2.5D green function accelerated calculation method and system based on key frame recognition

Abstract

The application provides a 2.5D Green function accelerated calculation method and system based on key frame identification, relates to the field of rail transit vibration evaluation, and comprises the following steps: a 2.5D numerical model for calculating the Green function is established, wave number domain coarse sampling calculation is performed on a target wave number discrete grid, and key frames and non-key frames are identified; 2.5D accurate solving is performed on the key frames, and the complex Green function of the key frames is obtained; complex spectrum reconstruction is performed on the non-key frames, and the complex Green function of the non-key frames is obtained; the Green function of the complete wave number domain is generated, and the spatial domain Green function is obtained; according to the spectrum distribution characteristics of the wave number domain Green function, the key frames are adaptively determined, and the non-key frame complex response is completed by using the key frame results, so that the number of complex linear equation system solving is significantly reduced, the calculation time and hardware resource demand are reduced, and the engineering applicability of rail transit environmental vibration prediction is improved.

Description

Technical Field

[0001] This invention relates to the field of vibration assessment in rail transit, specifically to a method and system for accelerating the calculation of 2.5D Green's function based on keyframe recognition. Background Technology

[0002] With the rapid expansion of urban rail transit and high-speed railway networks in my country, environmental vibration problems caused by train operation are becoming increasingly prominent, significantly impacting the use of precision instruments along the lines, the comfort of residents, and the preservation of historical buildings. Accurately predicting and assessing the site vibration response caused by rail transit during the planning, design, and environmental impact assessment stages of transportation lines is crucial for determining the scope of land acquisition and demolition, guiding the deployment of track vibration reduction measures, and optimizing route plans. Since even minor adjustments to route plans often involve economic costs of tens or even hundreds of millions of yuan, developing vibration prediction methods that combine high accuracy and high computational efficiency has become an urgent need in the fields of rail transit engineering design and environmental management.

[0003] In existing technologies, calculating the Green's function (i.e., the displacement response under unit force) is fundamental for vibration prediction. While three-dimensional numerical methods (such as 3D FEM) can accurately describe complex spatial structures, their models are large-scale and computationally time-consuming, making them unsuitable for the needs of multi-scheme comparison in engineering design. Therefore, 2.5D numerical methods are widely used. This method utilizes Fourier transform to decompose the three-dimensional problem along the track direction into a series of two-dimensional cross-sectional problems under independent wavenumbers, significantly reducing computational degrees of freedom and facilitating parallel computation. However, this method still faces significant efficiency bottlenecks in practical applications. First, traditional 2.5D methods treat all wavenumber frames equally, requiring the complete assembly of complex matrices and solution of linear equations. For sites with complex geological conditions, including layered soil and tunnel structures, the cost of a single solution is already high. As the number of wavenumber discrete points, frequency points, and source-receptor combinations increases, the total computational cost increases dramatically.

[0004] Secondly, existing methods rely heavily on experience to achieve uniformity or pre-defined encryption when discretizing wavenumbers. They fail to fully utilize the spectral distribution characteristics of the Green's function in the wavenumber domain, where the energy contribution is non-uniform, the dominant information is concentrated in the low and medium wavenumber regions, and the high wavenumber range decays rapidly. This results in a large amount of computational resources being consumed in wavenumber regions where the contribution is relatively small.

[0005] In addition, although there are improvement methods such as parallel computing, matrix optimization or directly reducing wavenumber sampling points, these methods either only improve the execution efficiency of a single frame without reducing the total number of solution frames, or sacrifice the accuracy of near-field and high-frequency response, making it difficult to effectively reduce the number of solutions and the overall computational cost while ensuring computational accuracy.

[0006] Therefore, existing technologies that solve complex linear equations frame by frame for all wavenumber frames suffer from high computational costs, long computation time, large memory consumption, and difficulty in meeting the needs of rapid prediction and multi-scheme comparison in rail transit environmental vibration engineering, thus limiting the efficiency of engineering applications. Summary of the Invention

[0007] To address the aforementioned issues, this invention proposes a 2.5D Green's function acceleration calculation method and system based on keyframe identification. The method adaptively determines keyframes based on the distribution characteristics of the wavenumber domain Green's function spectrum and uses the keyframe results to complete the complex responses of non-keyframes. This significantly reduces the number of solutions to complex linear equations while meeting preset calculation accuracy requirements, thereby lowering computation time and hardware resource requirements and improving the engineering applicability of vibration prediction in rail transit environments.

[0008] According to some embodiments, the present invention adopts the following technical solution: A method for accelerating the computation of the 2.5D Green's function based on keyframe recognition includes: The environmental data of the target rail transit is acquired, a 2.5D numerical model for calculating the Green's function is established, and the target wavenumber discrete grid containing the complete wavenumber domain is determined according to the requirements of rail transit environmental vibration prediction. Using a 2.5D numerical model, coarse sampling calculations in the wavenumber domain are performed on the discrete grid of the target wavenumber. Based on the calculation results, a wavenumber frame importance discrimination index is constructed to identify key frames and non-key frames. Perform a 2.5D exact solution on the keyframe to obtain the complex Green's function of the keyframe; Based on the complex Green's function of the keyframe, the complex spectrum of the non-keyframe is reconstructed to obtain the complex Green's function of the non-keyframe. Based on the complex Green's functions of keyframes and non-keyframes, a Green's function in the complete wavenumber domain is generated. By performing an inverse Fourier transform on the Green's function, a Green's function in the spatial domain is obtained, which can be used to calculate the vibration response at a specific spatial location.

[0009] According to some embodiments, the present invention adopts the following technical solution: A 2.5D Green's function acceleration system based on keyframe recognition includes: The model building module is configured to: acquire environmental data of the target rail transit, build a 2.5D numerical model for calculating the Green's function, and determine the target wavenumber discrete grid containing the complete wavenumber domain according to the requirements for predicting rail transit environmental vibration. The keyframe identification module is configured to: use a 2.5D numerical model to perform coarse sampling calculations in the wavenumber domain on the target wavenumber discrete grid, construct a wavenumber frame importance discrimination index based on the calculation results, and identify key frames and non-key frames. The keyframe solving module is configured to perform a 2.5D exact solution on the keyframe to obtain the complex Green's function of the keyframe; The complex spectrum reconstruction module is configured to: reconstruct the complex spectrum of non-key frames based on the complex Green's function of the key frames, and obtain the complex Green's function of the non-key frames; The complete domain generation module is configured to generate Green's functions in the complete wavenumber domain based on the complex Green's functions of keyframes and non-keyframes, and obtain the spatial domain Green's functions through the inverse Fourier transform of the Green's functions, so as to calculate the vibration response at a specific spatial location.

