Page 137 - The Combined Finite-Discrete Element Method

120 CONTACT DETECTION

Extension of the screening contact detection algorithm into 4D space or multi-
dimensional space is almost impossible due to RAM requirement.

3.10.4 Direct mapping contact detection algorithm

Extension of the direct mapping contact detection algorithm into 3D space involves sub-
division of the 3D space into identical cube shaped cells (Figure 3.55).
Mapping of discrete elements onto the cells is achieved through integerisation of x, y
and z coordinates of the centre of each discrete element:

int x i − x min
x i = 1 + Int (3.72)
d

int y i − y min
y i = 1 + Int
d

int z i − z min
z i = 1 + Int
d
In 2D the screening array was replaced by three sorting arrays. In 3D four arrays are
used; one array for each coordinate axis and one array for the discrete element number
(identifier). Thus for cells in the x-direction, the X array is used, for cells in the y-direction
the Y array is used, while for cells in the z-direction the Z array is used. Arrays X, Y
and Z are of size N,where N is the total number of discrete elements comprising the
discrete element system. The third array, D, indicates discrete element number, and is
also of size N.
After mapping of discrete elements onto the cells, sorting of arrays X and Y and Z is
performed to give spatial meaning to array D.
Contact detection is performed through binary search of arrays X, Y and Z for a
specific cell.
This algorithm is linear in terms of RAM in both 2D and 3D space. For the same
number of discrete elements, the CPU time for 3D contact detection is on average 30%
larger than the CPU time for 2D contact detection. This is due to the processing of the Z
array. The algorithm remains in 3D hyper-linear space in terms of CPU requirements.
Extensions to 4D and multidimensional spaces are possible, and these extensions pre-
serve the major CPU and RAM properties of 2D and 3D algorithms.

3.11 GENERALISATION OF MUNJIZA-NBS CONTACT DETECTION
ALGORITHM TO MULTIDIMENSIONAL SPACE

The contact mask for the NBS contact detection algorithm in 3D space is shown in
Figure 3.56.
There are 26 neighbouring cells in total. However, because each cell with at least one
discrete element mapped onto it is considered in turn as the central cell, all neighbouring
cells for a given central cell need not be considered. It is enough to consider cells as
shown in Figure 3.56.