Geology Reference
In-Depth Information
where the subscripts in parentheses indicate the
integration divisions.
or calculated rigorously. Either way, they are
defined such that a higher sensitivity number
represents higher efficiency.
Considering the definition of the objective
function in Equation (1.111), the sensitivities of
this function can be calculated as
9.1. Bidirectional Evolutionary
Structural Optimization (BESO)
The BESO method has been used in eigenfre-
quency optimization (see e.g. Yang et al. 1999b,
Huang et al. 2010) as well as in maximizing
energy absorption (see e.g. Huang et al. 2007,
Ghabraie et al. 2010). This method solves the
topology optimization problem in two steps. At
first, the optimization algorithm finds the most
and least efficient elements to be added and re-
moved respectively. Then, it adjusts the number
of adding and removing elements based on the
volume constraint.
This method is capable of totally removing the
elements, hence one does not require to represent
voids with a very soft material. This approach,
usually known as
hard kill
approach, results in
faster solutions for only solid elements remain in
the finite element model. Moreover, this approach
works well in geometrically non-linear problems
as it is not prone to the instabilities caused by soft
elements in the SIMP material model (Buhl et al.
2000, Yoon and Kim 2005). Zhou and Rozvany
(2001) showed that in certain cases the hard kill
approach may result in non-optimal solutions and
thus this approach need to be applied with care. In
shape optimization, however, using the hard kill
approach will not cause such problems.
Another advantage of the BESO method is
that the solutions will not contain any intermedi-
ate design variables (0 <
x
< 1) or grey areas. In
this method the boxing constraints 0 ≤
x
e
≤1,
e
=
1,…,
N
change to binary constraints of the form
x
e
T
T
∂
∂
Π
p
e
∂
∂
u
−
∂
u
)
∂
∂
p
+
∂
∂
p
1
2
n
(
)
+
(
i
( )
i
(
i
−
1
)
( )
i
e
(
i
−
1
)
=
lim
+
T
−
T
p
p
−
u
u
( )
i
(
i
1
)
( )
i
(
i
−
1
)
x
x
∂
x
x
x
n
→∞
=
1
e
e
e
(1.112)
The first term in the right hand side cancels out
because on the boundaries with essential boundary
conditions ∂
u
/∂
x
= 0 and elsewhere
p
= 0. Hence
the above equation reduces to
)
∂
+
∂
∂
∂
Π
p
p
p
1
2
n
(
i
( )
i
(
i
−
1
)
T
T
=
lim
−
u
u
( )
i
(
i
−
1
)
x
∂
x
∂
x
n
→∞
e
=
1
e
e
(1.113)
On the other hand, differentiating Equation
(1.106) and using Eqs. (1.107) to (1.109), we obtain
∂
∂
p
=
∂
∂
p
ˆ
=
G q
T
,
e
=
1
,
,
N
(1.114)
e
e
x
x
e
e
Substituting Equation (1.114) in Equation
(1.113), we can write
∂
∂
Π
p
1
2
n
(
)
(
)
∑
T
T
T
=
lim
u
−
u
G q
+
q
( )
i
(
i
−
1
)
e
e i
( )
e i
(
−
1
)
x
n
→∞
i
=
1
e
(1.115)
Using the trapezoidal numerical integration
scheme and recalling the definition of the strain
energy in Eqs. (1.110) and (1.111), the above
equation reduces to
{0,1},
e
= 1,…,
N
. This is particularly help-
ful if one wants to impose shape restrictions as
the boundaries of solids and voids can be clearly
defined in the black-white solutions of BESO.
In BESO, the so-called
sensitivity numbers
are
used to evaluate the efficiency of the elements.
Sensitivity numbers might be assigned intuitively
∈
∂
∂
Π
p
n
(
)
=
∑
=
lim
π
−
π
π
(1.116)
e i
( )
e i
(
−
1
)
e
x
n
→∞
i
=
1
e
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