WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

ρ=1
κ=1
1
R
Figure 1 - A one-dimensional acoustic wave in air is partially reflected (R) and transmitted (T )
at the interface to an acoustic medium.
and define continuous property interpolation functions ρ(̺) and κ(̺) that fulfill the
following conditions:
1, ̺ = 0
1, ̺ = 0
ρ(̺) =
ρ2
ρ1
T
, ̺ = 1 , κ(̺) =
ρ
κ
κ2,
̺ = 1 κ1
The introduction of continuous properties allows us to compute the gradients of the
objective functional with respect to the indicator function: dΦ/d̺, and we can thus use
a gradient based optimization strategy with ̺ as a topological design variable.
Clearly, we are only interested in binary values (0 or 1) of the indicator function
in our optimized design, and our optimization formulation must include a strategy to
avoid intermediate values of ̺ in the final design. Various strategies exist to ensure a
binary design (see e.g. [5]) and in this work we rely on the choice of the functions ρ(̺)
and κ(̺). To find suitable candidates we study a simple 1D acoustic system (Fig. 1).
A unit magnitude wave propagating in air is partially transmitted (T ) and partially
reflected (R) at the interface to an acoustic medium with the material properties ρ(̺)
and κ(̺). The amplitudes of the reflected and the transmitted wave are:
R =
√ κρ − 1
√ κρ + 1 , T =
(4)
2
√ κρ + 1 , (5)
so that R → 1, T → 0 when κρ → ∞ (a perfectly rigid solid), and R = 0, T = 1
when κρ = 1 (air).
We want to choose our interpolation functions so that the reflection from the
acoustic medium is a smooth and well-behaved function of ̺. One possible choice is
the polynomial form:
ρ(̺) = 1 + ̺ q1
ρ2
( − 1), κ(̺) = 1 + ̺ q2
κ2
( − 1). (6)
ρ1
The reflection versus ̺ is depicted in Fig. 2(left) for different values of q1 and q2
(ρ2/ρ1 = κ2/κ1 = 500). All curves have vanishing slope at ̺ = 1, which turns
out to make it difficult to obtain well defined solid regions in the design.
Instead, inspired by the Helmholtz equation, we use polynomial interpolation in
ρ −1 and κ −1 :
ρ(̺) −1 = 1 + ̺ q1
ρ2
(( ) −1 − 1), κ(̺) −1 = 1 + ̺ q1
κ2
(( ) −1 − 1). (7)
ρ1
3
κ1
κ1