Maria Bayard Dühring - Solid Mechanics

22 Chapter 4 Design of sound barriers by topology optimization [P1]
4.1 Method
The aim of the presented method is to design sound barriers by redistribution of
air and solid material (aluminum) in the design domain Ωd such that the noise is
reduced in the output domain Ωo behind the barrier. This is illustrated by figure 4.1.
The sound source, which is placed to the left of Ωd, is emitting harmonic waves with
the vibrational, particle velocity U. The ground is reflecting and the outer boundary
in the air is absorbing. The governing equation for a driving angular frequency ω is
the Helmholtz equation [1]
∂
∂xi
−1 ∂p
ρ + ω
∂xi
2 κ −1 p = 0, (4.1)
where p is the complex sound pressure amplitude, ρ is the mass density and κ is the
bulk modulus, which all depend on the position r. The design variable ξ takes the
value 0 for air and 1 for the solid material. The inverse density and bulk modulus
are interpolated linearly between the two material phases. The objective function
Φ is the average of the squared sound pressure amplitude in the output domain, Ωo.
The formulation of the optimization problem thus takes the form
1
min log(Φ) = log
ξ
Ωo dr
|p(r, ξ(r))|
Ωo
2
dr , objective function (4.2)
1
subject to
Ωd dr
ξ(r)dr − β ≤ 0, volume constraint (4.3)
Ωd
0 ≤ ξ(r) ≤ 1 ∀ r ∈ Ωd, design variable bounds (4.4)
Figure 4.1 The geometry employed for the sound barrier topology optimization problem
with the design domain Ωd, the output domain Ωo and the point source with the vibrational
velocity U.