A rapid calculation method and system for metro vibration Green's function
By coupling the fundamental solution of the infinite domain with the layered site, the contradiction between accuracy and efficiency, stability and practicality in subway vibration assessment is resolved, and rapid, stable and efficient vibration prediction is achieved, which is applicable to multi-condition analysis under complex geological conditions.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- SHANDONG JIANZHU UNIV
- Filing Date
- 2026-02-05
- Publication Date
- 2026-05-12
AI Technical Summary
Existing technologies for subway vibration assessment suffer from a trade-off between accuracy and efficiency, and stability and practicality. They are unable to quickly and accurately predict the impact of subway vibration on the surrounding environment, especially under complex geological conditions, where the computational scale is enormous and the results are unstable.
By employing the method of coupling the fundamental solution of the infinite domain with the layered site, a near-field finite element model of the subway tunnel is constructed and an infinite domain virtual vibration source is arranged externally. The coupling equation is established using the continuity condition of displacement and force to obtain the amplitude vector of the virtual vibration source independent of the layered site. The vibration response is then calculated using the propagation matrix of the layered site to obtain the Green's function of subway vibration.
It achieves rapid, stable, and efficient subway vibration assessment, reduces computational scale and time, improves prediction accuracy, can handle multi-condition analysis needs under complex geological conditions, and provides a practical engineering tool.
Smart Images
Figure CN121637937B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of subway vibration assessment technology, specifically to a method and system for rapid calculation of the Green's function of subway vibration. Background Technology
[0002] During operation, urban rail transit systems transmit vibrations and noise to the surrounding soil and surface buildings, which may lead to problems such as decreased structural comfort, interference with precision instruments, and noise complaints from residents. To assess the propagation law of subway vibrations and their impact on the surrounding built environment, it is usually necessary to obtain the Green's function of the track structure and site system, which is the propagation response of a unit excitation in the soil. The Green's function is the basis for soil vibration analysis, the establishment of vibration source-propagation-reception models, subway vibration prediction, and vibration reduction design.
[0003] The vibration problem of subway has the following typical characteristics: (1) The soil is a semi-infinite domain and has obvious layered structure, such as fill, silt, sand, rock, etc., and the density, wave velocity and damping of each layer are significantly different; (2) The vibration source (subway) has obvious spatial effects, and wave propagation needs to be considered on a two-dimensional plane, while frequency domain expression is more commonly used; (3) Under the condition of long tunnels, the engineering community generally adopts the idea of 2.5D solution model (i.e., Fourier transform in the longitudinal direction and modeling only in the cross section) to reduce the calculation cost.
[0004] In existing technologies, vibration prediction for rail transit environments mainly adopts the following approaches:
[0005] (1) Pure finite element (FEM) or finite element-boundary element coupled method: A large-scale three-dimensional model is established in a semi-infinite soil domain. The computation is large and difficult to reuse. It cannot quickly predict multiple working conditions and multiple lines.
[0006] (2) Simple layered foundation Green function method: It often uses analytical approximation or literature database, which is difficult to accurately describe the complexity of tunnel structure, local geological changes and irregular boundaries in actual engineering.
[0007] The above approach has two core shortcomings:
[0008] The first point is that there is an irreconcilable contradiction between high-precision methods and high-efficiency methods: if we want to accurately simulate the wave propagation effect of complex tunnel structures and layered soils (such as using the 2.5D finite element method or the boundary element method based on layered fundamental solutions), we must finely discretize the entire site or solve complex equations, resulting in a huge computational scale and extremely long time consumption, which cannot meet the needs of rapid analysis and multi-condition comparison in engineering practice; on the contrary, if we use computationally efficient infinite domain fundamental solution methods (such as the traditional source point method) to directly process layered sites, we cannot accurately describe the physical mechanism of multiple reflections and transmissions of waves between layers, resulting in distortion of far-field vibration prediction results, which is difficult to meet the accuracy requirements of environmental vibration assessment.
