WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Phononic band-gap optimization
2, . . . . . . . .
80 -
I.~ -20I~,,~ I{11 1 I I I I I -
V I I I I I I I I I
-10 ( ' ' ' '
( 40 60) 80 100
frequ6ency. /' (k t l/)
1009
Figure 5. Response of a 12 cm x 12 cm square structure subjected to periodic loading on the left
boundary, absorbing boundary conditions on the left and right edges and out-of-plane modelling.
High-contrast case.
ing two material phases in the design domain such that an objective function is
extremized subject to a number of constraints. The design variables are material
'densities' Xe C [0, 1] that interpolate the material properties for each element used
to discretize the structure. If the design variable takes a zero value we should have
pure phase 1 material, and if the design variable takes a value of one we should have
pure phase 2 material in the element. We choose a linear interpolation between the
phases, given as
p(Xe) = (1 - Xe)p + XeP2, (3.1)
/1(Xe) = (1 - Xe)Pl + XeL2, (3.2)
A(xe) = (1 - xe)Al + XeA2, (3.3)
where subscripts '1' and '2' denote the properties in materials 1 and 2, respectively.
The choice of linear interpolation stems from the observation that there appears
to be no need for penalizing intermediate 'densities', i.e. values of Xe other than zero
or one in the final design. The reason for this is believed to originate in the nature
of the band-gap phenomenon, where large contrasts between the involved material
phases is favoured.
(a) Material optimization
An obvious goal for the optimization of band-gap materials is to maximize the
relative band-gap size. In this way the range of prohibited frequencies will be wider
and more signals may be sent through a waveguide based on defects in the band-gap
material.
The goal of the optimization is to maximize the relative band-gap size between
bands j and j + 1, i.e. maximize the lowest value of the overlying bands and minimize
the maximum value of the underlying bands. This can written as a (double) max-min
objective
w2 - )
M ink1) 2
(J+k,x) - maxk w2(k,x) 3
max:
c(x)-=
() - .
x
(3.4)
W2 m(x) 0~~~~~)+ mink Wi (2 (k,x) j +maxk W j(k,x)
Phil. Trans. R. Soc. Lond. A (2003)