Method and system for tunnel shape optimization under anisotropic geological conditions

Abstract

The application discloses a multi-elliptical hole tunnel shape optimization method and system under anisotropic geological conditions. The method comprises the following steps: determining a tunnel surrounding rock stress field, surrounding rock orthogonal anisotropic parameters and an elliptical tunnel geometric shape, and constructing a plane unit model containing a multi-elliptical hole tunnel; obtaining a stress field calculation formula of the anisotropic multi-elliptical hole tunnel under any load through an analytic-semi-analytic stress model of the multi-elliptical tunnel in the anisotropic rock mass; defining an objective function, and determining a final mixed penalty function; using a DE algorithm, performing iteration on a design variable set of the mixed penalty function, obtaining an optimal design variable set, calculating maximum ring stresses of all tunnels, and completing the multi-elliptical hole tunnel shape optimization. The method realizes accurate solution of the stress field of the multi-elliptical hole tunnel in the anisotropic medium and optimization of the shape and spatial arrangement of the multiple tunnels, effectively improves the calculation precision, and can be widely applied to tunnel safety evaluation and stability analysis.

Description

Technical Field

[0001] This invention belongs to the field of tunnel engineering, specifically relating to a method and system for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions. Background Technology

[0002] In practical tunnel engineering, parallel twin-bore tunnels are becoming increasingly common due to the growing scarcity of urban land resources and increasing geographical constraints. In geotechnical engineering, many strata (such as layered rock masses and schist) exhibit significant anisotropy in their mechanical properties, leading to more complex deformation and stress responses of the tunnel surrounding rock under far-field tectonic stress. After a tunnel is put into operation, its cross-section often exhibits asymmetrical elliptical deformation under continuous anisotropic geostress. Simultaneously, dynamic loads such as trains passing through the tunnel apply non-uniformly distributed arbitrary surface stresses to the tunnel lining. Due to the anisotropy of geological conditions, the diversity of elliptical bore geometric parameters, and the complexity of external loads, the stress distribution of the tunnel surrounding rock exhibits high non-uniformity, easily leading to severe stress concentrations in localized areas, which can induce engineering disasters such as rock cracking and instability. Therefore, establishing an evaluation method applicable to anisotropic geological conditions and capable of accurately calculating the interaction stress of arbitrarily distributed twin elliptical bores under complex loads is of crucial guiding significance for the optimized design, strength verification, and long-term safety evaluation of parallel tunnels. Summary of the Invention

[0003] To address the shortcomings of existing technologies, one of the objectives of this invention is to provide a method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions, so that the boundaries of multiple tunnels are simultaneously in an optimal stress state, characterized by minimizing stress concentration and minimizing the range of the plastic zone around the tunnel, thereby improving the long-term stability and safety of the tunnel structure.

[0004] The second objective of this invention is to provide a system for implementing the method of optimizing the shape of multi-elliptical-hole tunnels under the aforementioned anisotropic geological conditions.

[0005] This invention provides a method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions, comprising the following steps:

[0006] S1. Determine the geostress field, orthogonal anisotropy parameters of the surrounding rock, and geometry of the elliptical tunnel; construct a planar element model of the tunnel with multiple elliptical holes; and define boundary conditions.

[0007] S2. The stress field calculation formula for anisotropic multielliptical tunnels under arbitrary loads is derived by using the analytical-semi-analytical stress model of multielliptical tunnels in anisotropic rock masses.

[0008] S3. Determine the set of design variables, define the objective function, determine the constraints for tunnel optimization, and determine the final hybrid penalty function;

[0009] S4. Using the DE algorithm, iterate the design variable set of the hybrid penalty function to obtain the optimal design variable set, calculate the maximum circumferential stress of all tunnels, and complete the shape optimization of the multi-elliptical-hole tunnel.

[0010] Step S1 includes the following steps:

[0011] Measure the stress field at the far end of the surrounding rock and the boundary conditions at the tunnel borehole edge, and construct a quasi-orthogonal anisotropic plane element model;

[0012] The equivalent compliance coefficient is calculated based on existing data, and the relevant calculation parameters are determined.

