Mathematical Physics - Volume II - Numerical Methods

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

192

Figure 5.15: Particle discretisation in SPH (MCM) of strain-softening bar.

The smoothing length, which determines the range over which particle velocities and stresses are smoothed, is defined as: h = λ · ∆ p (5.100) where λ is a factor, which relates the interparticle spacing to the smoothing length. For the B -Spline kernel, the smoothing domain has radius 2 h , which is user defined parameter in SPH simulations. The influence of the smoothing length on the results is illustrated with three numerical experiments summarised in Table 5.4. • The first experiment investigates the influence of variable h , where h was varied by changing the interparticle distance ∆ p while keeping parameter λ constant at λ = 1 . 3. This is a typical value used in SPH analyses. In this experiment, the number of neighbours if a given particle, i.e. particles that lie within a spherical domain of radius 2 h , was the same for all models. • The second experiment investigates the influence of h on the size of softening zone in the case of constant discretisation density (interparticle distance). The interparticle distance was fixed as ∆ p = 200 [ mm ] / 201, and λ varied. Parameter λ was given values of λ = 1 . 25 , 2 . 25 and 3.25, corresponding to 5, 9 and 13 neighbour particles in the loading direction, respectively. • The third experiment investigated a fixed smoothing length of h = 2 . 5 [ mm ] for different discretisation densities. In these tests, both the smoothing length parameter, λ and the interparticle distance ∆ p , were varied.

Interparticle distance ∆ p [ mm ] Support domain factor λ ( − ) Physical smoothing length ( h = λ ∆ p ) Particles through thickness ( y & z Directions) Experiment 1: Influence of interparticle distance, ∆ p =variable, λ = 1 . 3 = constant 200/101 260/101 5

101 2525 151 12231

200/151 200/201

260/151 260/201

9

1.3

11 Experiment 2: Influence of averaging over several neighbouring particles, ∆ p =variable, λ = 1 . 3 = variable 1.25 250/201 650/201 Experiment 3: Influence of averaging over several neighbouring particles, ∆ p =variable, λ = 1 . 3 = variable, h = 25 [ mm ] 2.25 2.25 150/67 11

200/101

201 24321

200/101 200/151 200/201

1.2625 1.8875 2.5125

5 9

101 2525 151 12231 201 24321

2.5

11

Table 5.4: Summary of the SPH discretisation parameters used in the three numerical experiments.