Considering the interaction between a sleeper, ballast layer, and substructure, a three-dimensional coupled discrete-finite element method for a ballasted railway track is proposed in this study. Ballast granules with irregular shapes are constructed using a clump model using the discrete element method. Meanwhile, concrete sleepers, embankments, and foundations are modelled using 20-node hexahedron solid elements using the finite element method. To improve computational efficiency, a GPU-based (Graphics Processing Unit) parallel framework is applied in the discrete element simulation. Additionally, an algorithm containing contact search and transfer parameters at the contact interface of discrete particles and finite elements is developed in the GPU parallel environment accordingly. A benchmark case is selected to verify the accuracy of the coupling algorithm. The dynamic response of the ballasted rail track is analysed under different train speeds and loads. Meanwhile, the dynamic stress on the substructure surface obtained by the established DEM-FEM model is compared with the

As a traditional railway transportation structure, ballasted railway tracks typically comprise steel rails, fastening systems, sleepers, ballast, sub-ballasts, embankments, and foundations. Owing to the advantages of low price, easy maintenance, sufficient drainage, and vibration reduction, ballasted railways are still the most typical used railway structure [

The finite element method (FEM) can be used to represent different ballasted railway structures as continuums on a macro scale. By adopting different element types and material properties, the continuums can be used to study the dynamic response between structures [

Proposed by Cundall et al. in 1979 [

As mentioned above, the sleepers and substructure in a ballasted railway structure can be considered as continuous media, whereas the ballast bed is a discrete medium. The interaction between sleeper, ballast bed and substructure can be regarded as a contact problem between continuum and discrete materials. The coupled DEM-FEM combined the advantages in simulating granular and continuum materials and avoided the disadvantages of DEM and FEM alone for the ballasted railway track studies. So, the coupled method can more reasonably and efficiently simulate the dynamic response of ballasted railways at the macro and meso levels.

For coupled DEM-FEM method, it is particularly important to develop coupling algorithms for contact interfaces of different media. Recently, to achieve a contact interaction between a continuum material and a discrete ballast material, interface elements [

With the development of computer hardware and the demand for large-scale engineering simulations, CUDA (compute unified device architecture) as a parallel computing platform is released by NVIDIA Company, which overcome the limitations of traditional GPU hardware architecture. It highlights the advantages of GPU in high-performance computing and reduces the difficulty of programming [

In this study, a coupled DEM-FEM with GPU-based parallel framework was proposed to analyse the dynamic characteristics of a sleeper, ballast bed and substructure (embankment and foundation). A parameter transfer algorithm was developed between the DEM and FEM in a GPU parallel environment. A benchmark case was selected to verify the precision and reliability of the proposed coupled algorithm. Under different traffic speeds and loads, the dynamic response of the sleeper, ballast layer and the substructure were analysed, which provided invaluable references for high-performance coupling algorithms and the dynamic characteristics of full scale ballasted railway tracks.

In ballasted railway structures, as natural ballasts appear in complex geometrical shapes, the interlocking effect between adjacent ballasts upon loading can significantly increase the carrying capacity of the ballast layer, thereby resulting in the permanent deformation of the ballast bed. Hence, the clump formed by overlapping spheres was used to simulate the approximate geometry of a ballast. Because the particles in contact with the large overlap in the clump can overlook its contact force, the clump can be regarded as an unbreakable rigid element of irregular shape.

Considering the randomness of the effective size of ballast, the ballast particles were classified firstly according to the Chinese railway special ballast grading standards. To precisely establish the discrete element model of a natural ballast, three-dimensional laser scanning technique was employed to extract and reconstruct the ballast appearance in high-speed railways because this method can accurately reproduce the irregular geometry, sharp edges, and rough surface texture of the scanned ballast. Due to the time-consuming laser scanning process, 143 representative ballasts with different textures and sizes were scanned to establish a ballast library containing clump model. As shown in