[0013] In the quasi-orthogonal anisotropic planar element model, the orthogonal elastic constant is determined by the longitudinal elastic modulus. and shear modulus Compared to Poisson Characterized by the presence of several elliptical tunnels within, the geometric features of which are defined by the major axis a, minor axis b, and azimuth angle. Sure;

[0014] The parametric equation for the k-th elliptical tunnel is expressed using the following formula: ;in, Let be the coordinates of the k-th elliptical tunnel; Let be the major axis of the k-th elliptical tunnel; Let be the minor axis of the k-th elliptical tunnel; For the angle parameter of the ellipse;

[0015] The boundary conditions of the tunnel bore are expressed using the following formula: ;in, The normal stress is the wall stress of the k-th elliptical tunnel. Let K be the shear stress of the wall of the k-th elliptical tunnel. This represents the boundary of the k-th elliptical tunnel. The circumferential stress at the edge of the k-th elliptical tunnel hole; Let be the tangential stress at the edge of the k-th elliptical tunnel hole.

[0016] Calculate the equivalent compliance coefficient using the following formula: ;in, The strain is the normal strain along the x-axis; The strain is the normal strain along the y-axis; For shear strain; The stress is the normal stress along the x-axis; The normal stress is along the y-axis; Shear stress; Here, is the equivalent coefficient, where and ; The equivalent compliance coefficient is related to the material properties;

[0017] Based on the measured material parameters, the relevant calculation parameters are determined using the following formula: ; ;in, The roots of the above fourth-order characteristic equation are denoted as . The number of conjugate roots characterizing anisotropy; This is the real part of the first pair of conjugate roots; This is the real part of the second pair of conjugate roots; This is the imaginary part of the first pair of conjugate roots; This is the imaginary part of the second pair of conjugate roots.

[0018] Step S2 specifically involves using the following formula to calculate the stress at any point A in the multi-elliptical tunnel structure within the anisotropic rock strata; ;

[0019] in , and (k=1,2) represent any point A generated by the interaction of any bielliptical loaded tunnel under far-field stress. Radial stress, circumferential stress and tangential stress at the location, Let A represent the radial stress caused by a unit radial force along the j-th arc of the k-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit tangential force along the j-th arc of the k-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit radial force along the j-th arc of the m-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit tangential force along the j-th arc of the m-th hole at any point A on the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by a unit radial traction force along the j-th arc of the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by the unit tangential traction force along the j-th arc of the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by a unit radial traction force along the j-th arc of the m-th hole. Let represent the circumferential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the m-th hole. Let A represent the tangential stress at any point A on the k-th hole caused by the unit radial traction force along the j-th arc of the k-th hole. Let represent the tangential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the k-th hole. Let represent the tangential stress at any point A on the k-th hole caused by the unit radial traction force along the j-th arc of the m-th hole. This represents the tangential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the m-th hole; M is the number of arcs divided by the elliptical boundary.

[0020] Step S3 includes the following steps:

[0021] For a system containing K elliptical tunnels, determine the set of design variables;

[0022] The multi-objective problem is standardized by adopting the minimax criterion and defining the objective function based on the maximum value of the critical stress measure at all tunnel boundaries;

[0023] The geometric engineering constraints for tunnel optimization are determined, and the objective function problem is transformed into an unconstrained optimization problem using the penalty function method.

[0024] An optimized mathematical model is used to represent the unconstrained optimization problem.

[0025] The set of design variables is represented by the following formula: The design variables for the k-th tunnel (k=1,2,...K) are: ;in, and These are the semi-major axis and the semi-minor axis, respectively. This is the azimuth angle (used to determine the direction of rotation of the local major axis relative to the global x-axis). Represents the global coordinates of the tunnel center.

[0026] The objective function is expressed using the following formula: ;in Show the angle position at the boundary of the kth tunnel. The circumferential stress at each tunnel boundary is a function of all design variables X. The optimization objective is to determine the value of X that minimizes the objective function, thereby minimizing the maximum circumferential stress at each tunnel boundary.

[0027] The geometric engineering constraints include size constraints, shape constraints, cross-sectional area constraints, and minimum spacing constraints;

[0028] The dimensional constraints are expressed using the following formula: ;in, The minimum major axis of the tunnel; The longest axis of the tunnel; It is the minimum minor axis of the tunnel; The maximum minor axis of the tunnel;

[0029] The shape constraint is expressed using the following formula: ;in, The maximum permissible elliptical tunnel eccentricity;

[0030] The cross-sectional area constraint is expressed using the following formula: ;in, To maintain the required internal area of ​​the tunnel;

[0031] The minimum spacing constraint is expressed using the following formula: ;in, Let r represent the boundary of the i-th ellipse, and r be the position vector of the boundary point. This indicates the minimum required width of the rock column, which may depend on the rock mass quality and in-situ stress.

