An underground high-pressure water tunnel lining crack analysis method

Abstract

A method for analyzing cracks in underground high-pressure water tunnel lining, which is characterized by a large range of crack-free areas and a local crack-sensitive area. The method uses the proportional boundary finite element method to discretize the crack-free area (only the boundary needs to be discretized) and the near-field dynamics to focus on the crack area. This avoids the high cost of full-domain discretization in pure near-field dynamics and overcomes the inefficiency of traditional methods, significantly improving analysis efficiency. The coupling is achieved based on the common point force balance condition, eliminating the need for a transition area or complex shape function in the lining "crack-free area-crack area", avoiding artificial setting errors, reducing modeling complexity, and enhancing the applicability of the method in actual tunnel projects. The method can accurately predict the pre-existing crack and spontaneous crack propagation behavior of underground high-pressure water tunnel lining, especially for stress concentration-type crack scenarios around holes, providing reliable support for lining design, anti-seepage arrangement, and operation and maintenance.

Description

Technical Field

[0001] This invention relates to the field of analysis technology for cracks in the lining of underground high-pressure water conveyance tunnels, and particularly to a method for analyzing cracks in the lining of underground high-pressure water conveyance tunnels. Background Technology

[0002] In the field of underground water conservancy engineering, underground high-pressure water conveyance tunnels are the core water conveyance carriers of long-distance water transfer and pumped storage systems. The integrity of their lining structure directly determines the safety of the project and the efficiency of water conveyance. The lining is subjected to high-pressure internal water pressure, temperature fluctuation stress, and localized maintenance loads over long periods, making it prone to crack initiation in stress concentration areas (such as construction joints, around openings, and on the inner surface). If these cracks continue to propagate, they will not only cause high-pressure water leakage and increase water conveyance energy consumption, but may also lead to softening of the surrounding rock and instability of the lining, seriously threatening the safe operation of the project. Therefore, accurately calculating the crack initiation threshold and propagation path of underground high-pressure water conveyance tunnel linings is a critical requirement for ensuring the safety of project design and operation.

[0003] Existing technologies for analyzing cracks in the lining of underground high-pressure water conveyance tunnels have significant limitations: While extended finite element method (EPM) and meshless methods can handle some crack problems, simulating spontaneous cracking under high-pressure internal water pressure and crack propagation at construction joints requires frequent mesh refinement or complex interpolation, making modeling cumbersome and prone to errors due to mesh distortion. Pure near-field dynamics methods can naturally capture the discontinuity of cracks, but due to their nonlocal nature, they require discretizing a massive number of material points, resulting in extremely high computational costs, and are prone to surface effect errors when handling lining boundary conditions. Simplified analytical methods, while computationally inexpensive, are based on idealized assumptions such as "homogeneous elastic body" and "regular cracks," failing to reflect the stress field around non-homogeneous interfaces and complex pores in actual engineering, and thus lacking sufficient accuracy in simulating spontaneous cracks and multiple crack propagation.

[0004] Existing coupling methods are also difficult to adapt to the needs of lining analysis: near-field dynamics-finite element coupling requires the setting of a transition region, but the lining cross-sectional space is limited, and the transition region easily occupies the effective analysis space, resulting in complex modeling and low efficiency; near-field dynamics-boundary element coupling relies on the fundamental solution of the material and calculates singular integrals, which limits its applicability when dealing with heterogeneous interfaces and cracks around pores. Scaled boundary finite element method ( SBFEM While this method can efficiently solve the mechanical response of large-scale crack-free regions by only requiring discrete boundaries, its coupling method with near-field dynamics has not yet been reported, failing to provide an integrated solution of "large-scale efficient calculation + local high-precision simulation" for lining crack analysis. Therefore, there is an urgent need to propose a low-cost, simple-to-model, and high-precision method for analyzing cracks in the lining of underground high-pressure water conveyance tunnels to overcome the limitations of existing technologies. Summary of the Invention

[0005] The main objective of this invention is to provide a method for analyzing cracks in the lining of underground high-pressure water conveyance tunnels, aiming to solve the technical problems of high computational cost, complex modeling, and insufficient accuracy in existing analyses of cracks in the lining of underground high-pressure water conveyance tunnels.

