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Unformatted text preview: though these difficulties also apply to 2D
simulations, they are much more acute in 3D. For
example, many defects in the CAD model can be
extremely difficult to detect in 3D and its repair can be an
extremely laborious, manualintervention process. This
can be particularly difficult when trying to automatically
assign boundary normals for SPH particles at surface
intersections.
The aim of this paper is to report on the progress of our
effort in developing a 3D SPH computer program for
industrial and engineering applications, and to present
results of a couple of 3D die filling simulations.
THE SPH METHOD
SPH is a Lagrangian method that uses an interpolation
kernel of compact support to represent any field quantity
in terms of its values at a set of disordered points (the
particles). The fluid is discretised, and the properties of
each of these elements are associated with its centre,
which is then interpreted as a particle. A particle b has
mass mb , position rb , density ρb and velocity vb . In
SPH, the interpolated value of any field A at position r is
approximated by:
A(r ) = ∑m b b Ab
W (r  rb , h)
ρb (1) where W is an interpolating kernel, h is the interpolation
length and the value of A at rb is denoted by Ab . The
sum is over all particles b within a radius 2h of rb . W(r,h)
is a spline based interpolation kernel of radius 2h. It is a
C 2 function that approximates the shape of a Gaussian
function and has compact support. This allows smoothed
approximations to the physical properties of the fluid to be
calculated from the particle information. The smoothing
formalism also provides a way to find gradients of fluid
properties. The gradient of the function A is then given
by:
A
mb b ∇W (r  rb , h)
(2)
∇A(r ) =
ρb ∑
b 437 In this way, the SPH representation of the hydrodynamic
governing equations can be built from the NavierStokes
equations. These equations of motion are given in Cleary
and Ha (1998). The simulation progresses by explicitly
integrating this system of ordinary differential equations.
These SPH equations at a particle position are made up of
terms similar to those of equations (1) and (2). The forms
of these equations are the same regardless of the
dimension of the governing equations. There is no
reference to a computational grid. The particle position is
the only geometric term in the equations. The
computation of the sums in the equations requires only the
identification of the particles' neighbours. Initial Setup
In a SPH calculation, one needs to initially specify particle
masses, positions, velocities and other necessary
quantities. All of these except the positions and masses
are usually straight forward to specify according to the
PDE initial conditions. Our strategy is to take the
geometric description of a casting component from
industry as input, to feed it through a commercial mesh
generator and to then produce the initial setup for SPH
simulation using an inhouse preprocessor operating on
the FEM mesh produced by the mesh gener...
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 Fall '13
 Cheng,M.

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