[0032] The objective function problem can be transformed into an unconstrained optimization problem using the following penalty function method: ;in, Inequality constraints (size constraints, shape constraints, and minimum spacing constraints can all be converted to) (in the form of); Representing equality constraints (cross-sectional area constraints can be considered as) (or included as an inequality with small tolerance); To apply the penalty parameters for inequality constraints that are progressively increased through an iterative sequence. It is an equality constraint penalty parameter that is gradually increased through an iterative sequence, used to force the solution to approach the feasible region; m represents the number of inequality constraints; p represents the number of equality constraints.

[0033] The optimized mathematical model is represented by the following formula: .

[0034] In step S4, the differential evolution algorithm is used to solve the optimization problem in step S3, including the following steps:

[0035] Initialization: Randomly generate within a preset boundary range containing... candidate design vectors The population, where each design vector represents a complete multi-tunnel configuration;

[0036] Stress analysis: for each candidate vector The stress model (analytical-semi-analytical stress model for multi-elliptical tunnels) was used to calculate the circumferential stress distribution of all tunnels. This step involves solving the integral equation of the interaction traction force and then superimposing the contributions of each stress.

[0037] Fitness evaluation: Calculate the penalty function value for each candidate vector based on its maximum circumferential stress and constraint violation. ;

[0038] Differential evolution operations (mutation, crossover, selection): New candidate solutions are generated through differential mutation and binomial crossover; the selection operator is based on... The value preserves a better solution between the target vector and the test vector;

[0039] Iteration: Repeat the stress analysis to differential evolution operation until the convergence criterion is met (such as the optimal fitness value stagnates or the maximum number of generations is reached).

[0040] The present invention also provides a system for implementing the multi-elliptical-hole tunnel shape optimization method under the anisotropic geological conditions, including a planar element model construction module, a stress field calculation module, an objective function optimization module, and a hybrid penalty function solution module.

[0041] The planar element model construction module measures the stress field of the surrounding rock, the orthogonal anisotropy parameters of the surrounding rock, and the geometry of the elliptical tunnel, constructs a planar element model containing a multi-elliptical tunnel, defines boundary conditions, and uploads the data to the stress field calculation module.

[0042] Based on the received data, the stress field calculation module derives the stress field calculation formula for anisotropic multi-elliptical tunnels under arbitrary loads using the analytical-semi-analytical stress model of multi-elliptical tunnels in anisotropic rock masses, and uploads the data to the objective function optimization module.

[0043] The objective function optimization module determines the set of design variables, defines the objective function, determines the constraints for tunnel optimization, determines the final hybrid penalty function, and uploads the data to the hybrid penalty function solving module based on the received data.

[0044] The data received by the hybrid penalty function solution module is used to iterate the design variable set of the hybrid penalty function using the DE algorithm to obtain the optimal design variable set, calculate the maximum circumferential stress of all tunnels, and complete the shape optimization of the multi-elliptical-hole tunnel.

[0045] This invention discloses a method and system for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions. It achieves accurate solutions for the stress field of multi-elliptical-hole tunnels in anisotropic media, as well as the shapes and spatial arrangements of multiple tunnels. Compared to traditional calculation methods, this technique not only expands the applicability of interaction analysis of multi-elliptical-hole tunnels in anisotropic rock strata but also effectively improves calculation accuracy, making it widely applicable to tunnel safety assessment and stability analysis. The final optimized output is the design vector X, which is obtained by minimizing... This method aims to optimize the shape, orientation, and layout of tunnel groups under specified complex load conditions. It seamlessly combines rigorous stress analysis with efficient optimization search, providing a practical tool for designing safe and efficient multi-tunnel systems in anisotropic rock masses. Attached Figure Description

[0046] Figure 1 This is a schematic diagram of the method flow of the present invention;

[0047] Figure 2 This is a schematic diagram of the system structure of the present invention;

[0048] Figure 3 This is a schematic diagram of the superposition of stress components interacting at any point A in an anisotropic rock stratum in the method of the present invention.

[0049] Figure 4 This is a schematic diagram of the double elliptical tunnel structure in an embodiment of the present invention;

[0050] Figure 5 This is a schematic diagram showing the location and shape of two tunnels in an embodiment of the present invention, and a schematic diagram showing the maximum circumferential stress of tunnel 2. Figure 5 A displays angles in different directions. The impact; Figure 5 B shows different half-shafts The impact;

[0051] Figure 6 This is a stress cloud diagram of the two tunnels under the change of azimuth angle of tunnel 2 in an embodiment of the present invention. Detailed Implementation

[0052] This invention provides a method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions, the flowchart of which is shown below. Figure 1 As shown, it includes the following steps:

[0053] S1. Determine the geostress field, orthogonal anisotropy parameters of the surrounding rock, and geometry of the elliptical tunnel; construct a planar element model of the tunnel with multiple elliptical holes; and define boundary conditions.