[0006] To achieve the above objectives, the present invention provides a method for analyzing cracks in the lining of underground high-pressure water conveyance tunnels, the method comprising the following steps:

[0007] S1. First, the elastic body is divided into two computational regions based on the crack distribution: For the structurally intact, crack-free computational region, a scaled boundary finite element method is used. SBFEM For modeling, scaled boundary finite element method only needs to discretize the boundary of the crack-free computational region to complete the representation of the crack-free computational region; for computational regions with existing cracks or potential crack paths, which will be collectively referred to as cracked computational regions below, near-field dynamics is used. PD By modeling, near-field dynamics naturally describes the initiation and propagation behavior of cracks through the nonlocal interactions between material points;

[0008] S2. By introducing a scaled boundary coordinate system, the momentum equation in the Cartesian coordinate system is transformed into a first-order ordinary differential governing equation in the scaled boundary coordinate system, thereby solving for the stiffness matrix of the crack-free computational region: First, an analytical method is used in the radial direction, while numerical discretization is used in the circumferential direction to convert the coordinates in the Cartesian coordinate system to those in the scaled boundary coordinate system. Then, based on the principle of virtual work, the momentum equation in the Cartesian coordinate system is derived, transforming it into a first-order ordinary differential governing equation in the scaled boundary coordinate system. Finally, by solving the eigenvalues ​​of the first-order ordinary differential governing equation, the stiffness matrix of the crack-free computational region is obtained.

[0009] S3. In the crack calculation region, the elastic body is discretized into a series of material points. Each material point is connected to other material points in its neighborhood through bonds, and the bond-basis peridynamics is based on peridynamics. BBPD Theoretically constructing the forces between material points. By matrixing the forces between material points, the micro-stiffness matrix between any two material points is derived. According to the finite element assembly rules, the micro-stiffness matrices of all material points are integrated into the overall stiffness matrix of the crack calculation region;

[0010] S4. At the common node at the boundary between the crack-free calculation region and the crack calculation region, the two calculation regions are directly coupled based on the force equilibrium condition: by dividing the stiffness matrices of the two regions into blocks according to the internal nodes and the common node respectively, and then superimposing and combining them on the stiffness matrix corresponding to the common node, a unified overall stiffness matrix is ​​finally formed.

[0011] S5. An implicit solution algorithm combined with load increment steps is used for nonlinear calculation: Within each load step, the elastic body displacement is first solved, and then all keys in the crack calculation region are traversed. The fracture status of the keys is determined based on the current displacement. When a key fracture is detected, the overall stiffness matrix of the elastic body is updated, and iterative recalculation is performed within the current load step until the elastic body reaches an equilibrium state and no new key fracture occurs. Then, the next load increment is applied. This process is repeated until the calculation of all specified load steps is completed.

[0012] The beneficial effects of this invention are:

[0013] (1) In view of the characteristics of “large-scale crack-free area + local crack-sensitive area” in the lining of underground high-pressure water conveyance tunnel, the crack-free area is discretized by proportional boundary finite element method (only boundary discretization is required) and the crack area is focused by near field dynamics. This avoids the high cost of pure near field dynamics global discretization, and overcomes the inefficiency of traditional methods, thus significantly improving the analysis efficiency.

[0014] (2) Coupling is achieved based on the common point force balance condition, without the need to set a transition region or complex shape function in the "crack-free zone-crack zone" of the lining, avoiding human error setting, reducing modeling complexity, and enhancing the applicability of the method in actual tunnel engineering;

[0015] (3) It can accurately predict the pre-existing crack and spontaneous crack propagation behavior of underground high-pressure water conveyance tunnel lining, especially suitable for stress concentration crack scenarios around the hole, providing reliable support for lining design, seepage prevention layout and operation and maintenance. Attached Figure Description

[0016] Figure 1 For the purposes of this invention SBFEM-PD A schematic diagram of the process for analyzing cracks in the lining of underground high-pressure water conveyance tunnels using implicit coupling.

[0017] Figure 2 for Figure 1 The schematic diagram shown below;

[0018] Figure 3 Model diagram for an example;

[0019] Figure 4 This is a schematic diagram of the crack patterns in a perforated square plate. Detailed Implementation

[0020] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. The realization of the invention's objectives, functional characteristics, and advantages will be further explained in conjunction with the embodiments and with reference to the accompanying drawings.

