Groundwater seepage boundary detection restoration method and system based on trefftz coupled ftim

Abstract

The application relates to a groundwater seepage boundary detection recovery method and system based on a Trefftz coupled FTIM, which comprises the following steps: based on the characteristics of a groundwater aquifer, establishing a groundwater seepage control equation, solving Trefftz base functions and an approximate solution meeting the control equation, and adopting a polar coordinate system to disperse known boundaries of a groundwater seepage boundary detection problem and to distribute points on a calculation domain; making the dispersed known boundary points meet the control equation and corresponding known boundary values, substituting point position information and boundary values of the known boundary points into the approximate solution, and solving a to-be-determined coefficient matrix; performing boundary detection on the groundwater seepage boundary, assuming an initial guess value of the distance between an unknown position boundary point and a source point, substituting the guess value into the approximate solution, and establishing a nonlinear equation; solving the nonlinear equation by adopting a quasi-time integration method FTIM; and when the calculation result reaches a convergence condition, detecting and recovering the unknown boundary position. The method and system are beneficial to accurately and efficiently detecting and recovering the groundwater seepage boundary position.

Description

Technical Field

[0001] This invention relates to the field of groundwater monitoring technology, specifically to a method and system for detecting and reconstructing groundwater seepage boundaries based on Trefftz coupled FTIM. Background Technology

[0002] Water plays an indispensable role in human survival and development, and is also a crucial energy resource. With the development of urban construction in my country and the exploration and application of various uses for underground space, the depth and scale of excavation are increasing, which brings about certain groundwater problems. The overall groundwater level can comprehensively reflect hydrogeological factors, not only reflecting the irregular changes in groundwater recharge, diffusion, and dissipation, but also serving as a key indicator for determining whether groundwater environmental problems exist and their severity.

[0003] Especially in areas rich in groundwater, fluctuations in groundwater levels can pose significant risks to various engineering projects during excavation. Low water levels are largely caused by human activities, such as unrestricted groundwater extraction or the construction of reservoirs upstream of water bodies, both of which can lead to varying degrees of groundwater level reduction. Furthermore, while the water pressure generated during groundwater diffusion poses little threat under natural conditions, construction processes can amplify its dynamic effects. If the pressure exceeds the tolerance of the supporting structure, it can cause damage to the affected area and related engineering projects, potentially leading to building collapse and soil subsidence. In recent years, the irrational extraction and management of groundwater have given rise to a series of ecological and geological environmental problems. Therefore, research on groundwater control and management, particularly water level management, and the understanding and optimization of the groundwater environment are crucial. Exploring methods for faster and more accurate hydrogeological exploration and groundwater monitoring in engineering projects is of practical significance. Groundwater seepage boundary detection falls under the category of boundary detection inverse problems. Inverse problems are often difficult to define, but generally refer to problems that use partially known information from the solution to find certain unknowns in a definitive solution; these are called mathematical physics inverse problems. Compared to traditional forward problems, mathematical physics inverse problems are a newly emerging research field. Due to advancements in science and technology, many inverse problems urgently needing solutions have emerged in practical applications such as resource exploration, aerospace engineering, geophysics, atmospheric measurement, ocean engineering, remote sensing technology, and quantum mechanics, making mathematical physics inverse problems a truly widely valued research area. Summary of the Invention

[0004] The purpose of this invention is to provide a method and system for detecting and restoring groundwater seepage boundaries based on Trefftz coupled FTIM. This method and system are beneficial for accurately and efficiently detecting and restoring the location of groundwater seepage boundaries.

[0005] To achieve the above objectives, the technical solution adopted by this invention is: a method for detecting and restoring groundwater seepage boundaries based on Trefftz coupled FTIM, comprising the following steps:

[0006] S1: Based on the characteristics of groundwater aquifers, the control equations for groundwater seepage under the influence of steady-state groundwater seepage are established. The Trefftz basis functions satisfying the control equations and the approximate solutions satisfying the control equations are solved. Using a polar coordinate system, the known boundary of the groundwater seepage boundary detection problem is discretized into N... b N points, and arrange N points in the computational domain. i One point;

[0007] S2: Let the known boundary points obtained by discretization satisfy the control equation and the corresponding known boundary values. Substitute the point information and boundary values ​​of the known boundary points into the approximate solution that satisfies the control equation, establish a system of equations with the form B = Fα, and solve for the undetermined coefficient matrix α.

