smoothing length in SPH
Nishant Ojal
James M. Kennedy
Dear Nishant,
The smoothing length (h) and the mesh characteristic length (Δx) are the two spatial parameters to determine spatial resolutions (only one spatial discretization parameter is used in finite elements).
In LS-DYNA, the initial smoothing length is constant for each part and is calculated as maximum value of all the minimum distance for each particle. For example, if we have three particle, A, B, C, then first the distance to the nearest particle is computed for all three particles. Let’s assume the shortest distances for the particles A, B, and C are AD, BD, and CD respectively. The default smoothing length DSML is then taken as CLSH*maximum of (AD, BD, and CD) where CLSH is a scale factor to the initially computed smoothing length DSML can vary between HMIN * DSML and HMAX * DSML as the simulation progresses. The computed smoothing length at time zero is based on per part basis such that each part will have one smoothing length for all the particles that belongs to that part. After determining the initial smoothing length, it is recomputed every cycle for each particle based on the divergence of the velocity and can vary with respect to time and space.
Because of the lack of a numerical grid, care must be employed in setting the initial particle masses and coordinates. The particle mesh needs to be enough regular. This means that all the particles of a given neighborhood need to have the same mass. As a consequence, the particles of a same material, which have the same initial density, need to have the same initial volume. To preserve this, they need to be distributed on a uniform mesh.
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The smoothing length is forced to be constant and large by setting HMIN=HMAX=3.0 which means that you are scaling the default smoothing length by taking on 300% of the initially computed value and also making it to be constant for the remainder of the simulation.
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Please see if this presentation is of some help:
Lee, S., and Hong, J.-W., “Parametric Studies on Smoothed Particle Hydrodynamic Simulations for Accurate Estimation of Open Surface Flow Force”, International Journal of Naval Architecture and Ocean Engineering, Vol. 12. Pp. 86-101.
https://www.sciencedirect.com/science/article/pii/S2092678219300652
The smoothing length parameter λ within the range from 1.05 to 1.3 has been widely used for the analysis (Hallquist, 2007). The value cannot be smaller than 1, but the calculation time increases if the value is larger than 1.3. For the fast calculation, it is not recommended to use larger value of the smoothing length parameter λ (Hallquist, 2006).
In general, the larger the interaction range, the more particles are used to calculate the governing equation causing the improvement of the accuracy. However, if the smoothing length is greater than a certain value, more particles are contained in the interaction range resulting in poor results. Since the computation time tends to increase as the smoothing length λ increases, the value between 1.05 and 1.2 could be used for effective calculation of hydrostatic problem.
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Sincerely,
James M. Kennedy
KBS2 Inc.
May 6, 2021
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