Method for constructing water-rich body boundary condition model for transient electromagnetic advance detection in karst tunnels based on finite element method

Abstract

The present invention discloses a method for constructing a water-rich body boundary condition model for transient electromagnetic advance detection of karst tunnels based on the finite element method. The method includes: using a cubic B-spline interpolation basis function as a global basis function in a global grid; using a linear Lagrange interpolation basis function as a local basis function in a local grid; establishing a Poisson equation boundary value problem for the vector magnetic potential of the entire target area, and sorting out an equivalent variational problem of the Poisson equation boundary value problem with Dirichlet boundary conditions for water-filled caves in a karst tunnel; rewriting the equivalent variational problem of the Poisson equation boundary value problem with Dirichlet boundary conditions into a finite element equation by a mathematical method, and converting the finite element equation into a coupled group of a linear algebraic equation group so as to calculate a numerical solution using a computer. The method of the present invention has the advantages of high local accuracy, short calculation time, and simple meshing process, thereby providing a better model construction method for the detection of circular isomers in front of the karst tunnel face.

Description

Technical Field

[0001] The invention discloses a method for constructing a water-rich body boundary condition model for transient electromagnetic advance detection in a karst tunnel based on a finite element method, and belongs to the field of tunnel construction safety engineering. Background Art

[0002] Transient Electromagnetic (TEM) is a geophysical exploration method used to detect changes in underground conductivity. Finite Element Method (FEM) is a numerical calculation method used to solve complex physical problems.

[0003] Applying the finite element method to transient electromagnetic detection means using the finite element method to establish a mathematical model to solve the underground electromagnetic response. However, the traditional finite element method has problems with the accuracy and calculation time of numerical integration, as well as the singularity of the matrix, which leads to poor convergence in solving linear equations. The former occurs because the global basis function has low continuity on the element boundary. In the traditional finite element method, Lagrangian functions are used as both global and local basis functions. Because these functions have L 0 -continuity, their derivatives show discontinuity on element boundaries. Therefore, when a local element contains several global elements, the integrand tends to become discontinuous and the accuracy of Gaussian quadrature decreases. The latter problem arises because the global basis functions and local basis functions cannot be guaranteed to be linearly independent of each other.

[0004] Although many existing studies have solved this problem by limiting the degrees of freedom, the methods used therein will generate extra computation time. The advantage of the present invention is that it can not only improve the accuracy of numerical integration but also save extra computation cost and technology, thereby achieving highly efficient exploration data simulation calculation. Summary of the invention

[0005] The technical problem to be solved by the present invention is: the present invention provides a method for constructing a water-rich body boundary condition model for transient electromagnetic advance detection of karst tunnels based on the finite element method, which can improve the accuracy of numerical integration and solve the problem of poor convergence of solving linear equations caused by the singularity of the matrix. The method also saves additional computing costs and technologies, so that the same or even higher accuracy can be obtained than the traditional finite element method in the same time, thereby providing a better model construction method for the detection of circular isomers in front of the karst tunnel face.

[0006] The technical solution of the present invention is: a method for constructing a water-rich body boundary condition model for transient electromagnetic advance detection of karst tunnels based on the finite element method, and the specific steps of the construction method are as follows:

[0007] Step 1: Based on the relevant knowledge of transient electromagnetic detection forward modeling and the simulated karst tunnel scene with rich water, the Galerkin method is used to solve the boundary value problem of the Poisson equation with Dirichlet boundary value conditions under the current situation;

[0008] Step 2: mesh the target area twice, using cubic B-spline interpolation basis function and linear Lagrange interpolation basis function as global basis functions respectively;

[0009] Step 3: Analyze the global grid and the local grid independently by using the principle of virtual work, establish the boundary value problem of the Poisson equation for the vector magnetic potential of the entire target area, and sort out the equivalent variational problem of the boundary value problem of the Poisson equation with Dirichlet boundary conditions for the water-filled cave in the karst tunnel;

[0010] Step 4: Rewrite the equivalent variational problem of the Poisson equation with Dirichlet boundary conditions into a finite element equation through mathematical methods, and convert the finite element equation into a coupled set of linear algebraic equations so that the numerical solution can be calculated by computer.

