Keywords: corrected smooth particle hydrodynamics, high pressure casting, meshless methods, computational modelling, complex fluid flow, SPH, large deformation.

This paper develops the numerical procedure for simulating the behaviour of liquid metals subjected to high pressure die casting. During such a process, liquid metal is injected under very high pressures through complex gate and runner systems and into the dye. The dynamic nature of the fluid flow leads to significant free surface fragmentation and droplet formation. Complex flow patterns of the molten metal in the die cavity is one of the most crucial factors influencing the quality of the casting and is not well understood in many applications of high pressure die casting. These uncertainties can produce very high rejection rates for cast components and consequently produce large waste. Problems encountered with high pressure die casting include trapped air bubbles, fine scale porosity, cold shuts and oxide inclusions in the casting which reduce the mechanical strength and produce variations in the surface properties.

Validating the numerical simulations with experimental data is very difficult because of the speeds, pressures, temperatures, thinness of the die cavity and the bulk of the die. Clearly, numerical modelling offers a better and inexpensive way to analyse these complex metal flows and to optimise the gating system and the geometry of the die. Eulerian based techniques such as the finite element and finite volume methods have been used to model low pressure die casing processes [9,10]. However, for high pressure, high velocity, die casting these methods are unable to accurate simulate the complex free surface behaviour.

The Lagrangian based Corrected Smooth Particle Hydrodynamics (CSPH) [11,12] method is used to model the numerical simulation of the high pressure die casting process. CSPH does not use a grid to compute partial derivatives. The variables determining the flow, such as pressure and velocity, are localised on a set of particles which move with the flow. This property enables the technique to simulate complex free surfaces including fragmentation and droplet formation. The CSPH method is based around the reproducing kernel function in which for any given function can be approximated by the following integral as,

where is the reproducing kernel function and is the smoothing length that determines the range of influence of the kernel. Based on the above technique SPH [1,2,3,4,5,6] interpolation of a function and its gradient are approximated in terms of the values of the function at a number neighbouring particles and a kernel function as,

where denotes a tributary volume associated to particle (typically calculated as the particle mass divided by the density). In the standard SPH method, the gradient vectors are simply . However, in the CSPH method gradient functions are amended to ensure that the gradient of general constant or linear function is correctly evaluated. This requirement leads to two simple conditions for these gradient vectors, namely

A number of authors have described various methodologies that can be adopted to fulfil the above conditions [8,11,12] . This paper focuses on the application of the CSPH method to simulate the high pressure die casting process. A detailed description of the formulation of the governing equations is presented with several numerical simulations to demonstrate the performance of the proposed computational model.

1

L.B. Lucy, "A numerical approach to the testing of the fission hypothesis", Astro. J., 82 1013, (1977). doi:10.1086/112164

2

R.A. Gingold and J.J. Monaghan, "Smooth Particle Hydrodynamics: Theory and Application to Non- Spherical Stars", Mon. Not. R. Astron. Soc., 181, 375, (1977).

3

J.J. Monaghan, "An introduction to SPH", Comput. Phys. Comm., 48, 89-96, (1988). doi:10.1016/0010-4655(88)90026-4

4

J.J. Monaghan, "Smooth Particle Hydrodynamics", Annu. Rev. Astron. Astrophys., 30, 543, (1992). doi:10.1146/annurev.aa.30.090192.002551

5

J.J. Monaghan, "Simulating Free Surface Flows with SPH", J. Comput. Phys., 110, 399, (1994). doi:10.1006/jcph.1994.1034

6

L.D. Libersky, A.G. Petschek, T.C. Carney, J.R. Hipp, F.A. Allahadi, "High Strain Lagrangian Hydrodynamics", J. Comput. Phys., 109, 67, (1993). doi:10.1006/jcph.1993.1199

7

W.K. Liu, S. Jun, Y.F. Zhang, "Reproducing Kernel Particle Methods", Int. J. Num. Meth. Engrg., bf 20 (1995). doi:10.1002/nme.1620381005

8

R.W. Randles, L.D. Libersky, "Smooth Particle Hydrodynamics: Some Recent Improvements and Applications", Comput. Methds Appl. Mech. Engrg., 139, 376, (1996). doi:10.1016/S0045-7825(96)01090-0

9

R.W. Lewis, K. Ravendran, "Finite element simulation of metal casting", Int. J. Num. Meth. Engrg, 47, 29- 59 (2000). doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<29::AID-NME760>3.0.CO;2-X

10

W-S. Hwang, R.A. Stoehr, "Modelling fluid flow", Metals handbook Casting ASM International, 15, 867- 876, (1988)

11

J. Bonet, T.S.L. Lok, "Variational and Momentum Aspects of Smooth Particle Hydrodynamics Formulations", Comput. Methds Appl. Mech. Engrg., 180, 97-115, (1999). doi:10.1016/S0045-7825(99)00051-1

12

J. Bonet, S. Kulasegarm, "Correction and Stabilization of Smooth Particle Hydrodynamics Method with Applications in Metal Forming Simulations", Int. J. Num. Meth. Engng., 47, 1189-1214, (2000). doi:10.1002/(SICI)1097-0207(20000228)47:6<1189::AID-NME830>3.0.CO;2-I

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