[0008] S3: For groundwater seepage boundary detection, assume an initial guess r0 for the distance between the unknown boundary point and the source point, substitute the guess into the approximate solution, and establish the nonlinear equation T(r i );

[0009] S4: Solving the nonlinear equation T(r) using the quasi-time integral method (FTIM). i );

[0010] S5: When the calculated result meets the convergence condition, i.e., T(r) i K )≤ε,r i K This refers to the unknown boundary location of the detected and reconstructed groundwater seepage.

[0011] Furthermore, in step S1, the groundwater aquifer includes unconfined aquifers and confined aquifers. The flow rate in the groundwater aquifer is in a steady state. The resistance to vertical flow is ignored, and only the resistance to horizontal flow is considered. The soil hydraulic conductivity T is constant. The soil layer is homogeneous and isotropic. The permeability coefficient is equal at any point within the groundwater aquifer. Therefore, the groundwater seepage control equation for this area is as follows:

[0012]

[0013] in, Let represent the Laplace operator, h be the total groundwater head, x and y be the coordinates of points in the computational domain, and S be the computational domain for groundwater seepage calculation.

[0014] Further, in step S1, the Trefftz method basis functions satisfying the groundwater seepage control equations are solved using the separation of variables method, and an approximate solution to the control equations satisfying the computational domain is obtained through the linear superposition and combination of the basis functions; expressed as a series, under the first kind of boundary conditions, the series expression of the approximate solution is as follows: Under the second kind of boundary conditions, the series expression for the approximate solution is as follows: Among them Ψ j This represents the Trefftz basis functions that satisfy the first type of boundary conditions. This represents the Trefftz basis functions that satisfy the second type of boundary conditions, when dealing with the same problem. This represents the coefficients to be determined.

[0015] Furthermore, in step S1, the governing equations are transformed into polar coordinates, as shown in the following expression:

[0016]

[0017] Where h is the total groundwater head, r is the radius, and θ is the polar angle.

[0018] Discretize the known boundary of the groundwater seepage boundary detection problem, and discretize the known boundary into N. b One point, This indicates the number of boundary points placed on the boundary, where and They represent the Γ lines that satisfy the first type of boundary conditions. D Boundary and Γ satisfying the second type of boundary conditions N The number of boundary nodes configured for the boundary.

[0019] Furthermore, in step S1, the computational domain for the groundwater seepage boundary detection problem is configured with points, and N... i A point is configured within the computational domain to calculate physical quantities within the computational domain. The distance between the computational node and the source point within the computational domain is r, and the physical quantity h represented by the node satisfies the Laplace equation.

[0020] Furthermore, in step S2, let matrix F be a matrix of size N constructed using Trefftz basis functions. b A matrix B is a 2m+1 × (2m+1) matrix, where matrix B is a matrix composed of N. b A boundary value matrix consisting of N boundary values, the size of which is N. b ×1, the undetermined coefficient matrix α is a matrix of size (2m+1)×1, where m is the order of the Trefftz basis functions;

[0021] Using the matrix left division form α = F -1 B, solve for the undetermined coefficient matrix α.

[0022] Furthermore, in step S3, the groundwater seepage boundary to be detected satisfies the governing equation, and its head value or flow rate value is a known condition. In the initial assumption, at any angle, the initial guess of the distance between the unknown location boundary point and the source point is assumed to be r0. The initial guess can be arranged within or outside the calculation domain as needed.

[0023] The guess r of the unknown location boundary point i Substituting into the Trefftz basis functions, we get the following expression:

[0024]

[0025] Where, r i To estimate the distance between the unknown location boundary point and the source point, θ i Let be the angle between the line connecting the unknown boundary point and the source point in the same positive polar coordinate direction, m be the order of the Trefftz basis function, and a0, b j d j Let α be the coefficients to be determined, and α = [a0, b0]. j ,d j ] T ;

[0026] The known head value at an unknown boundary point is represented as follows:

[0027] F′(r i )=h(r i ,θ i )

[0028] Where h(r) i ,θ i ) is located at (r i ,θ i The known head value of the unknown location boundary point at () is given, where i represents the number of unknown location boundary points;

[0029] Therefore, the nonlinear equation is established as follows:

[0030] T(r i )=|F′(r i )-F(r i )|.

