Wave Manipulation by Topology Optimization - Solid Mechanics

4.5 Topology optimized acoustic cloaks [P3] 29
(b)
(a)
Objective, Φ
(c) (d)
(a)
(c)
(b)
(e)
Iterations
Figure 4.8 A three-step MMOS based optimization approach for acoustic cloak design
with cylindrical aluminum inclusions in air. (a): The design is initialized using 306 aluminum
cylinders. (b): Optimized design where the cylinders can change in size and
position. (c): Cylinders with a radius smaller than 0.25 cm are discarded. (d): Optimized
design with a lower bound on the radii. (e) Radii of all cylinders are rounded off to 0.5
cm, 1.0 cm or 1.5 cm. (f): Final design after reoptimizing the positions of the cylinders.
neous and anisotropic mass density, which is not common in naturally occurring
fluids, thus making realizations very challenging. Equivalent to the electromagnetic
carpet cloak approach the acoustic problem can also be relaxed to that of hiding
objects on a reflecting surface[98] and a recent realization for airborne sound has
been reported[99]. A fully enclosing acoustic cloak for underwater ultrasound has
also been realized [100] based on an acoustic transmission line approach. As such
most of the reported work are concentrated on realizing the anisotropic material
parameters with engineered acoustic metamaterials[101, 102, 103]. However, our
goal in this problem is with a limited reformulation to use the initial methodology
of designing optical cloaks to design an acoustic cloak with isotropic material properties
to circumvent the problems of the anisotropic mass density.
For the acoustic case air and solid material (aluminum) are redistributed in the
cloak domain in order to minimize the norm of the scattered pressure field (p s ↔ E s z
in equation (6.2)) in the surrounding domain, Ωout. In the electromagnetic cloak
only one of the material properties (ɛr) is varied, whereas in the acoustic case both
density, ρ, and bulk modulus, κ are redistributed in the design process. Here the
inverse material properties are interpolated linearly.
(d)
(f)
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(f)