Hexagonally spaced SPH particles achieve the highest density when the particles represent a fluid under gravity.
-15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 0 5 10 15 20 25 E rr o r in B u o y a n c y F o rc e L / s 2D Square 2D Circle 3D Cube 3D Sphere
B K Cartwright, 2012. p.40 Setting up a simulation with hexagonally spaced particles has been done manually to date, as many pre-processors, including the one used here, do not automatically generate hexagonally arranged nodes to assign the SPH particles to.
The cases presented thus far that have used initially hexagonal spaced nodes were developed by a manual process that used a spreadsheet to calculate the location of the nodes. The nodal coordinates were exported to a text file that was then imported into the pre-processor to have SPH particles assigned.
For a large number of nodes, say greater than 64,000, this becomes quite tedious, as a single worksheet in a spreadsheet such as Microsoft Excel can only handle 64,000 lines of data. It is desirable to have a purpose-built tool to generate this distribution of nodes for simulations requiring large numbers of SPH particles, however that is beyond the scope of this research. An alternate approach is to use a conventional orthogonal distribution of nodes, as might be produced by many mesh generators or other FEA pre-processors.
When SPH particles on an orthogonal distribution are used, the centre-centre spacing varies between the adjacent nodes, depending on whether the adjacent nodes are on the major axis of the distribution, or on the diagonal. Consequently not all the neighbours of a single SPH particle are equi-spaced. This is shown in Figure 4-9a, where 4 of the 8 neighbours of any central particle will be a distance of ‘s’ away, and the other 4 neighbours on the diagonals will have a distance of √2 times the orthogonal distance of ‘s’.
If the SPH properties allow a sufficiently low viscosity behaviour, then the SPH particles will rearrange themselves as the simulation begins, although sometimes a slight “numerical nudge” is required to overcome the mathematically perfect nature of the initial balance of the forces. In the simulations here, the reaction of the particles due to contact forces at the walls, and the insertion of the box into the water surface, provided sufficient stimulus for the
rearrangement of the particles from their initial perfectly orthogonal distribution to the more stable hexagonal arrangement.
The theoretical change in spacing from an initially orthogonal distribution of particles to their stable hexagonal arrangement can be calculated based on constant area for 2D particles. In Figure 4-9a each SPH particle on the orthogonal distribution has an area of s2. When these elements have rearranged themselves into the hexagonal arrangement of Figure 4-9b, based on consistency of element area, the new nearest neighbour spacing, s’, can be shown to be 1.075s, with the next nearest neighbour being √3 multiplied by s’, or approximately 1.86s. To confirm this behaviour an initially orthogonal distribution of 1000 SPH particles was allowed to settle over time, and when settled, the neighbour distances of 100 centrally located SPH particles were measured. The 100 SPH particles were chosen from a region of the domain that showed a uniform particle arrangement, i.e. an arrangement that was not influenced by the irregular packing of particles necessary to accommodate the straight boundaries of the domain.
B K Cartwright, 2012. p.41
Figure 4-9a. Particle Spacing for Orthogonal Packing.
Figure 4-9b. Particle Spacing Hexagonal Packing.
Figure 4-10 and Figure 4-11 show the initial and settled states of the SPH particles. The 100 SPH particles that were studied have been omitted from the view of the Figure 4-10 to highlight their location in the central region of the domain. Figure 4-11 shows a detailed view of these central SPH particles at the same initial and settled states.
s’ =
s’√3
s
B K Cartwright, 2012. p.42
Figure 4-10. Initial and Orthogonal and settled hexagonal SPH particles. The hole in the middles is the location of the particles that were removed to study their neighbour distances.
(Outer rows of particles were held fixed for containment)
Figure 4-11. Close-up of the 100 SPH particles that were studied for neighbour distance.
The nodal coordinates of each SPH particle were exported to a text file and the distance to neighbours calculated using a spreadsheet. The neighbour distances were normalised by the initial spacing, s, and the frequency of neighbour distances was accumulated into distance ranges to produce Figure 4-12.
In Figure 4-12 the most common neighbour distances for the initial orthogonal condition are the initial spacing of 1.0, the diagonal spacing of √2 (approximately 1.4) and then twice the initial spacing of 2.0. A normalised distance of 2.0 is not the nearest neighbour but the next neighbour in the same direction. (Note there is a slight tolerance on the exact distances measured due to the distance range of each interval.)
B K Cartwright, 2012. p.43
Figure 4-12. Frequency of normalised neighbour distances for orthogonal and hexagonal distributions.
For the settled particles the most common normalised distances are at about 1.09 and about 1.9. (The presence of some distances in the adjacent lower distance bin implies the mean of the settled distances would be slightly lower than the quoted values of 1.09 and 1.9.)
These settled normalised neighbour distances compare favourably with the theoretical values shown in the Figure 4-9b of 1.075s and 1.86s presented earlier.
Following the earlier results, the recommended contact thickness distance, h, is 0.6 times the hexagonal spacing, or 0.55 of the orthogonal spacing.