[0009] Secondly, there are significant deficiencies in numerical stability and engineering practicality: In the process of attempting to directly couple the tunnel structure with the layered site in the inversion solution (for example, introducing layered basic solutions in the source point method or arranging virtual sources for each layer), the condition number of the system matrix deteriorates sharply with the increase of the number and frequency of strata, resulting in a highly ill-conditioned solution process, unstable results, and susceptibility to numerical noise interference, and even non-physical solutions. At the same time, existing high-precision methods have extremely high requirements for modeling skills, computational resources, and professional numerical knowledge, while simplified methods are unreliable due to insufficient accuracy or stability. There is a lack of a practical tool that can guarantee robustness and be efficiently integrated with engineering geological data and design processes, which seriously hinders the widespread application of advanced vibration prediction technology in actual subway engineering projects. Summary of the Invention
[0010] This invention proposes a fast calculation method and system for the Green's function of subway vibration. By coupling the fundamental solution of the infinite domain with the layered site, it overcomes the dual imbalance between "accuracy-efficiency" and "stability-practicability".
[0011] According to some embodiments, the present invention adopts the following technical solution:
[0012] A fast method for calculating the Green's function of subway vibration includes:
[0013] A near-field finite element model of a subway tunnel is constructed, and multiple infinite-domain virtual vibration sources are arranged outside the near-field finite element model. The vibration response of each virtual vibration source is described by the infinite-domain fundamental solution.
[0014] Based on the continuity condition of displacement and force, the coupling equation between the near-field finite element model and the infinite domain virtual vibration source array is established, and the coupling equation is solved for each frequency point to obtain the infinite domain virtual vibration source amplitude vector independent of the layered site.
[0015] Based on the physical parameters of each layer of the target layered site, a layered site propagation matrix is constructed from the virtual vibration source location to any site location.
[0016] The amplitude vector of the infinite domain virtual vibration source is input into the propagation matrix of the layered site to calculate the vibration response of the target layered site at the corresponding frequency, thereby obtaining the Green's function of subway vibration.
[0017] According to some embodiments, the present invention adopts the following technical solution:
[0018] A fast calculation system for the Green's function of subway vibration includes:
[0019] The virtual vibration source module is configured to: construct a near-field finite element model of a subway tunnel, and arrange multiple infinite-domain virtual vibration sources outside the near-field finite element model, wherein the vibration response of each virtual vibration source is described by the infinite-domain fundamental solution;
[0020] The vibration source amplitude module is configured to: establish the coupling equation between the near-field finite element model and the infinite domain virtual vibration source array based on the continuity condition of displacement and force, and solve the coupling equation for each frequency point to obtain an infinite domain virtual vibration source amplitude vector that is independent of the layered site.
[0021] The propagation matrix module is configured to: construct a layered site propagation matrix from the virtual vibration source location to any site location based on the physical parameters of each layer of the target layered site;
[0022] The vibration response module is configured to input the amplitude vector of the infinite domain virtual vibration source into the layered site propagation matrix, calculate the vibration response of the target layered site at the corresponding frequency, and obtain the Green's function of subway vibration.
[0023] According to some embodiments, the present invention adopts the following technical solution:
[0024] A computer program product includes a computer program that, when executed by a processor, implements the aforementioned method for rapidly calculating the Green's function of subway vibration.
[0025] According to some embodiments, the present invention adopts the following technical solution:
[0026] A non-transitory computer-readable storage medium is provided for storing computer instructions, which, when executed by a processor, implement the aforementioned method for fast calculation of the Green's function for subway vibration.
[0027] According to some embodiments, the present invention adopts the following technical solution:
[0028] An electronic device includes a processor, a memory, and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory to enable the electronic device to implement the aforementioned method for fast calculation of the Green's function of subway vibration.
[0029] Compared with the prior art, the beneficial effects of the present invention are as follows:
[0030] 1. This invention decouples the virtual source solution from the layered propagation—by coupling the infinite domain fundamental solution (rather than using the layered fundamental solution in the source-finding stage) with the tunnel FEM at the coupling interface, a virtual source amplitude vector independent of the layered structure is obtained. This avoids directly introducing complex layered site fundamental solutions during the virtual source inversion stage. This makes the virtual source amplitude vector independent of the layered site structure, and the size of its solution matrix no longer increases with the number of site layers. This fundamentally solves the problem of the virtual source solution size expanding rapidly due to inter-layer coupling in traditional methods, and greatly saves computational memory and time.