[0054] Step S1 includes the following steps:

[0055] Measure the stress field at the far end of the surrounding rock and the boundary conditions at the tunnel borehole edge, and construct a quasi-orthogonal anisotropic plane element model;

[0056] The equivalent compliance coefficient is calculated based on existing data, and the relevant calculation parameters are determined.

[0057] In the quasi-orthogonal anisotropic planar element model, the orthogonal elastic constant is determined by the longitudinal elastic modulus. and shear modulus Compared to Poisson Characterized by the presence of several elliptical tunnels within, the geometric features of which are defined by the major axis a, minor axis b, and azimuth angle. Sure;

[0058] The parametric equation for the k-th elliptical tunnel is expressed using the following formula: ;in, Let be the coordinates of the k-th elliptical tunnel; Let be the major axis of the k-th elliptical tunnel; Let be the minor axis of the k-th elliptical tunnel; For the elliptical tunnel parameter angle;

[0059] The boundary conditions of the tunnel bore are expressed using the following formula: ;in, The normal stress is the wall stress of the k-th elliptical tunnel. Let K be the shear stress of the wall of the k-th elliptical tunnel. This represents the boundary of the k-th elliptical tunnel. The circumferential stress at the edge of the k-th elliptical tunnel hole; Let be the tangential stress at the edge of the k-th elliptical tunnel hole.

[0060] Calculate the equivalent compliance coefficient using the following formula: ;in, The strain is the normal strain along the x-axis; The strain is the normal strain along the y-axis; For shear strain; The stress is the normal stress along the x-axis; The normal stress is along the y-axis; Shear stress; Here, is the equivalent coefficient, where and ; The equivalent compliance coefficient is related to the material properties;

[0061] Based on the measured material parameters, the relevant calculation parameters are determined using the following formula: ; ;in, The roots of the above fourth-order characteristic equation are denoted as . The number of conjugate roots characterizing anisotropy; This is the real part of the first pair of conjugate roots; This is the real part of the second pair of conjugate roots; This is the imaginary part of the first pair of conjugate roots; This is the imaginary part of the second pair of conjugate roots.

[0062] S2. The stress field calculation formula for anisotropic multielliptical tunnels under arbitrary loads is derived by using the analytical-semi-analytical stress model of multielliptical tunnels in anisotropic rock masses.

[0063] Specifically, a schematic diagram illustrating the superposition of stress components interacting at any point A within anisotropic rock strata in a multi-elliptical tunnel structure is shown below. Figure 3 As shown.

[0064] Step S2 specifically involves calculating the stress at any point A in the multi-elliptical tunnel structure within the anisotropic rock strata using the following formula: ;

[0065] in , and (k=1,2) represent any point A generated by the interaction of any bielliptical loaded tunnel under far-field stress. Radial stress, circumferential stress and tangential stress at the location, Let A represent the radial stress caused by a unit radial force along the j-th arc of the k-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit tangential force along the j-th arc of the k-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit radial force along the j-th arc of the m-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit tangential force along the j-th arc of the m-th hole at any point A on the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by a unit radial traction force along the j-th arc of the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by the unit tangential traction force along the j-th arc of the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by a unit radial traction force along the j-th arc of the m-th hole. Let represent the circumferential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the m-th hole. Let A represent the tangential stress at any point A on the k-th hole caused by the unit radial traction force along the j-th arc of the k-th hole. Let represent the tangential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the k-th hole. Let represent the tangential stress at any point A on the k-th hole caused by the unit radial traction force along the j-th arc of the m-th hole. This represents the tangential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the m-th hole; M is the number of arcs divided by the elliptical boundary.

[0066] S3. Determine the set of design variables, define the objective function, determine the constraints for tunnel optimization, and determine the final hybrid penalty function;

[0067] Step S3 includes the following steps:

[0068] For a system containing K elliptical tunnels, determine the set of design variables;

[0069] The multi-objective problem is standardized by adopting the minimax criterion and defining the objective function based on the maximum value of the critical stress measure at all tunnel boundaries;

[0070] The geometric engineering constraints for tunnel optimization are determined, and the objective function problem is transformed into an unconstrained optimization problem using the penalty function method.

[0071] An optimized mathematical model is used to represent the unconstrained optimization problem.

[0072] The set of design variables is represented by the following formula: The design variables for the k-th tunnel (k=1,2,...K) are: ;in, and These are the semi-major axis and the semi-minor axis, respectively. This is the azimuth angle (used to determine the direction of rotation of the local major axis relative to the global x-axis). Represents the global coordinates of the tunnel center.