[0010] According to some embodiments, the present invention adopts the following technical solution: A computer program product includes a computer program that, when executed by a processor, implements the aforementioned 2.5D Green's function acceleration calculation method based on keyframe recognition.

[0011] According to some embodiments, the present invention adopts the following technical solution: A non-transitory computer-readable storage medium is provided for storing computer instructions, which, when executed by a processor, implement the aforementioned 2.5D Green's function acceleration calculation method based on keyframe recognition.

[0012] According to some embodiments, the present invention adopts the following technical solution: 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 perform the 2.5D Green's function acceleration calculation method based on keyframe recognition.

[0013] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention fully utilizes the characteristics of the 2.5D Green's function of rail transit environmental vibration in the wavenumber domain, such as non-uniform energy contribution, concentrated dominant information, and high wavenumber tail attenuation. It transforms the traditional calculation mode of "solving each wavenumber frame accurately" into a calculation mode of "solving key wavenumber frames accurately and reconstructing and completing non-key wavenumber frames". This reduces the number of times the complex linear equation system is solved while ensuring the accuracy of the spatial domain vibration response after inverse Fourier transform.

[0014] This invention provides a 2.5D Green's function acceleration calculation method that can adaptively determine key wavenumber frames based on the distribution characteristics of the wavenumber domain Green's function spectrum and use the key frame results to complete the complex response of non-key frames. Under the condition of meeting the preset calculation accuracy requirements, it significantly reduces the number of times the complex linear equation system is solved, reduces the calculation time and hardware resource requirements, and improves the engineering applicability of vibration prediction in rail transit environment. Attached Figure Description

[0015] 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.

[0016] Figure 1 This is a flowchart of the method in Example 1.

[0017] Figure 2 This is a diagram showing the correspondence between wavenumber frames and energy contribution indicators in Example 1.

[0018] Figure 3 This is an example graph of the energy accumulation curve for Example 1. Figure 4 This is a structural diagram of the CVNN Green's function surrogate model in Example 1.

[0019] Figure 5 This is a comparison diagram of the methods in Example 1. Detailed Implementation

[0020] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0021] 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.

[0022] 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.

[0023] Example 1 One embodiment of the present invention provides a method for accelerating the calculation of a 2.5D Green's function based on keyframe recognition, comprising: Step 1: Obtain environmental data of the target rail transit, establish a 2.5D numerical model for calculating the Green's function, and determine the target wavenumber discrete grid containing the complete wavenumber domain according to the requirements for predicting rail transit environmental vibration. Step 2: Using a 2.5D numerical model, perform coarse sampling calculations in the wavenumber domain on the target wavenumber discrete grid. Based on the calculation results, construct a wavenumber frame importance discrimination index to identify key frames and non-key frames. Step 3: Perform a 2.5D exact solution on the keyframe to obtain the complex Green's function of the keyframe; Step 4: Based on the complex Green's function of the keyframe, perform complex spectrum reconstruction on the non-keyframe to obtain the complex Green's function of the non-keyframe. Step 5: Based on the complex Green's functions of keyframes and non-keyframes, generate the Green's function in the complete wavenumber domain. By performing an inverse Fourier transform on the Green's function, obtain the Green's function in the spatial domain, which can be used to calculate the vibration response at a specific spatial location.

[0024] As one embodiment, the 2.5D Green's function acceleration calculation method based on keyframe recognition of the present invention adaptively determines keyframes according to the distribution characteristics of the Green's function spectrum in the wavenumber domain, and uses the keyframe results to complete the complex response of non-keyframes. This significantly reduces the number of solutions to the complex linear equation system while meeting preset calculation accuracy requirements, lowers computation time and hardware resource requirements, and improves the engineering applicability of vibration prediction in rail transit environments. A detailed description follows: To address the problem that existing 2.5D Green's function calculation methods require solving complex linear equations frame-by-frame across all wavenumber frames, resulting in high computational costs, long computation times, and large memory consumption, which makes it difficult to meet the needs of rapid prediction and multi-scheme comparison in rail transit environmental vibration engineering, this embodiment proposes an accelerated 2.5D Green's function calculation method based on wavenumber domain keyframe identification and complex spectrum reconstruction. The "2.5D Green's function" refers to the complex displacement response function obtained at various wavenumbers after transforming the spatial response along the track direction or a certain horizontal direction to the wavenumber domain using Fourier transform in the 2.5D numerical calculation method; the "wavenumber" is denoted as β, representing the spatial response along that direction. The term "wavenumber frame" or "frame" refers to the solution of a first-order 2.5D complex equation and the resulting Green's function when a certain wavenumber β is fixed. The term "key frame" refers to a wavenumber frame that contributes significantly to the final spatial domain vibration response or has a representative and controlling effect on the distribution of the Green's function spectrum. The term "non-key frame" refers to a wavenumber frame that contributes less to the final response compared to the key frame, or can be reconstructed from the results of adjacent key frames. "Complex spectrum reconstruction" refers to the process of predicting, interpolating, or fitting the complex Green's function at non-key frames that have not been directly solved, based on the solved complex Green's function results at the key frames.

[0025] Specifically, this embodiment aims to solve the following technical problems: First, it addresses the problem that the existing 2.5D Green's function calculation method treats all wavenumber frames equally, resulting in significant computational redundancy.

[0026] Traditional methods typically discretize the entire wavenumber range according to a preset wavenumber step size and perform complete matrix assembly and complex linear equation solving for each wavenumber frame, without distinguishing the contribution of different wavenumber frames to the final vibration response. This embodiment establishes a wavenumber frame importance discrimination method to identify key wavenumber frames that contribute significantly to the spatial domain response, reducing the need for direct solving of wavenumber frames with smaller contributions.

[0027] Second, it addresses the problem that simply reducing wavenumber sampling points or directly truncating high wavenumber regions can easily lead to calculation errors.

[0028] While the wavenumber domain Green's function typically exhibits concentrated dominant information and attenuation at high wavenumber tails, the high wavenumber region can still influence the near-field response, local response, or high-frequency response. Therefore, simply deleting some wavenumber frames is insufficient. This embodiment reduces the number of direct solutions by reconstructing the complex spectrum of non-critical wavenumber frames, while maintaining the continuity, attenuation trend, and phase change characteristics of the wavenumber domain Green's function, thereby reducing the response error after the inverse Fourier transform.

[0029] Third, it addresses the problem that existing acceleration methods such as parallel computing and matrix optimization mainly improve the efficiency of single-frame computation, but are difficult to reduce the total number of solution frames.

