Geology Reference
In-Depth Information
where
π
e
is the total strain energy of element
e
upon completion of the load cycle. Huang and Xie
(2008) derived these sensitivities using the adjoint
method and verified it using a simple example.
We may now define the sensitivity numbers
for problem (1.105) as follows
numbers and the algorithm tries to move the
volume towards the volume limit gradually. Thus,
if the current volume is bigger than
v
the algorithm
will increase the number of removing elements
and vice versa. The procedure continues until no
further significant improvement can be achieved.
The algorithms to update the solutions in the
BESO method have been improved over time.
One of the most recent algorithms is proposed by
Huang and Xie (2007). In this algorithm, at each
iteration
k
, the target volume of the next iteration
is calculated using a small positive controlling
parameter called the
evolutionary volume ratio
(
R
v
)
=
∂
∂
Π
p
α
=
π
,
e
=
1 2
(1.117)
,
,
,
N
e
e
x
e
Based on this definition and using first order
approximation, we can write
N
(
)
∑
α
1
(
k
+
1
)
( )
k
( )
k
∆Π
=
∆
x
(1.118)
v
=
v
1
+
sign(
v
−
v R
)
v
p
e
e
e
=
(1.120)
Note that in Equation (1.118) adding element
e
will be reflected by ∆
x
e
= 1 - 0 = 1 and remov-
ing it results in ∆
x
e
= -1. Thus during the solution
procedure, if one introduces the
a
th element and
removes the
r
th element, the change in the objec-
tive function can be estimated as
where superscripts enclosed in parentheses in-
dicate the iteration number. Then the number of
adding and removing elements are calculated such
that the volume of the next design becomes equal
to
v
(
k
+1)
and the total number of added elements
do not exceed
v
add
= v
t
×
R
a
in which
R
a
is another
controlling parameter known as the
maximum
allowable admission ratio
.
If one starts BESO with an initial design vol-
ume equal to
v
, the volume will be kept constant
during the optimization procedure and the number
of adding and removing elements at each iteration
would be equal to each other. In this case
R
v
will
have no effect on the optimization procedure and
the maximum number of adding and removing
elements is only controlled by
R
a
. Thus with a
fixed volume, the effect of the
R
a
factor is similar
to the move limit
η
in Equation (1.28).
=
∂
∂
Π
−
∂
∂
Π
p
p
∆Π
= −
α
α
(1.119)
p
a
r
x
x
a
r
As we are interested in maximizing Π
p
it is
clear from Equation (1.119) that the elements with
highest sensitivity numbers should be added to
the design domain while the elements with lowest
sensitivity numbers should be removed.
9.2. Adding and Removing
the Elements
9.3. Mirroring and Filtering
Sensitivity Numbers
After ranking the efficiency of the elements, the
algorithm should select the number of elements
to be added and removed such that the volume
constraint is satisfied. Generally in BESO, one
starts the solution with an initial volume which
is not necessarily equal to the volume limit
v
.
The design is then updated using the sensitivity
Due to the nonlinear nature of Problem (1.105),
the loading sequence affects the mechanical re-
sponses. As a result, the optimal shape flips by
mirroring the loading sequence. In real life, how-
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