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