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Naive Neighbor
Collapse GCB Hash Homogeneous
19
26
30
36
37
Ta b l e 6 . 3 . Actual framerates produced by the original two-pass algorithm (Naive), the
original algorithm with neighbor lists (Neighbor), the collapsed SPH algorithm (Collapse),
the collapsed algorithm with a grid cell based hash (GCB Hash) and an implementation
with all previous improvements and the optimizations for homogeneous fluids (Homoge-
neous). The simulation consists of 8000 fluid particles at rest in a glass, with k r
= 750
g r = 100 and μ r
= =1 . 5 . All tests are performed on a single core of an Intel Xeon
W3520.
6.8 Conclusion
In this chapter, the elements for constructing an efficient and stable SPH simu-
lation have been discussed, applicable to both sequential and parallel algorithms.
This includes optimizing the SPH algorithm and equations, designing an efficient
spatial hash for nearest-neighbor searching, and obtaining a stable simulation.
Further work can focus on the implementation and optimization of the spatial
hash data structure—or other spatial data structures—for specific parallel plat-
forms. For an implementation on multiple processors, one can take a look at
Chapter 7, “Parallelizing Particle-Based Simulation on Multiple Processors” by
Takahiro Harada. Because current real-time SPH simulations still allow the fluid
to compress, another interesting topic of research is incompressible SPH for a
more convincing simulation of water, as described in [Koshizuka and Oka 96]
and [Edmond and Shao 02]. Lastly, efficient real-time visualization of fluids re-
mains an open problem despite the plethora of methods available and is also a
necessity for presenting a believable simulation to the end user.
6.9 Appendix: Scaling the Pressure Force
Scaling the fluid simulation—the fluid-particle positions x and smoothing kernel
size h —with a factor s does not necessarily have an influence on fluid behav-
ior. This section shows that for our choice of smoothing kernels, the fluid-particle
acceleration a scales linearly with the fluid simulation. Consequently, the fluid be-
havior resulting from pressure forces does not change relative to the scale of the
simulation. The only requirement is that the pressure constant k scale according
to s 2 . The proof follows from Equation (6.9) below, which defines the accelera-
tion of the scaled fluid particles i s in terms of the acceleration of the original fluid
particles i . The definition of a i
is taken from Equation (6.8):
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