WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

18 Chapter 3 Bandgap structures as optimal designs
In paper [8] the optimization problem is extended to the case of transverse vibrations
and waves in moderately thick plates. The paper deals with optimizing the
distribution of two elastic materials in order to suppress forced vibrations or guide
elastic waves in designated paths in the plate.
Paper [9] studies a closely related problem in a quite different physical setting.
Solid material is distributed in an acoustic chamber in order to minimize the transmission
of acoustic waves throughthe chamber. The problem isconsidered for waves
in a finite frequency range.
Main references
The design of phononic bandgap structures using a material distribution optimization
method is believed to be a novel contribution of this thesis. Topology optimization
methods had previously been used to design structures subjected to forced
vibrations in the lower frequency range by Ma et al. (1993) for 2D elastic problems
and by Soto and Diaz (1993) for plate structures. Tcherniak (2002) considered optimization
of resonating structures and Jog (2002) optimized plates for minimum
dynamic compliance by considering the dissipated energy. Optimization problems
related to maximization and minimization of eigenfrequencies had also been considered
using topology optimization, e.g. by Diaz and Kikuchi (1992) for 2D elastic
structures and by Pedersen (2000) for plates. Ma et al. (1994) and Osher and Santosa
(2001) considered the problem of separating eigenfrequencies. However, here
the separation of low order eigenfrequencies was examined with no special consideration
to bandgap structures.
Topology optimization of plate structures for maximum bandgaps was first explored
by the author, Ole Sigmund and Søren Halkjær (Halkjær et al., 2006). Diaz
et al. (2005) had previously studied maximization of bandgaps in similar beam grillage
structures.
The problem of using topology optimization to design acoustic structures was
proposed in two conference papers by the author, Ole Sigmund and co-workers
(Sigmund and Jensen, 2003; Sigmund et al., 2004).
3.1 Vibration-quenching structures
The primary optimization problem is:
What is the optimal distribution of two elastic materials in a structure
so that the forced vibration response is minimized?
Fig.3.1ashowsthemodelusedtocreatetheoptimizedstructures. Afreeplanartwodimensional
structure (plane-strain condition) is subjected to forced vibrations by a
harmonicload(withfrequency Ω) actingalongafreeedge. Thevibrationresponse is
minimized alongtheoppositeedge. Fig.3.1billustratestheoptimizeddistributionof