## Abstract

In this study, a non-local partial Peridynamic mesh-free incompressible method is proposed for simulating dry dense granular flows such as granular column collapse. In the method, a predictor-and-corrector time splitting scheme is used, and Peridynamic theory is incorporated only in the predictor to solve an integrated momentum equation, in which part of the pressure gradient force, viscous force and external force are included. An incompressible explicit solver is employed to obtain the pressure field, and the corrector allows for the remaining part of the pressure gradient force to be calculated and updates the flow field. The proposed method is then validated by simulating granular flows in several configurations, including a steady granular flow down an inclined slope, granular column collapses at both one side and two sides, and collision of two adjacent granular columns. The simulated velocity profiles are in good agreement with the analytical solution in the steady granular down an inclined slope in which sensitivity of the particle distance, a coefficient by incorporating Peridynamics, and Peridynamic horizon are examined. The simulation of the granular column collapses shows that the method can reproduce the final deposit, free surface, and velocity in the flows. The method can capture interface variations between two granular columns during their collision in good agreement with experimental observations.

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## Data availability

The data generated in the present study are available from the corresponding authors on reasonable request.

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## Appendices

### Appendix

### Flow chart of the numerical algorithm

To clearly illustrate the numerical method for simulating granular flows, a flow chart of the algorithm is shown in Fig.

16. The numerical algorithm is based on the predictor-and-corrector time splitting scheme. The predictor solves shear stress term, external gravity force term, and part of the pressure gradient term by using a coefficient *ε*. With the continuity equation, the pressure Poisson equation is obtained, which is solved by an explicit equation. After obtaining the pressure field, the remaining pressure gradient is solved in the correction and the particles are updated to new locations. To avoid the clustering of particles, a collision model is implemented. Then, the calculation proceeds to the next loop.

### Collision distance sensitivity investigation

The collision distance in the collision model is examined by varying the *α* value to simulate the granular column collapse. A schematic diagram for the numerical setup is shown in Fig. 5b in Sect. 4.2. Collision occurs when the distance of two particles is less than a threshold and the *α* value in the collision model is varied as *α* = 0.95, 0.9, and 0.8. In the investigation, *ε* = 0.75 and the particle distance *d* = 0.0025 m. The free surface profiles in the granular flow by using different *α* values is shown in Fig.

17. By using larger values of *α* in the collision model, the top part of the flow is higher, which can be observed for *α* = 0.9 and 0.8 at *t* = 0.22 s and 0.42 s. Meanwhile, the wave front part is shorter by using the two *α* values. By using *α* = 0.95, the free surface is closer to the experimental measurements.

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Xu, T., Li, S.S. Development of a non-local partial Peridynamic explicit mesh-free incompressible method and its validation for simulating dry dense granular flows.
*Acta Geotech.* (2022). https://doi.org/10.1007/s11440-022-01766-4

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DOI: https://doi.org/10.1007/s11440-022-01766-4

### Keywords

- Explicit scheme
- Granular flows
- Mesh-free method
- Peridynamics
- Pressure Poisson equation