[0011] Furthermore, the step 1 comprises:

[0012] According to the relevant knowledge of transient electromagnetic detection forward modeling and the simulated karst tunnel scene with rich water, the vector magnetic potential A is introduced. Z , solve the vector magnetic potential A with Dirichlet boundary conditions in the current case Z Boundary value problem of Poisson equation

[0013]

[0014] Ω is the target area, i.e., the water-rich area in the karst tunnel scene, Γ D is the boundary of the target area. Where μ represents the magnetic permeability, A Z is the introduced vector magnetic potential, J Z Represents the current density. Using the Galerkin method, first define the experimental function space and the test function space They are:

[0015]

[0016]

[0017] in, is the Sobolev space. Then the vector magnetic potential A with Dirichlet boundary conditions Z The weak solution of the Poisson equation boundary value problem obtained by the Galerkin method is as follows: For a given μJ Z , to find So that for all All satisfied

[0018] a Ω (W Z ,A Z )=L Ω (W Z )

[0019] in is a bilinear functional, L Ω (W Z )=∫ Ω W Z μJ Z dΩ is a linear functional. The following steps start from the basic idea of ​​step 1 and finally obtain the improved finite element equations.

[0020] Furthermore, the specific steps of step 2 are:

[0021] Step 2.1: Perform structural grid division on the entire target area, divide each unit into rectangular elements, and call this grid the global grid. Now assume that the global domain Ω G , whose boundary is Γ D . η G is a set of indices of all global basis functions, To include all D The set of indices of the non-zero global basis functions on Ω. G In each rectangular element, a two-dimensional cubic B-spline interpolation basis function is used a∈η G As a global basis function. First, let's look at the one-dimensional B-spline basis function. Given a set of non-decreasing coordinates in parameter space Ω, Ξ={ξ 1 ,ξ 2 ,...,ξ n+p+1}. Among them ξ i is the i-th node, p is the order of the B-spline basis function, and n is the number of B-spline basis functions. Then the B-spline basis function is derived from the Cox-deBoor recursive formula:

[0022]

[0023]

[0024] Thus, the one-dimensional cubic B-spline interpolation basis function is In higher dimensions, the B-spline basis functions are constructed by tensor products. Therefore, for a given set of two non-decreasing coordinates X = {x 1 ,x 2 ,...,x n+p+1},Y={y1 ,y 2 ,...,y n+p+1}, the two-dimensional cubic B-spline basis function is

[0025] Step 2.2: Based on the visualization results of transient electromagnetic detection, the abnormal area in the target area is divided into local grids with higher precision, and each unit is divided into rectangles, and this grid is called the local grid. In the scene of karst tunnels with rich water bodies, the center of the water-rich area is generally selected. Now assume that the local domain Ω L , whose boundary is Γ GL . η L is a set of indices containing all local basis functions in the local domain Ω L In each row element, a two-dimensional linear Lagrangian basis function is used a∈η L As a local basis function. We found that in step 2.1, when p = 1, we get the same result as the linear Lagrange interpolation basis function, so

[0026] Furthermore, the step 3 comprises:

[0027] Based on the principle of virtual work, the global grid and local grid are analyzed independently. L ∈Ω G , so that the test function A z Defined as:

[0028]

[0029] To ensure the continuity between the global domain and the local domain mesh, the following Dirichlet boundary conditions are adopted:

[0030]

[0031]

[0032] Similarly, based on the Galerkin method, the test function is defined as:

[0033]

[0034] In addition, the experimental function space should be redefined And the test function space

[0035]

[0036]

[0037]

[0038] Therefore, after the meshing in steps 2.1 and 2.2, the vector magnetic potential A with Dirichlet boundary conditions is Z The weak solution of the Poisson equation boundary value problem obtained by the Galerkin method is as follows: For a given μJ Z , to find So that for all All satisfy a′ Ω (W Z ,A Z ) = L′ Ω (W Z )

[0039] in

[0040] Furthermore, due to a Ω (·,·), L Ω The linear properties of (·) can be rewritten as

[0041] Furthermore, the step 4 comprises:

[0042] In order to meet the conditions of this step, it should be considered to be carried out in a finite-dimensional subspace, and these finite subspaces are defined as

[0043] Then we can easily know that for a given function If it satisfies Then for each and has a unique decomposition:

[0044] Therefore, the equivalent Galerkin weak solution a′ in step 3 is Ω (W Z ,A Z ) = L′ Ω (W Z ) is: Find as well as So that for all All meet the following requirements:

[0045] a′ Ω ((W Z ) h ,(V Z ) h ) = L′ Ω ((W Z ) h )-a′Ω ((W Z ) h ,g h )

