WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Γ in
H
design domain
ε
l
symmetry
L
Figure 3 - Half model of a channel with two symmetrically placed reflection chambers.
where pR and pI are the real and the imaginary parts of the complex pressure. The
sensitivities are then found as:
dΦ
d̺i
= ∂Φ
∂̺i
+ 2ℜ(λ T ( ∂S
∂̺i
h
Γout
p − ∂f
)). (11)
∂̺i
The vectors ∂Φ ∂Φ ∂Φ
∂f
, , , and the matrix (∂Sp−
) are obtained through the FEMLAB
∂̺ ∂pR ∂pI ∂̺ ∂̺
matrix assembly procedure (see e.g. [6]).
OPTIMIZATION OF A REFLECTION CHAMBER
We consider the model problem illustrated in Fig. 3. The goal is to distribute solid
material in a chamber in such a way that a propagating wave is reflected. The acoustic
waves propagate in air through the main channel of height 2H and length L. Two
reflection chambers with height h and length l are positioned symmetrically on the
main channel. Each chamber consists of an air-filled part and the design domain where
a favorable distribution of air and solid is to be found.
We apply the following boundary conditions:
n · (ρ −1 ∇p) + iω ρ −1 κ −1 p = 2iω ρ −1 κ −1 p0, Γin
n · (ρ −1 ∇p) + iω ρ −1 κ −1 p = 0, Γout
n · (ρ −1 ∇p) = 0, other boundaries.
This specifies an incoming plane wave with amplitude p0 at Γin (p0 set to unity in the
following), absorbing boundaries at Γin and Γout, traction free conditions on the outer
boundaries and a symmetry condition at the lower boundary.
As optimization objective we consider the squared amplitude of the acoustic pres-
sure averaged over the output boundary:
Φ(ω) = 1
H
Γin
(12)
|p(ω)| 2 dS, (13)
and minimize the maximum value of Φ(ωi) for a set of frequencies ωi, i = 1,..., M.
We now state the optimization problem as follows:
min max Φ(ωi) i = 1,..., M
subject to : S(̺)p = f
0 ≤ ̺j ≤ 1 j = 1,..., Nd
1
Nd
Nd
j=1 ̺j ≤ V
5
(14)