Vehicle-rail-soil strong coupling method based on finite element-time domain boundary element time cross coupling

Abstract

The invention provides a vehicle-rail-soil strong coupling method based on finite element-time domain boundary element time cross coupling, which comprises the following steps: carrying out internal loop iteration solution on vehicle-rail coupling based on a relaxation iteration prediction correction method, and obtaining FEM interface displacement of current external loop iteration after vehicle-rail coupling, the FEM interface displacement of the current outer loop iteration is converted into the TD-BEM interface displacement of the current outer loop iteration by using a time interleaving iteration coupler; solving the soil body subsystem through the TD-BEM based on the self-adaptive semi-infinite boundary element by taking the interface displacement of the TD-BEM of the current outer loop stack as a boundary condition, and obtaining the interface force of the TD-BEM of the current outer loop stack; the TD-BEM interface force of the current outer loop iteration is converted into FEM node force of the current outer loop iteration by using a time interleaving iteration coupler; and if the convergence condition is not met, correcting the FEM node force of the current outer loop iteration based on a relaxation correction method. According to the method, computing resources are saved, and the computing precision is improved.

Description

Technical Field

[0001] This invention relates to the technical field of rail transit engineering and environmental vibration prediction, and in particular to a vehicle-track-soil strong coupling method. Background Technology

[0002] The rapid development of rail transit, while bringing convenience, has also triggered environmental vibration problems. Vibrations generated by vehicle operation are transmitted to the foundation soil through the track structure and further propagate to the surrounding environment, potentially causing secondary vibrations and noise in nearby buildings, affecting the normal operation of precision instruments and the comfort of residents. Therefore, developing numerical models that can accurately predict the dynamic response of the vehicle-track-soil coupled system is crucial for the optimized design of rail transit, the scientific assessment of environmental impacts, and the effective formulation of vibration reduction measures.

[0003] Currently, numerical simulation methods for this system are mainly classified into the following categories: pure finite element method (FEM). FEM leverages its advantages in handling complex geometries, nonlinear materials, and behaviors to perform detailed modeling of vehicle and track structures. While FEM can effectively simulate the dynamic response of near-field structures, its fundamental theory requires a bounded computational domain. To simulate a semi-infinite soil domain, artificial boundaries (such as viscous boundaries or transmission boundaries) must be introduced to truncate the computational domain. This approach has inherent drawbacks: low-order artificial boundaries (such as viscous boundaries) have limited accuracy and cannot completely absorb reflected stress waves, leading to boundary reflections and reduced accuracy of the calculation results; high-order artificial boundaries (such as perfectly matched layers), while highly accurate, are complex to implement and often result in asymmetric system matrices, significantly increasing computational costs and storage requirements. Furthermore, to meet computational accuracy requirements, a sufficient number of elements need to be divided along the wave propagation direction, resulting in a large model size and low computational efficiency. The pure boundary element method (FEM) leverages the property that BEM automatically satisfies the far-field radiation conditions of soil by simply discretizing the boundaries, efficiently and accurately simulating wave propagation in semi-infinite soil. BEM has a natural advantage in handling uniform infinite domain problems, avoiding the need for artificial boundaries. However, BEM is very difficult and inefficient in handling nonlinear problems (such as wheel-rail Hertz contact nonlinearity and material nonlinearity). Furthermore, for systems like track structures with complex geometry and multiple material compositions, the flexibility and convenience of modeling with BEM are far inferior to those of FEM.

[0004] Traditional FEM-BEM coupling methods, such as those proposed by Zienkiewicz and Kelly (Zienkiewicz and Kelly, Coupling of finite element method and boundary solution procedures, Int. J.Numer. Meth. Eng. 1977,11(2):355-375.) and Galvín et al. (Galvín et al., Fully three-dimensional analysis of high-speed vehicle-track-soil-structure dynamic interaction, J. Sound Vib. 2010,329(24):5147-5163.), typically establish unified FEM-type or BEM-type governing equations to force the two subsystems to satisfy displacement continuity and force balance conditions at the coupling boundary. Theoretically, this method combines the advantages of both approaches and is an ideal direction for solving such problems. However, the inventors of this application have discovered two key bottlenecks in the traditional FEM-BEM coupling method: First, time matching constraints: requiring the FEM and BEM subsystems to use the exact same time step for propagation. Because of the significant differences in the dynamic characteristics of the vehicle-track system (rich in high-frequency components, requiring small time steps) and the soil system (with relatively low-frequency responses, allowing for larger time steps), this forced matching forces the entire system to adopt the smallest time step, resulting in a huge waste of computational resources. Second, the complexity of the unified equations: traditional methods require the construction and solution of complex unified coupling equations, which are difficult to implement numerically and are prone to affecting computational stability and accuracy due to deterioration in equation behavior. Summary of the Invention

[0005] To address the technical problems of existing FEM-BEM coupling methods, such as wasted computational resources due to time matching constraints, high computational difficulty due to complex unified equation construction, and insufficient computational stability and accuracy, this invention proposes a vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling. It constructs a coupling framework based on relaxation iterative prediction and correction method and adaptive semi-infinite boundary element, which solves the above problems.

[0006] To achieve the above objectives, the technical solution of the present invention is implemented as follows:

[0007] A vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling, comprising the following steps:

[0008] S1: The vehicle-track subsystem is modeled based on FEM, and the soil subsystem is modeled based on TD-BEM. Different time steps are used to solve the vehicle-track subsystem and the soil subsystem.

[0009] S2: Construct a time-interleaved iterative coupler;

[0010] S3: Based on the FEM node forces obtained from the previous outer loop iteration at the current time step, the vehicle-track coupling is solved by inner loop iteration based on the relaxation iterative prediction and correction method to obtain the FEM interface displacement of the current outer loop iteration after vehicle-track coupling, and the FEM interface displacement of the current outer loop iteration is converted into the TD-BEM interface displacement of the current outer loop iteration using the time-interleaved iterative coupler.

[0011] S4: Using the TD-BEM interface displacement of the current outer loop iteration as the boundary condition, the soil subsystem is solved by TD-BEM based on the adaptive semi-infinite boundary element method to obtain the TD-BEM interface surface force of the current outer loop iteration; and the TD-BEM interface surface force of the current outer loop iteration is converted into the FEM nodal force of the current outer loop iteration using the time-interleaved iterative coupler.

[0012] S5: Perform convergence judgment: If the convergence condition is not met, the FEM node force of the current outer loop iteration is corrected based on the relaxation correction method, and the outer loop iteration is restarted in step S3 until the convergence condition is met; if the convergence condition is met, the calculation of the next time step is entered, and steps S3-S5 are repeated until the calculation of all time steps is completed, and the dynamic response of all time steps is output.