[0021] like Figure 1 , Figure 2 As shown, Figure 1This invention discloses a method for analyzing cracks in the lining of underground high-pressure water conveyance tunnels. The detailed method includes the following steps:

[0022] S1. First, the elastic body is divided into two computational regions based on the crack distribution: For the structurally intact, crack-free computational region, a scaled boundary finite element method is used. SBFEM For modeling, scaled boundary finite element method only needs to discretize the boundary of the crack-free computational region to complete the representation of the crack-free computational region; for computational regions with existing cracks or potential crack paths, which will be collectively referred to as cracked computational regions below, near-field dynamics is used. PD By modeling, near-field dynamics naturally describes the initiation and propagation behavior of cracks through the nonlocal interactions between material points;

[0023] S2. By introducing a scaled boundary coordinate system, the momentum equation in the Cartesian coordinate system is transformed into a first-order ordinary differential governing equation in the scaled boundary coordinate system, thereby solving for the stiffness matrix of the crack-free computational region: First, an analytical method is used in the radial direction, while numerical discretization is used in the circumferential direction to convert the coordinates in the Cartesian coordinate system to those in the scaled boundary coordinate system. Then, based on the principle of virtual work, the momentum equation in the Cartesian coordinate system is derived, transforming it into a first-order ordinary differential governing equation in the scaled boundary coordinate system. Finally, by solving the eigenvalues ​​of the first-order ordinary differential governing equation, the stiffness matrix of the crack-free computational region is obtained.

[0024] S3. In the crack calculation region, the elastic body is discretized into a series of material points. Each material point is connected to other material points in its neighborhood through bonds, and the bond-basis peridynamics is based on peridynamics. BBPD Theoretically constructing the forces between material points. By matrixing the forces between material points, the micro-stiffness matrix between any two material points is derived. According to the finite element assembly rules, the micro-stiffness matrices of all material points are integrated into the overall stiffness matrix of the crack calculation region;

[0025] S4. At the common node at the boundary between the crack-free calculation region and the crack calculation region, the two calculation regions are directly coupled based on the force equilibrium condition: by dividing the stiffness matrices of the two regions into blocks according to the internal nodes and the common node respectively, and then superimposing and combining them on the stiffness matrix corresponding to the common node, a unified overall stiffness matrix is ​​finally formed.

[0026] S5. An implicit solution algorithm combined with load increment steps is used for nonlinear calculation: Within each load step, the elastic body displacement is first solved, and then all keys in the crack calculation region are traversed. The fracture status of the keys is determined based on the current displacement. When a key fracture is detected, the overall stiffness matrix of the elastic body is updated, and iterative recalculation is performed within the current load step until the elastic body reaches an equilibrium state and no new key fracture occurs. Then, the next load increment is applied. This process is repeated until the calculation of all specified load steps is completed.

[0027] The specific steps in step S1 are as follows: The two-dimensional elastic body is divided into a crack-free computational region (divided using proportional boundary finite element method) and a cracked computational region (divided using near-field dynamics), and the momentum equation, geometric equation, and constitutive equation expressions of the elastic body are given:

[0028]

[0029]

[0030]

[0031] in, For gradient operators, For stress, For external loads, the load is a volume force load. and surface force load composition, In response, For displacement, D It is the elasticity matrix.

[0032] The specific steps in step S2 are as follows: SBFEM By introducing a scaled boundary coordinate system within the crack-free computational region, the relationship between coordinates in the Cartesian coordinate system and coordinates in the scaled boundary coordinate system is as follows:

[0033]

[0034]

[0035] in, and This represents the coordinates of the point to be found in the Cartesian coordinate system. Represents the radial coordinates of the proportional boundary coordinate system. Represents the circumferential coordinates of the proportional boundary coordinate system; It is a shape function; and These are the nodal coordinates in the Cartesian coordinate system. The transformation equations of the differential operators in the Cartesian coordinate system and the proportional boundary coordinate system are expressed as:

[0036]

[0037]

[0038] Represented in matrix form as follows:

[0039]

[0040] Where J represents the Jacobian matrix;

[0041] Two-dimensional elastic differential operators are represented using scaled coordinates:

[0042]

[0043] in , All are coefficient matrices, specifically represented as:

[0044]

[0045]

[0046] in, and This means taking the partial derivative with respect to x and y, i.e. and ;