[0031] 2. Improved efficiency in frequency scanning and multi-point response calculation—This invention obtains a site-independent virtual source amplitude vector. Then, the layered site propagation matrix can be reused at any receiving point. Or repeat the calculation for different layered models. This completely avoids repeating large-scale coupled solutions at every frequency point or every receiving point, resulting in an order-of-magnitude improvement in computational efficiency for large-scale frequency scanning and multi-point response analysis in complex sites, thereby significantly reducing the computational load for multiple frequency points and multiple receiving points.
[0032] 3. Improved numerical stability: This invention limits the inversion / source finding to a relatively "well-determined" problem based on an infinite-domain kernel, making the solution condition number more controllable and effectively avoiding the numerical ill-conditioning problem caused by directly embedding complex hierarchical transfer matrices into the inversion process. This makes the solution results of the virtual source amplitude more stable and reliable, reduces the occurrence of non-physical solutions, and improves the numerical robustness of the entire calculation process.
[0033] 4. This invention provides an engineering-feasible Green's function calculation process. Under the premise of ensuring physical correctness (linear superposition and layered propagation can be expressed by a propagation matrix), this invention realizes an efficient and scalable process for coupling tunnel site to Green's function output. This process, while ensuring the correctness of the physical mechanism of wave propagation, has both high efficiency and scalability. It can conveniently handle the complex multi-layered geological conditions and large-scale parameter analysis needs commonly encountered in engineering, and provides a practical technical tool for the efficient prediction and assessment of the environmental impact of subway vibration. Attached Figure Description
[0034] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.
[0035] Figure 1 This is a flowchart of the method in Example 1.
[0036] Figure 2 This is a schematic diagram of the infinite domain virtual vibration source setup in Example 1.
[0037] Figure 3 This is a schematic diagram of the layered site propagation matrix in Example 1.
[0038] Figure 4 The graph shows the calculation results for Example 1. Detailed Implementation
[0039] The present invention will be further described below with reference to the accompanying drawings and embodiments.
[0040] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. 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 invention pertains.
[0041] 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 scope of exemplary embodiments according to the invention. 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.
[0042] Example 1
[0043] One embodiment of the present invention provides a method for fast calculation of the Green's function of subway vibration, the working principle of which is as follows:
[0044] (1) The virtual source amplitude is obtained by using the fundamental solution of a uniform infinite domain in the near field.
[0045] The tunnel and the surrounding soil within a few meters are considered as a homogeneous region, so an infinite-domain Green's function is used to construct the MFS external field; this avoids the high complexity of layered sites, making the virtual source solution a well-determined problem.
[0046] (2) The virtual source amplitude is an "equivalent external excitation".
[0047] The obtained virtual source magnitude vector Equivalent to: If an amplitude of [value] is placed in a stratified site The actual point source can reproduce the real vibration field near the tunnel boundary.
[0048] (3) Then use the layered site propagation matrix to obtain the far-field response.
[0049] By using a virtual source as input and the analytical propagation matrix of the layered half-space (with velocity, density, and thickness given by definition), the surface and far-field responses can be obtained efficiently.
[0050] Based on the above principles, the specific steps are as follows: Figure 1 As shown, it includes:
[0051] Step 1: Construct a near-field finite element model of the subway tunnel, and arrange multiple infinite-domain virtual vibration sources outside the near-field finite element model. The vibration response of each virtual vibration source is described by the infinite-domain fundamental solution.
[0052] Furthermore, the near-field finite element model for constructing the subway tunnel is specifically as follows:
[0053] Establish finite element models of the subway tunnel lining structure, surrounding rock, and track system, determine material parameters, set boundary nodes as coupling interfaces, and output the finite element stiffness matrix.
[0054] Specifically, the near-field finite element model (i.e., the tunnel FEM model) includes a set of near-field finite element elements for the tunnel (FEM substructure), containing node sets, element sets, material parameters, boundary conditions, and load interfaces. It is used to simulate the subway tunnel lining structure and its surrounding near-field soil layers. The steps are as follows:
[0055] (1) Establish finite element models of tunnel lining, surrounding rock, and track structure;
[0056] (2) Determine material parameters: elastic modulus Poisson's ratio ,density Loss factor ;
[0057] (3) Set the boundary nodes as coupling interfaces;
[0058] (4) Output the finite element stiffness matrix .