[0046] in

[0047]

[0048]

[0049]

[0050] The relevant global basis functions and local basis functions defined in steps 2.1 and 2.2 can be Respectively expressed as:

[0051]

[0052]

[0053] Similarly, we can Respectively expressed as:

[0054]

[0055]

[0056] Substituting these four new expressions into a′ Ω ((W Z ) h ,(V Z ) h ) = L′ Ω ((W Z ) h )-a′ Ω ((W Z ) h ,g h )get:

[0057]

[0058] because and Can be any value, then there must be:

[0059]

[0060]

[0061] The finite element equations of the boundary value problem of the Poisson equation with Dirichlet boundary conditions for water-filled caves in karst tunnels can be obtained:

[0062]

[0063]

[0064] Furthermore, the step 4 specifically includes:

[0065] for and c,d∈η L , we define:

[0066]

[0067]

[0068]

[0069]

[0070]

[0071]

[0072] K GG =[J ab ],

[0073] J GL =[J ac ],

[0074] J LG =[K db ],

[0075] K LL =[K dc ];

[0076] Where K GG and K LL are the sub-matrices of the global grid and the local grid, respectively, K GL and K LG is a submatrix representing the relationship between the grids.

[0077] F G ={F a},

[0078] F L ={F d};

[0079]

[0080]

[0081]

[0082]

[0083]

[0084] After the above definition, the finite element equations in this physical scenario can now be coupled into a linear algebraic equation system: Kd=F.

[0085] The beneficial effects of the present invention are:

[0086] 1. This invention solves the challenge of inaccurate numerical integration accuracy in traditional finite element methods, which occurs when the continuity of global basis functions across element boundaries is low. We use a cubic method with L 2 -Continuity as a cubic B-spline basis function of the global basis function, so Gaussian quadrature can be applied to the submatrix K GL and K LG , using Gaussian quadrature to compute with sufficient accuracy without the need for additional and computationally expensive techniques to improve accuracy;

[0087] 2. The present invention fundamentally solves the problem that the matrix becomes singular or almost singular due to the inability to ensure the linear independence of the global basis function and the local basis function, and the convergence of solving the simultaneous linear equations decreases. 2 Continuity is used as the cubic B-spline basis function for the global basis function while retaining the Lagrangian basis function as the local basis function. The method shows convergence with a significantly small number of iterations and guarantees the linear independence of the basis functions. Therefore, we believe that this method has the potential to reduce the computational time while maintaining accuracy; thus providing a better model construction method for the detection of circular isomers in front of the karst tunnel face. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] Figure 1 The scene diagram of the present invention for detecting a circular isomer in front of a karst tunnel face is shown as an example;

[0089] Figure 2 The schematic diagram of the division of the global grid and the local grid is shown as an example;

[0090] Figure 3 The flow chart of the present invention is shown by way of example. DETAILED DESCRIPTION

[0091] In order to make the purpose, technical scheme and advantages of the embodiments of the present invention clearer, the technical scheme in the embodiments of the present invention will be clearly and completely described in combination with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention. It should be noted that the embodiments in this application and the features in the embodiments can be combined with each other arbitrarily without conflict.

[0092] Since the traditional finite element method has problems such as inaccurate integral precision and matrix singularity that leads to decreased convergence of solving the equation system, we now consider improving the traditional finite element method.

[0093] The method for constructing a water-rich body boundary condition model for karst tunnel transient electromagnetic advance detection based on the finite element method proposed in the present invention is implemented according to the following steps.

[0094] like Figure 1-Figure 3 , a method for constructing a water-rich body boundary condition model for karst tunnel transient electromagnetic advance detection based on the finite element method, the specific steps of the construction method are as follows:

[0095] Step 1: Based on the relevant knowledge of transient electromagnetic detection forward modeling and the simulated karst tunnel scene containing rich water, the Galerkin method is used to solve the boundary value problem of the Poisson equation with Dirichlet boundary value conditions under the current situation; Step 1 includes:

[0096] According to the relevant knowledge of transient electromagnetic detection forward modeling and the simulated karst tunnel scene with rich water, the vector magnetic potential A is introduced. Z , solve the vector magnetic potential A with Dirichlet boundary conditions in the current case Z Poisson's boundary value problem:

[0097]