[0047] Displacement and stress are expressed in Cartesian coordinates as follows:

[0048]

[0049]

[0050] in, Displacement in Cartesian coordinates with respect to the proportional boundary coordinate system The function, Displacement Regarding coordinates The first-order partial derivative, Displacement in Cartesian coordinates with respect to the proportional boundary coordinate system and The function; Displacement in Cartesian coordinates with respect to the proportional boundary coordinate system and The function; Displacement in Cartesian coordinates with respect to the proportional boundary coordinate system and The function; , The strain matrix is ​​specifically expressed as:

[0051]

[0052] The virtual work form of the momentum equation is:

[0053]

[0054] in, Indicates variation, Indicates volume, Displacement in Cartesian coordinates with respect to the proportional boundary coordinate system The function, The external load in the Cartesian coordinate system is expressed with respect to the proportional boundary coordinate system. The function is simplified to obtain the first-order ordinary differential governing equations of the proportional boundary coordinate system:

[0055]

[0056]

[0057] in, Displacement in Cartesian coordinates with respect to the proportional boundary coordinate system The second-order partial derivative, superscript " " represents the transpose of a matrix. , , All are coefficient matrices, and their specific expressions are as follows:

[0058]

[0059]

[0060]

[0061] The matrix expression of the first-order ordinary differential governing equation is as follows:

[0062]

[0063] in, Represents state vector pairs The first-order partial derivative, Represents the state vector. The Hamiltonian matrix is ​​specifically represented as:

[0064]

[0065] The first-order ordinary differential governing equation is solved using the eigenvalue problem, specifically expressed as:

[0066]

[0067]

[0068]

[0069] in and For the eigenvalues ​​and eigenvectors, express The upper half; , , , express It is divided into four equal parts.

[0070] Finally, the stiffness matrix of the crack-free computational region:

[0071] .

[0072] The specific steps in step S3 are as follows: Using the bond-based near-field dynamics theory, the crack calculation region is assumed to consist of multiple material points, each material point... a Volume is V a Each material point a With the radius of the adjacent region δ h Other material points a' There are interactions between the cracks, and the set of all material points within the computational region is called the influence domain H. a Each material point a The momentum equation is as follows:

[0073]

[0074] in, For material points The displacement; For material points a The displacement; For material points a With material points a' The opposing forces between them; For material points a The volume force density, Material point Volume. In the theory of near-field dynamics of bond groups, The description is as follows:

[0075]

[0076] in, and They are material points a and material points The relative position and relative displacement. For microelastic brittle materials. Represented as:

[0077]

[0078] in, It is a historically relevant scalar-valued function; Represents the micromodulus function; It is a kernel function that characterizes the long-range force intensity between two material points; Let be the elongation of the bond, and the expressions for each term are as follows:

[0079]

[0080]

[0081]

[0082]

[0083] in, Indicates the elastic modulus; Indicates Poisson's ratio; s c This indicates the critical elongation.

[0084] If the bond elongation greater than the critical elongation of the bond s c If this happens, the bond between the two material points will break. The critical elongation of the bond... s c From fracture energy G 0 A decision can be written as:

[0085]

[0086] Damage is defined in PD as follows:

[0087]

[0088] The momentum equation is obtained by matrixing:

[0089]

[0090] in, and Indicates two material points; Material point volume; and Material point and The displacement; Material point The volumetric force density;

[0091] In order to achieve PD and SBFEM The coupling is addressed by proposing a microstiffness matrix in PD. k ,expression At the same time, it is represented in the local coordinate system as The forces between the global coordinate system and the local coordinate system and displacement The relationship is achieved through coordinate transformation matrix T express:

[0092]

[0093]

[0094]

[0095] in, l and m It is the coordinate transformation factor between the local coordinate system and the global coordinate system;

[0096] Therefore, material points in the global coordinate system Relative to material point The microstiffness matrix is:

[0097]

[0098] Material Point The stiffness matrix is ​​formed by combining the stiffness matrix with all material points in the neighborhood. And further combine the stiffness matrices of all material points. :

[0099]

[0100]

[0101] In step S4, the specific steps are as follows: First, in SBFEM and PD In the coupled model constructed in the model, the forces at the common points have the following relationship:

[0102]

