Mathematical Physics - Volume II - Numerical Methods

Chapter 5. Review of Development of the Smooth Particle Hydrodynamics (SPH) Method

182

SPH, is an interpolation of a set of discrete data, a constant field, can be approximated with a sinusoidal curve/surface if the order of the interpolation is high enough. If one would approximate the derivative of the velocity field shown in Figure 5.5 with a central difference formula: d v d x x = x i = f ( v i + 1 ) − f ( v i − 1 ) x i + 1 − x i − 1 = 0 (5.78) at all sampling points. Hence this mode can not be detected, and can grow un-resisted in other words this mode could grow to a level where it dominates the solution.

Figure 5.6: Velocity field that corresponds to a zero energy mode of deformation.

Zero energy or spurious modes are characterised by a pattern of nodal displacement that is not rigid body motion but still produces zero strain energy. One of the key ideas to reduce spurious oscillations is to compute derivatives away from the particles where kernel functions have zero derivatives. Randles [78] proposed a stress point method. Two sets of points are created for the domain discretisation, one carries velocity, and another carries stress. The velocity gradient and stress are computed on stress points, while stress divergence is sampled at the velocity points using stress point neighbours. According to Swegle et al. [87][88], these spurious modes can be eliminated by replacing the strain measure by a non-local approximation based on gradient approach. Beissel [11] proposed another way to stabilise nodal integration, the least square stabilisation method. 5.11 Non-Local properties of SPH SPH is by nature a nonlocal method, capable of overcoming difficulties related to material softening without any additional regularisation measures. In this chapter, a local damage model resulting in material strain-softening was used in a stable Total-Lagrange SPH code, Vignjevic at al. [101]. The investigation was done by considering a simple uniaxial wave propagation problem in a symmetrically loaded homogeneous bar, in presence of damage induced strain-softening, which is defined by Bažant and Belytschko [8]. They derived an exact solution for given initial and boundary conditions for stress wave propagation