[0030] This embodiment starts from the calculation process itself and reduces the overall calculation cost by reducing the number of wavenumber frames that need to be solved directly. It can be used in conjunction with existing acceleration methods such as parallel computing, matrix compression, and fast solvers to further improve the calculation efficiency in engineering applications.

[0031] Fourth, it addresses the problem of excessive repetitive calculations in vibration prediction for rail transit environments, which involves multiple frequencies, operating conditions, sources, and receptors.

[0032] This embodiment reduces the number of wavenumber frames to be solved for each frequency and operating condition by identifying key frames and reconstructing non-key frames. It is suitable for application scenarios that require a lot of repetitive calculations, such as line scheme comparison, vibration reduction measure evaluation, site parameter sensitivity analysis and environmental vibration impact prediction.

[0033] Therefore, the basic idea of ​​this embodiment is as follows: the traditional 2.5D Green's function calculation method requires solving the complex linear equation system frame by frame for all wavenumber frames. However, this embodiment divides the wavenumber frames into key frames and non-key frames based on the spectral distribution characteristics of the wavenumber domain Green's function. For key frames, the 2.5D finite element method, 2.5D finite element-boundary element method, 2.5D finite element-fundamental solution method, or other 2.5D numerical methods are still used for accurate solution. For non-key frames, the complex linear equation system is no longer solved frame by frame. Instead, the complex spectral reconstruction method is used to complete the solution based on the complex Green's function results of the key frames. Finally, the accurate solution results of the key frames and the reconstruction results of the non-key frames are combined to form the complete wavenumber domain Green's function, and the spatial domain vibration response is obtained through inverse Fourier transform.

[0034] Based on the above basic ideas, the technical solution of this embodiment is as follows: Figure 1 As shown, the specific steps are as follows: S1. Establish a 2.5D calculation model for environmental vibration of rail transit.

[0035] Based on target rail transit environment data such as rail transit lines, tunnel structures, track structures, geological parameters, receiver locations, and load locations, a 2.5D numerical model is established to calculate the site Green's function.

[0036] The 2.5D numerical model can be a 2.5D finite element model, a 2.5D finite element-boundary element model, a 2.5D finite element-fundamental solution model, or other 2.5D models capable of solving the Green's function in the wavenumber domain.

[0037] In this embodiment, a 2.5D finite element-fundamental solution model is adopted, using the 2.5D finite element-fundamental solution method, i.e., the 2.5D FEM-MFS method, as the original exact solution method; the calculation object is the site environmental vibration caused by subway or high-speed rail operation. A 2.5D FEM-MFS numerical calculation model is established based on site soil parameters, tunnel or track structure parameters, load location, receiver location, and calculation frequency range.

[0038] Among them, 2.5D FEM-MFS refers to a semi-analytical numerical method that combines the finite element method with the fundamental solution method and performs wavenumber domain expansion along the track direction. The finite element domain is used to describe tunnel structures, track structures, or locally complex sites; the fundamental solution method is used to simulate wave radiation effects in finite outer domains or semi-infinite sites.

[0039] Spatial coordinates are denoted along the route. A Fourier transform is performed in this direction to decompose the three-dimensional dynamic response into a series of different wavenumbers. The problem of two-dimensional sections. For the first... For each wavenumber frame, the governing equation can be written as:

[0040] In the formula, For wavenumber frame index, For frequency index, For the first Discrete wavenumbers For the first The frequency of a circle; For the 2.5D FEM-MFS model at wavenumber Circular frequency The complex dynamical matrix formed below, This is the corresponding wavenumber domain complex displacement response vector. This is the corresponding wavenumber domain unit load vector.

[0041] In one specific implementation, the complex dynamical matrix can be expressed as:

[0042] In the formula, The imaginary unit satisfies italics This only represents the wavenumber frame index; For the first Discrete wavenumbers; For frequency index, For the first The frequency of a circle; , and The stiffness correlation matrix is ​​obtained after 2.5D wavenumber domain transformation. For the quality matrix, Here is the damping matrix. The basic solution method in wavenumber Circular frequency External domain coupling terms or boundary coupling terms introduced below.

[0043] For a given unit load direction At the source location Apply a unit load and solve for the location of the receptor. First Complex displacement response in each direction The wavenumber domain Green's function can be obtained as follows:

[0044] in, Index of unit load direction, For displacement response direction index For receptor indexing, For source point index; For the first One receptor location, For the first The location of each source point; For direction Under the load, the first The first receptor at the first Wavenumber domain complex displacement response in each direction; For direction The unit load amplitude; For direction The unit load in the direction The wavenumber domain complex Green's function generated above. If the unit load has been normalized, it should be further specified. .

[0045] S2. Determine the frequency range and wavenumber discrete range.

[0046] Based on the vibration prediction requirements of the rail transit environment, the calculation frequency range is determined, for example, 1–100 Hz. For each calculation frequency, the range of wavenumber β and the target wavenumber discrete grid are determined. The target wavenumber discrete grid refers to the complete set of wavenumber points used for the final inverse Fourier transform, containing the complete wavenumber domain, and its corresponding total wavenumber frames include keyframes and non-keyframes.

[0047] Specifically, for each calculation frequency The corresponding angular frequency is:

[0048] The wavenumber range was then determined:

[0049] And establish within this scope including Target wavenumber discrete grid for each wavenumber frame:

[0050] Traditional 2.5D FEM-MFS methods require processing all... The complex linear equations are solved frame by frame for each wavenumber frame; in this embodiment, the target wavenumber frame is divided into key frames and non-key frames. Only the 2.5D FEM-MFS is used to solve the key frames accurately, and the trained complex value neural network frame interpolation model is used to predict the non-key frames.

[0051] Key frames are those that contribute significantly to the final spatial domain vibration response in the wavenumber domain or have a controlling effect on the distribution of the Green's function spectrum. Non-key frames are those that contribute less energy to the wavenumber frames and can be predicted and completed by complex-valued neural network interpolation models.

[0052] In one embodiment, the target wavenumber discrete grid includes The number of wavenumber frames is only an example; in practical applications, it can be determined based on the frequency range, site wave velocity, receiver distance, and inverse Fourier transform accuracy requirements.

[0053] S3. Perform coarse sampling calculation in the wavenumber domain.

[0054] A small number of initial sampling frames are selected within the discrete grid of the target wavenumber. These initial sampling frames are then precisely solved using a 2.5D numerical method to obtain the complex Green's function response at the corresponding wavenumber. The initial sampling frames may include low-wavenumber sampling frames, medium-wavenumber sampling frames, high-wavenumber tail sampling frames, and several wavenumber frames preset based on experience. The output of this step is used in the next step to preliminarily identify the amplitude distribution, phase change, and tail attenuation characteristics of the Green's function in the wavenumber domain.