[0013] Furthermore, the vehicle-track subsystem includes a multi-degree-of-freedom vehicle multibody dynamics model and a multi-degree-of-freedom slab track Euler beam model;

[0014] The soil subsystem is constructed based on the three-dimensional elastic dynamic time-domain boundary integral equation with embedded adaptive semi-infinite boundary elements.

[0015] Furthermore, the multi-degree-of-freedom vehicle multibody dynamics model includes: the heave and pitch motion of the vehicle body; the heave and pitch motion of the front and rear bogies respectively; the heave motion of the four wheelsets; and the vehicle components are connected by spring-damping units of the primary and secondary suspensions.

[0016] The system dynamic equations of the multi-body dynamics model of the multi-degree-of-freedom vehicle are:

[0017] ;

[0018] in, , , These are the vehicle's lumped mass matrix, lumped damping matrix, and lumped stiffness matrix, respectively. , , These are the lumped displacement, lumped velocity, and lumped acceleration vectors of the vehicle, respectively. This is the wheel-rail force vector;

[0019] The multi-degree-of-freedom slab track Euler beam model includes: modeling the rail, track slab, and concrete base plate as Euler beams; simplifying the fasteners, cement asphalt mortar layer, and concrete support layer as discrete spring-damping elements; and including the vertical displacement and rotation at both ends of the rail, the vertical displacement and rotation at both ends of the track slab, and the vertical displacement and rotation at both ends of the concrete base plate from top to bottom.

[0020] The system dynamic equations of the multi-degree-of-freedom slab track Euler beam model are as follows:

[0021] ;

[0022] in, , , These are the lumped mass matrix, lumped damping matrix, and lumped stiffness matrix of the track, respectively. and These represent the lumped displacement vector and the lumped nodal force vector, respectively.

[0023] Furthermore, when modeling the vehicle-track subsystem, the vehicle and track are coupled through Hertz contact theory. Based on the system dynamics equations of the multi-degree-of-freedom vehicle multibody dynamics model and the system dynamics equations of the multi-degree-of-freedom slab track Euler beam model, the vehicle-track coupled system equations are established:

[0024] ;

[0025] in, , , These represent the displacement, velocity, and acceleration vectors of the entire vehicle-track coupling system; and the lumped nodal force vector. Including the wheel-rail contact force acting on the vehicle and external force .

[0026] Furthermore, the adaptive semi-infinite boundary element adopts a quadrilateral element, including inner nodes and outer nodes. The inner nodes are fixed and connected to the finite domain, and the coordinates of the outer nodes are dynamically determined according to the stress wavefront: c1(t−τ), where c1 is the P-wave velocity and τ is the excitation time of the wave source.

[0027] Furthermore, the three-dimensional elastic dynamic time-domain boundary integral equation of the embedded adaptive semi-infinite boundary element is:

[0028] ;

[0029] in, Here are the free term coefficients related to the boundary geometry, P is the source point, Q is the field point, and t is the current calculation time. The moment when the stress wave originates at the source point P. Let be the displacement component of the source point P along direction i at time t. Let field point Q be at time... Displacement component along direction i, Let field point Q be at time... Surface force components in direction i, For the fundamental solution of surface forces in the time domain, This is the fundamental solution in the displacement time domain, where k and i represent different directions, and S is the TD-BEM boundary.

[0030] In displacement components and displacement components The displacement field is embedded with adaptive semi-infinite boundary elements:

[0031]

[0032] in, For the displacement field of the adaptive semi-infinite boundary element, Coordinates in the natural coordinate system It is a quadrilateral unit shape function. It is an internal node The interface displacement is D, where D is the displacement decay function.

[0033] Furthermore, the time-interleaved iterative coupler performs displacement transfer between the vehicle-track subsystem and the soil subsystem based on displacement continuity, performs force transfer between the vehicle-track subsystem and the soil subsystem through a force transformation matrix based on force balance conditions, and performs time step alignment between the vehicle-track subsystem and the soil subsystem based on a time interpolation formula.

[0034] Furthermore, in step S3, based on the FEM nodal forces obtained from the previous iteration, the vehicle-track coupling is solved using the relaxation iterative prediction and correction method to obtain the current FEM interface displacement after vehicle-track coupling, including:

[0035] S31: Input vehicle and track parameters, and initialize the system matrix of the vehicle-track subsystem;

[0036] S32: Predict the overall displacement, velocity, and acceleration of the vehicle-track coupling system at the current time step using explicit integration from the overall displacement, velocity, and acceleration of the vehicle-track coupling system at the previous time step;

[0037] S33: The overall displacement, velocity, and acceleration of the vehicle-track coupling system at the predicted current time step, along with the corrected FEM nodal forces obtained from the previous outer loop iteration at the current time step. Using Newmark implicit integration as input, the vehicle-track coupled system equations are solved to obtain the initial wheel-track time step. ;

[0038] S34: In the inner loop iteration of the vehicle-track subsystem, in each iteration, the interface displacement obtained from the previous inner loop iteration at the current time step is used. And the corrected FEM nodal force obtained from the previous outer loop iteration at the current time step. , Using the outer loop iteration number as an example, the vehicle-track coupled system equations are solved using Newmark implicit integration to obtain the wheel-rail forces at the current time step in the k-th inner loop iteration. ;

[0039] S35: Introducing a relaxation factor wheel-rail force Make corrections: ,in, The wheel-rail force for the k-th inner loop iteration at the current time step, after correction;

[0040] S36: Wheel-rail force based on the k-th inner loop iteration of the corrected current time step. And the corrected FEM nodal force obtained from the previous outer loop iteration at the current time step. By using Newmark implicit integration to solve the vehicle-track coupled system equations, the interface displacement of the k-th inner loop iteration at the current time step is obtained. ;

[0041] S37: Calculate the difference in interface displacement between two consecutive inner loop iterations. If it is less than the set convergence tolerance The iteration ends if the iteration ends; otherwise, return to step S34 and continue iterating until convergence, outputting the FEM interface displacement of the s-th outer loop iteration at the current time step. , This is the convergence iteration number of the inner loop iteration.

[0042] Furthermore, the FEM interface displacement of the current outer loop iteration is converted into the TD-BEM interface displacement of the current outer loop iteration using a time-interleaved iterative coupler, including: converting the FEM interface displacement using a time interpolation formula. Converted to TD-BEM interface displacement Achieve cross-domain time step alignment.

