Environmental Engineering Reference
In-Depth Information
Fig. 8.3 Applying torus boundary condition in a rectangular grid by connecting opposite edges
where their neighbourhoods can be different from those situated in the inside of the
grid. There are alternative ways in which boundary conditions can be specified.
To this end the following solutions are frequently taken:
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Setting a different neighbourhood at the boundaries, taking into consideration
that cells at the boundary have a different number of neighbours compared to the
other cells and therefore require an according adaptation of the rule-set.
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The grid can be framed by a number of outer cells that maintain a particular state
without being updated.
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In case of a rectangular grid, boundary cells can take the cells of the opposite
boundary as their neighbours (i.e. the Eastern edge of a grid connects to the
Western, the Northern connects to the Southern edge). Topologically this yields
a torus (a doughnut like shape), as shown in Fig.
8.3
.
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Grid extension: in the case of a homogeneous background state the grid could be
dynamically extended. This solution, however, is possible only to the limits of
processing capacity.
8.3 An Easy Example: Conway's Game of Life
Conway's Game of Life (Gardner 1970) is an excellent example to familiarize
with the concept of CAs and with the process of updating the grid cells. Since
the rules are rather simple, it is even possible to solve smaller grid iterations on
paper. In more complex models, this process can of course be done only by a
computer.
The Game of Life CA uses a two-dimensional rectangular grid. The cells can
have two states, either “alive” (black) or “dead” (white). The state they take in a
succeeding iteration (the rule set) depends on their own state and the states of their
eight adjacent neighbours (Moore neighbourhood):
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A white cell becomes black (alive) if exactly three cells in its neighbourhood are
black.
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A black cell remains black if two or three neighbours are black.