For the contact law of ballast clump model, the linear spring model [

where

Without considering the viscous force, the tangential contact force between particles can be calculated incrementally based on the Mohr-Coulomb friction law, as follows [

where

The normal and tangential stiffness of the particles

where _{A}_{B}

During the computational process of the discrete element, the formula for calculating the maximum time step

where

where the coefficient

In the DEM simulation, the central difference method was used to solve the dynamic equation. During the movement of the irregular ballast, the ballast mass and moment of inertia are important parameters. The finite segmentation method can be used to obtain the ballast mass accurately. Meanwhile, the movement parameters, such as the moment and angular velocity of the ballast, can be calculated and updated using the quaternion method, which can convert between global and local coordinate systems [

It is well known that neighbour search and contact judgment between particles in DEM is the key parts which affect calculating efficiency. In discrete element simulation based on GPU parallel computing, the cell list method is used to make the contact judgment between particles [

Determine space cell of computing domain and cell label where particles are located, as shown in

Obtain sorted particle label based on the cell label and recorded the minimum and maximum particle labels

Create a neighbour particle array

The prefix sum

Obtain contact candidate pair list

Compute contact force between contacting particles;

Update particle information (displacement, velocity, etc.)

Concrete sleepers and substructures in railway structures demonstrate excellent rigidity and bearing capacity. Macroscopically, because both of them can be regarded as continuous media, based on the FEM, a 20-node hexahedral solid element was applied to construct the sleeper and substructure (embankment and foundation), as shown in

Besides, the Newmark method [

Regarding the simulation calculation of the coupled discrete-finite element model of the ballasted railway, the most critical aspect is to contact search and transfer the calculation parameters on the contact interface.

During coupling, as the force boundary condition for the dynamic calculation, the contact force

For the coupling model between the ballast grain and hexahedral solid element in three dimensions, the corresponding coupling interfaces must first be extracted, as shown in

During coupling, contact information between particles and element faces can be obtained, such as the number of contacts, contact force, contact position and the serial number of element face in a GPU parallel computation. The information must be transferred accurately to the finite element model and a dynamic calculation of the continuum zone must be performed in a CPU. However, the calculation results in DEM are distributed randomly in GPU memory, which increases the difficulty in transferring the contact parameters accurately. In the study, the

For the discrete-continuous model of the ballasted railway, when the ballast grains are in contact with the quadrilateral surface of the solid element, the contact point is typically not located on the node of the element face. Therefore, the contact force must be equivalent to each node on the contact surface. For the isoparametric element with 20 nodes, the appropriate shape function is required for the calculation of the equivalent nodal force on the coupling interface. By referring to the method for solving the shape function of triangular elements by area coordinates, the area coordinate applied to the quadrilateral face with eight nodes [

As shown in

where

The area coordinates of the contact point

where

The relationship between the contact force at contact point

where

The shape function of the four corner points on the quadrilateral face can be expressed as:

The shape function of the midpoint on the sides is

In the coupling calculation of the discrete–finite element model, the global coordinates

To verify the reliability of the coupled algorithm when the particles in the discrete zone contact with element face in the continuum zone, a benchmark case [_{z}

The mass of the granular ensemble is kept constant. So the upper surface of the plate is always subjected to a constant stress load

where _{s}

DEM | FEM | ||
---|---|---|---|

Parameters | Values | Parameters | Values |

Density of particles (kg/m^{3}) |
1745 | Density (kg/m^{3}) |
2800 |

Young’s Modulus of particles (Pa) | 5e9 | Young’s modulus (Pa) | 220e6 |

Poisson Ratio of particles | 0.3 | Poisson ratio | 0.3 |

Friction Coefficient | 0.6 | Friction coefficient | 0.6 |

Number of particles | 1000 | Number of elements | |

The radius of particles (mm) | 84 |

Based on the proposed DEM-FEM coupling algorithm with GPU-based parallel computing, a calculation model of a ballasted railway track is established, as shown in ^{−/+} directions) and the front and back faces (^{−/+} directions) of the model were constrained with zero displacements in the normal directions. A displacement constraint in the

