Although these difficulties also apply to 2d

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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, manual-intervention 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 Navier-Stokes 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 Set-up 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 set-up for SPH simulation using an in-house pre-processor operating on the FEM mesh produced by the mesh gener...
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