[0043] Furthermore, based on the adaptive semi-infinite boundary element method, the soil subsystem is solved using TD-BEM to obtain the TD-BEM interface surface forces of the current outer loop iteration, including: TD-BEM interface displacements. As boundary conditions, the displacement field of the adaptive semi-infinite boundary element is embedded into the three-dimensional elastic dynamic time-domain boundary integral equation, and the surface forces at the TD-BEM interface are obtained by solving the equation. ;

[0044] The TD-BEM interface surface forces of the current outer loop iteration are converted into FEM nodal forces of the current outer loop iteration using a time-interleaved iterative coupler, including:

[0045] The force transformation matrix is ​​used to convert the surface forces of the TD-BEM interface. Convert to current FEM node force without time step alignment : The force of the current FEM node that has not been time-step aligned is obtained through a time interpolation formula. Transformed into FEM nodal forces Achieve cross-domain time step alignment.

[0046] The beneficial effects of this invention are as follows:

[0047] Improved computational efficiency: By employing a time-staggered algorithm, the vehicle-track subsystem (FEM subdomain) can use a smaller time step to ensure the accuracy of high-frequency dynamic response, while the soil subsystem (TD-BEM subdomain) can use a larger time step to save computational resources. Based on the proposed RI-PCM method, by introducing iteration and relaxation techniques, the vehicle-track subsystem (FEM subdomain) can achieve higher efficiency than traditional PCM.

[0048] Significantly enhanced simulation accuracy and physical realism: The adaptive semi-infinite boundary element method can physically and dynamically simulate the propagation and energy radiation of stress waves, fundamentally avoiding the reflection errors caused by artificial boundaries, making far-field vibration prediction more reliable.

[0049] Improved numerical stability and robustness: The iterative coupling framework avoids constructing and solving potentially ill-conditioned unified system matrices, reducing the difficulty of numerical implementation and improving the stability of the computation process. The introduction of relaxation iteration techniques and convergence criteria effectively controls error accumulation at the coupling interface, ensuring the reliability of long-term simulation results. Attached Figure Description

[0050] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0051] Figure 1 This is the overall technical roadmap of the present invention.

[0052] Figure 2 This is a schematic diagram of the vehicle multibody dynamics model of the present invention.

[0053] Figure 3 The diagram shows the plate track structure of the present invention. (a) is a cross-sectional view, and (b) is a computational dynamic model.

[0054] Figure 4 This is a schematic diagram of the vehicle-track coupled dynamics model of the present invention.

[0055] Figure 5 This is a detailed flowchart of the implementation of the Relaxational and Iterative Predictor-Corrector Method (RI-PCM) of the present invention.

[0056] Figure 6 This is a schematic diagram of the adaptive semi-infinite boundary element of the present invention.

[0057] Figure 7 This is a schematic diagram of the Vehicle-Track-Soil (VTS) strongly coupled system of the present invention.

[0058] Figure 8 This is a schematic diagram illustrating the transmission of vibration between the vehicle-track and soil subsystems in the track foundation according to the present invention.

[0059] Figure 9 This is a schematic diagram illustrating the interaction between the track foundation beam segment unit and the soil unit of the present invention.

[0060] Figure 10 This is a schematic diagram illustrating the allocation of track length according to the present invention.

[0061] Figure 11 This is a schematic diagram of the soil TD-BEM mesh model of the present invention.

[0062] Figure 12The figure shows a comparison of the displacement calculation results of the RI-PCM of the present invention and the conventional PCM at point A on the vehicle body. (a) is the vehicle body displacement calculated by RI-PCM, and (b) is the vehicle body displacement calculated by PCM.

[0063] Figure 13 To compare the displacement calculation results of the RI-PCM of the present invention and the conventional PCM at point B on the track, (a) is the rail displacement calculated by RI-PCM, and (b) is the rail displacement calculated by PCM.

[0064] Figure 14 This is a schematic diagram of the on-site setup for the freight train induced vibration test according to the present invention.

[0065] Figure 15 The results of time history acceleration and measured acceleration at the calculated point on the rail in the VTS strongly coupled model of the present invention are shown in (a) and (b) respectively.

[0066] Figure 16 The results of the track-ground contact point force calculated by the VTS strong coupling model and the vehicle-track coupling model of the present invention are shown in (a) and (b) respectively. (a) is the track-ground contact point force calculated by the VTS strong coupling model and (b) is the track-ground contact point force calculated by the vehicle-track coupling model.

[0067] Figure 17 The following are cloud maps showing the vertical displacement of the ground surface caused by train vibration at different speeds according to the present invention: (a) is the case with a train speed of 70 m / s, (b) is the case with a train speed of 92 m / s, and (c) is the case with a train speed of 120 m / s.

[0068] Figure 18 The following are top views of the vertical displacement of the ground surface caused by train vibration at different speeds according to the present invention: (a) is the case with a train speed of 70 m / s, (b) is the case with a train speed of 92 m / s, and (c) is the case with a train speed of 120 m / s. Detailed Implementation

[0069] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0070] A vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling is proposed. The invention adopts the following overall technical solution: Figure 1As shown, its core lies in constructing a system consisting of a vehicle-track subsystem (FEM subdomain), a soil subsystem (TD-BEM subdomain), and a time-staggered iterative coupler. This coupler is responsible for transferring and iteratively updating the physical quantities (displacement, force) at the coupling boundary between the two subsystems without forcing their time steps to be completely consistent.

[0071] S1: The vehicle-track subsystem is modeled based on FEM, and the soil subsystem is modeled based on TD-BEM. Different time steps are used to solve the vehicle-track subsystem and the soil subsystem.

[0072] In this embodiment of the application, the vehicle-track subsystem is modeled based on FEM, including:

[0073] The vehicle model uses a 10-DOF multibody dynamics model, such as... Figure 2 As shown, this specifically includes: the sinking and floating of the vehicle body ( ) and nod ( ) movement; the rise and fall of the front and rear bogies respectively ( , ) and nod ( , ) motion; the rising and falling motion of the four wheelsets ( , , , Vehicle components are connected by spring-damping units in the primary and secondary suspension systems.

[0074] The system dynamics equations of the vehicle model are:

[0075] ;

[0076] in, , , These are the vehicle's lumped mass matrix, lumped damping matrix, and lumped stiffness matrix, respectively. , , These are the lumped displacement, lumped velocity, and lumped acceleration vectors, respectively, containing all 10 degrees of freedom of the vehicle. This is the wheel-rail force vector.