[0073] The objective function is expressed using the following formula: ;in Show the angle position at the boundary of the kth tunnel. The circumferential stress at each tunnel boundary is a function of all design variables X. The optimization objective is to determine the value of X that minimizes the objective function, thereby minimizing the maximum circumferential stress at each tunnel boundary.

[0074] The geometric engineering constraints include size constraints, shape constraints, cross-sectional area constraints, and minimum spacing constraints;

[0075] The dimensional constraints are expressed using the following formula: ;in, The minimum major axis of the tunnel; The longest axis of the tunnel; It is the minimum minor axis of the tunnel; The maximum minor axis of the tunnel;

[0076] The shape constraint is expressed using the following formula: ;in, The maximum permissible elliptical tunnel eccentricity;

[0077] The cross-sectional area constraint is expressed using the following formula: ;in, To maintain the required internal area of ​​the tunnel;

[0078] The minimum spacing constraint is expressed using the following formula: ;in, Let r represent the boundary of the i-th ellipse, and r be the position vector of the boundary point. This indicates the minimum required width of the rock column, which may depend on the rock mass quality and in-situ stress.

[0079] The objective function problem can be transformed into an unconstrained optimization problem using the following penalty function method: ;in, This represents inequality constraints (size constraints, shape constraints, and minimum spacing constraints can all be converted to the form g(X)≥0); Representing equality constraints (cross-sectional area constraints can be considered as) (or included as an inequality with small tolerance); To apply the penalty parameters for inequality constraints that are progressively increased through an iterative sequence. It is an equality constraint penalty parameter that is gradually increased through an iterative sequence, used to force the solution to approach the feasible region; m represents the number of inequality constraints; p represents the number of equality constraints.

[0080] The optimized mathematical model is represented by the following formula: .

[0081] S4. Using the DE algorithm, iterate the design variable set of the hybrid penalty function to obtain the optimal design variable set, calculate the maximum circumferential stress of all tunnels, and complete the shape optimization of the multi-elliptical-hole tunnel.

[0082] In step S4, the differential evolution algorithm is used to solve the optimization problem in step S3, including the following steps:

[0083] Initialization: Randomly generate within a preset boundary range containing... candidate design vectors The population, where each design vector represents a complete multi-tunnel configuration;

[0084] Stress analysis: for each candidate vector The stress model (analytical-semi-analytical stress model for multi-elliptical tunnels) was used to calculate the circumferential stress distribution of all tunnels. This step involves solving the integral equation of the interaction traction force and then superimposing the contributions of each stress.

[0085] Fitness evaluation: Calculate the penalty function value for each candidate vector based on its maximum circumferential stress and constraint violation. ;

[0086] Differential evolution operations (mutation, crossover, selection): New candidate solutions are generated through differential mutation and binomial crossover; the selection operator is based on... The value preserves a better solution between the target vector and the test vector;

[0087] Iteration: Repeat the stress analysis to differential evolution operation until the convergence criterion is met (such as the optimal fitness value stagnates or the maximum number of generations is reached).

[0088] The present invention also provides a system for implementing the method for optimizing the shape of multi-elliptical-hole tunnels under the aforementioned anisotropic geological conditions, the structural schematic diagram of which is shown below. Figure 2 As shown, it includes a planar element model construction module, a stress field calculation module, an objective function optimization module, and a hybrid penalty function solution module.

[0089] The planar element model construction module measures the stress field of the surrounding rock, the orthogonal anisotropy parameters of the surrounding rock, and the geometry of the elliptical tunnel, constructs a planar element model containing a multi-elliptical tunnel, defines boundary conditions, and uploads the data to the stress field calculation module.

[0090] Based on the received data, the stress field calculation module derives the stress field calculation formula for anisotropic multi-elliptical tunnels under arbitrary loads using the analytical-semi-analytical stress model of multi-elliptical tunnels in anisotropic rock masses, and uploads the data to the objective function optimization module.

[0091] The objective function optimization module determines the set of design variables, defines the objective function, determines the constraints for tunnel optimization, determines the final hybrid penalty function, and uploads the data to the hybrid penalty function solving module based on the received data.

[0092] The data received by the hybrid penalty function solution module is used to iterate the design variable set of the hybrid penalty function using the DE algorithm to obtain the optimal design variable set, calculate the maximum circumferential stress of all tunnels, and complete the shape optimization of the multi-elliptical-hole tunnel.

