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