Mathematical Physics - Volume II - Numerical Methods

5.1 Introduction

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the locations other than the SPH particles (non-collocational SPH). The results achieved in 1D were encouraging but a rigorous stability analysis was not performed. A 2D version of this approach was investigated by Vignjevic and Campbell [97], based on the normalised version of SPH. This investigation showed that extension to 2D was possible, although general boundary condition treatment and simulation of large deformations would require further research. Monaghan [64] showed how the instability can be removed by using an artificial stress which, in the case of fluids, is an artificial pressure. When used in simulation of solids this artificial, in other words non-physical stress, may result in an unrealistic material strength and therefore has to be used with caution. In spite of these developments, the crucial issue of convergence in a rigorous mathe matical sense and the links with conservation have not been well understood. Encouraging preliminary steps in this direction have already been made by Moussa and Vila [67], who proved convergence of their meshless scheme for non-linear scalar conservation laws; see also Moussa [66]. This theoretical result appears to be the first of its kind in the context of meshless methods. Furthermore, Moussa and Vila, proposed an interesting new way to stabilise normalised SPH and allow for treatment of boundary conditions by using approximate Riemann solvers and up-winding, an approach usually associated with finite volume shock-capturing schemes of the Godunov type. This work developed a strong following which include: Parshikov et al. [68] also uses the Riemann solver to calculate the numerical flux between pair of interacting particles; Cha and Whitworth [20], who have applied the Riemann solver of van Leer [93][94] to isothermal hydrodynamics; and Inutsuka [36] who proposed an interesting but fairly complex approach to obtain second-order accurate SPH in 1 D . The improvements in accuracy and stability achieved by kernel re-normalisation or correction do not come for free; now it is necessary to treat the essential boundary conditions in a rigorous way. The approximations in SPH do not have the property of strict interpolants, so that in general they are not equal to the particle value of the dependent variable, i.e. u h ( x j ) = ∑ I phi I ( x j ) u I̸ = u J . Consequently it does not suffice to impose zero values for at the boundary positions to enforce homogeneous boundary conditions. Another issue with this approach is that in conventional SPH the boundary is diffuse. In the case of normalised SPH particles do lie on the domain boundary which is in this case precisely defined. The treatment of boundary conditions and contact could be and was neglected in the conventional SPH method. If the imposition of the free surface boundary condition (stress free condition) is simply ignored, then conventional SPH behaves in an approximately correct manner, giving zero pressure for fluids and zero surface stresses for solids, because of the deficiency of particles at the boundary. This is the reason why conventional SPH gives physically reasonable results at free surfaces. Contact between bodies, in conven tional SPH, is treated by smoothing over all particles neighbouring the contact interface, regardless of material types in contact (for instance contact between a solid body and a fluid). Although simple this approach can give physically incorrect results, such as tensile forces between the bodies in contact. Campbell et al. [19] made an early attempt to introduce a more systematic treatment of boundary condition by re-considering the original kernel integral estimates and taking