[0098] Ω is the target area, i.e., the water-rich area in the karst tunnel scene, Γ D is the boundary of the target area. Where μ represents the magnetic permeability, A Z is the introduced vector magnetic potential, J Z Represents the current density. Using the Galerkin method, first define the experimental function space and the test function space They are:

[0099]

[0100]

[0101] in is the Sobolev space. Then the vector magnetic potential A with Dirichlet boundary conditions Z The weak solution of the Poisson equation boundary value problem obtained by the Galerkin method is as follows: For a given μJ Z , to find So that for all All meet the following requirements:

[0102] a Ω (W Z ,A Z )=L Ω (W Z )

[0103] in is a bilinear functional, L Ω (W Z )=∫ Ω W Z μJ Z dΩ is a linear functional. The following steps start from the basic idea of ​​step 1 and finally obtain the improved finite element equations.

[0104] Step 2: mesh the target area twice, using cubic B-spline interpolation basis function and linear Lagrange interpolation basis function as global basis functions respectively;

[0105] Furthermore, the specific steps of step 2 are:

[0106] Step 2.1: Perform structural grid division on the entire target area, divide each unit into rectangular elements, and call this grid the global grid. Now assume that the global domain Ω G , whose boundary is Γ D . η G is a set of indices of all global basis functions, To include all D The set of indices of the non-zero global basis functions on Ω. G In each rectangular element, a two-dimensional cubic B-spline interpolation basis function is used a∈η G As a global basis function. First, let's look at the one-dimensional B-spline basis function. Given a set of non-decreasing coordinates in parameter space Ω, Ξ={ξ 1 ,ξ 2 ,...,ξ n+p+1}. Among them ξ i is the i-th node, p is the order of the B-spline basis function, and n is the number of B-spline basis functions. Then the B-spline basis function is derived from the Cox-deBoor recursive formula:

[0107]

[0108]

[0109] Thus, the one-dimensional cubic B-spline interpolation basis function is In higher dimensions, the B-spline basis functions are constructed by tensor products. Therefore, for a given set of two non-decreasing coordinates X = {x 1 ,x 2 ,...,x n+p+1},Y={y 1 ,y 2 ,...,y n+p+1}, the two-dimensional cubic B-spline basis function is

[0110] Step 2.2: Based on the visualization results of transient electromagnetic detection, the abnormal area in the target area is divided into local grids with higher precision, and each unit is divided into rectangles, and this grid is called the local grid. In the scene of karst tunnels with rich water bodies, the center of the water-rich area is generally selected. Now assume that the local domain Ω L , whose boundary is Γ GL . η L is a set of indices containing all local basis functions in the local domain Ω L In each row element, a two-dimensional linear Lagrangian basis function is used a∈η L As a local basis function. We found that in step 2.1, when p = 1, we get the same result as the linear Lagrange interpolation basis function, so

[0111] Step 3: Analyze the global grid and the local grid independently by using the principle of virtual work, establish the boundary value problem of the Poisson equation for the vector magnetic potential of the entire target area, and sort out the equivalent variational problem of the boundary value problem of the Poisson equation with Dirichlet boundary conditions for the water-filled cave in the karst tunnel;

[0112] Furthermore, the step 3 comprises:

[0113] Step 3: Based on the principle of virtual work, the global grid and local grid are analyzed independently. L ∈Ω G , so that the test function A Z Defined as:

[0114]

[0115] To ensure the continuity between the global domain and the local domain mesh, the following Dirichlet boundary conditions are adopted:

[0116]

[0117]

[0118] Similarly, based on the Galerkin method, the test function is defined as:

[0119]

[0120] In addition, the experimental function space should be redefined And the test function space

[0121]

[0122]

[0123]

[0124] Therefore, after the meshing in steps 2.1 and 2.2, the vector magnetic potential A with Dirichlet boundary conditions is Z The weak solution of the Poisson equation boundary value problem obtained by the Galerkin method is as follows: For a given μJ Z , to find So that for all All satisfy a′ Ω (W Z ,A Z ) = L′ Ω (W Z );

[0125] in

[0126] Furthermore, due to a Ω (·,·), L Ω The linear properties of (·) can be rewritten as follows:

[0127] Step 4: Rewrite the equivalent variational problem of the Poisson equation boundary value problem with Dirichlet boundary value conditions into a finite element equation by mathematical methods, and convert the finite element equation into a coupled set of linear algebraic equations so as to calculate the numerical solution by computer. Step 4 includes:

[0128] In order to meet the conditions of this step, it should be considered to be carried out in a finite-dimensional subspace, and these finite subspaces are defined as

