WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Chapter 5
Advanced optimization procedures
As mentioned in the previous chapters the optimization studies on phononic and
photonic bandgap structures have inspired the development of new optimization
procedures. Thischapterwillhighlightthreesuchprocedureswhichcanbeidentified
as independent contributions. As a common featurethey have beenused tooptimize
bandgap structures but are also applicable to a broader class of problems that deal
with the optimization of dynamic systems either based on steady-state or transient
computations.
Thesis papers [18]–[21]
Paper [18] introduces an efficient scheme for optimization of dynamical systems in
finitefrequency rangesbasedontheuseofPadéapproximants. Adetailedderivation
of the computation of Padé approximants is provided along with derivation of the
design sensitivities. The method is demonstrated on topology optimization of two
dynamical structures subjected to forcedvibrations with the aimto createfrequency
ranges with a low vibrational response.
In paper [19] the one-dimensional mass-spring structure with attached nonlinear
oscillators is reconsidered (the same system was analyzed in paper [3]). Here, the
four independent oscillator parameters, natural frequency, mass, damping ratio, and
nonlinear stiffness, are individually optimized in order to reduce the transmission
of waves through the chain. The optimization procedure is based on a transient
optimization formulation for a nonlinear system.
In papers [20] and [21] the transient topology optimization method is extended
to allow for optimized structures where the material distributions vary in time. In
these papers, the necessary methodology is developed and expressions for the design
sensitivities are derived. Paper [21] also demonstrates the importance of choosing
the proper time-integration scheme. The papers present examples in which dynamic
bandgap structures and pulse shaping structures are designed.
Main references
Optimization of the frequency response using Padé approximants has not received
much attention. Webb (2002) marks an exception. Here an electromagnetic component,
parameterized by a few design variables, was optimized.
Transient topology optimization has previously been used for the design of dynamic
systems, e.g. by Min et al. (1999) for applications in structural mechanics,
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