[0031] Furthermore, in step S4, the quasi-time integral method uses the antecedent difference method to solve the nonlinear equation system by introducing the quasi-time parameter ω and the control parameter μ. Its numerical time integral expression is as follows:

[0032]

[0033] in, η is the approximate time step; K is the Kth discrete step; η is the time step size.K For the accumulated pseudo-time, i.e.

[0034] Furthermore, in step S5, ε is the convergence criterion; when T(r) i K When ε ≤ ε, the position of the unknown location boundary point detected is determined to be the position of the known head value; otherwise, it is substituted back into FTIM to continue detection and calculation.

[0035] The present invention also provides a groundwater seepage boundary detection and restoration system based on Trefftz coupled FTIM, including a memory, a processor, and computer program instructions stored in the memory and capable of being executed by the processor. When the processor executes the computer program instructions, it can implement the above-mentioned method steps.

[0036] Compared with existing technologies, the present invention has the following beneficial effects: The present invention provides a method and system for detecting and restoring groundwater seepage boundaries based on Trefftz coupled FTIM. This method establishes groundwater seepage control equations, solves the Trefftz basis functions that satisfy the control equations and the approximate solutions that satisfy the control equations, and then uses the quasi-time integral method FTIM to solve the nonlinear equations established based on the approximate solutions. This method can accurately and efficiently detect and restore groundwater seepage boundaries, thereby enabling further understanding and analysis of groundwater evolution patterns and groundwater coverage boundaries, providing a basis for the construction and exploration of underground spaces, and thus reducing the risks of groundwater construction and saving construction costs. Attached Figure Description

[0037] Figure 1 This is a flowchart illustrating the method implementation of an embodiment of the present invention;

[0038] Figure 2 This is a typical cross-sectional view of the dam in an embodiment of the present invention;

[0039] Figure 3 This is a distribution diagram of the dam boundary points and internal points in an embodiment of the present invention;

[0040] Figure 4 This is a schematic diagram of the dam seepage line detected in an embodiment of the present invention;

[0041] Figure 5 This is a schematic diagram of the total head of the dam in an embodiment of the present invention. Detailed Implementation

[0042] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0043] It should be noted that the following detailed descriptions are exemplary and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0044] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0045] like Figure 1 As shown, this embodiment provides a groundwater seepage boundary detection and restoration method based on Trefftz coupled FTIM, including the following steps:

[0046] S1: Based on the characteristics of groundwater aquifers, the control equations for groundwater seepage under the influence of steady-state groundwater seepage are established. The Trefftz basis functions satisfying the control equations and the approximate solutions satisfying the control equations are solved. Using a polar coordinate system, the known boundary of the groundwater seepage boundary detection problem is discretized into N... b N points, and arrange N points in the computational domain. i One point.

[0047] S2: Let the known boundary points obtained by discretization satisfy the control equation and the corresponding known boundary values. Substitute the point information and boundary values ​​of the known boundary points into the approximate solution that satisfies the control equation, establish a system of equations with the form B = Fα, and solve for the undetermined coefficient matrix α.

[0048] S3: For groundwater seepage boundary detection, assume an initial guess r0 for the distance between the unknown boundary point and the source point, substitute the guess into the approximate solution, and establish the nonlinear equation T(r i ).

[0049] S4: Solving the nonlinear equation T(r) using the quasi-time integral method (FTIM). i ).

[0050] S5: When the calculated result meets the convergence condition, i.e., T(r) i K )≤ε,r i K This refers to the unknown boundary location of the detected and reconstructed groundwater seepage.

[0051] The following example, using a seepage simulation of a homogeneous earth-rock dam on an impermeable foundation, further illustrates the relevant content of this invention. Figure 5This is a schematic diagram of the total head of the dam in this embodiment.

[0052] In step S1, the groundwater aquifer includes unconfined and confined aquifers. The flow rate in the groundwater aquifer is in a steady state. Vertical flow resistance is ignored; only horizontal flow resistance is considered. The soil hydraulic conductivity T is constant. The soil layer is homogeneous and isotropic. The permeability coefficient is equal at any point within the groundwater aquifer. Therefore, the groundwater seepage control equation for this area is as follows:

[0053]

[0054] in, Let represent the Laplace operator, h be the total groundwater head, x and y be the coordinates of points in the computational domain, and S be the computational domain for groundwater seepage calculation.

[0055] The Trefftz method basis functions satisfying the groundwater seepage control equations are obtained by using the separation of variables method, and an approximate solution of the control equations satisfying the computational domain is obtained by linear superposition and combination of the basis functions.

