WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

paper extends on this work. For an outline of the general method as well as examples
of various applications see e.g. [5].
We start out by describing the governing equation and discuss the basic material
interpolation scheme on the basis of a simple acoustic model. We solve the model equations
using a FEM procedure, and move on to illustrating the optimization algorithm
for a sample problem. Lastly, we give conclusion and directions for further work.
ACOUSTIC MODEL
We consider a plane model of steady-state acoustic wave propagation, modelled by the
2D Helmholtz equation:
∇ · (ρ −1 ∇p) + ω 2 κ −1 p = 0. (1)
In Eq. (1), p is a complex pressure amplitude, ρ and κ are the position dependent density
and bulk modulus of the acoustic medium. The wave frequency is denoted ω.
The optimization problem is to distribute two materials (air and solid), in a design
domain Ωd, so that some functional Φ(ω,ρ,κ, ∇p,p) is extremized. The two materials
have the following set of material properties: (ρ,κ) = (ρ1,κ1) (air), and (ρ,κ) =
(ρ2,κ2) (solid). It should be emphasized that the present pure-acoustic model neglects
any acoustic-structure interaction. For large values of ρ2κ2 the solid is practically a
perfectly rigid and thus fully reflecting, whereas for smaller values it represents an
acoustic medium.
We rescale the equations by introducing the new variables:
1, air
1, air
˜ρ = ρ
ρ1
=
and inserting them into Eq. (1):
ρ2
ρ1
, solid , ˜κ = κ
κ1
=
κ2,
solid κ1
(2)
∇ · (˜ρ −1 ∇p) + ˜ω 2 ˜κ −1 p = 0, (3)
where ˜ω = ω/c is a scaled wave frequency and c = κ1/ρ1 is the speed of sound in
air. In the following the tildes will be omitted for simplicity.
We may apply different boundary conditions to simulate incoming waves, as well
as reflecting and absorbing boundaries. Additionally, perfectly matching layers can be
added to the model such that low reflection at radiating boundaries is ensured for all
angles of incidence.
DESIGN VARIABLES AND MATERIAL INTERPOLATION
The basis of the topology optimization method is to let the "material" properties ρ and κ
take not only the discrete values of air and solid (cf. Eq. (2)), but also any intermediate
values. To facilitate this, we introduce a continuous material indicator field 0 ≤ ̺ ≤ 1,
2