[0077] Track Model: A 12-DOF (degrees of freedom) slab track Euler beam model is adopted, with the rails, track slab, and concrete base slab all modeled as Euler beams, considering their vertical displacement and rotation degrees of freedom. Fasteners, cement asphalt mortar layers, and concrete support layers are simplified as discrete or continuous spring-damped elements. A typical track element contains 12 degrees of freedom, including the vertical displacements at both ends of the rail from top to bottom. , ) and corner ( , Vertical displacement at both ends of the track slab , ) and corner ( , Vertical displacement at both ends of the concrete base slab , ) and corner ( , ),like Figure 3 As shown.

[0078] The system dynamics equations of the orbital model are:

[0079] ;

[0080] in, , , These are the lumped mass matrix, lumped damping matrix, and lumped stiffness matrix, respectively, which are composed of the correlation matrices of each discrete track element according to the "matching" principle. , , These are the lumped displacement, lumped velocity, and lumped acceleration vectors, respectively, containing all 12 degrees of freedom of the orbit. This represents the lumped displacement vector and the lumped nodal force vector of the track.

[0081] The vehicle and track are coupled using Hertz contact theory, and the established vehicle-track coupling model is as follows: Figure 4 As shown.

[0082] The equations for the vehicle-track coupled system are:

[0083] ;

[0084] in, and These are the wheel-rail contact forces acting on the vehicle and the wheel-rail contact forces acting on the rail, respectively. It is a nonlinear function of wheel displacement and rail displacement, which can be obtained through Hertz contact theory. , , These represent the displacement, velocity, and acceleration vectors of the entire vehicle-track coupled system. The wheel-rail contact forces are a pair of equal-sized, opposite-direction interaction forces acting on the wheelset and rail, respectively. They can be considered as external load excitations of the system, and their corresponding dynamic responses can be solved using the dynamic equations of their respective systems. F h The external force term reflects the vibration feedback of the soil sub-model to the vehicle-track coupled system.

[0085] In this embodiment of the application, the soil subsystem is modeled based on TD-BEM, including:

[0086] The three-dimensional elastic dynamics time-domain boundary element model is constructed based on the boundary integral equation, discretizing only the soil boundary rather than the entire domain. Ignoring initial conditions and volume forces, the three-dimensional elastic dynamics time-domain boundary integral equation is:

[0087] ;

[0088] in, Here are the free term coefficients related to the boundary geometry, P is the source point, Q is the field point, and t is the current calculation time. The moment when the stress wave originates at the source point P. Let be the displacement component of the source point P along direction i at time t. Let field point Q be at time... Displacement component along direction i, Let field point Q be at time... Surface force components in direction i, For the fundamental solution of surface forces in the time domain, This is the fundamental solution in the displacement time domain, where k and i represent different directions, and S is the TD-BEM boundary. The fundamental solution of the surface force in the time domain has the physical meaning of: at time... When a unit displacement along the k direction is applied at the source point P, the surface force response along the i direction generated at the field point Q at time t is... The fundamental solution in the displacement-time domain has the physical meaning of: at time t = 0. A unit surface force along the k direction is applied at the source point P. The displacement response along the i direction produced at the field point Q at time t is as follows: and The solution is known.

[0089] Traditional TD-BEM uses fixed finite boundaries for the far-field. To address the far-field simulation problem, this invention constructs an adaptive semi-infinite quadrilateral boundary element at the outer boundary of a finite domain (e.g., the Earth's surface), such as... Figure 6 As shown. The internal nodes of this element. Connected to the finite domain (i.e., the coupling surface between the soil and the track), internal nodes Fixed coordinates; external nodes The coordinates are determined dynamically based on the stress wavefront, with its x-coordinate set as c1(t−τ), where c1 is the P-wave velocity and τ is the excitation time of the wave source. This allows the range of the integration domain S to adaptively expand with the wave propagation process, rather than having a fixed geometric boundary. This is a fundamental extension of the traditional integration domain. For field point displacement... Traditionally, fixed interpolation is used. This application introduces attenuation functions and dynamic shape functions, which are embedded into the field point displacements of the integral equation. The fundamental solution of displacement in the integral equation. Basic solution of dough kneading force The value of is directly related to the spatial relative positions of the source point P and the field point Q. Within the adaptive semi-infinite boundary element region, the physical coordinates of the field point Q contain a dynamic term c1(t−τ), making the calculation of the fundamental solution no longer dependent on fixed spatial coordinates, but dynamically updated with the excitation time τ of the wave source and the wave velocity P and wave velocity c1. That is, the coordinates of the independent variable Q of the fundamental solution are time-varying. This correlation is embedded into the calculation of the fundamental solution of the integral equation through the node coordinate rules of the adaptive boundary element. Solving the time-domain boundary integral equation of three-dimensional elastic dynamics is a process of numerically processing the time-domain boundary integral equation and transforming the integral equation into a series of solvable algebraic equations. First, the spatial boundary is discretized using isoparametric elements, and the time domain is divided into equal step sizes, approximating the integral in the equation as a time-step summation. Then, a time-domain recursive algorithm is used to solve the discretized algebraic equations step by step. In the context of vehicle-track-soil coupling, the boundary displacement of the track structure is transmitted in real time by the vehicle system dynamics as a known condition. At each time step, this displacement boundary condition is combined with the discrete equations to solve for the unknown surface forces and the global dynamic response. The calculation logic for solving the time-domain boundary integral equation of three-dimensional elastic dynamics remains unchanged. Those skilled in the art can solve it based on the above integral equation, so it will not be elaborated here.

[0090] Displacement field of adaptive semi-infinite quadrilateral boundary element ( (where η is the coordinate in the natural coordinate system) is represented as:

[0091] ;

[0092] in, It is a quadrilateral unit shape function. It represents the internal node interface displacement, and D is the linear displacement decay function, defined as follows: , Indicates from internal node Radial distance to the relevant source point Indicates internal node This structure automatically satisfies the boundary condition that the far-field displacement tends to zero, accurately simulating radiation damping. When the source point P is located inside the adaptive element, i.e. Figure 6 Middle node A i At this time We can obtain D=1; when the source point P is located on the outer side of the adaptive unit ( Figure 6 In the middle, at point B, where r approaches infinity, we can obtain D=0. Therefore, it can be said that the inner nodes of the adaptive element are in the conventional discrete case ( Figure 6In the finite field portion, the outermost node is an inner node, and its displacement is zero. Therefore, no special numerical processing is required for the adaptive element during the solution process; only a displacement attenuation function needs to be introduced. Without increasing the discrete workload, it can guarantee the accuracy of the solution without increasing the workload. In addition, other forms of displacement attenuation functions can also be used to describe the physical attenuation of far-field displacement.

