Maria Bayard Dühring - Solid Mechanics

Chapter 3
Topology optimization applied to time-harmonic
propagating waves
In the previous chapter, wave propagation problems for time-harmonic elastic and
optical waves have been introduced. It has been explained that they are governed
by second order differential equations and can be solved in a similar way by the
finite element method. As discussed, different kinds of wave propagation is utilized
in an increasing number of applications and it is therefore important to improve the
performance for a certain frequency or frequency range. The aim is either to hinder
wave propagation, for instance minimization of the sound power at a certain position
in a room, or waveguiding, as maximization of the power flow through a point
of a structure. To optimize these kinds of problems different types of optimization
methods can be applied. There are at least two fundamentally different approaches
to solve an optimization problem. The simplest strategy is to take the best solution
from a number of trial solutions, perhaps generated by stochastic methods or simply
from a parameter study where one or more parameters are varied. The other
approach is to use an iterative method where an intermediate solution is obtained in
each iteration step, which finally converges to an optimum. This can be done by a
variety of algorithms suitable for different types of problems, for example for linear
or nonlinear, constrained or unconstrained problems. To determine the direction
and the length of the next step in the iteration, information from the objective and
constraint functions as function values, gradient and curvature information can be
employed.
In this work, wave propagation problems are either investigated by parameter
studies, where geometry parameters are varied, or by topology optimization, which
is an iterative optimization method based on repeated sensitivity analysis of the
objective and constraint functions and mathematical programming steps. In the
following, the method of topology optimization is introduced and a description of
the way it is applied to the wave propagation problems studied in this work is given.
3.1 Topology optimization and propagating waves
By the method of topology optimization, material is freely distributed in a design
domain and the aim is to get a structure that consists of well defined regions of solid
material and void such that the objective function is optimized. Thus, the layout of
the structure with the number, position and shape of the holes is determined. The
method therefore offers higher design freedom than other known methods as size
and shape optimization, which are restricted to a predefined number of holes in the
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