Because the embankment strength is much lower than those of the sleeper and foundation, the Drucker–Prager yield criterion was used for the elastoplastic analysis of the embankment. Furthermore, the linear elastic model was applied to the response behaviour of the sleeper and foundation. In the initialization of coupled simulation, the timestep in DEM was determined firstly (the value is generally about 10^{−6} s on the time scale). And then given the calculation efficiency and communication between discrete and finite zone, the timestep in FEM was taken as integer multiples of that in DEM (40 times in the paper). Therefore, in the coupled process, the exchanged time interval of contact information on the coupled interface is determined by timestep in FEM. The calculation parameters in the coupling model are shown in

Parameters | Variables | Units | Values |
---|---|---|---|

Clump density [ |
kg/m^{3} |
2600 | |

Friction coefficient | – | 0.7 | |

The porosity of the ballast layer | – | 0.365 | |

Elastic modulus | GPa | 5 | |

Poisson ratio | 0.22 | ||

The average diameter of ballast | m | 0.035 | |

Coefficient of restitution | – | 0.5 |

Parameters | Variables | Units | Embankment | Foundation | Sleeper |
---|---|---|---|---|---|

Elastic modulus | MPa | 25 | 300 | 3000 | |

Poisson’s ratio | – | 0.3 | 0.3 | 0.15 | |

Density | kg/m^{3} |
2000 | 2700 | 2200 | |

Cohesion | kN/m^{3} |
10 | – | – | |

Internal friction angle | 15 | – | – |

The high-speed train load impacts on the rails in the form of wheel-rail forces and is then transmitted to the granular ballast bed through a fastening system and sleeper. In this study, five kinds of the train speeds with 160, 200, 250, 300 and 350 km/h and four cyclic loads with 16, 18, 20 and 22 t were considered and applied to the sleeper. The time-history curve of the cyclic load in a cycle was obtained using a coupled dynamic model of a high-speed train-ballasted track, as shown in

The macroscopic degradation and deformation of the ballast bed are closely related to the dynamic characteristic of the ballast at the mesoscopic scale. To study the vibration response of the ballast layer under a train load, six clumps with similar sizes and shapes were selected as the monitoring points in the discrete element model of the ballast bed, as shown in

^{2}/s. As the ballast was far from the bottom of the sleeper, the ballast acceleration amplitude decreased gradually. Also, the ballast acceleration amplitudes at monitoring points 4–6 were small because the points were far from the trapezoidal compression zone beneath the sleeper. Furthermore, the smallest amplitude was exhibited by point 6 located at the bottom of the slope.

The vibration excitation generated by train operation will be transmitted to the substructure through the sleeper and ballast layer. The study of the substructure response when the train wheel passes is very important for subgrade design and evaluation.

Besides, Studies [

To further visualise the response of the ballasted railway,

As shown by the displacement distribution of the coupled model in

As shown in

In this paper, a three-dimensional ballasted railway track model is established based on coupled DEM-FEM. To improve computational efficiency, a GPU-based parallel framework was applied to the discrete element simulation of ballast bed. Meanwhile, an algorithm containing contact search and parameter transfer were developed and verified in the GPU parallel environment. Finally, the dynamic characteristics of the ballasted railway were analyzed at the macro–meso scale. The relevant conclusions are as follows:

The established coupling model can reflect the settlement and vibration characteristics of sleeper under train load. With the increasing train speed and load, the vertical acceleration of sleeper gradually increases.

The vibration acceleration of ballast near the sleeper bottom was the largest (i.e., approximately 40 m^{2}/s at 250 km/h train speed and 16 t load) and then decreased gradually with the increasing depth of the ballast layer. Increasing train speed and axle load will lead to an increase in the vertical acceleration of the ballast. When the train speed exceeds 250 km/h, the growth rate of the vertical acceleration increases obviously.

The maximum dynamic stress on substructure changes in the range of 50–70 kPa and increases linearly with the increasing train speed and load. The simulation results of dynamic stress obtained from the coupled DEM-FEM model were consistent with the previous

The stress and displacement concentration area in the substructure model were located on the embankment surface below steel rail. Owing to the discrete nature of the granular ballast bed, the ballast grains established a point-facet contact with the upper surface of the embankment, resulting in an unevenly distributed contact force on the upper surface of the embankment.