[0093] Construction of a vehicle-track-soil integrated coupled system:

[0094] When a high-speed train is running, it generates vibrational energy at the wheel-rail contact point. This energy is mainly transmitted to the soil below through the superstructure of the track. As a key interface between railway infrastructure and the natural foundation, the soil plays a dual role: it must bear the static and dynamic loads of the train and facilitate the transfer of energy. Figure 7 This demonstrates a multi-level vehicle-track-soil (VTS) interaction system, which includes vehicles, tracks, and the surrounding soil.

[0095] In the VTS interaction system, the vehicle-track subsystem and the soil subsystem communicate through their common interface ( Figure 2 The foundation beam segments are connected to each other. This connection enforces equilibrium and continuity conditions, allowing the vehicle-track dynamic response to be transmitted to the soil subsystem, while soil vibrations can also be fed back to the vehicle-track subsystem. In solving the VTS interaction system, the role of the foundation beam segments is as follows: Figure 8 As shown.

[0096] Figure 9 The connection relationship between the track foundation unit and the soil unit is shown, where, as referenced... Figure 4 Additional nodes C and D on the soil element correspond to nodes A and B on the track foundation beam segment element in the vehicle-track subsystem, respectively. These are the interaction forces between the node pairs (AC and BD). If the continuity and equilibrium conditions between the node pairs (AC and BD) are satisfied, the vehicle-track and soil subsystems will be coupled.

[0097] S2: Construct a time-interleaved iterative coupler. The time-interleaved iterative coupler is used to transfer displacement between the vehicle-track subsystem and the soil subsystem based on displacement continuity, to transfer force between the vehicle-track subsystem and the soil subsystem through a force transformation matrix based on force balance conditions, and to align the time steps between the vehicle-track subsystem and the soil subsystem based on a time interpolation formula.

[0098] In this invention, the vehicle-track subsystem is modeled using FEM, and the soil subsystem is modeled using TD-BEM. The overall vehicle-track-soil coupled system is solved based on the FEM / TD-BEM coupling mechanism. The physical conditions below the coupler operate at the track-soil interface.

[0099] Displacement continuity: ;

[0100] Force balance: ;

[0101] Wherein, the subscripts FI and BI represent the coupling interface of the FEM domain (vehicle-track part) and the TD-BEM domain (soil part), respectively; u is displacement; f is FEM nodal force; p is TD-BEM interface surface force; M is force transformation matrix, used to convert the surface force integral into nodal force through shape function and Jacobian matrix.

[0102] Time Interleaving and Interpolation Algorithms:

[0103] Allowing the FEM domain and TD-BEM domain to use different time steps Δt F (FEM time step, i.e., the calculation time step of the vehicle-track subsystem) and Δt B (TD-BEM time step, i.e., the calculation time step of the soil subsystem). When it is necessary to transfer variables from one subsystem to another, time interpolation or extrapolation is used. For example, in the nth step of FEM, TD-BEM needs to be performed at t= The displacement at time t, while the solution of TD-BEM only exists at time t. If the time is known, it can be calculated using time interpolation or extrapolation formulas:

[0104] ;

[0105] in, For interpolation functions, such as linear interpolation, Represents the interface variables (displacement or surface force) of the source domain. Interface variables representing the target domain, The time step of the source domain. For the time step corresponding to the target domain, For source domain time identifier, The target domain is identified by time, and m and n are the time step numbers.

[0106] S3: Based on the FEM node forces obtained from the previous outer loop iteration at the current time step, perform an inner loop iteration to solve the vehicle-track coupling based on the relaxation iterative prediction and correction method, obtain the FEM interface displacement of the current outer loop iteration after vehicle-track coupling, and use the time-interleaved iterative coupler to convert the FEM interface displacement of the current outer loop iteration into the TD-BEM interface displacement of the current outer loop iteration.

[0107] In this embodiment, the vehicle and track are coupled via Hertz contact theory. To efficiently solve this nonlinear system, this invention proposes a relaxation iterative prediction correction method (RI-PCM), such as... Figure 5 As shown:

[0108] S31: Input vehicle and track parameters, and initialize the system matrix of the vehicle-track subsystem;

[0109] S32: Predict the overall displacement, velocity, and acceleration of the vehicle-track coupling system at the current time step using explicit integration from the overall displacement, velocity, and acceleration of the vehicle-track coupling system at the previous time step;

[0110] S33: The overall displacement, velocity, and acceleration of the vehicle-track coupling system at the predicted current time step, along with the corrected FEM nodal forces obtained from the previous outer loop iteration at the current time step. As input, with As the external force term, the vehicle-track coupled system equations are solved using Newmark implicit integration to obtain the initial wheel-track parameters at the current time step. ;

[0111] S34: In the inner loop iteration of the vehicle-track subsystem, in each iteration, the interface displacement obtained from the previous inner loop iteration at the current time step is used. And the corrected FEM nodal force obtained from the previous outer loop iteration at the current time step. , Using the outer loop iteration number as an example, the vehicle-track coupled system equations are solved using Newmark implicit integration to obtain the wheel-rail forces at the current time step in the k-th inner loop iteration. ;

[0112] S35: Introducing a relaxation factor wheel-rail force Make corrections: ,in, The wheel-rail force for the k-th inner loop iteration at the current time step, after correction;

[0113] S36: Wheel-rail force based on the k-th inner loop iteration of the corrected current time step. And the corrected FEM nodal force obtained from the previous outer loop iteration at the current time step. By using Newmark implicit integration to solve the vehicle-track coupled system equations, the interface displacement of the k-th inner loop iteration at the current time step is obtained. ;

[0114] S37: Calculate the difference in interface displacement between two consecutive inner loop iterations. If it is less than the set convergence tolerance The iteration ends if the iteration ends; otherwise, return to step S34 and continue iterating until convergence, outputting the FEM interface displacement of the s-th outer loop iteration at the current time step. , This is the convergence iteration number of the inner loop iteration.

[0115] The FEM interface displacement of the current outer loop iteration is converted into the TD-BEM interface displacement of the current outer loop iteration using a time-interleaved iterative coupler, including:

[0116] The displacement of the FEM interface is determined by time interpolation or extrapolation formulas. Converted to TD-BEM interface displacement Achieving cross-domain time step alignment:

[0117] .

