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The discrete element method (DEM) and smoothed particle hydrodynamics (SPH) can be adopted to simulate the granular materials and fluid media respectively. The DEM-SPH coupling algorithm can be developed for the dynamic interaction between the two media. When the particle material is simulated by polyhedral element, a fluid-solid coupling interface would lead to the complex geometry between the granular particle and the fluid. The boundary particle method is traditionally used for the fluid-solid interface but with low computational efficiency. In this paper, the dilated polyhedral element is constructed based on Minkowski sum theory, while the contact force between the elements is calculated by Hertzian contact model. Accordingly the dilated polyhedra based DEM is established. The weakly compressible SPH is adopted to simulate the fluid medium, while the interaction on the geometrically complex fluid-solid interface is evaluated with the repulsive force model which can be determined by the contact detection between SPH particles and solid particles in geometry. This method avoids the storage and calculation of a large number of boundary particles, which can be potentially applied for the complex fluid-solid boundary. In order to improve the computational efficiency, a GPU-based parallel algorithm is employed to achieve high performance computation of SPH. The acceleration of the parallel algorithm is evaluated by the cases of dam break. The numerical simulation of the impact of dam break on cubes is implemented. The simulation results are verified with the corresponding experimental and simulation results. Therefore, the rationality and accuracy of the DEM-SPH coupling method for numerical simulation of the interaction between granular materials and fluid media are illustrated. This method is then adopted for the impact of falling rocks on underwater pipeline. The force of water and rocks on the pipeline is analyzed. This method can be further applied for real engineering problems.

The coupling of granular materials and fluid media is widely existed in geological hazards, geotechnical engineering, ocean engineering, and process engineering [

The dilated polyhedral element is generated by the Minkowski sum of a sphere and an arbitrary polyhedron, which can be used effectively for the numerical simulation of irregular granular materials [

Traditionally the mesh-grid CFD methods have the advantages of solid theoretical basis and mature application in engineering. A lot of open source and commercial software can be used for the DEM-CFD coupling simulations [

The smoothed particle hydrodynamics (SPH) based on Lagrangian particle method was first proposed by Lucy, Gingold and Managhan in 1977 [

In the fluid-solid coupling simulation of granular materials and fluid media, SPH and DEM have highly consistent data structure and operation logic in the implementation of computer program, which can be employed to simulate the fluid-solid coupling problem effectively and suitable for parallel computational environment [

In this paper, the dilated polyhedra based discrete element method according to the Minkowski sum theory is established. The weakly compressible SPH is adopted for the fluid simulation. The interaction between DEM and SPH is calculated by the repulsive force model. The DEM-SPH coupling algorithm is implemented in the GPU-based parallel environment. This method is validated with results of several corresponding examples to prove the reliability and accuracy.

For any two geometric bodies

Dilated polyhedral element based on the Minkowski sum theory.

Because the geometry of dilated polyhedron is complex, it is complicated to calculate its mechanical parameters precisely, such as mass, centroid, moment of inertia, etc. Therefore, the minimum envelope polyhedron approximation of the dilated polyhedron is used instead of the dilated polyhedron itself to calculate the mechanical parameters.

In the contact force model between dilated polyhedral elements, the normal contact force considers both elastic force and viscous force. It can be written as follows:

The tangential force considers the Mindline model and the Mohr-Coulomb friction law. It can be expressed as follows:

According to the surface geometric characteristics of dilated polyhedron, the contact between dilated polyhedrons is divided into six categories: sphere-sphere, sphere-face, sphere-cylinder, cylinder-cylinder, cylinder-face, and face-face contact, as shown in Figure

Contact types of dilated polyhedral elements.

Sphere-sphere

Sphere-face

Sphere-cylinder

Cylinder-cylinder

Cylinder-face

Face-face

SPH approximates the field function by the smoothing function in a certain range of space domain. The related physical quantities are represented by the particle approximation. Thus, the partial differential equations can be transformed into time-dependent ordinary differential equations. The variation of the field variables of each particle with time can be obtained by the time integration. For a continuous function in any field _{0} is the initial density of the fluid;

Ignoring the viscous stress of fluid, the momentum conservation equation can be written as follows:

The fluid-solid boundary is regarded as the boundary condition of SPH in this paper. Meanwhile, the SPH particles are subject to the repulsive force from the boundary, while the repulsive force will react on the DEM elements at the same time. Thus the coupling between the two media can be achieved. The traditional SPH repulsive force boundary model is composed of particles, which is not conducive to the complex shaped boundary [

The repulsive force model at the boundary between solid and fluid.

