WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

a thick solid was considered. A few years earlier the
use of a material distribution method was suggested by
Cox and Dobson (1999) for optimizing repetitive unit
cells for optical wave reflecting structures (photonic
crystals).
The study in Sigmund and Jensen (2003) was extended
in two papers (Halkjær et al. 2005, 2006) to
deal with propagation of bending waves in moderately
thick plates. The relatively low frequency of bending
waves makes the plate structures attractive candidates
for applications in the audible frequency range. The papers
considered the optimization of repetitive unit cells
and in Halkjær et al. (2005) bi-material designs were
generated, whereas in Halkjær et al. (2006) designs
with one material (and void) were generated. The latter
paper included a comparison of the theoretical model
with experimental results obtained for a specimen with
10 by 10 optimized unit cells.
Several related studies that deal with design of elastic
structures for optimized wave propagation characteristics
have appeared recently. Hussein et al. (2007)
considered the design of 1D layered structures for longitudinal
wave propagation and Rupp et al. (2007) considered
design of 3D structures for optimized surface
wave propagation. In these papers bi-material designs
were generated for wave reflection and wave guiding.
In Jensen (2007b) bi- and triple-material designs were
generated for minimizing the transmission and absorption
of elastic waves through slabs of material.
These papers build on numerous works that have
considered topology optimization of steady-state
forced vibrations of elastic structures. The problem
was considered e.g. by Ma et al. (1995), Jog (2002) and
recently by Du and Olhoff (2007), Olhoff and Du
(2008) for plate structures.
The results in the present paper extends the work in
Halkjær et al. (2005, 2006) on plates. Instead of considering
the unit cell optimization problem, this paper
considers the structural problem that was initiated in
Sigmund and Jensen (2003) for a 2D plane strain model,
in which the material in the entire plate can be distributed
freely not relying on periodicity. Additionally, we
consider the possibility to create novel wave propagation
control in plates by directing the propagation of
waves in certain directions and also the possibility of
creating ring wave devices that function purely due to
material contrast differences.
The paper is organized as follows: section two
presents the plate model and the formulas describing
energy transport in plates are presented in section
three. In section four the problem is discretized and
in section five the optimization problems are defined.
Section six and seven show two examples: in the first
A.A. Larsen et al.
example the method is used to design structures with
minimized vibrational response in finite frequency
ranges and the second example demonstrates the generation
of energy transporting devices.
2 Plate model
Plate models are well established but to provide the basis
for the energy flow derivations presented in the next
section we review the basic equations. The equilibrium
equations for an infinitesimal small plate element
ρh 3
12
ρh 3
12
d2ψx dt2 = Tx
∂ Mx ∂ Mxy
− −
∂x ∂y
d2ψy dt2 = Ty
∂ My ∂ Mxy
− −
∂y ∂x
ρh d2w ∂Tx ∂Ty
= +
dt2 ∂x ∂y
(1)
(2)
(3)
govern the out-of-plane deflection w and the angles
of rotation ψx and ψy (see Fig. 1). The moments are
denoted Mx, My and Mxy, the shear forces Tx and Ty,
and ρ is the mass density and h is the plate thickness.
We use the Mindlin plate theory to account for the
transverse shear deformation introduced in moderately
thick plates. Thus the moments and shear forces are
defined in terms of the deflection and rotations as
⎧ ⎫ ⎧
⎫
⎪⎨
⎪⎩
Mx
My
Mxy
Tx
Ty
⎪⎬ ⎪⎨
=−D
⎪⎭
⎪⎩
ψx,x
ψy,y
ψx,y + ψy,x
ψx − w,x
ψy − w,y
⎪⎬
⎪⎭
(4)
where the subscript (), denotes differentiation with
respect to the spatial variables and the stiffness matrix
is defined as
⎡
D ν D 0 0 0
⎤
⎢ ν D D
⎢
D = ⎢ 0 0
⎢
⎣ 0 0
0 0
1 − ν
D 0
2
0 kGh
0
0
0
⎥
⎦
0 0 0 0 kGh
(5)
where D = Eh3
12(1−ν2 is the flexural rigidity, G is the shear
)
modulus, ν is Poissons’ ratio, and k is a shear correction
factor which takes the value 5
for the plate.
6
Assuming harmonic excitation of the form feiωt ,
with angular frequency ω, the resulting displacements