[0059] Furthermore, when arranging virtual vibration sources outside the near-field finite element model, the virtual vibration sources are located within the annular or surface envelope region outside the tunnel structure.
[0060] Specifically, such as Figure 2 As shown, an infinite-domain virtual vibration source array (MFS point source set) is constructed, containing... There are several virtual vibration sources, each of which is a frequency-domain point force source with an amplitude of the vector to be determined. The response of each point source is described by the fundamental solution of the infinite field, which includes the force operator of the fundamental solution of the infinite field. and the fundamental solution shift operator for infinite fields The virtual vibration sources in the array are set within the surface envelope region outside the tunnel FEM.
[0061] In this embodiment, the optimal number of virtual sources is calculated using the following formula:
[0062]
[0063]
[0064] in, The minimum wavelength is often chosen in engineering because it is the minimum wavelength required for the calculation process. , For shear wave velocity, This represents the maximum frequency during the calculation process.
[0065] Step 2: Based on the continuity condition of displacement and force, establish the coupling equation between the near-field finite element model and the infinite domain virtual vibration source array, and solve the coupling equation for each frequency point to obtain the infinite domain virtual vibration source amplitude vector independent of the layered site.
[0066] Infinite Domain Virtual Vibration Source Array and Tunnel FEM Boundary Nodes via Infinite Domain Fundamental Solution Displacement Operator The connection here uses the displacement operator of the infinite domain fundamental solution to represent the displacement Green's function in a uniform elastic infinite domain. This is used for coupling the MFS virtual source with the free field at the tunnel boundary. This coupling produces an equivalent virtual source, ensuring displacement-force continuity between the FEM displacement field and the MFS external field at the boundary. Therefore, a displacement / force continuity condition is applied to the tunnel interface. Using the infinite domain fundamental solution, the FEM–MFS coupling equation is constructed. The specific form of the coupling equation is as follows:
[0067]
[0068] The first formula represents the force equilibrium condition, and the second formula represents the displacement continuity condition. The left side of both formulas represents the finite element solution domain, while the right side represents the solution domain of the fundamental solution method. This is the stiffness matrix of the finite element part. Let be the displacement vector of the finite element nodes on the boundary. For the fundamental solution force operator of an infinite field, For the fundamental solution of the infinite field, the shift operator. The amplitude of the virtual vibration source array. This is the vector to be determined in this step.
[0069] Specifically, after arranging the infinite-domain virtual vibration source array within the outer envelope region of the tunnel FEM, the infinite-domain fundamental force operator is used to solve the problem. Amplitude of the virtual vibration source array The dot product is used to calculate the forces in the solution domain at the boundary using the fundamental solution method. Based on the force equilibrium condition, the forces in the solution domain at the boundary using the fundamental solution method are in equilibrium with the forces at the boundary of the finite element solution domain. The displacement operator of the infinite domain fundamental solution is then used. Amplitude of the virtual vibration source array The dot product is used to calculate the displacement of the solution domain at the boundary using the fundamental solution method. According to the displacement continuity condition, the displacement of the solution domain at the boundary using the fundamental solution method is equal to the displacement of the boundary of the solution domain using the finite element method.
[0070] After constructing the coupling equations, equivalent equilibrium conditions are applied at the tunnel boundary, and Gaussian elimination, LU decomposition, and Krylov subspace method are used to solve for each frequency. virtual source amplitude During the solution process, Tikhonov regularization or QR decomposition is used to improve stability, i.e., given a tunnel FEM model and the surrounding environment... An infinite-domain virtual source, solving for the frequency. The virtual source amplitude vector.
[0071] The "infinite domain" here does not contain any hierarchical propagation effect, therefore, the virtual source magnitude vector here... It is obtained through the fundamental decoupling equations of finite element to infinite domain, and is independent of the number of layers.
[0072] Step 3: Based on the physical parameters of each layer of the target layered site, construct the layered site propagation matrix from the virtual vibration source location to any site location;
[0073] Furthermore, the layered site propagation matrix is constructed using the Thompson-Haskell or Kausel-Roesset recursive algorithm, with input parameters including the shear wave velocity, compression wave velocity, density, thickness, and frequency-dependent damping coefficient of each layer.
