Maria Bayard Dühring - Solid Mechanics

3.1 Topology optimization and propagating waves 13
reflection chambers [47], sound barriers designed by genetic algorithms [48] and
acoustic-structure interaction [49]. These optimization problems usually result in
non-intuitive designs with holes in the structures and parts not attached to each
other, which would not be possible by size and shape optimization.
Parallel to the work about elastic wave propagation, topology optimization methods
concerning propagating optical waves appeared. First planar electromagnetic
waves were studied for photonic-bandgap materials in [50, 51] and then structures
such as bends and splitters [52, 53, 54, 55] based on band-gap materials were designed
for single frequencies or frequency intervals. Planar photonic-crystal waveguide
components with low loss as well as high bandwidth and transmission have
been fabricated and characterized in various papers [56, 57, 58]. The tailoring of
dispersion properties are considered in [59] for photonic-crystal waveguides and for
optical fibers in [60]. In [61] a coupled magnetomechanical analysis is carried out in
order to use topology optimization to maximize the elastic wave transduction of a
magnetostrictive patch-type transducer. Lately, the method of topology optimization
has been extended to transient wave propagation for one-dimensional problems
[62, 63] and the concept can be applied to both elastic and optical waves.
Topology optimization is now a well established method and has been employed
to improve problems in a range of different fields. However, when the method is
applied to a new type of problem it takes some trial and error to obtain applicable
designs and overcome problems as badly defined boundaries of the designs, appearance
of intermediate densities and non-unique solutions. It is first of all important to
formulate the optimization problem with the objective function and constraints such
that the wanted physical effect is obtained. In order to obtain mesh-independent
designs and reduce the appearance of intermediate materials, filtering methods and
appropriate interpolation functions must be employed. For problems with nonunique
solutions, convergence to a local optima can be prevented by continuation
methods. Finally, it is important to interpret the optimized designs physically and
compare them with intuitive designs.
In the present work, the method of topology optimization is extended to three
new types of wave propagation problems. The first is mathematically the simplest
and the aim is to minimize noise from acoustic waves in air by designing structures
as sound barriers. The next is concerned with optical waves and here the
purpose is to maximize the energy flow in a hollow photonic-crystal fiber. Finally,
the method is applied to propagating surface acoustic waves in a piezoelectric material
in order to improve the acousto-optical interaction with an optical wave in
a waveguide. The three different problems are governed by similar equations and
the way topology optimization is applied to them is therefore essentially the same.
In the following sections, a generic optimization problem for the wave propagation
problems is introduced with design variables, objective and constraint functions as
well as the sensitivity analysis. The different initiatives applied to prevent problems
as intermediate materials and mesh-dependent solutions are explained.