Geology Reference
In-Depth Information
T
ϕ
M =
ϕ
δ ,
n r
,
=
1 
,
,
N
(1.6)
ing (or minimizing) any combination of natural
frequencies. A practically useful example of such
extensions will be briefly addressed in section 6.
It is worth noting that maximizing fundamental
frequency results in structures with a reasonable
stiffness against static loads in general (Bendsøe
and Sigmund 2003).
Damping effects are ignored and linear elastic
material behavior is assumed in this section. Also
all material parameters are taken as deterministic
quantities. Random variability of material strength
parameters can significantly affect the ductility
and energy absorption capacities of structures
subjected to seismic loading (Kuwamura and Kato
1989, Elnashai and Chryssanthopoulos 1991).
Uncertainties of variables can be considered
in structural optimization by integrating Reli-
ability Analysis (see e.g. Kharmanda et al. 2004
and Papadrakakis et al. 2005) or through Robust
Optimization (see e.g. Beyer and Sendhoff 2007).
Using the finite element discretization and the
SIMP approach we introduce the following mate-
rial interpolation scheme to express the Young's
modulus E e of element e in terms of its relative
density x e as
n
r
nr
d
where δ nr is the Kroneker's delta which equals 1
for n = r and 0 otherwise. Premultiplying Equa-
tion (1.5) by ϕ T and using Equation (1.6) we get
ϕ ϕ
n
T
K =
ω δ
2
,
n r
,
=
1
,
,
N
(1.7)
r
r nr
d
which means the modes are also K -orthogonal.
Using Equation (1.3) in Equation (1.1) and pre-
multiplying by ϕ T we obtain
N
N
N
d
d
d
T

T
T
T
ϕ
M
ϕ
q
+
ϕ ϕ
C
q
+
ϕ ϕ
K
q
=
ϕ
p
n
r
r
n
r
r
n
r
r
n
r
=
1
r
=
1
r
=
1
(1.8)
We now make use of M -orthonormality of the
modes and the classical damping Equation (1.2)
to simplify Equation (1.8) to

2
T
q
+
2
ζ ω
q
+
ω
q
=
ϕ p
(1.9)
n
n
n n
n n
n
a
a
1
where ζ
=
M
+
K
ω
is the damping ratio
n
n
2
ω
2
n
of the n -th mode (Chopra 1995).
According to Equation (1.9), the response of
a structure under a dynamic load depends on its
natural frequencies ω n and damping ratios ζ n .
p
E x
) =
x E
(1.10)
(
e
e
e
where E is the Young's modulus of the base
isotropic material. The power p > 1 is known as
the penalty factor and is introduced to push the
solutions towards a solid-void topology. A typical
value for the penalty factor is p = 3 (Bendsøe and
Sigmund 1999). The relative densities are chang-
ing in the range 0 ≤ x e ≤ 1 in which x e = 1 represents
solids and x e = 0 represents void areas. In order
to avoid singularities in the stiffness matrix of the
system, Equation (1.10) may be replaced by
4. MAXIMIZING EIGENFREQUENCIES
IN FREE VIBRATION
As seen in the previous section, controlling the
response of structures can involve eigenfrequency
optimization. In this section we address the prob-
lem of maximizing the fundamental frequency of
a structure in free vibration. This problem was
initially addressed by Díaz and Kikuchi (1992)
using the homogenization method. Here, we for-
mulate the problem using the SIMP approach. This
formulation can be simply extended to maximiz-
E x
E x E E
p
(
)
= +
(
)
(1.11)
e
e
e
in which E is a small elastic modulus assigned
to voids. Based on Equation (1.10), the (local
 
Search WWH ::




Custom Search