[0129] Then we can easily know that for a given function If it satisfies Then for each and has a unique decomposition:

[0130]

[0131] Therefore, the equivalent Galerkin weak solution a′ in step 3 is Ω (W Z ,A Z ) = L′ Ω ( Z ) is: Find as well as So that for all All meet the following requirements:

[0132] a′ Ω ((W Z ) h ,(V Z ) h ) = L′ Ω ((W Z ) h )-a′ Ω ((W Z ) h ,g h )

[0133] in

[0134]

[0135]

[0136]

[0137] The relevant global basis functions and local basis functions defined in steps 2.1 and 2.2 can be Respectively expressed as:

[0138]

[0139]

[0140] Similarly, we can Respectively expressed as:

[0141]

[0142]

[0143] Substituting these four new expressions into a′ Ω ((W Z )h ,(V z ) h ) = L′ Ω ((W Z ) h )-a′ Ω ((W Z ) h ,g h )get:

[0144]

[0145] because and Can be any value, then there must be:

[0146]

[0147]

[0148] The finite element equations of the boundary value problem of the Poisson equation with Dirichlet boundary conditions for water-filled caves in karst tunnels can be obtained:

[0149]

[0150]

[0151] Furthermore, the step 4 specifically includes:

[0152] for and c,d∈η L , we define:

[0153]

[0154]

[0155]

[0156]

[0157]

[0158]

[0159] K GG =[K ab ],

[0160] K GL =[K ac ],

[0161] K LG =[K db ],

[0162] K LL =[K dc ];

[0163] Where K GG and K LL are the sub-matrices of the global grid and the local grid, respectively, K GL and K LG is a submatrix representing the relationship between the grids.

[0164] F G ={F a},

[0165] F L ={F d};

[0166]

[0167]

[0168]

[0169]

[0170]

[0171] After the above definition, we can now couple the finite element equations in this physical scenario into a linear algebraic equation system: Kd=F.

[0172] The specific embodiments of the present invention are described in detail above with reference to the accompanying drawings, but the present invention is not limited to the above scenarios, and various changes can be made within the knowledge scope of ordinary technicians in the field without departing from the purpose of the present invention.

Claims

1. A method for constructing a water-rich body boundary condition model for transient electromagnetic advance detection in karst tunnels based on the finite element method, characterized by: The specific steps of the construction method are as follows: Step 1: Based on the relevant knowledge of transient electromagnetic detection forward modeling and the simulated karst tunnel scene with rich water, the Galerkin method is used to solve the boundary value problem of the Poisson equation with Dirichlet boundary value conditions under the current situation; Step 2: mesh the target area twice, using cubic B-spline interpolation basis function and linear Lagrange interpolation basis function as global basis functions respectively; Step 3: Analyze the global grid and the local grid independently by using the principle of virtual work, establish the boundary value problem of the Poisson equation for the vector magnetic potential of the entire target area, and sort out the equivalent variational problem of the boundary value problem of the Poisson equation with Dirichlet boundary conditions for the water-filled cave in the karst tunnel; Step 4: Rewrite the equivalent variational problem of the Poisson equation with Dirichlet boundary value conditions into a finite element equation by mathematical methods, and convert the finite element equation into a coupled set of linear algebraic equations so as to calculate the numerical solution by computer; The step 1 comprises: According to the relevant knowledge of transient electromagnetic detection forward modeling and the simulated karst tunnel scene with rich water, the vector magnetic potential A is introduced. Z , solve the vector magnetic potential A with Dirichlet boundary conditions in the current case Z Poisson's boundary value problem: Ω is the target area, i.e., the water-rich area in the karst tunnel scene, Γ D is the boundary of the target area, where μ represents the magnetic permeability, A Z is the introduced vector magnetic potential, J Z Represents the current density; Using the Galerkin method, first define the experimental function space and the test function space They are: in, is a Sobolev space, then the vector magnetic potential A with Dirichlet boundary conditions Z The weak solution of the Poisson equation boundary value problem obtained by the Galerkin method is as follows: For a given μJ Z , to find So that for all All meet the following requirements: a Ω (W Z ,A Z )=L Ω (W Z ) in is a bilinear functional, L Ω (W Z )=∫ Ω W Z μJ Z dΩ is a linear functional.

