WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Topological material layout in plates for vibration suppression and wave propagation control
where ˜ Qe x and ˜ Qe y are the element matrices Qex and
Qe y evaluated at the center of the elements. For a fine
mesh the error introduced with this simplification is
negligible.
A finite subdomain of the plate s is composed of
Ns elements with Nx and Ny elements in the two
directions respectively. The energy transported in the
two directions averaged over s, denoted ¯W, has the
components:
¯Wx = 1
Wxd
x
= 1
Nx
s
Ns
W
e=1
e x
¯Wy = 1
Wyd
y s
= 1
Ny
Ns
W
e=1
e y
2bπ
= ℜ id
Nx
T Qxd ˜ ∗
2aπ
= ℜ id
Ny
T Qyd ˜ ∗
(39)
(40)
the global matrices ˜ Qx and ˜ Qy are assembled from the
element matrices ˜ Qe x and ˜ Qe y in the usual way.
5 Optimization problem
We consider two different optimization problems. The
first is to minimize the global response of a plate
subjected to time-harmonic loading. Different formulations
for the dynamic compliance for steady-state
response have been suggested, e.g. Ma et al. (1995),
Jog (2002). We choose a formulation similar to Du and
Olhoff (2007) that ensures that the global response of
the plate is minimized. In discretized notation it may be
written as
Φ1 = d T Md ∗
(41)
in which M is the mass matrix. The objective function
is thus proportional to the kinetic energy of the plate.
Alternatively, we could have chosen to consider the
potential energy or a weighted average between kinetic
and potential energy, but with time-harmonic loading
the difference is minimal.
The second optimization problem considered is the
maximization of energy transport through a vertical
or horizontal line in the structure, Ɣ, which has the
normal vector n. The objective function is then written
in continuous and discretized form as
Φ2 = n · WdƔ ∼ Re id T Q˜ jd ∗
(42)
Ɣ
where subscript j indicates either the x-direction or ydirection
depending on the orientation of the line Ɣ and
the factor appearing in (39)–(40) has been omitted for
simplicity. In the case of lines not aligned with either
the x- ory-direction the formulas (39)–(40) canbe
modified in a straightforward way.
For the gradient-based optimization procedure we
need sensitivities of the objective function with respect
to the design variables. In the following ϱe will denote
the element-wise continuous design variable that may
take values between 0 and 1. The adjoint method is
used (Bendsøe and Sigmund 2003; Sigmund and Jensen
2003) and the sensitivities become
dΦ1 T ∂M
= d d
dϱe ∂ϱe
∗
T ∂S
+ 2ℜ λ d
(43)
∂ϱe
in which the Lagrange multipliers λ are calculated from
the equation
S T λ =−M T d ∗
(44)
Since S is symmetric and already factorized during the
solution of the equation of motion λ can be computed
without much computational effort.
For the second objective function the sensitivities are
found as:
dΦ2
=−ℜ
dϱe
id T ∂ ˜ Q j
d∗
∂ϱe and
S T λ =− i
Q
2
T j − Q
j d ∗
T ∂S
+ 2ℜ λ d
∂ϱe (45)
(46)
In this paper we consider plates built from two different
materials so that the design variable ϱe determines
the fraction of one material in element e. The
stiffness and mass density of the material is interpolated
between the two materials with stiffness E 0 and E min
and mass density ρ 0 and ρ min , respectively, based on
the value of ϱe. WeusetheRAMPscheme(Stolpeand
Svanberg 2001) for the interpolation of stiffness
E(ϱe) = E min +
ϱe 0 min
E − E
1 + q(1 − ϱe)
and a linear interpolation for the mass density
ρ(ϱe) = ρ min 0 min
+ ϱe ρ − ρ
(47)
(48)
This formulation is used in order to penalize intermediate
densities and at the same time it allows us to use two
different materials. If q = 0 then (47) reduces to linear
interpolation.