[0093] The method of the present invention will be further described below with reference to an embodiment:

[0094] This embodiment takes the optimization of the shape of a double elliptical hole tunnel in anisotropic materials as an example, and includes the following steps:

[0095] The first step is to define the external loads, anisotropic rock strata parameters, and the geometric parameters of the double-elliptical tunnel. For example... Figure 4 As shown, the loading conditions consist of two parts to simulate real deep-buried conditions: 1) Far-field static stress: , 2) Tunnel 1 surface tension: Asymmetric pressure is applied to the tunnel wall: In this example, it is assumed that the shear tension of tunnel 1 is zero. The anisotropic rock mass is used with the following elastic parameters: Young's modulus. , shear modulus Poisson's ratio The anisotropy stems from the assumption that the sedimentary layers or bedding planes are inclined at a 10-degree angle relative to the horizontal direction. The geometry and location of Tunnel 1 (left side) are fixed, and its geometric parameters are: semi-major axis... short half shaft Azimuth The angle is 90° (horizontal direction of the major axis). To simplify calculations, its center point is set at the origin of the global coordinate system. The optimization objective is tunnel 2 (right side), and its design variables need to be determined.

[0096] The second step involves deriving the stress field calculation formula for anisotropic multi-elliptical tunnels under arbitrary loads using an analytical-semi-analytical stress model of multi-elliptical tunnels in anisotropic rock masses. The accurate stress field is then obtained by superimposing the fundamental solutions of concentrated and distributed forces.

[0097] The third step is to determine the set of design variables, define the objective function, determine the constraints for tunnel optimization, and determine the final hybrid penalty function.

[0098] Determine the design variables for Tunnel 2: semi-major axis short half shaft Azimuth and center coordinates These parameters determine the horizontal center spacing *l*. In this example, the design variables during the optimization process are: .

[0099] Define the objective function: To clarify the control objective for key stress concentration points in the dual-tunnel system, fix the parameters of tunnel 1, and minimize the maximum circumferential stress at the boundary between the two tunnels through collaborative optimization. Therefore, the objective function is: Determine geometric constraints: In practical applications, the following constraints must be met: ① Dimensional constraints: , ② Shape constraints (aspect ratio): ③ Area constraint: (Minimum clearance area must be maintained); ④ Minimum spacing constraint: To ensure the stability of the rock pillar, the minimum distance between the boundaries of Tunnel 1 and Tunnel 2 must be greater than 4m; ⑤ Layout constraint: The center of Tunnel 2 must be located within the rectangular area to the right of Tunnel 1.

[0100] Determine the mixed penalty function This is constructed by introducing penalty terms into the original objective function. These penalty terms increase as the severity of constraint violation increases. ;in Inequality constraints (size constraints, shape constraints, and minimum spacing constraints can all be converted to) (in the form of). Representing equality constraints (cross-sectional area constraints can be considered as) (or included as an inequality with small tolerance). To apply the penalty parameters for inequality constraints that are progressively increased through an iterative sequence. It is an equality constraint penalty parameter that is gradually increased through an iterative sequence, used to force the solution to approach the feasible region; m represents the number of inequality constraints; p represents the number of equality constraints.

[0101] Therefore, the penalty function method is used to solve this optimization problem, and the mathematical model of the optimization is expressed as follows: ;

[0102] The fourth step is to use the DE algorithm to find the optimal design variables and then verify them using numerical simulation.

[0103] The optimal design vector X, with specific parameters shown in Table 1.

[0104] Table 1 Tunnel Optimization Results

[0105] To verify the accuracy of the optimization results, this paper conducts numerical simulation verification based on Abaqus software. Table 2 shows the maximum circumferential stress of the two tunnels as a function of the tunnel azimuth angle. The numerical simulation results show the changes in the maximum circumferential stress of the two tunnels as a function of the tunnel's two axes. Table 3 presents the results. The changing numerical simulation results, Figure 5 This demonstrates the correspondence between the shape and spatial arrangement of Tunnel 2 and the change in its maximum circumferential stress. Figure 6 This shows the azimuth angle of the tunnel. Stress cloud diagrams of the two tunnels under varying conditions.

[0106] Table 2 Azimuth Variation

[0107] Table 3 Minor axis variation

[0108] This case study successfully demonstrates the application effectiveness of the proposed porous optimization framework. This method transforms the complex problem of designing multiple interacting tunnels from a traditional trial-and-error process into a systematic, computationally driven optimal solution search process. The method efficiently handles anisotropic characteristics, complex load conditions (including combinations of far-field forces and arbitrary wall tensions), and multiple geometric constraints. Although this case study uses only two tunnels as examples for illustrative purposes, the methodology has inherent scalability and can be directly applied to the layout design of large tunnel groups in anisotropic foundations, providing a powerful tool for the rational and economical design of deep underground engineering projects.