[0055] Specifically, a small number of initial sampling frames are selected to form a coarse sampling wavenumber set:

[0056] in, , This is the initial number of coarse sampling frames. The target wavenumber is the total number of frames.

[0057] The initial sampling frames are used to initially identify the energy distribution, amplitude variation, phase variation, and attenuation trend at high wavenumber tails of the Green's function in the wavenumber domain. The initial sampling frames can be arranged non-uniformly, with denser arrangement in the low wavenumber region, appropriate arrangement in the medium and high wavenumber region, and a small number of sampling points reserved at the high wavenumber tail.

[0058] For each initial sampling frame, the 2.5D FEM-MFS method in S1 is used to solve for the complex Green's function at the corresponding wavenumber.

[0059] In one embodiment, the target wavenumber frame number is: The initial number of sampling frames can be set to 1. In other words, we first perform 2.5D FEM-MFS exact solution only on about 10% of the wavenumber frames to identify the wavenumber domain spectral energy distribution characteristics.

[0060] S4. Construct wavenumber frame importance discrimination index and identify key wavenumber frames.

[0061] Based on the complex Green's function result obtained from coarse sampling, an index for judging the importance of wavenumber frames is constructed. The index may include one or more of the following: energy contribution index, local change index, and tail attenuation index.

[0062] Among them, the energy contribution index is used to describe the contribution of the complex Green's function corresponding to a certain wavenumber frame to the overall response. For example, for the i The energy contribution index for each wavenumber frame can be expressed as:

[0063] In the formula, For the first Energy contribution index per wavenumber frame; This refers to the fixed direction of the unit load in this indicator; , and These are the receptor, source, and response component indices, respectively. , and These are the total number of receptors, the total number of source sites, and the total number of response components, respectively. For the first The first receptor, the first The source point and the first The weighting coefficients corresponding to each response component; Let be the complex Green's function in the wavenumber domain at a given angular frequency. For a given , The frequency of a circle The following will Abbreviated as .

[0064] Local change indicators include amplitude change indicators, phase change indicators, and local curvature indicators: The amplitude variation index is used to characterize the degree of change in the Green's function amplitude between adjacent wavenumber frames, and is represented by the amplitude difference between adjacent wavenumber frames:

[0065] In the formula, Wavenumber interval The relative amplitude change rate on and These are the weighted spectral amplitudes at adjacent wavenumber frames. This is a preset positive small quantity with the same dimensions as the spectral amplitude.

[0066] No. i Weighted spectral amplitude at each wavenumber frame Defined as: In the formula, For the first Energy contribution index per wavenumber frame.

[0067] The phase change index is used to characterize the severity of the phase change in the complex response between adjacent wavenumber frames, and is expressed as the phase difference between adjacent wavenumber frames: In the formula, For the first The and the first Phase change index between wavenumber frames; Indicates taking the complex phase. This indicates that the phase sequence is expanded to eliminate adjacent phases. Jump; , , , For fixed unit load direction; This represents the maximum absolute value of the phase difference among all combinations of receptor, source, and response components; this formula is applicable to... .

[0068] The local curvature index is used to characterize the degree of curvature of the Green's function in the wavenumber domain as the wavenumber changes.

[0069] For non-uniform wavenumber grids, the local curvature index is defined as the second difference of the spectral amplitude with respect to the wavenumber: In the formula, For the first Local curvature index at each wavenumber frame; , and These are the weighted spectral amplitudes at three adjacent wavenumber frames; , and For the corresponding discrete wavenumber; A pre-defined positive small quantity having the same dimensions as the spectral amplitude; The target wavenumber is the total number of frames; since this formula uses adjacent frames, the index range is... .

[0070] The tail decay index is used to characterize the trend of Green's function amplitude decaying with increasing wavenumber in the high wavenumber region. For the high wavenumber tail region... , can be defined as: In the formula, This is the set of indices for the high wavenumber tail frames. For wavenumber frame index, For the first Discrete wavenumbers The preset tail start wave number; all that meet the following conditions All wavenumber frames belong to the tail set. .

[0071] for The local tail attenuation index between adjacent wavenumber frames can be defined as: In the formula, For the first Wave number intervals Local tail attenuation index; and These are the weighted spectral amplitudes of the wavenumber frames at both ends of the interval; and For the corresponding discrete wavenumber; This is a pre-defined positive small quantity with the same dimensions as the spectral amplitude; this formula is only used for... and The adjacent tail frame.

[0072] In another implementation, an exponential decay fit can be used to define the overall tail decay index in the high wavenumber tail region.

[0073] Assume that the spectral amplitude at the high wavenumber tail follows an approximately exponential decay relationship: In the formula, wave number The fitted spectral amplitude at that point, This is the leading magnitude of the tail exponent fit. This represents the overall attenuation coefficient at the high wavenumber tail. The absolute value of the wave number; and Taking the logarithm of the spectral amplitude in the tail region, we have: In the formula, For the first Weighted spectral amplitude at each tail wavenumber frame A pre-defined positive small quantity with the same dimensions as the spectral amplitude. This is the leading magnitude of the tail exponent fit. The overall tail attenuation coefficient, For the first Discrete wavenumbers For the set of trailing frame indices; this formula applies to all Established.

[0074] Therefore, the overall tail attenuation coefficient can be obtained through least squares fitting: In the formula, The overall tail attenuation coefficient is obtained using the least squares method; For the set of tail frame indices; For the wavenumber frame index in the tail set, For the first Discrete wavenumbers; The average of the absolute values ​​of wavenumbers within the tail set; For the first Weighted spectral amplitude at each tail wavenumber frame To and Pre-defined positive small quantities with the same dimensions; This represents the average value of the logarithmic spectrum amplitude within the tail set; This indicates the number of wavenumber frames contained in the tail set.

[0075] Based on the above indicators, wavenumber frames that meet one of the following conditions are identified as keyframes: 1) Energy contribution indicators Wavenumber frames with energy values ​​greater than a preset energy threshold; 2) Wavenumber frames located in the low wavenumber main energy region; 3) Wavenumber frames where the amplitude change index or phase change index is greater than the preset change threshold; 4) Wavenumber frames where the local curvature index is greater than the preset curvature threshold; 5) Anchor frames located at the high wavenumber tail, used to control the tail attenuation trend; 6) Boundary frames or supplementary frames located at the boundary of the wavenumber range, used to ensure the stability of the inverse Fourier transform.