[0103] in, Indicates the distance between two material points;

[0104] The result obtained in S2 SBFEM stiffness matrix according to SBFEM The unique points and common points are divided into blocks as follows:

[0105]

[0106]

[0107] in, The rows and columns are all corresponding SBFEM Unique features; The corresponding row SBFEM Unique points, corresponding common points; The rows correspond to common points, and the columns correspond to... SBFEM Unique features; The rows and columns all correspond to common points. express SBFEM Unique displacement; Indicates the displacement of the common points; express SBFEM Unique strength; Forces representing common points;

[0108] S3 PD The stiffness matrix is ​​divided into blocks as follows:

[0109]

[0110]

[0111] in, The rows and columns all correspond to common points; The rows correspond to common points, and the columns correspond to... PD Unique features; The corresponding row PD Unique points, corresponding common points; The rows and columns are all corresponding PD It has a unique feature. Indicates the displacement of the common points; express PD Unique displacement; This represents the volumetric force density at the common point; express PD Unique volumetric force density.

[0112] The global stiffness matrix of the computational domain is calculated by... SBFEM and PD The stiffness matrix is ​​assembled to obtain:

[0113]

[0114] in, yes and General term, i.e. = = .

[0115] The specific steps in step S5 are as follows: After assembling the overall stiffness matrix obtained in S4, apply displacement constraints and load boundaries. Set a load step counter. m =0, initial load increment P =0. The load step calculation process is as follows:

[0116] S5.1, Settings m = m +1, and update the load increment. P ;

[0117] S5.2 Solving the system of linear equations Get all nodes (including SBFEM Nodes and PD Displacement of the material point;

[0118] S5.3 Calculate the elongation of the point bonds in the crack calculation region using formula S3. ;

[0119] S5.4, if <s c Proceed to step S5.5; otherwise, update the stiffness matrix of the crack calculation region and return to step S5.1.

[0120] S5.5 If all load steps have been calculated, exit the analysis; otherwise, return to step S5.4.

[0121] This embodiment uses the crack analysis of a porous structure in the lining of an underground high-pressure water conveyance tunnel as an example to verify the "proportional boundary finite element method" proposed in this invention. SBFEM - Perifield Dynamics PD The effectiveness of the "implicit coupling method" in predicting cracks around lining holes.

[0122] To address the issue of crack initiation and propagation in the lining of underground high-pressure water conveyance tunnels under high internal water pressure, a 50m × 50m underground structure was selected as the research object, and a diameter [not specified] was set at its center. D =8m circular hole, width of SBFEM area on both sides l 1 = 10m and l 3=10m, middle PD area l 2=30m. By simulating the crack initiation and propagation process of a square plate under tension, the crack development law around the hole is analyzed, providing numerical support for the crack-resistant design of the lining with holes.

[0123] like Figure 3As shown, the square plate is made of an elastic material with an elastic modulus of [missing value]. E= 30 GPa, Poisson's ratio v= 1 / 3 (default setting for plane stress problems with near-field dynamics of bonded bases), critical elongation of the crack computational region s c =8×10 -4 (Derived from material fracture energy). The square plate is divided into two crack-free computational regions and one cracked computational region. The material point spacing in the cracked computational region is... x=0.5m, radius of influence zone =4 x. The displacement load for each step is 2 × 10. -8 m.

[0124] Crack patterns in perforated square plates are as follows Figure 4 As shown, it can be observed that the cracks in the perforated square plate initiate at the stress-sensitive points above and below the circular holes, and the propagation direction is basically perpendicular to the load application direction. This is consistent with the actual propagation law of cracks around the holes in the lining of underground high-pressure water conveyance tunnels, indicating that the coupling method proposed in this invention can effectively handle the crack analysis problem of perforated lining structures.

