WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Z. Kristallogr. 220 (2005) 895–905 895
# by Oldenbourg Wissenschaftsverlag, München
Inverse design of phononic crystals by topology optimization
Søren Halkjær, Ole Sigmund and Jakob S. Jensen*
Technical University of Denmark, Department of Mechanical Engineering 2800 Kgs. Lyngby, Denmark
Received July 13, 2004; accepted December 9, 2004
Phononic crystal / Band gaps / Inverse design /
Topology optimization
Abstract. Band gaps, i.e frequency ranges for which
waves cannot propagate, can be found in most elastic
structures if the material or structure has a specific periodic
modulation of material properties. In this paper, we
maximize phononic band gaps for infinite periodic beams
modelled by Timoshenko beam theory, for infinite periodic,
thick and moderately thick plates, and for finite thick
plates. Parallels are drawn between the different optimized
crystals and structures and several new designs obtained
using the topology optimization method.
Introduction
Phononic or sonic crystals (SCs) [1] offer a great variety of
interesting and important design problems, e.g. how do we
design crystals for maximum size of band gaps, i.e. the frequency
ranges for which waves cannot propagate through
the crystals, or how many crystals are needed in order to
obtain the desired properties of a phononic crystal structure.
Such design problems can usually be formulated as inverse
problems: find the crystal or structure that satisfies some
specified requirements. Determination of SC characteristics
in form of e.g. band structures involves complex computations
which typically makes the inverse design problem
non-trivial. Most examples of solving this problem are
based on size optimization using a few design variables and
typically using genetic algorithms or other heuristic approaches.
In this paper we solve three inverse design problems
by using the topology optimization method which is
based on a gradient-based algorithm with a large number of
design variables and practically unlimited design freedom.
Topology optimization has in the last decade evolved
as a popular design tool in structural and material mechanics
[2]. The first design problem to be solved was to
find the optimal distribution of a restricted amount of material
in a given domain that gives the stiffest possible
structure [3]. Another design problem was to identify the
material that has the lowest (negative!) Poisson’s ratio [4].
The method is based on repeated material re-distributions
* Correspondence author (e-mail: jsj@mek.dtu.dk)
using a large number of continuous design variables, typically
one for each element in the corresponding discretized
model, and the use of computationally in-expensive
analytical sensitivity analysis and advanced mathematical
programming tools. Lately the method has been applied to
other problems in alternative physics settings as well, ranging
from MEMS and fluid dynamics to electromagnetics
(see [2] for an overview).
In the closely related research area of photonic crystals
(PhCs) inverse design methods have received an increased
focus [5] due to the potential large application possibilities
for using PhCs in optical circuits. Cox and Dobson [6]
used a material distribution method to design two-dimensional
square photonic crystals with maximum band gaps.
The references [7, 8] applied the topology optimization
method to design photonic crystal waveguides. More traditional
studies of photonic crystal structures using a few design
variables and optimization based on either numerical
sensitivity analysis, genetic algorithms, simulated annealing
or a combination of these are seen in Refs. [9, 10].
This study extends the results published in a recent paper
[11] which presented some results for maximizing the
band gap size of square lattice two-dimensional SCs and
optimization of finite structures subjected to periodic loading.
Here, we present results for maximizing the band gap
size for bending and longitudinal waves in rods modelled
by Timoshenko beam theory 1 , we optimize 2D crystals
with quadratic and hexagonal base cells for inplane and
bending waves, and finally present examples of optimized
finite-size structures subjected to in-plane polarized waves.
In previous work [13] we have performed experiments on
simple longitudinal waves propagating in a 1D SC composed
of PMMA and Aluminum. Since we plan to extend
these experiments to plate and shell structures, we will
throughout this paper make use of these two materials.
The results of this paper will be used in the planning of
the experiments.
Inverse design by topology optimization
The topology optimization method is a gradient-based optimization
algorithm that is used to find optimal material
1 A modelling study has previously been performed based on
Bernoulli-Euler beam theory [12].