[0074] Specifically, the target layered field consists of n layers. For each layer k, 1 <= k <= n, the propagation matrix of layer k is calculated using layer k as the set of receiving points. ,like Figure 3 As shown, the steps are as follows:
[0075] (1) Establish models of each individual stratum in the layered site, i.e., models of each layer, and obtain the material parameters of each individual stratum: shear wave velocity, compression wave velocity, density, thickness and frequency-related damping coefficient.
[0076] (2) Based on the material parameters of each individual stratum, the vibration propagation matrix of each individual stratum is established using the Thompson-Haskell or Kausel-Roesset recursive algorithm. ;
[0077] (3) Establish the vibration propagation matrix of layer k. .
[0078] By integrating all the layered vibration propagation matrices, the vibration propagation matrix of the target layered site is obtained. .
[0079] The receiving point here is the point where we ultimately want to calculate the vibration response, i.e., the vibration receiving point; the Green's function connects two points, one is the source point and the other is the receiving point, which is the displacement at the receiving point when the force is applied at the source point. Therefore, we set the receiving point in the field and solve for the vibration response in the field (such as the ground surface).
[0080] Step 4: Input the amplitude vector of the infinite domain virtual vibration source into the layered site propagation matrix, calculate the vibration response of the target layered site at the corresponding frequency, and obtain the Green's function of subway vibration.
[0081] Furthermore, the vibration response includes at least one of displacement, velocity, or acceleration.
[0082] Specifically, the input for this step is the virtual source amplitude. The output is the vibration response at any receiving point in the target layered field, which is the virtual source amplitude. propagation matrix of layered sites Continuing to propagate outwards, we obtain the Green's function of the tunnel's response to surface vibrations, which can be expressed by the formula:
[0083]
[0084] in, Let Green's function be the tunnel's response to surface vibrations. For frequency, By using a layered field propagation matrix, the displacement, velocity, or acceleration of any receiving point can be obtained in the frequency domain.
[0085] This embodiment demonstrates the effectiveness of the method through an experiment. A circular single-line shield tunnel was used, with a tunnel arch depth of 16.2m, a track surface depth of 20.7m, an inner diameter of 5.4m, an outer diameter of 6.0m, and a segment thickness of 0.3m. The material parameters are shown in Table 1.
[0086] Table 1 Model material parameters
[0087] Material Depth (m) Elastic modulus (MPa) Poisson's ratio Density (kg / m3) Loss factor concrete 32500 0.3 2450 0.04 Miscellaneous fill 1.8 95.8 0.4 1650 0.08 silt 5.5 325.7 0.35 1800 0.08 Heavy silty clay 8.1 304.8 0.4 1950 0.08 fine sand 10.3 376.0 0.35 1850 0.06 pebble 14.8 714.8 0.30 2100 0.06 Silty clay 20.0 357.0 0.40 1900 0.08
[0088] The program was written in MATLAB 2025b, and the computing platform was AMD 7950X. The code execution speed was optimized for parallel computation. The vibration of the Earth's surface under an 80 Hz moving force was calculated. The calculation results are shown below. Figure 4 As can be seen, the calculation results of the method proposed in this embodiment are almost the same as those of FEM-MFS based on the fundamental solution of infinite field, with calculation times of 3408 s and 301 s respectively, which improves the calculation efficiency by 10 times.
[0089] In summary, the fast calculation method for the Green's function of subway vibration based on FEM-MFS multi-domain coupling proposed in this embodiment, through innovative technical approaches such as coupling finite and infinite domains, calculating layered site propagation matrices, and constructing virtual source arrays, has the following main advantages:
[0090] (1) Significantly reduce the overall computational scale and achieve fast solution.
[0091] This embodiment uses finite element modeling for the near-field region of the tunnel, while representing the far-field region with a meshless "virtual source column-basic solution substructure". The modeling region and the number of meshes are then solved by the FEM-MFS coupled equations, which greatly reduces the modeling region and the number of meshes. The solution time for a single frequency point is reduced from "hours" to "minutes" or even "seconds", which can support high computational density tasks such as large-scale, multi-frequency, and sensitivity analysis.
[0092] (2) A reusable Green's function database is formed, which greatly reduces the amount of repetitive calculations.
