block through the mobility theorem; then uses the kinematic theorem to further analyze 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 entire process is based on geometric calculation and vector analysis, and can be automatically implemented through a computer program to reduce human judgment errors.

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

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

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

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

[0044] Figure 3 It is the morphology and stereographic projection diagram of the block in Example 3 described in the present invention.

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

[0046] Figure 5 It is the morphology and stereographic projection diagram of the block in Example 5 described in the present invention.

[0047] Figure 6 It is the morphology and stereographic projection diagram of the block in Example 6 described in the present invention. DETAILED DESCRIPTION

[0048] To make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and implementation methods. It is understood that the specific implementation methods described herein are only used to explain the relevant content, rather than to limit the present invention.

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

[0050] In the absence of conflict, the embodiments and features of the embodiments of the present invention may be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0051] Embodiment 1

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

[0053] S1. Obtaining structural surface parameters constituting the block, the structural surface parameters including direction parameters, dip angle, inclination angle and internal friction angle.

[0054] The number of structural faces constituting the block is not less than 2 groups, usually 3 or 4 groups. 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 planes of the block, n ≥ 2. Structural plane occurrence parameter α i and β i The internal friction angle of the structure surface is obtained by compass field testing. Determined by laboratory testing or engineering analogy.

[0055] S2. Based on the structural surface parameters, a normal vector matrix describing the direction of block movement is constructed. The method for constructing the normal vector matrix includes:

[0056] 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;

[0057] Arrange the normal vectors of all structural surfaces in rows to form the structural surface normal vector matrix A.

[0058] S3. Determine the unit vector of the block's motion direction in three-dimensional space by constructing the objective function and constraints of quadratic programming, where 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's motion direction in three-dimensional space, and the constraints include the quadratic norm constraint on 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 constraint of the unit vector of the block's motion direction.

[0059] The objective function is: maximizer T x, the constraints are: Where x = [x 1 x 2 x 3 ] 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.

[0060] In general, the components x i The value is between -1 and 1. In addition, x must 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 net external force acting on the block. When the net external force acting on the block is only gravity, the unit vector r = [0 0 -1] T ; When the total external force acting on the block includes water pressure or seismic inertia force, its components are recalculated under the condition that ‖r‖=1.

[0062] S4. Solve the unit vector of the block's motion direction.

[0063] 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;

[0064] 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, that is, jump to S5 after updating the unit vector x;

[0065] If |‖x‖-1|<ε, the unit vector x of the block motion direction is directly obtained to determine the failure mode, that is, jump directly to S5.

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

[0067] Methods for determining block types include:

[0068] If ‖x‖<ε, ε is an integer and ε→0 + , that is, ε is a minimum 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 movable and does not require friction to remain a stable block;

[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, the block type is a free-fall block, otherwise the block type is a sliding block.

[0072] If the discriminant vector T includes only one 0 element and the rest of the elements are all 1, the block type is a single-sided sliding block, and the structural surface corresponding to the row number of the 0 element is a sliding surface of single-sided sliding.

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

[0074] S6. Determine the block safety factor based on the block type.

[0075] If the block is an immovable block or a movable block that can remain stable without friction, the block safety factor FoS = +∞; that is, the block is absolutely stable under the action of the combined external force.

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

[0077] If the block is a sliding block, the block safety factor and the structural surface parameter (inclination angle α i , inclination angle β i and internal friction angle ) is related to the calculation of the safety factor using the limit equilibrium formula for single-sided sliding or double-sided sliding.

[0078] 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;

[0079] 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 It is the sliding force per unit volume.

[0080] in, N i and N j are the normal forces on structural surfaces i and j, and are the internal friction angles of structural surfaces 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 of the i-th and j-th structural surfaces pointing to the inside of the block respectively;

[0082]

[0083] Embodiment 2

[0084] This embodiment provides a single-sided sliding block having five planes.