Traditional particle boundary

Geometric boundary

In order to balance the pressure in different depths of fluid, the pressure correction term

In fact, the interaction force between dilated polyhedral particle and SPH particle only considers the normal direction ignoring the tangential direction. It is more like free-slip condition. The no-slip condition can be conducted by considering the tangential force between solid and fluid particles in the future.

Generally, the polyhedral surface is treated as SPH boundary in DEM-SPH coupling. The force between SPH particles and DEM elements satisfies Newton's third law. The detection between SPH particles and DEM elements is solved in geometry to determine the minimum distance between them which is utilized to calculate the repulsive force. By this simplified boundary treatment approach, the fluid-solid coupling of DEM-SPH algorithm for dilated polyhedral elements and fluid can be established.

The dam break flow is simulated by sequential and GPU-based parallel programs, respectively, and the acceleration ratio of GPU parallel computing is analyzed. In order to verify the reliability of DEM-SPH coupling method, the dam break flow impacting against single cube and three cubes is simulated, while the displacement of cubes is validated with the corresponding tests and simulations.

Figure

Sketch of dam break.

Figure

SPH simulation of dam break.

Position of flow front versus time and the comparison with other approaches [

To study the computational efficiency of GPU, the program is compiled in CPU serial and GPU-based parallel frameworks to simulate the dam break process of fluid with different number of SPH particles. The physical model is the same with that in Figure

Acceleration ratio between GPU parallel and CPU serial algorithm.

In order to validate the SPH-DEM coupling method, the fluid-solid interaction process of dam break flow impacting against cubes is simulated. Referring to the work of Canelas et al. (2016) [

Simulation parameters of dam break flow in DEM-SPH method.

Parameter | Value |
---|---|

Water tank: length × width × height | 8 m × 0.7 m× 1 m |

Cube size | 0.15 m × 0.15 m × 0.15 m |

Dilated radius of cube | 0.01 m |

Density of cube | 800 kg/m^{3} |

Distance between cube to water | 1.7 m |

Friction coefficient of water tank | 0.35 |

Friction coefficient of cube | 0.45 |

Poisson’s ratio of cube | 0.3 |

Water size | 3.5 m×0.7 m×0.4 m |

Smoothing length | 0.02 m |

Number of SPH particles | 119 000 |

Setup of DEM-SPH numerical simulations of cubes in break flow.

Single cube

Three cubes

Figure

Dam break flow impacts on single cube with the proposed DEM-SPH method.

The displacement in

Figure

Dam break flow impacts on three cubes simulated with the DEM-SPH method.

The photo of dam break flow on three cubes with the proposed DEM-SPH method and its comparison with the DCDEM-SPH and experimental results: DEM-SPH simulation, experimental results, and DCDEM-SPH simulation from left to right [

In the simulation of dam break flow on three cubes, the displacements of bottom particles in

Displacements of the three cubes in

Displacement of bottom cube in

Displacement of top cube in

Displacement of top cube in

The rocks are usually used to fix the pipeline for oil transportation under sea [

Simulation parameters of impact of falling rocks on underwater pipeline.

Parameter | Value |
---|---|

Water tank: length × width × height | 11.2 m × 6 m × 2 m |

Diameter of pipeline | 0.508 m |

Cube length | 0.15 m |

Density of cube | 3000 kg/m^{3} |

Elastic modulus of cube | 4.76 GPa |

Poisson’s ratio of cube | 0.3 |

Number of cubes | 600 |

Density of sea water | 1035 kg/m^{3} |

Height of sea water | 1.1 m |

Smoothing length | 0.05 m |

Number of SPH particles | 560 000 |

The simulation is shown in Figure

Impact of falling rocks on underwater pipeline simulated by the proposed DEM-SPH method.

The impact force on the pipeline from the rocks and water.

In this paper, the discrete element method (DEM) of complex particles is realized by developing the dilated polyhedral element and its contact model. The hydrodynamic simulation is implemented by the weakly compressible SHP method. A simplified boundary repulsive force model is employed to regard the coupling between dilated polyhedral element and SPH particles as a boundary problem of SPH. Hence, the DEM-SPH coupling method for the coupling simulation of granular materials and fluid is established. This method saves a lot of computing resources and improves the computational efficiency because it does not need to construct solid boundary particles.

The proposed DEM-SPH coupling method is used to simulate the dam break flow and the impact process on cube. The results are in good agreement with the corresponding numerical simulation and experimental results. Meanwhile, the method is applied to simulate the impact of falling rocks on underwater pipeline, while the forces from water and rocks are analyzed. Generally, the proposed method is validated in computational accuracy and can be applied for analyzing real engineering problems.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This study is financially supported by the National Key Research and Development Program of China (Grants numbers 2018YFA0605902, 2016YFC1401505, and 2016YFC1402705) and the National Natural Science Foundation of China (Grants numbers 11572067 and 11772085).