[0056] Using a series expression, under the first kind of boundary conditions, the approximate solution can be expressed as:

[0057]

[0058] Under the second type of boundary conditions, the series expression for the approximate solution can be expressed as:

[0059]

[0060] in:

[0061]

[0062]

[0063] Among them, Ψ j This represents the Trefftz basis functions that satisfy the first type of boundary conditions. This represents the Trefftz basis functions that satisfy the second type of boundary conditions, when dealing with the same problem. R0 represents the undetermined coefficients, and R0 is the characteristic length, used to speed up convergence and ensure computational accuracy.

[0064] Transforming the governing equations into polar coordinates, the expression is as follows:

[0065]

[0066] Where h is the total groundwater head, r is the radius, and θ is the polar angle.

[0067] Discretize the known boundary of the groundwater seepage boundary detection problem, and discretize the known boundary into N. b =120 boundary points, This indicates the number of boundary points placed on the boundary, where and They represent the Γ lines that satisfy the first type of boundary conditions. D Boundary and Γ satisfying the second type of boundary conditions N The number of boundary nodes in the boundary configuration. The seepage line boundary detection problem of earth-rock dams is a mixed boundary problem involving both Type I and Type II boundary conditions. The seepage surface Γ2 on the upper part of the dam and the wetted surface Γ1 between the overflow point and the downstream head together constitute the free liquid surface of the dam. The upper and lower reaches of the dam satisfy Type I boundary conditions, with head heights of 18 and 8 respectively. The bottom of the dam is an impermeable boundary, which satisfies Type II boundary conditions. Figure 2 This is a typical cross-sectional view of the dam in this embodiment.

[0068] In the problem of detecting the seepage line boundary of an earth-rock dam, the computational domain is set up with points, and N... i = 39996 points are configured within the computational domain to calculate physical quantities within the computational domain. The distance between the computational nodes and the source point is r, and the physical quantity h represented by the nodes satisfies the Laplace equation. Figure 3 This is a distribution diagram of the dam boundary points and internal points in this embodiment.

[0069] In step S2, the wetted surface Γ1 satisfies the first type of boundary condition:

[0070]

[0071] The seepage surface Γ2 satisfies the second type of boundary condition:

[0072]

[0073] Both upstream and downstream of the dam body satisfy the first type of boundary conditions:

[0074]

[0075]

[0076] The dam base is an impermeable boundary, satisfying the second type of boundary condition:

[0077]

[0078] In the diagram, boundaries Γ1 and Γ2 are unknown boundaries, with a total of 180 unknown boundary points, while boundaries Γ3, Γ4, and Γ5 are known boundaries, with a total of 120 known boundary points.

[0079] Establish a system of equations with expressions of the form B = Fα.

[0080]

[0081] α=[a0,b0…b j ,d0…d j ,c0…c j ,t0…t j ] T

[0082]

[0083] Where matrix F is a matrix of size N constructed using Trefftz basis functions. b A matrix B is a 2m+1 (120×61) matrix, where N is the number of elements in the matrix. b A boundary value matrix consisting of N boundary values, the size of which is N. b ×1 (120×1 in this embodiment), the undetermined coefficient matrix α is a matrix with a size of (2m+1)×1 (61×1 in this embodiment), and m=30 is the order of the Trefftz basis function.

[0084] Using the matrix left division form α = F -1 B. Solve for the undetermined coefficient matrix α. Using left matrix division eliminates the need to calculate the matrix inverse, avoiding matrix singularities and improving both speed and accuracy.

[0085] By substituting any point within the computational domain into the approximate solution obtained above, the head value h(x) at any point can be obtained from B = Fα. i ,y i ), can be represented as:

[0086]

[0087] In step S3, the unknown location boundary to be detected satisfies the governing equation. Initially, at any angle, an initial guess r0 is assumed to be the distance between the unknown location boundary point and the source point. This initial guess can be placed within or outside the computational domain as needed. In this embodiment, the initial guess is arranged as follows: Figure 4 The diagram shows the dam seepage line obtained from the detection.

[0088] The guess r of the unknown location boundary point i Substituting these values ​​into the Trefftz basis functions, we obtain the following expression:

[0089]

[0090] Where, r i To estimate the distance between the unknown location boundary point and the source point, θ iLet be the angle between the line connecting the unknown boundary point and the source point in the same positive polar coordinate direction, m be the order of the Trefftz basis function, and a0, b j d j Let α be the coefficients to be determined, and α = [a0, b0]. j ,d j ] T .

