Geology Reference
In-Depth Information
7.1. Objective Function and
Problem Formulation
as the objective function to be minimized. Min et
al. (1999) minimized the dynamic compliance for
structures under impulsive loads. The topology
design of structures under periodic loads has been
studied more extensively by Jog (2002) where he
proposed a new positive-definite definition of
dynamic compliance as the average input power
over a cycle. Jog (2002) also studied the problem
of minimizing the vibration amplitude at certain
control points. Topology design of structures
subjected to design-dependent dynamic loads
(e.g. hydrodynamic pressure loading) has been
addressed by Olhoff and Du (2005). In more recent
publications in this area, alternative approaches
in using topology optimization to control the
structural responses in a frequency interval has
been studied by Jensen (2007) and Yoon (2010b).
Consider a harmonic external force of form
Under static loads, the compliance defined as
T
=
p u
c
(1.65)
is proportional to the strain energy of the struc-
ture and is a typical objective function used in
topology optimization of structures. Minimizing
the compliance maximizes the stiffness of the
structure. Under dynamic loads, the value of
c
defined in Equation (1.65) varies with time. For
structures subjected to periodic loads, we consider
the average of
c
over a cycle, i.e.
Ω
2
π
/
Ω
∫
c
=
p u
d
T
t
(1.66)
2
π
0
p p
=
cos
Ω
t
(1.61)
C
as the objective function to be minimized. Here
T
= 2π/Ω is the time period. Note that this measure
is not always positive, and thus, in problem for-
mulation, one should consider the absolute value
(or square) of average compliance as the objective
function. Otherwise, for
c
<0
, the optimization
algorithm will push the structure towards resonance.
In absence of damping,
u
=
u
C
cosΩ
t
. Using this
and
p
=
p
C
cosΩ
t
in Equation (1.66), we can write
where
p
C
does not depend on time. The equation
of motion for a discretized undamped system in
forced vibration takes the form
Mu Ku
+
=
p
cos
Ω
t
(1.62)
C
To solve this problem, we consider
u
=
u
C
cosΩ
t
using which in Equation (1.62) gives
T
Ω
1
2
p u
2
/
π
Ω
∫
c
C
C
cos
2
t dt
T
=
Ω
=
p u
(
)
C
C
K
Ω
2
M u
p
2
π
−
=
C
(1.63)
0
C
(1.67)
or
The average compliance minimization problem
can now be formulated as follows
(
)
=
Ω
2
K
−
M u
p
(1.64)
T
min
,
c
=
2
c
=
p u
m
C
C
x x
,
,
x
1
2
N
Comparing with equilibrium equation in
static state,
K
(
)
2
such that
K
−
Ω
M u
=
p
C
C
−Ω
2
can be termed as “dy-
namic stiffness”. Note that unlike the static stiff-
ness, the dynamic stiffness matrix is not neces-
sarily positive definite.
M
N
∑
x v
≤
v
e e
e
=
1
0
<
x
≤ ≤
x
1
,
e
=
1 2
,
,
,
N
min
e
(1.68)
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