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task depends heavily on the purpose of the simulation and the target platform of
the implementation.
This chapter does not focus on the visualization of the metaballs. We will
solely focus on the SPH simulation—the algorithms forming the basis of SPH
and their implementation. Also, we will not focus on an implementation specific
to a particular target platform, as the current emergence of parallel architectures
in their multitudinous embodiments advances at a pace that swiftly depreciates
any such dedicated implementation effort. Instead, in this chapter we discuss
an efficient representation of the SPH equations, followed by an optimization
suitable for any target platform. As do many real-time applications, our version of
the SPH equations trades in accuracy for performance. Therefore, we also discuss
differences in stability and behavior compared to the standard implementation
from [Muller et al. 03]. Additionally, we discuss an optimization to the data
structure used for evaluation of the SPH equations, along with its performance
compared to the original method.
Section 6.2 gives an overview of the basic SPH equations, followed by a de-
scription of the basic algorithm used to evaluate these equations in Section 6.3.
Section 6.4.1 discusses the choice of data structure used to accelerate the eval-
uation of the SPH equations, followed by an optimization of the chosen data
structure in Section 6.4.2. Section 6.5 introduces an optimization for the SPH
algorithm. We evaluate the stability and behavior of the optimized SPH algorithm
in Section 6.6 and test the performance of the original and optimized algorithm
and data structure in Section 6.7.
6.2 The SPH Equations
In this section, we will briefly describe the SPH model, to establish the definitions
and highlight the choices specific to our implementation. We define a set of fluid
, with positions x i , velocities v i , accelerations a i ,and
masses m i constant over time. Also, every particle has a density ρ i , which is the
fluid density at x i . SPH consists of a set of equations that ultimately define a force
f i applied to every particle i . The positions and velocities of the fluid particles at
any time t can be determined by integration of f i using an integration scheme,
such as the semi-implicit Euler method [Vesely 01].
In general, every particle attribute can be seen as a quantity A defined at the
particle positions. The SPH model, as mentioned in Monaghan's original pa-
per [Monaghan 88], states a general equation that can be applied to interpolate
any such quantity A at an arbitrary location x from the values of A at all particle
i :1
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