Claims

1. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel, characterized by, The method comprises the following steps: S1, first, the elastomer is divided into two calculation regions according to the crack distribution: for the intact crack-free calculation region, a scaled boundary finite element (SBFEM) is used for modeling, and the scaled boundary finite element only needs to discretize the boundary of the crack-free calculation region to complete the representation of the crack-free calculation region; for the calculation region with existing cracks or potential crack paths, the calculation region with existing cracks or potential crack paths is collectively referred to as a crack calculation region in the following, and a peridynamics (PD) is used for modeling, which naturally describes the initiation and propagation behavior of cracks through non-local interaction between material points; S2, the momentum equation in the Cartesian coordinate system is converted into a first-order ordinary differential control equation in the scaled boundary coordinate system by introducing the scaled boundary coordinate system, and then the stiffness matrix of the crack-free calculation region is solved: first, the analytical method is used in the radial direction, and the numerical discretization is used in the circumferential direction, and the coordinates in the Cartesian coordinate system are converted into the coordinates in the scaled boundary coordinate system; then, the momentum equation in the Cartesian coordinate system is derived based on the virtual work principle, and is converted into the first-order ordinary differential control equation form in the scaled boundary coordinate system; finally, the stiffness matrix of the crack-free calculation region is obtained by solving the eigenvalue of the first-order ordinary differential control equation; S3, in the crack calculation region, the elastomer is discretized into a series of material points, each material point is connected to other material points within the neighborhood range through a bond, and the counterforce between the material points is constructed based on the bond-based peridynamics theory of peridynamics; by matrix processing the counterforce between the material points, the micro-stiffness matrix between any two material points is derived; according to the finite element assembly rule, the micro-stiffness matrices of all material points are integrated into the overall stiffness matrix of the crack calculation region; S4, at the common nodes between the crack-free calculation region and the crack calculation region, the direct coupling of the two calculation regions is realized based on the force balance condition: by block processing the stiffness matrices of the two regions according to the internal nodes and the common nodes respectively, and then superimposing and combining the stiffness matrices corresponding to the common nodes, a unified overall stiffness matrix is finally formed; S5, an implicit solution algorithm is used in combination with load increment steps for nonlinear calculation: in each load step, first, the displacement of the elastomer is solved, then all the bonds in the crack calculation region are traversed, and the fracture of the bond is judged according to the current displacement; when it is detected that the bond is fractured, the overall stiffness matrix of the elastomer is updated and iterative re-computation is performed in the current load step until the elastomer reaches an equilibrium state and no new bond fracture occurs, and then the next load increment is applied; this process is repeated until the calculation of all specified load steps is completed.

2. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 1, characterized in that, In the step S1, the two-dimensional elastomer is divided into a crack-free calculation region and a crack calculation region, and expressions of momentum equations, geometric equations and constitutive equations of the elastomer are given: where is the gradient operator, is the stress, is the external load, consisting of volume force loads and surface force loads , is the strain, is the displacement, D is the elasticity matrix.

3. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 2, characterized in that, The specific steps of the step S2 are as follows: SBFEM By introducing the proportional boundary coordinate system in the crack-free calculation area, the relationship between the coordinates in the Cartesian coordinate system and the coordinates in the proportional boundary coordinate system is: wherein and denotes the coordinates of the point to be found in the Cartesian coordinate system, denotes the radial coordinate of the proportional boundary coordinate system, denotes the circumferential coordinate of the proportional boundary coordinate system; is a shape function; and are the node coordinates in the Cartesian coordinate system; the transformation equation of the differential operator in the Cartesian coordinate system and the proportional boundary coordinate system is represented as: which is expressed in matrix form as: Wherein, J represents the Jacobian matrix; The two-dimensional elastic differential operator is expressed in scaled coordinates: wherein , are coefficient matrices, and are specifically represented as: wherein and denotes the partial derivative with respect to x and y, i.e. and ; The displacement and stress are expressed in the Cartesian coordinate system as: wherein denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system , denotes the first order partial derivative of the displacement with respect to the coordinate , denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; denotes the function of the displacement in the Cartesian coordinate system with respect to the scaled boundary coordinate system and ; , denotes the strain matrix, which is expressed as: The virtual work form of the momentum equation is: wherein, denotes the variation, denotes the volume, denotes the function of the displacement in the Cartesian coordinate system with respect to the proportional boundary coordinate system, denotes the function of the external load in the Cartesian coordinate system with respect to the proportional boundary coordinate system, simplifying to the first-order ordinary differential control equation of the proportional boundary coordinate system:​ wherein denotes the second order partial derivative of the displacement in the Cartesian coordinate system with respect to the second order partial derivative of the displacement in the proportional boundary coordinate system, the upper index denotes the transposed symbol of the matrix, are coefficient matrices, the specific expressions of which are:​​​ The matrix form of the first-order ordinary differential control equation is wherein denotes the first order partial derivative of the state vector with respect to the state vector denotes the state vector is the Hamiltonian matrix, which is given by The first-order ordinary differential control equation is solved by using an eigenvalue problem, and is specifically expressed as wherein and eigenvalues and eigenvectors of denotes the upper half of , , , denotes four equal parts Finally, the stiffness matrix of the crack-free calculation region is 。 4. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 3, characterized in that, The specific steps in step S3 are as follows: Using the bond-based near-field dynamics theory, the crack calculation region is assumed to consist of multiple material points, each material point... a Volume is V a Each material point a With the radius of the adjacent region δ h Other material points a' There are interactions between the cracks, and the set of all material points within the computational region is called the influence domain H. a Each material point a The momentum equation is as follows: where is the displacement of material point ; is the displacement of material point a ; is the force on material point a between material points a' ; is the body force density on material point a , denotes the volume of material point ; in the key-based peridynamic theory, is described as follows: wherein, and are the relative position and relative displacement of material points a and material points respectively; for a micro-elastic-brittle material is expressed as: wherein is a history-dependent scalar value function; denotes a micro-modulus function; is a kernel function characterizing the strength of long-range forces between two material points; is the elongation of the bond, the expressions for the individual terms being as follows: wherein, represents the modulus of elasticity; represents the Poisson's ratio; s c represents the critical elongation; If the elongation of the bond is greater than the critical elongation of the bond s c then the bond between the two material points will break; the critical elongation of the bond s c is determined by the breaking energy G 0 which can be written as: The damage in the PD is defined as follows: The momentum equation is matrixed to obtain wherein and denotes two material points; denotes a material point volume; and denotes a material point and displacement of denotes a material point volume force density of To achieve PD and SBFEM coupling, a micro-stiffness matrix k is proposed in the PD, the expression is expressed as ; the relationship between the force and displacement between the global coordinate system and the local coordinate system is represented by the coordinate transformation matrix T : wherein, l and m are coordinate transformation factors between the local coordinate system and the global coordinate system; Thus, the material point in the global coordinate system has a micro stiffness matrix relative to the material point ​ material points composing stiffness matrix with all material points in the neighborhood and further combining stiffness matrices of all material points : 。 5. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 4, characterized in that, The step S4 is implemented as follows: first, in SBFEM and PD a coupling model is constructed, and the force at the common point has the following relationship: wherein, represents the distance between two points of material; The stiffness matrix of S2 obtained in SBFEM is partitioned into individual and common points as follows: SBFEM ​ wherein the rows and columns of the matrix correspond to SBFEM unique points; the rows of the matrix correspond to SBFEM unique points, the columns correspond to common points; the rows of the matrix correspond to common points, the columns correspond to SBFEM unique points; the rows and columns of the matrix correspond to common points; denote SBFEM displacements of unique points; denote displacements of common points; denote SBFEM forces of unique points; denote forces of common points; In S3 PD The stiffness matrix is ​​divided into blocks as follows: wherein, the rows and columns of the matrix correspond to common points; the rows of the matrix correspond to common points, the columns correspond to PD unique points; the rows of the matrix correspond to PD unique points, the columns correspond to common points; the rows and columns of the matrix correspond to PD unique points; denotes displacement of a common point; denotes PD displacement of a unique point; denotes volume force density of a common point; denotes PD volume force density of a unique point; The overall stiffness matrix of the calculation region is obtained by assembling the stiffness matrices of SBFEM and PD ​ wherein is and collectively, i.e. = = .

6. A method of analyzing a lining crack of an underground high-pressure water conveyance tunnel according to claim 5, characterized in that, The specific steps of the step S5 are as follows: after the overall stiffness matrix in S4 is assembled, displacement constraints and load boundaries are applied; a load step counter is set m =0, initial load increment P =0; the load step calculation process is as follows: S5.1, Set up m = m +1, and update the load delta P ; S5.2, solving linear equations obtain the displacement of all nodes; S5.3, calculating the elongation of the crack calculation region material point key by the formula of S3 ; S5.4, if <s c enter step S5.5; otherwise, update the stiffness matrix of the crack calculation region and return to step S5.1; S5.5、If all load steps are calculated, exit the analysis; otherwise, return to step S5.4.

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