Geology Reference
In-Depth Information
A population of particles is initialized with
random positions and velocities. Every time in-
stances
k
of the PSO, the velocities of the particles
are changed (accelerated) towards the
b
i
( )
and
the
g
i
( )
and the particles are moved to new
positions according to the following formulae:
The initial values of the elements
v
i
,
( 0
of the
velocity vector
v
i
( 0
are calculated from the fol-
lowing formula:
v
r C
,
=
3
ε
(5)
i j
d
0
where
r
3
is the random number taken from the
range (0, 1) and
ε
0
is a low-value number which
assure that initial velocities are not too large (here
ε
0
v
(
k
+ =
1
)
w k
( )
v
( )
k
i
i
c
(
)
1
R
( )
k
b
( )
k
p
( )
k
+
−
∆
t
1
i
i
=
.
). The initial values of elements of the
vector
p
i
( 0
are determined from the following
relationship:
0 05
c
(
)
+
∆
R
2
( )
k
g
( )
k
−
p
( ) ,
k
2
i
i
t
p
(
k
+ =
1
)
p
( )
k
+
v
(
k
+
1
∆
)
t
(3)
rC
i
i
i
c
,
( 0
i
d
=
(6)
d i
m
∑
r
where
∆
t
is the time step (here
∆
t
=
1 sec
),
p
i
( )
is the position of the
i
i-th particle at the
k
-th
iteration,
v
i
( )
is the corresponding velocity vec-
tor,
R
1
(
k
,
R
2
(
k
are the diagonal matrices of
independent random numbers, uniformly distrib-
uted in the range (0, 1);
w
( )
is the inertia factor
providing balance between exploration and ex-
ploitation,
c
1
is the individuality constant, and
c
2
is the sociality constant. To speed up convergence,
the inertia weight was linearly reduced from
w
max
to
w
min
, i.e.:
j
j
=
1
where
r
i
is the random number taken from the
range (0, 1). The above choices assure that all of
the assumed initial approximations of damper
parameters, i.e., vectors
p
i
( 0
and
v
i
( 0
fulfil the
optimization constraints (2).
The way of handling the constraints introduced
in the optimization problem is an important part
of the PSO algorithm. The following very simple
procedure is used here to fulfil the constraints (2):
•
If non-admissible values
c
d
,
(
+
1 0
result from the relationship (3), then
c
k
(
w
−
w
)
w k
(
+ =
1
)
w
−
max
min
k
(4)
max
k
max
(
k
+ =
1
c
is artificially introduced,
d
,
min
where
k
max
denotes the maximal number of itera-
tions.
A new velocity, which moves the particle in
the direction of a potentially better solution, is
calculated based on its previous value, and the
particle location at which the best fitness so far
has been achieved.
•
In order to fulfil the constraint (2.1), ele-
ments of the vector
p
i
(
+
1
are normal-
)
ized in such a way that
c
d i
,
c
=
C
(7)
d i
,
m
d
∑
1
c
d j
,
j
=
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