[0091] The known head value at an unknown boundary point can be expressed as:

[0092] F′(r i )=h(r i ,θ i )

[0093] Where h(r) i ,θ i ) is located at (r i ,θ i The known head value of the unknown location boundary point at () is given, where i represents the number of unknown location boundary points;

[0094] Therefore, the nonlinear equation is established as follows:

[0095] T(r i )=|F′(r i )-F(r i )|

[0096] In step S4, the Fictitious Time Integration Method (FTIM) adopts the concept of a dynamical system. By using the antecedent difference method, it introduces parameters such as the finite-time parameter ω (ω = 0.1 in this embodiment) and the control parameter μ (μ = 1 in this embodiment) to solve the nonlinear equation system. Its numerical time integral can be expressed as:

[0097]

[0098] in, η is the approximate time step; K is the Kth discrete step; η is the time step size. K For the accumulated pseudo-time, i.e.

[0099] In step S5, ε is the convergence criterion, and ε = 10. -2 When T(r) i K ≤10 -2 If the position of the unknown boundary point detected is determined to be the position of the known head value, then the point is re-substituted into FTIM to continue detection and calculation. Figure 4 This is a schematic diagram of the dam seepage line detected in this embodiment.

[0100] Because the free surface is a nonlinear, movable boundary, the seepage overflow point must first be determined. At the beginning of the calculation, the boundary above the waterline on the right side of the dam is calculated first to locate the overflow point, i.e., to determine the wetted surface Γ1. The calculation accuracy is set to 10. -2 When x = 24.14 and y = 10.40, the value jumps out of the precision range for the first time, and the previous position point, x = 24.14 and y = 10.28, is the desired overflow point location. Simultaneously, the wetted surface Γ1 can be determined, and from this, the position and shape of the seepage surface Γ2 can be calculated. Figure 4 It can be clearly seen that the results obtained in this embodiment are in high agreement with the results of previous studies. We selected a few points where there was a significant discrepancy for comparison. The specific comparison results are shown in Table 1.

[0101] Table 1 Comparison of Immersion Line Location Results

[0102]

[0103]

[0104] Next, based on the calculated free surface position, the total head distribution within the earth-rock dam is determined. The specific distribution is as follows: Figure 5 As shown in the figure, the total head below the water surface on the left and right sides of the dam body is 18m and 8m respectively. As the water flows through the dam body, the total head gradually decreases from 18m to 8m due to energy loss. The overall trend is consistent with the physical phenomena within the earth-rock dam body under real seepage problems. Figure 5 This is a schematic diagram of the total head of the dam in this embodiment.

[0105] This embodiment also provides a groundwater seepage boundary detection and restoration system based on Trefftz coupled FTIM, including a memory, a processor, and computer program instructions stored in the memory and executable by the processor. When the processor executes the computer program instructions, it can implement the above-mentioned method steps.

[0106] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0107] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0108] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0109] These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable apparatus for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0110] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A method for detecting and reconstructing groundwater seepage boundaries based on Trefftz coupled FTIM, characterized in that, Includes the following steps: S1: Based on the characteristics of groundwater aquifers, the governing equations for groundwater seepage under the influence of steady-state groundwater seepage are established. The Trefftz basis functions satisfying the governing equations and approximate solutions satisfying the governing equations are solved. Using polar coordinates, the known boundaries of the groundwater seepage boundary detection problem are discretized into... Points, and arrangement of computational domains. One point; S2: Let the discretized known boundary points satisfy the governing equations and the corresponding known boundary values. Substitute the positional information of the known boundary points and the boundary values ​​into the approximate solution that satisfies the governing equations to establish an expression of the form... Solve the system of equations and the matrix of undetermined coefficients. ; S3: Perform boundary detection for groundwater seepage boundaries, assuming an initial guess of the distance between the unknown boundary point and the source point. By substituting guesses into approximate solutions, nonlinear equations are established. ; S4: Solving nonlinear equations using the quasi-time integral method (FTIM). ; S5: When the calculated result meets the convergence condition, that is... , This refers to the location of the unknown boundary of groundwater seepage detected and reconstructed. This is a convergence criterion; In step S1, the groundwater aquifer includes unconfined aquifers and confined aquifers. The flow rate in the groundwater aquifer is in a steady state. The resistance to vertical flow is ignored, and only the resistance to horizontal flow is considered. The soil hydraulic conductivity T is constant. The soil layer is homogeneous and isotropic. The permeability coefficient is equal at any point within the groundwater aquifer. The groundwater seepage control equation for the region is as follows: in, Let represent the Laplace operator, h be the total groundwater head, x and y be the coordinates of points in the computational domain, and S be the computational domain for groundwater seepage calculation. In step S1, the Trefftz method basis functions satisfying the groundwater seepage control equations are solved using the separation of variables method, and an approximate solution to the control equations satisfying the computational domain is obtained through linear superposition and combination of the basis functions; expressed as a series, under the first kind of boundary conditions, the approximate solution is expressed as follows: Under the second type of boundary conditions, the series expression for the approximate solution is as follows: ,in This represents the Trefftz basis functions that satisfy the first type of boundary conditions. This represents the Trefftz basis functions that satisfy the second type of boundary conditions, when dealing with the same problem. , represents an undetermined coefficient; In step S4, the quasi-time integration method introduces a quasi-time parameter by employing the antecedent difference method. Control parameters Solving the nonlinear system of equations, its numerical time integral form is as follows: in, Let K be the approximate time step; K is the Kth discrete step. For the accumulated pseudo-time, i.e. .