2. The method for constructing a water-rich body boundary condition model for karst tunnel transient electromagnetic advance detection based on the finite element method according to claim 1 is characterized by: The specific steps of step 2 are: Step 2.1: Perform structural grid division on the entire target area, divide each unit into rectangular elements, and call this grid the global grid; Now assume that the global domain Ω G , whose boundary is Γ D ; η G is a set of indices of all global basis functions, To include all D The set of indices of the non-zero global basis functions on ; in the global domain Ω G In each rectangular element, a two-dimensional cubic B-spline interpolation basis function is used As a global basis function; first look at the one-dimensional B-spline basis function, given a set of non-decreasing coordinates in the parameter space Ω as Ξ={ξ1,ξ2,...,ξ n+p+1 }, where ξ i is the i-th node, p is the order of the B-spline basis function, and n is the number of B-spline basis functions; then the B-spline basis function is derived from the Cox-deBoor recursive formula: Thus, the one-dimensional cubic B-spline interpolation basis function is In high dimensions, the B-spline basis functions are constructed by tensor products; thus, for a given set of two non-decreasing coordinates X = {x1, x2, ..., x n+p+1 },Y={y1,y2,...,y n+p+1 }, the two-dimensional cubic B-spline basis function is Step 2.2: According to the visualization results of transient electromagnetic detection, the abnormal area in the target area is divided into local grids with higher precision, and each unit is divided into rectangles, and this grid is called the local grid; in the scene of karst tunnels with rich water bodies, the center of the water-rich area is selected; now assume that the local domain Ω L , whose boundary is Γ GL ; η L is a set of indices containing all local basis functions in the local domain Ω L In each row element, a two-dimensional linear Lagrangian basis function is used As a local basis function; in step 2.1, when p = 1, the same result as the linear Lagrange interpolation basis function is obtained, so 3. The method for constructing a water-rich body boundary condition model for karst tunnel transient electromagnetic advance detection based on the finite element method according to claim 2 is characterized by: The step 3 comprises: Based on the principle of virtual work, the global grid and local grid are analyzed independently. L ∈Ω G , so that the test function A Z Defined as: To ensure the continuity between the global domain and the local domain mesh, the following Dirichlet boundary conditions are adopted: Similarly, based on the Galerkin method, the test function is defined as: In addition, the experimental function space should be redefined And the test function space Therefore, after the meshing in steps 2.1 and 2.2, the vector magnetic potential A with Dirichlet boundary conditions is Z The weak solution of the Poisson equation boundary value problem obtained by the Galerkin method is as follows: For a given μJ Z , to find So that for all All satisfy a′ Ω (W Z ,A Z ) = L′ Ω (W Z ); in Due to a Ω (·,·), L Ω The linear properties of (·) can be rewritten as follows:

4. The method for constructing a water-rich body boundary condition model for karst tunnel transient electromagnetic advance detection based on the finite element method according to claim 3 is characterized by: The step 4 comprises: Consider a finite-dimensional subspace and define these finite subspaces as Then we know that for a given function If it satisfies Then for each and has a unique decomposition: Therefore, the equivalent Galerkin weak solution a in step 3 is ′ Ω (W z ,A Z )=L ′ Ω (W Z ) is: Find as well as So that for all All meet the following requirements: a ′ Ω ((W Z ) h ,(V Z ) h )=L ′ Ω ((W Z ) h )-a ′ Ω ((W Z ) h ,g h ); in The relevant global basis functions and local basis functions defined in steps 2.1 and 2.2 can be Respectively expressed as: Similarly, Respectively expressed as: Substituting these four new expressions into a ′ Ω ((W Z ) h ,(V Z ) h )=L ′ Ω ((W Z ) h )-a ′ Ω ((W Z ) h ,g h )get: because and Can be any value, then there must be: The finite element equations of the boundary value problem of the Poisson equation with Dirichlet boundary conditions for water-filled caves in karst tunnels can be obtained:

5. The method for constructing a water-rich body boundary condition model for karst tunnel transient electromagnetic advance detection based on the finite element method according to claim 4 is characterized by: The step 4 specifically includes: for and c,d∈η L ,definition: K GG =[K ab ], K GL =[K ac ], K LG =[K db ], K LL =[K dc ]; Where K GG and K LL are the sub-matrices of the global grid and the local grid, respectively, K GL and K LG is a submatrix representing the relationship between the grids; F G ={F a }, F L ={F d }; Through the above definition, the finite element equations in this physical scenario can be coupled into a linear algebraic equation system: Kd=F.

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