Geology Reference
In-Depth Information
7.2. Sensitivity Analysis
= −
∂
∂
c
x
D
m
−
Γ
v
=
γ
,
e
=
1
,
,
N
e
e
e
e
In order to calculate the sensitivities, we rewrite
the dynamic compliance by adding an (arbitrary)
adjoint vector multiplied by a zero function
(1.74)
with additional conditions similar to (1.21). The
negative sign for
∂
c
m
in Equation (1.74) is
added because it relates to a minimization problem.
∂
(
)
−
T
T
Ω
2
c
m
=
p u
+
u K
−
M u
p
C
C
C
C
(1.69)
7.3. Examples
Differentiating with respect to the design
variables and rearranging the terms, we obtain
The frame in example 3 is considered under
forced vibration. It is assumed that 6 identical
horizontal periodic loads of magnitude
p
1
= 500
kN and frequency of Ω are applied at locations of
the concentrated masses as shown in Figure 10a.
Note that for linear elastic materials, changing the
force magnitude will only change the values of
the objective function but the evolution pattern of
the objective function values and the final topol-
ogy will remain unchanged. This is not true for
nonlinear problems.
Deformed shapes of optimum topologies ob-
tained for different load frequencies are shown in
Figure 10b-e. It can be seen that for high frequen-
cies, the optimum material distribution tends to
form damping masses.
When the input frequency is greater than the
fundamental frequency of the initial structure, the
optimization algorithm reduces the fundamental
frequency. This increases the static compliance
and can lead to disintegrated designs (clearly
observable in Figure 10e). To prevent this disin-
tegration, one can introduce an upper bound
condition on static compliance in the problem
formulation (Olhoff and Du 2005). One can also
include the static compliance in the objective
function as shown in Figure 10.
When the structure is subjected to a number
of loads with different frequencies, one can define
a multi-objective optimization problem to handle
the case. This is of practical importance, for ex-
ample when one approximates a periodic load
using Fourier series. In the following example,
u
C
(1.70)
∂
∂
c
x
∂
∂
u
∂
∂
K
∂
∂
M
(
)
m
e
=
sign( )
p
c
T
+
u K
T
−
Ω
2
M
C
+
u
T
−
Ω
2
C
x
x
x
where sign() is the sign function. Sensitivities of
the dynamic compliance can now be written as
c
x
∂
∂
∂
∂
K
∂
∂
M
u
u
T
Ω
2
m
=
−
(1.71)
C
x
x
e
in which the adjoint vector is selected such that
(
)
= −
Ω
2
K
−
M u
sign( )
c
p
C
(1.72)
Comparing Equation (1.72) with Equation
(1.63), the adjoint vector is found to be
u
C C
which can be substituted
in Equation (1.71) to simplify the latter to
= −
sign(
T
)
p u u
C
∂
∂
c
x
∂
∂
K
∂
∂
M
u
T
Ω
2
m
= −
sign( )
u
c
−
(1.73)
C
C
x
x
e
Having the sensitivities calculated, an ap-
propriate solution method such as the method of
moving asymptotes (MMA) can be employed to
solve the minimization problem. One can also
use the OC-based solution procedure proposed
in section 4.2. The optimality criteria to solve
Problem (1.68) can be expressed as
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