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A shift model based on particle collisions – preserving kinetic energy and potential energy in a constant force field – to avoid particle clustering in SPH

Lyu, Xipeng; Kruisbrink, Arno; Jefferson-Loveday, Richard


Xipeng Lyu

Arno Kruisbrink


In Smoothed Particle Hydrodynamics (SPH), the motion of particles is based on symmetric inter-particle forces, such that the conservation of momentum is guaranteed. Inter-particle forces, however, can not prevent particle clustering. Clustering may occur for several reasons. A fundamental issue is the tensile instability, which is caused by the properties of the kernel gradient. Clustering may also be caused by discontinuities in the pressure (e.g. due to surface tension) and the pressure gradient (e.g. due to gravity), which may lead to instabilities around the interface between two fluids (Kruisbrink et al., 2018 [1]). Wall penetration is also a form of particle clustering. To suppress particle clustering, the use of kinematic conditions (motion) rather than dynamic conditions (forces) was previously explored in the concept of particle collisions by Kruisbrink et al. (2018) [1]. In this concept, the conservation of momentum is always satisfied, whilst the conservation of energy is only satisfied for fully elastic collisions. They conclude that inelastic collisions are more stable than elastic collisions, but that energy is dissipated. In the present paper, the particle collision model is further explored and extended to a particle collision shift model, which in itself is non-dissipative, i.e. it preserves kinetic energy as well as potential energy in a constant force field, like gravitation. This is achieved by changing the particle positions due to inter-particle collisions, but not their velocities. Other popular and commonly used particle shift methods, such as the Fickian shift method of Lind et al. (2012) [2], are based on the particle concentration. They are non-conservative and as such dissipative to some extent. In this paper, the new particle collision shift model is introduced and explored in several case studies. This model is compared with another stabilization method, the Fickian Shift model of Lind et al. (2012) [2]. The particle collision shift model performs slightly better in terms of particle stability and reduction of energy dissipation in most of the case studies. However, it is simple (one algorithm only with few coefficients, no special treatment at the free surface) and efficient (no kernel approximation, less CPU time). This work is based on incompressible SPH (ISPH). The shift model may however also be applied to weakly compressible SPH.


Lyu, X., Kruisbrink, A., & Jefferson-Loveday, R. (2022). A shift model based on particle collisions – preserving kinetic energy and potential energy in a constant force field – to avoid particle clustering in SPH. Journal of Computational Physics, 469, Article 111530.

Journal Article Type Article
Acceptance Date Aug 6, 2022
Online Publication Date Aug 30, 2022
Publication Date Nov 15, 2022
Deposit Date Sep 2, 2022
Publicly Available Date Aug 31, 2023
Journal Journal of Computational Physics
Print ISSN 0021-9991
Electronic ISSN 1090-2716
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 469
Article Number 111530
Keywords Computer Science Applications; Physics and Astronomy (miscellaneous); Applied Mathematics; Computational Mathematics; Modeling and Simulation; Numerical Analysis
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