2. The method for detecting and reconstructing groundwater seepage boundaries based on Trefftz coupled FTIM according to claim 1, characterized in that, In step S1, the governing equations are transformed into polar coordinates, and the expression is as follows: Where h is the total groundwater head, r is the radius, and θ is the polar angle; Discretize the known boundary of the groundwater seepage boundary detection problem, and discretize the known boundary into... One point, This indicates the number of boundary points placed on the boundary, where and They represent the boundaries along the first type of boundary conditions. Boundaries and those satisfying the second type of boundary conditions The number of boundary nodes configured for the boundary.

3. The groundwater seepage boundary detection and reconstruction method based on Trefftz coupled FTIM according to claim 2, characterized in that, In step S1, the computational domain for the groundwater seepage boundary detection problem is set up with points, and N... i A point is configured within the computational domain to calculate physical quantities within the computational domain. The distance between the computational node and the source point within the computational domain is r, and the physical quantity h represented by the node satisfies the Laplace equation.

4. The method for detecting and reconstructing groundwater seepage boundaries based on Trefftz coupled FTIM according to claim 3, characterized in that, In step S2, let the matrix The size of the matrix constructed using Trefftz basis functions is The matrix, the matrix For the reason The boundary value matrix consists of n boundary values, and its size is 1. matrix of undetermined coefficients The matrix size is The matrix, where Let be the order of the Trefftz basis functions; Using matrix left division Solve the matrix of undetermined coefficients .

5. The method for detecting and reconstructing groundwater seepage boundaries based on Trefftz coupled FTIM according to claim 4, characterized in that, In step S3, the groundwater seepage boundary to be detected satisfies the governing equation, and its hydraulic head or flow rate is a known condition. Initially, at any angle, an initial guess is made regarding the distance between the unknown boundary point and the source point. The initial guesses can be placed within or outside the computational domain as needed; Guessing the boundary points of unknown locations Substituting into the Trefftz basis functions, we get the following expression: in, To estimate the distance between the unknown boundary point and the source point. The angle between the line connecting the unknown location boundary point and the source point in the same positive polar coordinate direction. Let be the order of the Trefftz basis functions. , , For undetermined coefficients, ; The known head value at an unknown boundary point is represented as follows: in, For located The known head value at the unknown location boundary point. Indicates the number of boundary points at unknown locations; Therefore, the nonlinear equation is established as follows: 。 6. The method for detecting and reconstructing groundwater seepage boundaries based on Trefftz coupled FTIM according to claim 5, characterized in that, In step S5 To determine the convergence criteria, when If the position of the unknown boundary point detected is determined to be the position of the known head value, then the point is re-substituted into FTIM to continue detection and calculation.

7. A groundwater seepage boundary detection and reconstruction system based on Trefftz coupled FTIM, characterized in that, It includes a memory, a processor, and computer program instructions stored in the memory and executable by the processor. When the processor executes the computer program instructions, it can implement the groundwater seepage boundary detection and restoration method based on Trefftz coupled FTIM as described in any one of claims 1-5.

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