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part cannot change value (the altitude, for instance). So the height of the central cell is
the sum of two terms p+q [ 0 ] ; q [ i ] , 1
m-1 is the height of the i -th adjacent cell of
the neighbourhood, where the distinction between mobile and fixed part is not
necessary, taking in account that only the mobile part of the central cell may be
distributed. The flow from the central cell to the i -th neighbouring cell will be denoted
by f [ i ], 0
i
i<m , where f [ 0 ] is the part of p which is not distributed. Let q' [ i ] =q [ i ] +f [ i ] ,
0
m-1 be the sum of the content of a neighbouring cell, plus the flow from the
central cell, and let q' _ min be the minimum value for q' [ i ].
Thus, the determination of outflows, from the central cell to the adjacent cells, is
based on the local minimisation of the differences in “height”, given by the following
expression:
i
m
=
1
( 1 )
(
q
[
i
]
q
_
min
)
i
0
The “minimisation” algorithm, i.e. the algorithm for the minimisation of
differences, and correlated theorems are not treated here, but they can be found in [4].
Furthermore, the minimum “imbalance” conditions cannot be always achieved in a
CA step, so a relaxation rate, depending on both the cell size and the duration of the
CA step, must be considered. The relaxation rate p r , specified by a multiplicative
factor, can assume values between 0 and 1 : 0<p r •1 .
This mechanism involves particular care in the space and time settlement: the size
of the cell limits at the top the CA step, because the outflow rate may not be so rapid
that the outflow overcomes the neighbourhood boundaries in a step.
3 The CA Model SCIDDICA for Debris Flows
SCIDDICA (Simulation through Computational Innovative methods for the Detection
of Debris flow path using Interactive Cellular Automata - to be read “'she:ddre: CA ”,
as the acronym was devised to mean “it slides” in Sicilian), is a CA model developed
in order to simulate the behaviour of landslides that can be typologically defined as
“flows” [8]. These are a good application field for CA , as they can be considered to
evolve in terms of local processes.
The new release S3hex with hexagonal tessellation is here presented.
3.1 The Problem of Modelling Debris Flows
Analytical solutions to the differential equations (e.g. the Navier-Stokes equations)
governing debris flows are a hopeless challenge, except for few simple, not realistic,
cases. The possibility to successfully apply numerical methods for the solution of
differential equations have been elevated considerably because of the continuous rise
of computing power, even if there are still difficulties to obtain high performances
regarding their implementation on parallel computing machines to exploit the
maximum computational power. It must be said that CA are a particular model that
can be easily implemented in parallel computing [2], [9].
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