Page 121 - The Combined Finite-Discrete Element Method

104 CONTACT DETECTION

3.8.2 Mapping of discrete elements onto cells

Mapping from the set of discrete elements

E ={1, 2, 3, 4, 5,..., N} (3.57)
to the set of cells

(1, 1), (1, 2), (1, 3), . . . (1,n y ),
 
 
 (2, 1), (2, 2), (2, 3), . . . (2,n y ), 




C = (3, 1), (3, 2), (3, 3), (3, 4), (1,n y ), (3.58)
... ... ... . .. . ..
 
 
 
 
(n x , 1), (n x , 2), (n x , 3), . . . (n x ,n y )
is introduced and defined in such a way that each discrete element is assigned to one
and only one cell. This is done through each particular discrete element with coordinates
(x, y) being assigned to the cell (i x ,i y ),where

x − x min
i x = Int (3.59)
d

y − y min
i y = Int
d
In essence, i x and i y are integerised relative coordinates of the centre of the bounding
circle for each discrete element, and are thus referred to as integerised coordinates. An
example is given in Figure 3.40, where for instance discrete element 1 is assigned to cell
(4,7), while discrete element 2 is assigned to cell (5,8), and discrete element 3 is assigned
to cell (4,6).

3.8.3 Mapping of discrete elements onto rows and columns of cells

In addition to the mapping of discrete elements onto cells, mapping of discrete elements
onto columns and rows of cells is also introduced.
A discrete element is said to be mapped to a particular row of cells if it is mapped
to any cell from that row. For instance, discrete element 1 is mapped to row 7 of cells,
discrete element 2 is mapped to row 8, and discrete element 3 is mapped to row 6 of
cells (Figure 3.40).
In a similar way, a discrete element is said to be mapped to a particular column of cells
if it is mapped to any cell from that column. For instance, discrete element 1 is mapped
to the column 4 of cells, discrete element 2 is mapped to column 5 and discrete element
3 is mapped to column 4 of cells (Figure 3.40).

3.8.4 Representation of mapping

In the previous sections, the binary tree, screening arrays and sorting arrays have all been
used to represent different types of mapping between discrete elements and cells. In the