[0076] The low wavenumber main energy region refers to the wavenumber interval where the energy contribution index is large and plays a major role in the final spatial domain vibration response, within the range of low or medium wavenumbers. The high wavenumber tail refers to the wavenumber interval where the Green's function amplitude generally shows a decreasing trend as the wavenumber increases. For the high wavenumber tail, this embodiment does not directly delete all wavenumber frames in this region, but retains a small number of key frames as tail anchor frames to constrain the attenuation trend during the reconstruction process of non-key frames.

[0077] The remaining wavenumber frames that were not identified as keyframes are treated as non-keyframes. For keyframes, the 2.5D complex linear equations are solved precisely; for non-keyframes, the complex linear equations are not solved directly, but the complex spectrum is reconstructed using a complex-valued neural network frame interpolation model based on the keyframe calculation results.

[0078] A specific method for keyframe identification based on energy contribution index is provided: The energy contribution index of each initial coarse-sampled wavenumber frame is calculated using the above formula. Let the total number of initial coarse-sampled wavenumber frames be... ,by This represents the local index of the initial coarse-sampled wavenumber frame, the first... The initial coarse-sampled wavenumber frames are denoted as follows: Its corresponding energy contribution index is marked as .

[0079] The energy contribution index of the initial coarse-sampled wavenumber frame is normalized:

[0080] In the formula, This is the local index for the initial coarse-sampled wavenumber frame; This represents the total number of initial coarse-sampling wavenumber frames; For the first Energy contribution index of the initial coarse-sampled wavenumber frames; Contribute to the corresponding normalized energy; This is the summation index in the normalized denominator. After normalization, it satisfies... ， and .

[0081] After normalization, we get the following: Figure 2 Each initial coarse-sampled wavenumber frame shown Normalized energy contribution index Based on this, construct as follows Figure 3 The energy accumulation curve shown:

[0082] In the formula, the wavenumber frames are arranged in ascending order of discrete wavenumber. To accumulate from the first wavenumber frame to the second wavenumber frame Energy accumulation per wavenumber frame; For the first Energy contribution index of the initial coarse-sampled wavenumber frames; And when the normalization covers all target wavenumber frames, there is .

[0083] Keyframes are identified using energy accumulation curves and the quantile method. Specifically, wavenumber intervals where the cumulative energy percentage reaches a preset proportion are determined as the main energy region. The preset proportion can be 80%, 85%, 90%, or other values ​​determined according to engineering accuracy requirements. In a preferred embodiment, wavenumber intervals with a cumulative energy of not less than 80% are designated as the main energy region, and wavenumber frames within this interval are identified as keyframes.

[0084] In one experimental embodiment, the target wavenumber frames numbered 256. Approximately 13.3% of the key wavenumber points, or about 34 key wavenumber frames, were identified using energy accumulation curves and quantile methods. These key frames covered more than 80% of the accumulated energy of the complex Green's function in the wavenumber domain. The remaining approximately 86.7% of the wavenumber frames were treated as non-key frames and were predicted and completed by a subsequent complex-valued neural network frame interpolation model.

[0085] To facilitate engineering implementation, a small number of boundary frames, tail anchor frames, or spot check frames can be added to the key frames as needed for error back-judgment.

[0086] S5. Perform 2.5D exact solution for the key wavenumber frames.

[0087] For the identified keyframes, a 2.5D numerical method is used to assemble the complex matrix frame by frame and solve the complex linear equations to obtain the complex Green's function results at the keyframes. Since the number of keyframes is less than the total number of frames in the target wavenumber discrete grid, the method in this embodiment can reduce the number of times the complex linear equations are solved compared with the traditional frame-by-frame solution method.

[0088] Specifically, for the first Solve for each keyframe:

[0089] In the formula, For keyframe index set, The total number of target wavenumber frames; Indicates the first Each wave number frame is a key frame. The discrete wavenumber corresponding to this frame; For frequency index, For the first The frequency of a circle; This is the complex dynamical matrix corresponding to the keyframe. This is the wavenumber domain complex displacement response vector at the keyframe. This is the corresponding wavenumber domain unit load vector.

[0090] The keyframe solution is not processed by the complex spectrum reconstruction below, but is directly retained as part of the final wavenumber domain Green's function. Therefore, this embodiment can ensure that the principal energy region, which accounts for more than 80% of the cumulative energy in the wavenumber domain, maintains the original 2.5DFEM-MFS calculation accuracy, avoiding the dominant response distortion that may be caused by the full surrogate model.

[0091] S6. Perform complex value neural network frame interpolation reconstruction on non-critical wavenumber frames.

[0092] For non-key frames that are not directly solved, a complex-valued neural network frame interpolation model is used to reconstruct the complex spectrum based on the complex Green's function results of the solved key frames, thereby obtaining the complex Green's function response at the non-key frames.

[0093] The complex-valued neural network frame interpolation model is used to learn the continuous spectrum characteristics of the complex Green's function as a function of wavenumber β in the wavenumber domain. Its input is the complex Green's function result at one or more keyframes, and its output is the complex Green's function prediction result at one or more non-keyframes between adjacent keyframes.

[0094] Since the Green's function is in complex form, in a preferred embodiment, the real and imaginary parts of the complex Green's function at the keyframe are used as two input channels, and the real and imaginary parts of the complex Green's function at the non-keyframe are used as two output channels. After the network outputs, the real and imaginary parts are recombined to obtain the complex Green's function at the non-keyframe.

[0095] In another preferred approach, the amplitude and phase of the complex Green's function at keyframes are used as two input channels, and the amplitude and phase of the complex Green's function at non-keyframes are used as two output channels. For cases where the phase changes abruptly, the phase can be expanded before being input into the complex neural network frame interpolation model.

[0096] The complex-valued neural network frame interpolation model can employ a one-dimensional convolutional network, residual network, recurrent neural network, Transformer network, or a combination of the above network structures to characterize the wavenumber domain complex response mapping relationship between key frames and non-key frames. This model does not change the accurate solution results of key frames, but is only used to complete the results of non-key frames that are not directly solved.

[0097] In this embodiment, the traditional method transforms the process of directly solving all wavenumber frames into a process of accurately solving only key frames and reconstructing non-key frames using neural network frame interpolation, thereby reducing the number of times complex linear equations need to be solved.