Claims

1. A method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions, characterized in that, Includes the following steps: S1. Determine the geostress field, orthogonal anisotropy parameters of the surrounding rock, and geometry of the elliptical tunnel; construct a planar element model of the tunnel with multiple elliptical holes; and define boundary conditions. S2. The stress field calculation formula for anisotropic multielliptical tunnels under arbitrary loads is derived by using the analytical-semi-analytical stress model of multielliptical tunnels in anisotropic rock masses. S3. Determine the set of design variables, define the objective function, determine the constraints for tunnel optimization, and determine the final hybrid penalty function; S4. Using the DE algorithm, iterate the design variable set of the hybrid penalty function to obtain the optimal design variable set, calculate the maximum circumferential stress of all tunnels, and complete the shape optimization of the multi-elliptical-hole tunnel.

2. The method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions according to claim 1, characterized in that, Step S1 includes the following steps: Measure the stress field at the far end of the surrounding rock and the boundary conditions at the tunnel borehole edge, and construct a quasi-orthogonal anisotropic plane element model; The equivalent compliance coefficient is calculated based on existing data, and the relevant calculation parameters are determined.

3. The method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions according to claim 2, characterized in that, In the quasi-orthogonal anisotropic planar element model, the orthogonal elastic constant is determined by the longitudinal elastic modulus. and shear modulus Compared to Poisson Characterized by the presence of several elliptical tunnels within, the geometric features of which are defined by the major axis a, minor axis b, and azimuth angle. Sure; The parametric equation for the k-th elliptical tunnel is expressed using the following formula: ;in, Let be the coordinates of the k-th elliptical tunnel; Let be the major axis of the k-th elliptical tunnel; Let be the minor axis of the k-th elliptical tunnel; For the angle parameter of the ellipse; The boundary conditions of the tunnel bore are expressed using the following formula: ;in, The normal stress is the wall stress of the k-th elliptical tunnel. Let K be the shear stress of the wall of the k-th elliptical tunnel. This represents the boundary of the k-th elliptical tunnel. The circumferential stress at the edge of the k-th elliptical tunnel hole; Let be the tangential stress at the edge of the k-th elliptical tunnel hole.

4. The method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions according to claim 2, characterized in that, Calculate the equivalent compliance coefficient using the following formula: ;in, The strain is the normal strain along the x-axis; The strain is the normal strain along the y-axis; For shear strain; The stress is the normal stress along the x-axis; The normal stress is along the y-axis; Shear stress; Here, is the equivalent coefficient, where and ; The equivalent compliance coefficient is related to the material properties; Based on the measured material parameters, the relevant calculation parameters are determined using the following formula: ; ;in, The roots of the above fourth-order characteristic equation are denoted as . The number of conjugate roots characterizing anisotropy; This is the real part of the first pair of conjugate roots; This is the real part of the second pair of conjugate roots; This is the imaginary part of the first pair of conjugate roots; This is the imaginary part of the second pair of conjugate roots.

5. The method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions according to claim 1, characterized in that, Step S2 specifically involves using the following formula to calculate the stress at any point A in the multi-elliptical tunnel structure within the anisotropic rock strata; ; in , and (k=1,2) represent any point A generated by the interaction of any bielliptical loaded tunnel under far-field stress. Radial stress, circumferential stress and tangential stress at the location, Let A represent the radial stress caused by a unit radial force along the j-th arc of the k-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit tangential force along the j-th arc of the k-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit radial force along the j-th arc of the m-th hole at any point A on the k-th hole. Let A represent the radial stress caused by a unit tangential force along the j-th arc of the m-th hole at any point A on the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by a unit radial traction force along the j-th arc of the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by the unit tangential traction force along the j-th arc of the k-th hole. Let A represent the circumferential stress caused at any point A on the k-th hole by a unit radial traction force along the j-th arc of the m-th hole. Let represent the circumferential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the m-th hole. Let A represent the tangential stress at any point A on the k-th hole caused by the unit radial traction force along the j-th arc of the k-th hole. Let represent the tangential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the k-th hole. Let represent the tangential stress at any point A on the k-th hole caused by the unit radial traction force along the j-th arc of the m-th hole. This represents the tangential stress at any point A on the k-th hole caused by the unit tangential traction force along the j-th arc of the m-th hole; M is the number of arcs divided by the boundary of the ellipse.