[0118] S4: Using the TD-BEM interface displacement of the current outer loop iteration as the boundary condition, the soil subsystem is solved by TD-BEM based on the adaptive semi-infinite boundary element method to obtain the TD-BEM interface surface force of the current outer loop iteration; and the TD-BEM interface surface force of the current outer loop iteration is converted into the FEM nodal force of the current outer loop iteration using a time-interleaved iterative coupler.

[0119] In this embodiment of the application, the complete calculation process of step S4 is as follows:

[0120] S41, displacement of TD-BEM interface Using the TD-BEM interface force as the boundary condition and the TD-BEM interface force as the solution objective, the displacement field of the adaptive semi-infinite boundary element is embedded into the three-dimensional elastic dynamic time-domain boundary integral equation to obtain the TD-BEM interface force. .

[0121] Regarding the embedding of the displacement field of the adaptive semi-infinite boundary element into the three-dimensional elastic dynamic time-domain boundary integral equation to obtain the surface force of the TD-BEM interface. Explanation:

[0122] In a boundary element system, every point on the soil boundary acts as a source point P. When one point is a source point, all other points are field points Q. Therefore, the relationship between source points and field points is only relative. The vehicle-track subsystem (the finite element FEM part) transmits the displacement data of the track-soil coupling (contact) interface to the soil subsystem (the TD-BEM part). At this point, the track-soil coupling interface within the TD-BEM boundary can be understood as... Figure 6 In the finite field part of the equation, the displacements of points P and Q are known. For the adaptive semi-infinite boundary element, its inner nodes and the outermost nodes of the finite field of the soil ( The nodes are connected and located at the coupling interface between the track and the soil, so their displacement is also known. The outer nodes of the adaptive semi-infinite boundary element ( The stress wave will automatically extend as it propagates, and dissipate at the furthest point, naturally resulting in no displacement. Therefore, the displacement of the outer nodes is defined as 0. In summary, the displacements of points within the finite domain (coupled interface) and the inner nodes of the adaptive semi-infinite boundary element, whether source points or field points, are known. Since the displacements of the outer nodes of the adaptive semi-infinite boundary element are zero, it means that all displacements in the three-dimensional elastic dynamics time-domain boundary integral equation are known, and therefore the unknown interface forces can be calculated.

[0123] Therefore, the unknowns in the time-domain boundary integral equation of three-dimensional elastic dynamics are only... Therefore, based on the three-dimensional elastic dynamics time-domain boundary integral equation, we can obtain... That is, the surface force of the TD-BEM interface. .

[0124] Using a time-interleaved iterative coupler to apply surface forces at the TD-BEM interface Convert to current FEM node force without time step alignment : Using a time-interleaved iterative coupler, the forces of the current FEM nodes that have not been time-step aligned are... Transformed into FEM nodal forces Achieve cross-domain time step alignment.

[0125] S5: Perform convergence assessment:

[0126] If the convergence condition is not met, the FEM node forces of the current outer loop iteration are corrected based on the relaxation correction method, and the outer loop iteration is restarted in step S3 until the convergence condition is met.

[0127] If the convergence condition is met, proceed to the next time step and repeat steps S3-S5 until all time steps are calculated and the dynamic response of all time steps is output.

[0128] In this embodiment of the application, the method for determining convergence is as follows: calculate the residual of the TD-BEM interface surface force after two iterations. ,like Then convergence, This is a relative error. This represents the absolute error.

[0129] If the convergence condition is not met, the current FEM nodal forces are corrected based on the relaxation correction method, including:

[0130] ;

[0131] in, The FEM nodal force is corrected in the s-th iteration at the current time step, and the relaxation factor is... Take a value of 0.5 to 0.8.

[0132] Experimental verification:

[0133] In one embodiment: Vibration analysis of a high-speed train running on a slab track is performed.

[0134] Objective: To verify the accuracy and efficiency of the model of this invention under high-speed, ballastless track conditions.

[0135] Model parameters:

[0136] Vehicles: Type A metro vehicles; Track: Slab track. Vehicle and track dynamic parameters are shown in Table 1.

[0137] Table 1. Vehicle and track dynamic parameters.

[0138]

[0139] Soil: Considered as a homogeneous elastic half-space with an elastic modulus of 170 MPa, a density of 1800 kg / m³, and a Poisson's ratio of 0.36.

[0140] Operating conditions: Train speed 252 km / h (70 m / s), track irregularities are assessed using the American FRA Class 6 spectrum.

[0141] Implementation steps:

[0142] 1) Subsystem modeling:

[0143] Establish a 10-DOF vehicle FEM model and a 300-element track FEM model (total length 150m), such as... Figure 10 As shown.

[0144] 2) Establish a TD-BEM model of the soil, and arrange adaptive semi-infinite boundary elements on its outer boundary, such as... Figure 11 As shown.

[0145] 3) Coupling settings:

[0146] Set the time step of the FEM subsystem to ΔtF = 0.0005 s and the time step of the TD-BEM subsystem to ΔtB = 0.001 s.

[0147] Set the coupling iteration convergence tolerance: relative tolerance ϵrel = 1.0 × 10−4.

[0148] 4) Solve:

[0149] The vehicle-track subsystem is solved using RI-PCM with a relaxation factor λ of 0.5.

[0150] Start the time-interleaved coupled solver to perform full-time domain analysis.

[0151] Results and Analysis:

[0152] Accuracy: The calculation results for two monitoring points, point A at the center of the vehicle body and point B (65 m, 0, 0) on the rail in the track assembly structure, are as follows: Figure 12 and Figure 13 As shown, the calculated track displacement response is in high agreement with the results obtained using the conventional predictive correction method (PCM), verifying the correctness of RI-PCM.

[0153] Efficiency: Table 2 shows that the RI-PCM method proposed in this invention significantly reduces the computation time of the vehicle-track subsystem from 4.5 hours to 0.4 hours, improving efficiency by approximately 91%. Table 3 shows that the simulation of the VTS coupled system takes approximately 3 hours. In comparison, the traditional time-matched coupling method (with a uniform time step of 0.00001s) takes approximately 7.5 hours. This invention improves efficiency by 60%.

[0154] Table 2 Comparison of computational efficiency between traditional PCM and the RI-PCM proposed in this invention in the vehicle-track subsystem.

[0155]

[0156] Table 3. Comparison of computational efficiency between time-interleaved coupling method and traditional time-matched coupling method in vehicle-track-soil coupled system.

[0157]

[0158] In another embodiment: Vibration testing of freight trains on ballasted tracks is conducted, with the site setup as follows: Figure 14 As shown:

[0159] Objective: To verify the applicability of the model of this invention in actual engineering by comparing it with field measurement data.