[0098] In this embodiment, a complex neural network, namely CVNN, is used to construct a Green's function surrogate model. CVNN is an abbreviation for Complex-Valued Neural Network, which refers to a neural network model whose network input, weights, activation functions, and outputs can all process complex data. The specific steps are as follows: S601. Resample non-critical wavenumber frames and construct CVNN training data. Because the energy distribution of the Green's function in the wavenumber domain is significantly non-uniform, directly training the neural network with the full wavenumber data can easily lead to the model being dominated by a large proportion of low-energy, non-critical data, resulting in distorted fitting of the high-energy, critical wavenumber intervals. To avoid this problem, this embodiment does not directly train the surrogate model with the full wavenumber frames, but instead processes key frames and non-key frames separately.

[0099] Specifically, the 2.5D FEM-MFS exact solution results are retained for keyframes and are not used in CVNN surrogate prediction; training data is constructed for non-keyframes, and frame interpolation and resampling are performed proportionally to balance the distribution of training samples across different wavenumber intervals. Frame interpolation and resampling refers to supplementing, balancing, or rearranging the training samples of non-keyframes based on their distribution density, amplitude variations, and phase variations in the wavenumber domain, enabling the CVNN model to more fully learn the variation of the complex Green's function with wavenumber within the non-keyframe region.

[0100] The training data can be represented as:

[0101] in, For the first The input to each training sample contains a sequence of complex Green's functions at adjacent or nearby keyframes; The corresponding output contains a sequence of complex Green's functions at non-keyframe locations between keyframes; , The total number of CVNN training samples; the output labels should be accurate reference results for the corresponding non-keyframes.

[0102] In one implementation, the input to the CVNN model includes only the complex Green's function results at the keyframes, without additional input of soil parameters, structural parameters, or load parameters. The working condition information such as soil, structure, and load is implicitly reflected through the complex response of the keyframes. This allows the CVNN model to focus on learning the wavenumber domain complex response mapping relationship between keyframes and non-keyframes.

[0103] In another engineering implementation, basic parameters such as soil layer parameters, frequency range, wavenumber interval, and spatial location can be used as inputs to the entire calculation process to generate a 2.5D FEM-MFS model, determine keyframes, and organize results, but are not required inputs to the CVNN model.

[0104] S602. Predict non-critical wavenumber frames using the CVNN Green's function surrogate model. For non-critical frames that are not directly solved, a trained CVNN Green's function surrogate model is used for prediction. The CVNN Green's function surrogate model is used to replace the process of solving the 2.5D FEM-MFS complex linear equations on non-critical wavenumber frames.

[0105] Unlike real neural networks that independently fit the real and imaginary parts, CVNN uses complex weights, complex biases, complex activation functions, and complex backpropagation mechanisms to perform integrated feature extraction and fitting of the real and imaginary parts of the complex Green's function. This allows the model to maintain the coupling relationship between the real and imaginary parts during training and inference, thereby preserving physical information such as vibration amplitude, phase, propagation delay, and propagation attenuation.

[0106] In a preferred embodiment, such as Figure 4 As shown, the CVNN Green's function surrogate model includes an input layer, three complex residual blocks, and an output layer. Each complex residual block includes a complex linear transformation layer, a complex activation function layer, and residual connections. Residual connections are used to alleviate the gradient vanishing problem in deep complex feature training and improve the model's ability to fit complex responses in the wavenumber domain. The complex activation function can be the CReLU function, which is used for complex inputs. CReLU can be represented as:

[0107] in, For complex number input, for The real part, for The imaginary part, It is the imaginary unit and satisfies ; It is a linear rectified function; This represents the complex activation function obtained by performing ReLU operations on the real and imaginary parts of the complex input respectively and then recombinating them.

[0108] The prediction process of CVNN can be represented as follows:

[0109] In the formula, For the completed CVNN Green's function surrogate model; Model parameters; The complex Green's function is obtained by exact solution of 2.5D FEM-MFS at the keyframe. The complex Green's function for non-keyframes predicted by CVNN.

[0110] To improve the generalization ability of the model, L2 regularization, Dropout, and early stopping mechanisms can be used during training. L2 regularization is used to constrain the size of the model weights, Dropout is used to reduce the risk of overfitting, and early stopping mechanisms are used to stop training when the validation error no longer decreases.

[0111] In a preferred embodiment, the CVNN model employs a composite loss function, starting from the real part... virtual part Amplitude and phase Four factors constrain prediction error:

[0112] In the formula, For CVNN, it is the composite loss function; For CVNN prediction sequences of non-keyframes, This is a precise reference sequence for non-keyframes corresponding to each element of the predicted sequence; and These represent the real part, imaginary part, and phase of a complex number, respectively. This indicates taking the modulus of the complex number element by element; Represents the L2 norm; This is the parameter vector for the CVNN model; , , , and All are non-negative weighting coefficients, used to adjust the relative weights of the real part error term, imaginary part error term, magnitude error term, phase error term, and model parameter regularization term, respectively.

[0113] In this way, CVNN not only fits the numerical magnitude of the complex Green's function, but also constrains its real part, imaginary part, amplitude, and phase, thereby improving the physical rationality and numerical integrity of vibration response prediction.

[0114] S7. Perform accuracy back-judgment and adaptive frame interpolation.

[0115] To avoid the impact of non-keyframe reconstruction errors on the final spatial domain response, this embodiment sets up a precision back-judgment step. The precision back-judgment refers to determining whether the current number of keyframes meets the preset precision requirements based on the wavenumber domain response errors before and after reconstruction.

[0116] If the error meets the preset threshold, proceed to the next step; if the error exceeds the preset threshold, new keyframes are added to the wavenumber range with larger errors, and the newly added keyframes are solved with 2.5D precision, and then the complex spectrum is reconstructed. This step can control the calculation error while reducing the amount of computation.

[0117] Specifically, to verify the prediction accuracy of the CVNN Green's function surrogate model, this embodiment uses the coefficient of determination. Evaluation is based on metrics such as mean absolute percentage error (MAPE).

[0118] Coefficient of determination The measure of consistency between predicted and reference values ​​can be expressed as:

[0119] In the formula, The coefficient of determination; To evaluate the sample index, The total number of samples is used to evaluate accuracy; For the first Precise reference values ​​for non-keyframes of each evaluation sample. This corresponds to the CVNN predicted value; The sample mean of all precise reference values; It represents the square of the modulus of a complex number.

[0120] The mean absolute percentage error (MAPE) can be expressed as:

[0121] In the formula, Mean absolute percentage error; To evaluate the sample index, The total number of samples is used to evaluate accuracy; For the first The CVNN predicted values ​​for each evaluation sample. For the corresponding non-keyframe precise reference value; Indicates taking the modulus of a complex number; A preset positive small quantity with the same dimensions as the Green's function amplitude, used to prevent the denominator from being zero; multiplied by Used to express the result as a percentage.

