Maria Bayard Dühring - Solid Mechanics

z [m]
3
2.5
2
3
y [m]
2
1
by Helmholtz equation for a homogeneous medium. The material properties for air in the room are ra and ka
and it is assumed that there is no damping effect in the air. The material on the boundaries is inhomogeneous
and the optimized design is a distribution of the usual reflecting material described by r2 and k2 and an
absorbing material with the properties r1 ¼ 3:04 kg m 3 and k1 ¼ 7:90 10 5 Nm 2 . The absorbing material
has an absorption coefficient equal to 0.1 which could be realized in practice by a cork sheet with the thickness
of a few millimeters. It is then convenient to use the new variables
8
8
~r ¼ r
¼
ra ><
>:
r1 ra r2 ra absorbing;
reflecting;
~k ¼ k
ka
><
¼
>:
With this rescaling the acoustic model for the problem takes the form
n r^p ¼
ARTICLE IN PRESS
M.B. Dühring et al. / Journal of Sound and Vibration 317 (2008) 557–575 567
2
1
0 0x [m]
k1
ka
k2
ka
absorbing;
reflecting:
(19)
r 2 ^p þ ~o 2 ^p ¼ 0 Helmholtz equation, (20)
pffiffiffiffiffiffiffiffiffi
i ~o kara ^p; b:c: for surface with impedance Z (21)
ZðrÞ
n r^p ¼ i ~o ffiffiffiffiffiffiffiffiffi p
karaU; b:c: for pulsating surface. (22)
The inhomogeneities pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
on the walls are described in the boundary condition (21) by the impedance boundary
ZðrÞ ¼raca ~kðrÞ~rðrÞ.
The impedance boundary condition is only strictly valid for plane waves of normal
incidence but is used for simplicity in this work. The material interpolation functions for ~rðxÞ and ~kðxÞ must
now satisfy the requirements
8
r1 >< ra ~rðxÞ ¼ r2 >:
x ¼ 0;
x ¼ 1;
8
k1
>< ;
ka
~kðxÞ ¼ k2
>: ;
x ¼ 0;
x ¼ 1;
(23)
r a
3
4
30 40 50
frequency, f [Hz]
60 70
and again interpolation functions in the inverse material properties are used
~rðxÞ 1 ¼ r1 ra 1
þ x
r2 ra 1
r1 ra !
1
, (24)
objective function, Φ [dB]
130
120
110
100
90
80
70
60
50
40
ka
initial guess
optimized design
frequency borders
Fig. 8. Results of the optimization for the frequency interval [40.5;45.5] Hz with four target frequencies: (a) the optimized design and (b)
the frequency response for the initial design and the optimized design.