[0085] like Figure 2 As shown in (a), the block contains 5 planes, specifically 4 structural planes J1-J4 and 1 free surface EF. The inclination and dip angle parameters of the structural planes and free surface are shown in Table 1. Please note that the method proposed in the present invention does not require the free surface parameters. Figure 2 (a) The block morphology and perspective view show that the blocks are located above the structural plane J1, above the structural plane J2, below the structural plane J3 and above the structural plane J4, so the direction parameter D 1 =1,D 2 =1,D 3 =-1, D 4 = 1. The total external force only considers gravity, that is, r = [0 0 -1] T All structural surfaces have the same internal friction angle

[0086] Table 1 Structural surface and free surface parameters of the block in Example 1

[0087]

[0088] Figure 2 (b) is the stereographic projection of the block, which shows the projection of the block cone (empty set does not exist) and the joint cone (shaded area). The relevant theorems of block theory can be used to identify the failure mode and movement direction of the block from the stereographic projection. Figure 2 (b) Display (The projection of the resultant external force r on the structural surface J4) is the movement direction of the block, the failure mode is single-sided sliding, and the sliding surface is the structural surface J4. The results obtained from the stereographic projection 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 motion 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, it is necessary to introduce a minimum value ε as a criterion for judging whether the solution is 0. Here, ε = 1e-4.

[0090] (1) The structural surface parameters required for calculation are given in Table 1.

[0091] (2) Calculate the structural surface normal vector v pointing to the interior of the block from the structural surface 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 get the matrix

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

[0098] maximize T x

[0099]

[0100] 0≤Ax≤inf

[0101] -1≤x i ≤1,i=1,2,3,

[0102] (4) The global optimal solution of the motion direction is calculated using the interior point method: x = [-0.1632, -0.9254, -0.3420]. |‖x‖-1| = 3.9135e-09 < ε, and the next step is directly entered.

[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] T contains only one 0 element and the rest of the elements are all 1. The failure mode of the block is single-sided sliding. The row number of the 0 element corresponds to the structural surface J4, which is the sliding surface of single-sided sliding. It can be seen that the results of this method are consistent with the stereographic projection results.

[0105] (6) Based on the results of the previous step, select the formula for single-sided sliding to calculate the safety factor

[0106] Embodiment 3

[0107] It is also a convex block with 5 planes. The morphology and stereographic projection of the block are shown in Figure 3 The block sliding direction obtained by stereographic projection is (The projection of the resultant external force r on the intersection of structural surfaces J1 and J2), the failure mode is double-sided sliding, and the sliding surfaces are structural surfaces J1 and J2.

[0108] (1) The parameters of the structural surfaces J1-J3 and the free surface of the block in the third embodiment are the same as those of the block in the second embodiment, except that the parameters of the structural surface J4 are different, 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 the matrix

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

[0111] (4) The global optimal solution of the motion direction is calculated using the interior point method: x = [-0.1296, -0.8375, -0.5309]. |‖x‖-1| = 5.2219e-10 < ε, and the next step is directly entered.

[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 0 elements and the rest are all 1. The failure mode of the block is double-sided sliding. The row number of the 0 element corresponds to the structural planes J1 and J2, which are the sliding planes of double-sided sliding. It can be seen that the results of this method are consistent with the results of the stereographic projection.

[0114] (6) Based on the results of the previous step, select the double-sided sliding formula to calculate the safety factor

[0115]

[0116] FoS=F as / F s＝ 1.2533.

[0117] Embodiment 4

[0118] The convex block body also includes 5 planes. The morphology and stereographic projection of the block body in Example 4 are shown in FIG. Figure 4 According to the stereographic projection, it can be seen that the joint cone of the block is an empty set, so the block is an immovable block.

[0119] (1) The parameters of the structural surfaces J1-J3 and the free surface of the block in the fourth embodiment are the same as those of the block in the second embodiment, except that the parameters of the structural surface J4 are different, specifically α 4 =20°,β 4 =20° and D 4 =1.

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

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

[0122] (4) The interior point method is used to calculate the global optimal solution of the motion direction x = [-3.8368e-11, 3.4362e-11, 3.7597e-11].