[0122] In one experimental embodiment, the CVNN Green's function surrogate model predicts the Green's function of the site using a coefficient of determination. It can reach 0.98, with a mean absolute percentage error (MAPE) of 21.8%, which can meet the requirements of prediction accuracy for engineering assessment of vibration in rail transit environment, environmental impact assessment and comfort assessment.

[0123] If the error evaluation result does not meet the preset threshold, you can return to S4 to adjust the keyframe ratio, energy threshold, non-keyframe resampling ratio, or CVNN model parameters. You can also add keyframes in the wavenumber range with large errors and perform 2.5D FEM-MFS exact solution on the newly added keyframes.

[0124] S8. Perform inverse Fourier transform to obtain the spatial domain Green's function.

[0125] The precise solution results of keyframes and the complex spectrum reconstruction results of non-keyframes are combined to form a complete wavenumber domain Green's function, and the spatial domain Green's function is obtained through inverse Fourier transform. The spatial domain Green's function can be further used for vibration prediction in rail transit environments, route selection, vibration reduction measure evaluation, site parameter sensitivity analysis, and evaluation of the impact of vibration on track-borne receptors, specifically: S801. Concatenate the precise results of keyframes with the prediction results of non-keyframes to form a complete wavenumber domain Green's function: The precise solution results of keyframes and the CVNN prediction results of non-keyframes are concatenated according to the target wavenumber grid order to form a complete wavenumber domain Green's function sequence:

[0126] In the formula, For the complete wavenumber domain Green's function sequence at the th The value at each wavenumber frame; These are the exact solution values ​​obtained from 2.5D FEM-MFS at the keyframes. These are the predicted values ​​obtained by CVNN at non-keyframe locations; and These are the sets of keyframe and non-keyframe indices, respectively, which are mutually exclusive and have a union of . ; and These are the indices for the unit load direction and the response direction, respectively. To simplify the formula, the common receptor location is omitted from this equation. Source location and a given circular frequency .

[0127] In one embodiment, keyframes comprise approximately 13.3% of all wavenumber frames and cover over 80% of the cumulative energy in the wavenumber domain; non-keyframes comprise approximately 86.7% of all wavenumber frames and are predicted by a CVNN surrogate model. Thus, the complete wavenumber domain Green's function sequence is composed of "the original accurate results of keyframes + the CVNN prediction results of non-keyframes".

[0128] S802. Perform an inverse Fourier transform on the complete wavenumber domain Green's function obtained in S801 to obtain the spatial domain Green's function, which can be used to calculate the vibration response at a specific spatial location:

[0129] In the formula, For the spatial domain Green's function, For the corresponding wavenumber domain Green's function; and These are the indices for the unit load direction and the response direction, respectively. and These are the coordinates of the receptor and the source along the Fourier transform direction, respectively. This represents the difference in coordinates between the two in that direction; and These are the receptor and source location vectors, respectively; For continuous wavenumber, and These are the lower and upper limits of the integral wavenumber range, respectively; Given a circular frequency; It is the imaginary unit and satisfies .

[0130] In the discrete wavenumber grid, it can be written as:

[0131] In the formula, The total number of target wavenumber frames, For wavenumber frame index, For the first Discrete wavenumbers For the first The integral weights of wavenumber points in the discrete inverse Fourier transform; For the spatial domain Green's function, For the first Green's function in the wavenumber domain at each wavenumber frame; and These are the indices for the unit load direction and the response direction, respectively. and These are the coordinates of the receptor and the source along the Fourier transform direction, respectively. ; and These are the receptor and source location vectors, respectively; Given a circular frequency; It is the imaginary unit and satisfies For a uniform wavenumber grid, each... Use the same wavenumber step size; for non-uniform wavenumber grids, this must be specified separately in the text. The specific calculation rules.

[0132] Furthermore, the spatial domain Green's function By combining the rail transit load spectrum, wheel-rail force, or equivalent excitation, the vehicle-induced environmental vibration response at the ground surface, building, precision instrument location, or other receptor location can be obtained.

[0133] This implementation plan establishes a standardized process of "parameter input—precise keyframe solution—CVNN model inference—result stitching—spatial domain reconstruction." In engineering applications, only basic environmental parameters such as site soil parameters, frequency range, wavenumber interval, load location, and receptor location need to be input to quickly solve the Green's function for multiple working conditions and predict vehicle-induced environmental vibration response. Through the above steps, such as... Figure 5 As shown, this embodiment transforms the traditional 2.5D method, which solves all wavenumber frames frame by frame, into a computation mode that accurately solves key frames and reconstructs and completes non-key frames. This reduces the number of times complex linear equations are solved while ensuring spatial domain response accuracy, thereby reducing computation time and hardware resource requirements.

[0134] Compared to the existing 2.5D FEM-MFS frame-by-frame solution method, this embodiment avoids repeatedly performing complex matrix assembly and linear equation solving for the vast majority of non-critical wavenumber frames. Actual measurements show that the overall site Green's function solution time in this embodiment is reduced by 62.5% compared to the existing 2.5D FEM-MFS method, compressing the single-case computation time from tens of hours to less than ten hours. In multi-point, multi-parameter batch scenarios, the overall efficiency improvement is even more significant because the CVNN surrogate model can be reused for prediction of a large number of non-critical frames.

[0135] In terms of accuracy, experimental results show that the prediction determination coefficient of the CVNN Green's function surrogate model is high. It can reach 0.98, with a mean absolute percentage error (MAPE) of 21.8%. Further calculations show that the predicted frequency level is less than 3dB, which can meet the accuracy requirements for environmental vibration engineering assessment, environmental impact assessment, vibration reduction measure selection, and comfort assessment in rail transit.

[0136] Therefore, this implementation scheme retains the original accuracy of the main energy key wavenumber frames while entrusting a large number of low-energy non-key wavenumber frames to the CVNN surrogate model, effectively balancing computational efficiency and prediction accuracy. It is suitable for rapid screening of large-scale site vibrations along subway and high-speed rail lines, prediction of sensitive points, optimization of vibration reduction scheme parameters, and batch analysis of multiple working conditions.

