Page 145 - The Combined Finite-Discrete Element Method

128 CONTACT DETECTION

12. loop over all discrete elements from the list neighbouring
to the central z-list according to contact mask
{ remove corresponding y-lists
}
}
}
13. Loop over discrete elements (k=0;k≤N;)
{ remove corresponding z-list
}
RAM requirements of the NBS algorithm in 3D are given by

M = 3N + n z + 2n x + 2n y (3.80)
which represents a negligible increase in comparison to the RAM requirements of the
NBS contact detection algorithm in 2D. In a similar way, the CPU requirements are
greater in 2D than in 3D for the same number of discrete elements. Both RAM and CPU
requirements increase linearly with the increase in the number of discrete elements.
In theory, similar extensions of NBS to 4-dimensional and multi-dimensional spaces
are relatively easy to implement. By generalising the NBS contact detection algorithm
to multi-dimensional space, the contact mask changes and the arrays to store the singly
connected lists change accordingly. At the top is always a set of singly connected lists
containing all discrete elements. All of these lists are represented by two arrays H[n cel ]
and D n . These lists are obtained by discretising the space along the nth dimension. A
loop over each such hyper-layer is performed, a non-empty hyper-layer is detected and
discrete elements are put into singly connected lists representing hyper-rows of cells. The
process is continued until the last level representing individual hyper-cells is reached,
when contact interaction is processed using the contact mask. Both the number of nested
loops and number of arrays needed to represent all the lists increase with an increase
in the dimension of space. Thus, the Munjiza-NBS algorithm in multi-dimensional space
requires more RAM space and more CPU time to process the same number of discrete
elements. However, the most important property of RAM and CPU linearity remains
regardless of the dimension of the space.

3.12 SHAPE AND SIZE GENERALISATION–WILLIAMS C-GRID
ALGORITHM

There are two approaches to shape generation. The first is to employ a discretised contact
solution similar to the discretised contact solution employed for contact interaction. The
basic idea is very simple; each discrete element is discretised into finite elements or grid
squares, and contact detection is performed on the finite element level or grid square level.
The contact detection algorithm therefore detects the finite elements or grid squares that
are likely to be in contact, regardless of what discrete element they may belong to. The
shape and size of finite elements is governed by the accuracy of deformability analysis,
which makes finite elements less elongated and more uniform in size than the discrete
elements. In this way, complexities associated both with the shape and size variation are