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SPH survival kit
Although SPH is an excellent method in many cases, it is fairly hard to find a set of
parameters that give stable and realistic simulations, and it can also be difficult to find
parameters that rescale a simulation (e.g. from 0.1 l of water to 10 l of water).
Here, we propose some improvements that will give more stable simulations.
However, note that it is indeed possible to get stable simulations with the previously
given parameters, but the timestep must be held perhaps too small for the simulation
to be interactive.
Parameters
Although one would like the SPH parameters to have direct correspondence to the
real world, this is in general not the case. The main problem when simulating water is
that the speed of sound is usually set to a value much smaller than the real value (e.g.
1 m/s instead of 1500 m/s). This results in a compressible system, and in turn this
results in water with a too high density.
Therefore, for a given sound velocity and timestep, try to lower the mass per particle
so that the computed average density becomes closer to 1000 kg/m3, i.e. so that the
pressure, p = c
2
(
ρ
–
ρ
0
) fluctuates around zero.
This will also affect the resulting volume, and thus the density has a non-linear
dependence on the particle mass, but by testing a couple of times you should at least
find a much better value.
Just to complicate this a bit ☺
If the density is lowered and particles are more sparse, one might need to use a larger
smoothing length. This will also affect the computed density, since, in particular, the
self contribution from the center particle depends a lot on the smoothing length (short
smoothing length = large self contribution, and there is also a lower bound, i.e. the
self contribution to the density should not be larger than ρ
0
).
Different ways of computing field differentials
When computing forces/acceleration in SPH, Newton’s third law tells us that in order
to conserve linear momentum (“rörelsemängd”), pairwise forces must be equal in size
with opposite sign. Therefore the differentials in Navier Stokes equations that give
rise to these forces, must also be symmetrized. This can be done in several ways,
leading to different types of averages, and in the end it can affect the simulation, in
particular in extreme situations, e.g. near a boundary, or at very short distances.
Referring to the SPH lecture notes, the gradient in the SPH pressure force can thus
take at least the two following different forms (in fact, there is a very recent article
looking into this![1]):
tiger-tiger
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