[0137] Example 2 One embodiment of the present invention provides a 2.5D Green's function acceleration computation system based on keyframe recognition, comprising: The model building module is configured to: acquire environmental data of the target rail transit, build a 2.5D numerical model for calculating the Green's function, and determine the target wavenumber discrete grid containing the complete wavenumber domain according to the requirements for predicting rail transit environmental vibration. The keyframe identification module is configured to: use a 2.5D numerical model to perform coarse sampling calculations in the wavenumber domain on the target wavenumber discrete grid, construct a wavenumber frame importance discrimination index based on the calculation results, and identify key frames and non-key frames. The keyframe solving module is configured to perform a 2.5D exact solution on the keyframe to obtain the complex Green's function of the keyframe; The complex spectrum reconstruction module is configured to: reconstruct the complex spectrum of non-key frames based on the complex Green's function of the key frames, and obtain the complex Green's function of the non-key frames; The complete domain generation module is configured to generate Green's functions in the complete wavenumber domain based on the complex Green's functions of keyframes and non-keyframes, and obtain the spatial domain Green's functions through the inverse Fourier transform of the Green's functions, so as to calculate the vibration response at a specific spatial location.

[0138] Example 3 One embodiment of the present invention provides a computer program product, including a computer program that, when executed by a processor, implements the aforementioned 2.5D Green's function acceleration calculation method based on keyframe recognition.

[0139] Example 4 In one embodiment of the present invention, a non-transitory computer-readable storage medium is provided for storing computer instructions. When the computer instructions are executed by a processor, they implement the 2.5D Green's function acceleration calculation method based on keyframe recognition.

[0140] Example 5 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, 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 execute the 2.5D Green's function acceleration calculation method based on keyframe recognition.

[0141] 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 1A device that provides the functions specified in one or more boxes.

[0142] 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.

[0143] 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 accelerating the calculation of 2.5D Green's function based on keyframe recognition, characterized in that, include: The environmental data of the target rail transit is acquired, a 2.5D numerical model for calculating the Green's function is established, and the target wavenumber discrete grid containing the complete wavenumber domain is determined according to the requirements of rail transit environmental vibration prediction. Using a 2.5D numerical model, coarse sampling calculations in the wavenumber domain are performed on the discrete grid of the target wavenumber. Based on the calculation results, a wavenumber frame importance discrimination index is constructed to identify key frames and non-key frames. Perform a 2.5D exact solution on the keyframe to obtain the complex Green's function of the keyframe; Based on the complex Green's function of the keyframe, the complex spectrum of the non-keyframe is reconstructed to obtain the complex Green's function of the non-keyframe. Based on the complex Green's functions of keyframes and non-keyframes, a Green's function in the complete wavenumber domain is generated. By performing an inverse Fourier transform on the Green's function, a Green's function in the spatial domain is obtained, which can be used to calculate the vibration response at a specific spatial location. The wavenumber frame importance discrimination index includes one or more of the following: energy contribution index, amplitude change index, phase change index, local curvature index, and tail attenuation index; wherein, the energy contribution index is used to characterize the magnitude of the contribution of the wavenumber frame to the final spatial domain response, and the amplitude change index and phase change index are used to characterize the severity of amplitude and phase changes between adjacent wavenumber frames, respectively. The energy contribution index is expressed by the formula: In the formula, For the first Energy contribution index per wavenumber frame; For wavenumber frame index, For receptor indexing, For the source point index, For response component index, This refers to the fixed unit load direction in this energy index; , and These are the total number of receptors, the total number of source sites, and the total number of response components, respectively. For the first The first receptor, the first The source point and the first The weighting coefficients corresponding to each response component; For the first Discrete wavenumbers For the first One receptor location, For the first The location of each source point. Given a circular frequency; For direction The unit load in the direction The wavenumber domain complex Green's function generated on the surface.

2. The method for accelerating the calculation of 2.5D Green's function based on keyframe recognition as described in claim 1, characterized in that, The specific steps for performing coarse sampling calculations in the wavenumber domain on the target wavenumber discrete grid are as follows: Initial sampling frames are selected in the target wavenumber discrete grid, including low wavenumber sampling frames, medium wavenumber sampling frames, high wavenumber tail sampling frames, and several wavenumber frames preset based on experience. The 2.5D numerical method is used to accurately solve the selected initial sampling frame to obtain the complex Green's function response at the corresponding wavenumber.

3. The method for accelerating the calculation of 2.5D Green's function based on keyframe recognition as described in claim 1, characterized in that, The complex spectrum reconstruction of non-key frames specifically involves: A complex-valued neural network frame interpolation model is used to learn the continuous spectral characteristics of the complex Green's function in the wavenumber domain as a function of wavenumber. The complex-valued neural network frame interpolation model takes the complex Green's function result at the key frame as input and the complex Green's function at the non-key frame as output.

4. The method for accelerating the calculation of 2.5D Green's function based on keyframe recognition as described in claim 3, characterized in that, During training, the complex-valued neural network frame interpolation model employs a composite loss function to constrain the prediction error. This composite loss function includes a real part loss term, an imaginary part loss term, an amplitude loss term, and a phase loss term.

5. The method for accelerating the calculation of 2.5D Green's function based on keyframe recognition as described in claim 1, characterized in that, Also includes: After completing the complex spectrum reconstruction, precision back-judgment and adaptive frame interpolation are performed, specifically as follows: The accuracy of complex spectrum reconstruction is judged by the coefficient of determination and the mean absolute percentage error. If the accuracy does not meet the preset accuracy requirements, new key frames are added in the wavenumber interval where the error exceeds the preset threshold. After performing 2.5D accurate solution on the new key frames, complex spectrum reconstruction is performed again.

6. A 2.5D Green's function acceleration computing system based on keyframe recognition, characterized in that, The method for accelerating the calculation of the 2.5D Green's function based on keyframe recognition as described in any one of claims 1-5 includes: The model building module is configured to: acquire environmental data of the target rail transit, build a 2.5D numerical model for calculating the Green's function, and determine the target wavenumber discrete grid containing the complete wavenumber domain according to the requirements for predicting rail transit environmental vibration. The keyframe identification module is configured to: use a 2.5D numerical model to perform coarse sampling calculations in the wavenumber domain on the target wavenumber discrete grid, construct a wavenumber frame importance discrimination index based on the calculation results, and identify key frames and non-key frames. The keyframe solving module is configured to perform a 2.5D exact solution on the keyframe to obtain the complex Green's function of the keyframe; The complex spectrum reconstruction module is configured to: reconstruct the complex spectrum of non-key frames based on the complex Green's function of the key frames, and obtain the complex Green's function of the non-key frames; The complete domain generation module is configured to generate Green's functions in the complete wavenumber domain based on the complex Green's functions of keyframes and non-keyframes, and obtain the spatial domain Green's functions through the inverse Fourier transform of the Green's functions, so as to calculate the vibration response at a specific spatial location.

7. 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 the 2.5D Green's function acceleration calculation method based on keyframe recognition as described in any one of claims 1-5.

8. 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 perform the 2.5D Green's function acceleration calculation method based on keyframe recognition as described in any one of claims 1-5.

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