[0093] This embodiment encapsulates the response results obtained by finite-infinite domain coupling into a Green's function database, and combines it with a layered foundation propagation matrix algorithm, so that the responses under different working conditions and different excitations can be quickly obtained through convolution superposition without resolving the three-dimensional dynamic model, thus improving the reusability rate, generating once and calling multiple times; reducing the cost of multi-working-condition prediction by more than 80%; and shortening the overall calculation cycle of the project from 2-4 weeks to 2-4 days.
[0094] (3) The propagation characteristics of stratified sites are accurately reflected, and the prediction accuracy is significantly improved.
[0095] This embodiment uses the propagation matrix method, which can accurately handle dynamic characteristics such as any number of strata, frequency-dependent damping, and complex modulus.
[0096] Multiple reflections and refractions of vertically and horizontally coupled waves; almost identical to the FEM-MFS calculation results based on the fundamental solution of the infinite field; capable of reflecting the influence of complex geology such as high-rise buildings, soft soil layers, and fill on the propagation path.
[0097] (4) It is easy to promote in engineering projects and can directly serve environmental impact assessment and vibration reduction design.
[0098] By utilizing the Green's function database, this embodiment can be directly applied to ground and building vibration prediction, optimization of vibration reduction track beds, steel spring floating slabs, and decision-making processes such as route selection and construction scheme comparison; thereby improving the efficiency of scheme comparison and reducing engineering rework.
[0099] Example 2
[0100] One embodiment of the present invention provides a fast calculation system for the Green's function of subway vibration, comprising:
[0101] The virtual vibration source module is configured to: construct a near-field finite element model of a subway tunnel, and arrange multiple infinite-domain virtual vibration sources outside the near-field finite element model, wherein the vibration response of each virtual vibration source is described by the infinite-domain fundamental solution;
[0102] The vibration source amplitude module is configured to: establish the coupling equation between the near-field finite element model and the infinite domain virtual vibration source array based on the continuity condition of displacement and force, and solve the coupling equation for each frequency point to obtain an infinite domain virtual vibration source amplitude vector that is independent of the layered site.
[0103] The propagation matrix module is configured to: construct a layered site propagation matrix from the virtual vibration source location to any site location based on the physical parameters of each layer of the target layered site;
[0104] The vibration response module is configured to input the amplitude vector of the infinite domain virtual vibration source into the layered site propagation matrix, calculate the vibration response of the target layered site at the corresponding frequency, and obtain the Green's function of subway vibration.
[0105] Example 3
[0106] One embodiment of the present invention provides a computer program product, including a computer program that, when executed by a processor, implements the aforementioned method for fast calculation of the Green's function of subway vibration.
[0107] Example 4
[0108] In one embodiment of the present invention, a non-transitory computer-readable storage medium is provided for storing computer instructions, which, when executed by a processor, implement the aforementioned method for fast calculation of the Green's function of subway vibration.
[0109] Example 5
[0110] One embodiment of the present invention provides an electronic device, including: a processor, a memory, and a computer program; wherein, the processor is connected to the memory, and the computer program is stored in the memory. When the electronic device is running, the processor executes the computer program stored in the memory, so that the electronic device performs the aforementioned method for fast calculation of the Green's function of subway vibration.
[0111] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. 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 illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.
[0112] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment 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.
[0113] While the specific embodiments of the present invention have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.