[0160] Model parameters:

[0161] Vehicle: Freight train; Track: Ballast track.

[0162] Soil: Considered as a homogeneous elastic half-space with an elastic modulus of 170 MPa, a density of 1800 kg / m³, and a Poisson's ratio of 0.36.

[0163] Operating conditions: Train speed 60 km / h, track irregularities are assessed using FRA Level 6 spectrum.

[0164] Implementation steps:

[0165] 1. On a freight line section from K21+300 to K21+900, magnetic accelerometers (such as...) are installed on the lower flange of the rails and on the sleepers. Figure 14 ).

[0166] 2. The VTS strongly coupled model of this invention is established using the same procedure as that used for vibration analysis of high-speed trains running on slab tracks, with parameters set as shown in Table 4.

[0167] 3. Run the model and output the acceleration time history of the rail monitoring points.

[0168] 4. Compare the simulation results with the actual measured data on site.

[0169] Table 4. Parameters of truck vehicle and track model.

[0170]

[0171] Results and Analysis:

[0172] Figure 15 (a) The peak acceleration calculated by the weak coupling of VTS is 25.2 m / s2, while the peak rail acceleration obtained by the strong coupling simulation of VTS is 30.5 m / s2.

[0173] Figure 15 (b) The peak acceleration obtained from the on-site measurement is 32.5 m / s2.

[0174] The peak acceleration calculated by the VTS strongly coupled model is more consistent with the field measured value, with an error of less than 6%, proving that the model of this invention has good engineering accuracy. The simulated value is slightly lower, mainly because the model assumes that the soil is linearly elastic and does not consider the nonlinear dissipation of the actual soil.

[0175] from Figure 16 It can be seen that in the VTS strongly coupled model, after coupling the soil subsystem, the track-ground interaction force increases from 48.7 kN to 76.5 kN, an increase of approximately 57%. The track-soil interaction force obtained after considering the influence of the soil system is closer to the measured value. Therefore, when studying the performance of the soil under the track and vibration propagation under vehicle loads, it is necessary to establish a VTS strongly coupled model; otherwise, it will cause significant errors.

[0176] In yet another embodiment, a Mach effect analysis under high-speed operation is performed.

[0177] Objective: To investigate the severe vibration phenomenon (Mach effect) caused by a vehicle speed exceeding the Rayleigh wave velocity of the soil using the model of this invention.

[0178] Model parameters:

[0179] Vehicle: High-speed train, parameters are shown in Table 5 (car body mass 63807 kg, etc.).

[0180] Table 5. Dynamic parameters of high-speed trains.

[0181]

[0182] Track: Ballastless track; rail mass 60kg / m, under-rail pad stiffness 6×107N / m, under-rail pad damping 7.5×104, elastic modulus of CA mortar under the track slab is 300Mpa, and the mass per unit length of track is 1115.62kg / m.

[0183] Soil: Parameters are shown in Table 6, with particular attention to its Rayleigh wave velocity CR=92.1 m / s.

[0184] Table 6. Main parameters of the soil.

[0185]

[0186] Operating conditions: Calculate the three vehicle speeds respectively: v = 70 m / s (<CR),v= m / s (=CR),v=120 m / s (> CR). Track irregularities are not considered.

[0187] Implementation steps:

[0188] 1. Establish VTS strongly coupled models for three vehicle speeds respectively.

[0189] 2. Run the simulation to calculate the displacement response of the ground area.

[0190] 3. Extract and plot the ground vertical displacement field contour map (e.g., Figure 17 , Figure 18 ).

[0191] Results and Analysis:

[0192] When the vehicle speed c=70 m / s, the ground vibration is mainly concentrated near the load application point, and the range of influence is limited.

[0193] When the vehicle speed c=92m / s, the displacement cloud map begins to show a "V" shaped pattern similar to the bow wave, indicating that the Mach effect begins to appear and the vibration influence range begins to expand.

[0194] When the vehicle speed c=120m / s, the Mach effect is very significant, the "V" shaped wavefront is clear, the vibration energy is more concentrated in the direction of the wavefront, and the propagation range is much greater than that at low speeds.

[0195] Conclusion: The model of this invention can clearly capture and quantify the Mach effect, providing crucial theoretical basis and simulation methods for the rational setting of high-speed railway operating speeds to control environmental vibrations. It is recommended that in practical engineering, train operating speeds should be kept as low as possible below the Rayleigh wave velocity of the railway track foundation.

[0196] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling, characterized in that, The steps are as follows: S1: The vehicle-track subsystem is modeled based on FEM, and the soil subsystem is modeled based on TD-BEM. Different time steps are used to solve the vehicle-track subsystem and the soil subsystem. S2: Construct a time-interleaved iterative coupler; S3: Based on the FEM node forces obtained from the previous outer loop iteration at the current time step, the vehicle-track coupling is solved by inner loop iteration based on the relaxation iterative prediction and correction method to obtain the FEM interface displacement of the current outer loop iteration after vehicle-track coupling, and the FEM interface displacement of the current outer loop iteration is converted into the TD-BEM interface displacement of the current outer loop iteration using the time-interleaved iterative coupler. S4: Using the TD-BEM interface displacement of the current outer loop iteration as the boundary condition, the soil subsystem is solved by TD-BEM based on the adaptive semi-infinite boundary element method to obtain the TD-BEM interface surface force of the current outer loop iteration; and the TD-BEM interface surface force of the current outer loop iteration is converted into the FEM nodal force of the current outer loop iteration using the time-interleaved iterative coupler. S5: Perform convergence judgment: If the convergence condition is not met, the FEM node force of the current outer loop iteration is corrected based on the relaxation correction method, and the outer loop iteration is restarted in step S3 until the convergence condition is met; if the convergence condition is met, the calculation of the next time step is entered, and steps S3-S5 are repeated until the calculation of all time steps is completed, and the dynamic response of all time steps is output.

2. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to claim 1, characterized in that, The vehicle-track subsystem includes a multi-degree-of-freedom vehicle multibody dynamics model and a multi-degree-of-freedom slab track Euler beam model; The soil subsystem is constructed based on the three-dimensional elastic dynamic time-domain boundary integral equation with embedded adaptive semi-infinite boundary elements.

3. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to claim 2, characterized in that, The multi-degree-of-freedom vehicle multibody dynamics model includes: the heave and pitch motion of the vehicle body; the heave and pitch motion of the front and rear bogies; the heave motion of the four wheelsets; and the vehicle components are connected by spring-damping units of the primary and secondary suspensions. The system dynamic equations of the multi-body dynamics model of the multi-degree-of-freedom vehicle are: ； in, , , These are the vehicle's lumped mass matrix, lumped damping matrix, and lumped stiffness matrix, respectively. , , These are the lumped displacement, lumped velocity, and lumped acceleration vectors of the vehicle, respectively. This is the wheel-rail force vector; The multi-degree-of-freedom slab track Euler beam model includes: modeling the rail, track slab, and concrete base plate as Euler beams; simplifying the fasteners, cement asphalt mortar layer, and concrete support layer as discrete spring-damping elements; and including the vertical displacement and rotation at both ends of the rail, the vertical displacement and rotation at both ends of the track slab, and the vertical displacement and rotation at both ends of the concrete base plate from top to bottom. The system dynamic equations of the multi-degree-of-freedom slab track Euler beam model are as follows: ； in, , , These are the lumped mass matrix, lumped damping matrix, and lumped stiffness matrix of the track, respectively. and These represent the lumped displacement vector and the lumped nodal force vector, respectively.

4. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to claim 3, characterized in that, When modeling the vehicle-track subsystem, the vehicle and track are coupled through Hertz contact theory. Based on the system dynamics equations of the multi-degree-of-freedom vehicle multibody dynamics model and the system dynamics equations of the multi-degree-of-freedom slab track Euler beam model, the vehicle-track coupled system equations are established as follows: ； in, , , These represent the displacement, velocity, and acceleration vectors of the entire vehicle-track coupling system; and the lumped nodal force vector. Including the wheel-rail contact force acting on the vehicle and external force .

5. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to any one of claims 2 to 4, characterized in that, The adaptive semi-infinite boundary element adopts a quadrilateral element, including inner nodes and outer nodes. The inner nodes are fixed and connected to the finite domain. The coordinates of the outer nodes are dynamically determined according to the stress wavefront: c1(t−τ), where c1 is the P-wave velocity and τ is the excitation time of the wave source.

6. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to claim 5, characterized in that, The three-dimensional elastic dynamic time-domain boundary integral equation of the embedded adaptive semi-infinite boundary element is: ； in, Here are the free term coefficients related to the boundary geometry, P is the source point, Q is the field point, and t is the current calculation time. The moment when the stress wave originates at the source point P. Let be the displacement component of the source point P along direction i at time t. Let field point Q be at time... Displacement component along direction i, Let field point Q be at time... Surface force components in direction i, For the fundamental solution of surface forces in the time domain, This is the fundamental solution in the displacement time domain, where k and i represent different directions, and S is the TD-BEM boundary. In displacement components and displacement components The displacement field is embedded with adaptive semi-infinite boundary elements: in, For the displacement field of the adaptive semi-infinite boundary element, Coordinates in the natural coordinate system It is a quadrilateral unit shape function. It is an internal node The interface displacement is D, where D is the displacement decay function.

7. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to claim 6, characterized in that, The time-interleaved iterative coupler performs displacement transfer between the vehicle-track subsystem and the soil subsystem based on displacement continuity, performs force transfer between the vehicle-track subsystem and the soil subsystem through a force transformation matrix based on force balance conditions, and performs time step alignment between the vehicle-track subsystem and the soil subsystem based on a time interpolation formula.

8. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to claim 7, characterized in that, In step S3, based on the FEM nodal forces obtained from the previous iteration, the vehicle-track coupling is solved using the relaxation iterative prediction and correction method to obtain the current FEM interface displacement after vehicle-track coupling, including: S31: Input vehicle and track parameters, and initialize the system matrix of the vehicle-track subsystem; S32: Predict the overall displacement, velocity, and acceleration of the vehicle-track coupling system at the current time step using explicit integration from the overall displacement, velocity, and acceleration of the vehicle-track coupling system at the previous time step; S33: The overall displacement, velocity, and acceleration of the vehicle-track coupling system at the predicted current time step, along with the corrected FEM nodal forces obtained from the previous outer loop iteration at the current time step. Using Newmark implicit integration as input, the vehicle-track coupled system equations are solved to obtain the initial wheel-track time step. ; S34: In the inner loop iteration of the vehicle-track subsystem, in each iteration, the interface displacement obtained from the previous inner loop iteration at the current time step is used. And the corrected FEM nodal force obtained from the previous outer loop iteration at the current time step. , Using the outer loop iteration number as an example, the vehicle-track coupled system equations are solved using Newmark implicit integration to obtain the wheel-rail forces at the current time step in the k-th inner loop iteration. ; S35: Introducing a relaxation factor wheel-rail force Make corrections: ,in, The wheel-rail force for the k-th inner loop iteration at the current time step, after correction; S36: Wheel-rail force based on the k-th inner loop iteration of the corrected current time step. And the corrected FEM nodal force obtained from the previous outer loop iteration at the current time step. By using Newmark implicit integration to solve the vehicle-track coupled system equations, the interface displacement of the k-th inner loop iteration at the current time step is obtained. ; S37: Calculate the difference in interface displacement between two consecutive inner loop iterations. If the value is less than the set convergence tolerance, the iteration ends; otherwise, return to step S34 to continue iterating until convergence, and output the FEM interface displacement of the s-th outer loop iteration at the current time step. , This is the convergence iteration number of the inner loop iteration.

9. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to claim 8, characterized in that, The FEM interface displacement of the current outer loop iteration is converted into the TD-BEM interface displacement of the current outer loop iteration using a time-interleaved iterative coupler, including: converting the FEM interface displacement using a time interpolation formula. Converted to TD-BEM interface displacement Achieve cross-domain time step alignment.

10. The vehicle-track-soil strong coupling method based on finite element-temporal boundary element time-interleaved coupling according to claim 9, characterized in that, Based on the adaptive semi-infinite boundary element method, the soil subsystem is solved using TD-BEM to obtain the TD-BEM interface surface forces in the current outer loop iteration, including: TD-BEM interface displacements. As boundary conditions, the displacement field of the adaptive semi-infinite boundary element is embedded into the three-dimensional elastic dynamic time-domain boundary integral equation, and the surface forces at the TD-BEM interface are obtained by solving the equation. ; The TD-BEM interface surface forces of the current outer loop iteration are converted into FEM nodal forces of the current outer loop iteration using a time-interleaved iterative coupler, including: The force transformation matrix is ​​used to convert the surface forces of the TD-BEM interface. Convert to current FEM node force without time step alignment : The force of the current FEM node that has not been time-step aligned is obtained through a time interpolation formula. Transformed into FEM nodal forces Achieve cross-domain time step alignment.

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