Geology Reference
In-Depth Information
As all the constraints are time-dependent the
consideration of all the constraints requires an
enormous amount of computational effort. Here,
the conventional method (Arora, 1999) is em-
ployed to deal with time-dependent constraints.
In this method the time interval is divided into
n
gp
subintervals and the time-dependent constraints
are imposed at each time grid point. Let the
j
th
time-dependent constraint be written as:
space and time are discrete, and physical quanti-
ties are taken from a finite set of discrete values.
In its basic form, a cellular automaton consists of
a regular uniform grid of sites or cells with a
discrete variable in each cell which can take on a
finite number of states. The state of the cellular
automaton is then completely specified by the
values
s
i
=
s
i
(
t
) of the variables at each cell
i
. Dur-
ing time, cellular automata evolve in discrete time
steps according to a parallel state transition de-
termined by a set of local rules: the variables
s
g X Z t
(
,
( ),
Z t
( ),
Z t
( ),
t
)
≤
0 0
,
≤ ≤
t
t
j
i
k
+
1
( )
at each site i at time tk+1 are up-
dated synchronously based on the values of the
variables
s
k
c
in their nc neighborhood at the pre-
ceding time instant
t
k
. The neighborhood
n
c
of a
cell
i
is typically taken to be the cell itself and a
set of adjacent cells within a given radius
r
. Thus,
the dynamics of a cellular automaton can be for-
mally represented as follows (Biondini
et. al
.,
2004):
=
s t
(5)
i
i
k
+
where
t
i
is time interval over which the constraints
need to be imposed.
Because the total time interval is divided into
n
gp
subintervals, the constraint (5) is replaced by
the constraints at the
n
gp
+
1 time grid points as:
g X Z t
(
,
(
),
Z t
(
),
Z t
(
),
t
)
≤
0
;
α
=
0
,
,
n
j
α
α
α
α
gp
(6)
s
k
+
=
1
θ
(
s
k
,
s
k
),
i
r
n
i
r
− ≤ ≤ +
(8)
i
i
n
c
c
The above constraint function can be evalu-
ated at each time grid point after the structure has
been analyzed. The objective function of con-
strained structural optimization problems is de-
fined in Box 2.where
∆
and
r
p
are the feasible
search space, and an adjusting penalty factor re-
spectively.
where the function
θ
is the evolutionary rule of
the automaton.
A proper choice of the neighborhood plays
a crucial role in determining the effectiveness
of such a rule. In this chapter, the widely used
Moore neighborhood (Von Neumann, 1966) of
interaction, by
r
=1, is adopted.
The CA technique can be combined with the
evolutionary algorithms to solve optimization
problems. One of the most popular evolution-
ary algorithms is GA. In the field of structural
Cellular Genetic Algorithm
Cellular automata (CA) represents simple math-
ematical idealizations of physical systems in which
Box 2.
f X
(
)
( )
=
f X
n
gp
if
X
∈
∆
m
(
)
2
∑
∑
(7)
f X
(
)
+
r
[
max(
g X Z t
(
,
(
),
Z t
(
),
Z t
(
),
t
), ) ]
0
P
j
α
α
α
α
α
=
0
j
=
1
otherwise
Search WWH ::
Custom Search