Claims
1. A method for fast calculation of the Green's function of subway vibration, characterized in that, include: A near-field finite element model of a subway tunnel is constructed, and multiple infinite-domain virtual vibration sources are arranged outside the near-field finite element model. The vibration response of each virtual vibration source is described by the infinite-domain fundamental solution. Based on the continuity condition of displacement and force, the coupling equation between the near-field finite element model and the infinite domain virtual vibration source array is established, and the coupling equation is solved for each frequency point to obtain the infinite domain virtual vibration source amplitude vector independent of the layered site. Specifically, the virtual vibration source solution is decoupled from the layered propagation. By coupling the infinite domain fundamental solution with the tunnel FEM at the coupling interface, a virtual source amplitude vector independent of the layered structure is obtained. The coupling equations are constructed using fundamental solvers for infinite fields, specifically in the form of: In the formula, This is the stiffness matrix of the finite element part. Let be the displacement vector of the finite element nodes on the boundary. For the fundamental solution force operator of an infinite field, For the fundamental solution of the infinite field, the shift operator. The amplitude of the virtual vibration source array; Based on the physical parameters of each layer of the target layered site, a layered site propagation matrix is constructed from the virtual vibration source location to any site location. The target layered site consists of n layers. For each layer k, 1 <= k <= n. The propagation matrix of layer k is calculated using layer k as the set of receiving points. The steps are as follows: Establish models of each individual stratum in the layered site, i.e., models of each layer, and obtain the material parameters of each individual stratum; based on the material parameters of each individual stratum, establish the vibration propagation matrix of each individual stratum. Establish the vibration propagation matrix of layer k. ; By integrating all the layered vibration propagation matrices, the vibration propagation matrix of the target layered site is obtained. The receiving point is the point where the vibration response is ultimately calculated, i.e., the vibration receiving point; the Green's function connects two points, one is the source point and the other is the receiving point, which is the displacement at the receiving point when the force is applied at the source point. Therefore, the receiving point is set in the field to solve for the vibration response in the field. The amplitude vector of the infinite domain virtual vibration source is input into the propagation matrix of the layered site to calculate the vibration response of the target layered site at the corresponding frequency, and the Green's function of subway vibration is obtained. Specifically, the input is the virtual source amplitude. The output is the vibration response at any receiving point in the target layered field, which is the virtual source amplitude. propagation matrix of layered sites Continuing to propagate outwards, we obtain the Green's function of the tunnel's response to surface vibrations, which can be expressed by the formula: in, The Green's function is the tunnel's response to surface vibrations. For frequency, By using a layered field propagation matrix, the displacement, velocity, or acceleration of any receiving point can be obtained in the frequency domain.
2. The method for fast calculation of the Green's function of subway vibration as described in claim 1, characterized in that, The near-field finite element model for constructing the subway tunnel is specifically as follows: Establish finite element models of the subway tunnel lining structure, surrounding rock, and track system, determine material parameters, set boundary nodes as coupling interfaces, and output the finite element stiffness matrix.
3. The method for fast calculation of the Green's function of subway vibration as described in claim 1, characterized in that, When a virtual vibration source is arranged outside the near-field finite element model, the virtual vibration source is located in the annular or surface envelope region outside the tunnel structure.
4. The method for fast calculation of the Green's function of subway vibration as described in claim 1, characterized in that, The layered site propagation matrix is constructed using the Thompson-Haskell or Kausel-Roesset recursive algorithm, with input parameters including shear wave velocity, compression wave velocity, density, thickness, and frequency-dependent damping coefficient for each layer.
5. The method for fast calculation of the Green's function of subway vibration as described in claim 1, characterized in that, The vibration response includes at least one of displacement, velocity, or acceleration.
6. A fast calculation system for the Green's function of subway vibration, characterized in that, The method for fast calculation of the Green's function of subway vibration as described in any one of claims 1-5 includes: The virtual vibration source module is configured to: construct a near-field finite element model of a subway tunnel, and arrange multiple infinite-domain virtual vibration sources outside the near-field finite element model, wherein the vibration response of each virtual vibration source is described by the infinite-domain fundamental solution; The vibration source amplitude module is configured to: establish the coupling equation between the near-field finite element model and the infinite domain virtual vibration source array based on the continuity condition of displacement and force, and solve the coupling equation for each frequency point to obtain an infinite domain virtual vibration source amplitude vector that is independent of the layered site. The propagation matrix module is configured to: construct a layered site propagation matrix from the virtual vibration source location to any site location based on the physical parameters of each layer of the target layered site; The vibration response module is configured to input the amplitude vector of the infinite domain virtual vibration source into the layered site propagation matrix, calculate the vibration response of the target layered site at the corresponding frequency, and obtain the Green's function of subway vibration.
7. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the fast calculation method for the Green's function of subway vibration as described in any one of claims 1-5.
8. A non-transitory computer-readable storage medium, characterized in that, The non-transitory computer-readable storage medium is used to store computer instructions, which, when executed by a processor, implement a method for fast calculation of the Green's function of subway vibration as described in any one of claims 1-5.
9. An electronic device, characterized in that, include: The device includes a processor, a memory, and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory to enable the electronic device to implement a method for fast calculation of the Green's function of subway vibration as described in any one of claims 1-5.