6. The method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions according to claim 1, characterized in that, Step S3 includes the following steps: For a system containing K elliptical tunnels, determine the set of design variables; The multi-objective problem is standardized by adopting the minimax criterion and defining the objective function based on the maximum value of the critical stress measure at all tunnel boundaries; The geometric engineering constraints for tunnel optimization are determined, and the objective function problem is transformed into an unconstrained optimization problem using the penalty function method. An optimized mathematical model is used to represent the unconstrained optimization problem.

7. The method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions according to claim 6, characterized in that, The set of design variables is represented by the following formula: The design variables for the k-th tunnel (k=1,2,...K) are: ;in, and These are the semi-major axis and the semi-minor axis, respectively. This is the azimuth angle (used to determine the direction of rotation of the local major axis relative to the global x-axis). The objective function represents the global coordinates of the tunnel center. ;in Show the angle position at the boundary of the kth tunnel. The circumferential stress at each tunnel boundary is a function of all design variables X. The optimization objective is to determine the value of X that minimizes the objective function, thereby minimizing the maximum circumferential stress at each tunnel boundary. The geometric engineering constraints include size constraints, shape constraints, cross-sectional area constraints, and minimum spacing constraints; The dimensional constraints are expressed using the following formula: ;in, The minimum major axis of the tunnel; The longest axis of the tunnel; It is the minimum minor axis of the tunnel; The maximum minor axis of the tunnel; The shape constraint is expressed using the following formula: ;in, The maximum permissible elliptical tunnel eccentricity; The cross-sectional area constraint is expressed using the following formula: ;in, To maintain the required internal area of ​​the tunnel; The minimum spacing constraint is expressed using the following formula: ;in, Let r represent the boundary of the i-th ellipse, and r be the position vector of the boundary point. This indicates the minimum required width of the rock column, which depends on the rock mass quality and in-situ stress.

8. The method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions according to claim 6, characterized in that, The objective function problem can be transformed into an unconstrained optimization problem using the following penalty function method: ;in, Indicates inequality constraints; Indicates equality constraints; To apply the penalty parameters for inequality constraints that are progressively increased through an iterative sequence. It is an equality constraint penalty parameter that is gradually increased through an iterative sequence, used to force the solution to approach the feasible region; m represents the number of inequality constraints; p represents the number of equality constraints; The optimized mathematical model is represented by the following formula: .

9. The method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions according to claim 1, characterized in that, In step S4, the differential evolution algorithm is used to solve the optimization problem in step S3, including the following steps: Initialization: Randomly generate within a preset boundary range containing... candidate design vectors The population, where each design vector represents a complete multi-tunnel configuration; Stress analysis: for each candidate vector The stress model (analytical-semi-analytical stress model for multi-elliptical tunnels) was used to calculate the circumferential stress distribution of all tunnels. This step involves solving the integral equation of the interaction traction force and then superimposing the contributions of each stress. Fitness evaluation: Calculate the penalty function value for each candidate vector based on its maximum circumferential stress and constraint violation. ; Differential evolution operation: generates new candidate solutions through differential mutation and binomial crossover; the operator is selected based on... The value preserves a better solution between the target vector and the test vector; Iteration: Repeat the stress analysis step to the differential evolution operation until the convergence criterion is met.

10. A system for implementing the method for optimizing the shape of multi-elliptical-hole tunnels under anisotropic geological conditions as described in any one of claims 1 to 9, characterized in that, It includes a planar element model construction module, a stress field calculation module, an objective function optimization module, and a hybrid penalty function solution module; The planar element model construction module measures the stress field of the surrounding rock, the orthogonal anisotropy parameters of the surrounding rock, and the geometry of the elliptical tunnel, constructs a planar element model containing a multi-elliptical tunnel, defines boundary conditions, and uploads the data to the stress field calculation module. Based on the received data, the stress field calculation module derives the stress field calculation formula for anisotropic multi-elliptical tunnels under arbitrary loads using the analytical-semi-analytical stress model of multi-elliptical tunnels in anisotropic rock masses, and uploads the data to the objective function optimization module. The objective function optimization module determines the set of design variables, defines the objective function, determines the constraints for tunnel optimization, determines the final hybrid penalty function, and uploads the data to the hybrid penalty function solving module based on the received data. The data received by the hybrid penalty function solution module is used to iterate the design variable set of the hybrid penalty function using the DE algorithm to obtain the optimal design variable set. The maximum circumferential stress of all tunnels is then calculated to complete the shape optimization of the multi-elliptical-hole tunnel.

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