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Note that this does not include the ghost particles of p 1 from p 1 . From Equa-
tions (7.1) and (7.2), the particles that have to be sent to p 1 are
SP t t
0 1
EP t t
0 1 + GP t t
0 1
x t t
g, x i
To send the data, SP t t
0 1 has to be selected from all the particles in the mem-
ory of a processor. Flagging particles in the region and using prefix sums, which
is often used in algorithms on the GPU [Harris et al. 07] to compact them to a
dense memory, adds some computation cost which may seem negligible, but not
for high-frequency applications like our problem. Most of the processors have
to select two sets of particles on each sides for two neighbors if more than two
processors are used. This means we have to run these kernels twice.
Instead, the grid constructed for efficient neighbor search is reused to select
the particles in our implementation. The data can be directly used to select the
particles so that we can avoid increasing the cost. The particles that have to be sent
to C 0 are particles in voxels with x v >m
g . However, the grid constructed in
this simulation step cannot be used directly because particles have changed their
positions in the time step. To avoid the full build of the grid, we used a simulation
condition to restrict the particles we want to find. We used the distinct element
method (DEM) to calculate force on a particle by placing springs and dampers.
DEM is an explicit method but is not unconditionally stable. It has to restrict the
size of the time step according to the velocity to maintain stability. Thus, we need
v Δ t/l 0 <c ,where v, Δ t, l 0 , and c are particle velocity, time-step size, particle
diameter, and Caulant number, respectively, which have to be less than one. This
condition guarantees that the motion of any particle is below its diameter. Since
we set the side length of a voxel equal to the particle diameter, particles do not
move more than l 0 , which is the side length of a voxel. From these conditions,
we find SP t t
in simulation time t are the particles (let the x -coordinate of this
be x )in
S 0 1 =
l 0 <x
and especially when d = r e ,
S 0 1 =
2 d<x
A buffer has to be prepared to store these selected particles. When a uniform
grid is used and g = r e , two voxel widths in the direction of the space split have
to be sent. The buffer size can be calculated from the configuration of the grid.
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