[0123] ‖x‖=6.3769e-11<ε, let r=-r, return 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] Embodiment 5

[0127] Provides a convex body with 4 movable planes that can remain stable without friction.

[0128] Example 5 is a convex block body including 4 planes. The morphology and stereographic projection of the block body are shown in Figure 5 According to the stereographic projection, the block has a joint cone, so the block is movable, but it does not meet the necessary and sufficient conditions for free fall, single-sided sliding and double-sided sliding, so it is movable and can remain stable without friction.

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

[0130] Table 2 Structural surface and free surface parameters of the block in Example 4

[0131]

[0132] (2) Calculate the structural surface normal vector v pointing to the interior of the block from the structural surface parameters i

[0133] v1 =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 get the matrix

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

[0138] (4) The interior point method is used to calculate the global optimal solution of the motion direction x = [-6.1503e-10, -9.3540e-10, 1.7729e-09]. ‖x‖ = 2.0968e-09 < ε, let r = -r, return to (3) and recalculate x = [-8.5216e-06, -0.3420, 0.9397], and then proceed to the next step.

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

[0140] (6) The safety factor of a movable block that can remain stable without friction is +∞.

[0141] Embodiment 6

[0142] Provides a convex block with 6 planes. The morphology and stereographic projection of the block can be seen in Figure 6According to the stereographic projection, it can be seen that the failure mode of the block is free fall, and the direction of movement is the same as the direction of the resultant external force.

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

[0144] Table 3 Structural surface and free surface parameters of the block in Example 6

[0145]

[0146] (2) Calculate the structural surface normal vector v pointing to the interior of the block from the structural surface 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 ·sinβ5 ,sinα 5 ·cosβ 5 ,cosα 5 )=(0.3214,-0.8830,-0.3420)

[0152] Further get the matrix

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

[0154] (4) The global optimal solution of the motion direction is calculated using the interior point method x = [-6.4677e-6, 1.557e-5, -1.0000]. |‖x‖-1| = 8.8307e-8 < ε, and the next step is directly entered.

[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 a free-falling block is 0.

[0157] Embodiment 7

[0158] A convex optimization terminal for determining a block safety factor comprises 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 a block safety factor as described above is implemented.

[0159] The memory can be used to store software programs and modules. The processor executes various functional applications and data processing of the terminal by running the software programs and modules stored in the memory. The memory can mainly include a program storage area and a data storage area, wherein the program storage area can store an operating system, an execution program 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 may include a high-speed random access memory, and may also include a non-volatile memory, such as at least one disk storage device, a flash memory device, or other volatile solid-state storage devices.

[0161] A computer program product comprises a computer program / instructions which, when executed by a processor, implement the convex optimization method for determining the block safety factor as described above.

[0162] A computer program product includes a computer program or set of instructions for performing specific tasks or implementing specific functions. These programs or instructions are designed to be executed by a processor to implement 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 disk or other forms of digital storage devices. It may exist in the form of compiled binary code or in the form of scripts or bytecodes that can be executed by an interpreter. The program product uses carefully designed algorithms and logical instructions to enable 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 the terms "one embodiment / method", "some embodiments / methods", "example", "specific example", or "some examples" etc. means that the specific features, structures, materials or characteristics described in conjunction with the embodiment / method or example are included in at least one embodiment / method or example of the present application. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment / method or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments / methods or examples in a suitable manner. In addition, those skilled in the art may combine and combine the different embodiments / methods or examples described in this specification and the features of the different embodiments / methods or examples, unless they are contradictory.

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

[0165] It should be understood by those skilled in the art that the above embodiments are only for the purpose of clearly illustrating 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 may be made based on 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 based on the block type.

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: maximize r 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: According to the unit vector x in the direction of block movement, x=[x1x2x3] T , and 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, 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.

7. A convex optimization method for determining a block safety factor according to claim 6, 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.

8. A convex optimization method for determining a block safety factor according to claim 7, characterized in that: 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 It is the sliding force per unit volume. 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; 9. 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 1-8 is implemented.

10. 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 8 is implemented.

Images