WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

896 S. Halkjær, O. Sigmund and J. S. Jensen
distributions. The design variables are spatial material densities.
By using analytical sensitivity analysis and mathematical
programming tools it is possible efficiently to handle
a large number of design variables. We base our
implementation on Finite Element (FE) modelling and typically
choose one continuous design variable per element
or node:
z e 2 R ; e ¼ 1; ...; N ; ð1Þ
where N is the number of finite elements or nodes in the
design domain. It should be emphasized that the method
can be used with a finite difference discretization just as
easily and that more design variables can be assigned to
each element in order to distribute more than two materials
in the design domain.
The goal for the optimization algorithm is to distribute
two materials in a domain such that some cost function is
minimized (or maximized). The continuous design variables
control the material distribution by defining the element-wise
constant material properties in each element. If
r e denotes an element-wise property we can write:
r e ¼ð1 z eÞ r 1 þ r 2 z e ð2Þ
where subscript 1 and 2 refers to the properties of material
1 and material 2, respectively. In this case we let z e vary
continuously between 0 and 1, such that for z e ¼ 0 the
corresponding element takes the material properties of material
1 and for z e ¼ 1 those of material 2.
Denoting the cost function F we can now formulate
the optimization problem as follows:
min:
F ¼ Fðw; u; zÞ ; ð3Þ
z
s:t:: giðzÞ g* i ; ð4Þ
0 ze 1 ; e ¼ 1; ...; N ; ð5Þ
ðK w 2 MÞ u ¼ f ; ð6Þ
where Eq. (6) is the discretized FE equation of the timeharmonic
wave propagation problem. The vector u contains
the discretized nodal values of the complex amplitudes
of the displacement field, and K and M are the stiffness
and mass matrices, respectively. The load vector f
comes from external wave loading. In (4) a number of
additional constraints can be specified, e.g. specifying a
maximum amount of a single material component. For
more details about computational procedures, sensitivity
analysis and more, the reader is referred to [2].
The advantage of using continuous design variables is
that it allows for the use of efficient gradient-based optimization
algorithms. However, it also implies that we may
end up with values of z e that are neither 0 or 1, but an
intermediate value that does not correspond to any of the
two materials. There are several ways to penalize the appearance
of intermediate density solutions (see e.g. [2]),
however, it is our experience that intermediate densities
seldomly remain in the optimized band gap designs since
band gaps are favoured by maximum contrast in the material
properties. Therefore we have not used any penalization
techniques in this work.
As for all topology optimization problems with spatial
material distribution the results depend on the mesh. How-
ever, as opposed to conventional stiffness design problems
where the objective is improved with mesh-refinement, it
appears that the finite length of the elastic waves imposes
a length-scale for the present band gap design problems
and thus we do usually not experience a significant meshdependence.
Therefore, we do not use regularization techniques
in this paper. For more discussions of this issue,
the reader is referred to [2]. Concerning the convergence
of the finite element model, we require discretizations that
at least have 10 elements pr. wave-length. Due to the reasons
discussed above, we do not expect to find significantly
different design solutions if the mesh is further refined.
Design of 1D structures:
the infinite periodic elastic beam
In this section, we study elastic wave propagation in a
periodic beam of infinite length. The beam consists of an
infinite number of copies of a base cell with a specific
geometry and material distribution. The periodicity introduces
frequency band gaps for both longitudinal and bending
waves, preventing waves with frequencies in these
gaps from propagating in the beam. In general, the band
gap frequency ranges for the two wave types are different,
but for certain geometries and material distributions of the
base cell, band gaps for the two wave types overlap, preventing
both of the two wave types from propagating in
the beam. Such studies are relevant in design problems
where vibration insulation and filtering are important objectives.
Theory
Let the beam axis be directed along the x-axis. Longitudinal
wave propagation in the beam is described using the
usual 1D theory
@
@x
@u
E
@x ¼ q @2u @t2 ð7Þ
where u ¼ uðx; tÞ denotes the displacement along the x-axis.
q denotes the mass density and E is Young’s modulus.
Bending waves in the beam are described by Timoshenko
theory
@
@x
@
@x
ksGA @w
@x þ q ¼ qA @2w ; ð8Þ
@t2 EI @q
@x
ksGA @w
@x þ q ¼ qI @2q @t2 ð9Þ
where w ¼ wðx; tÞ and q ¼ qðx; tÞ denote the transverse
displacement and angle of rotation of the cross-section respectively.
A, I, G and ks denote the cross-sectional area,
area moment of inertia, shear modulus and the shear correction
coefficient, respectively. Here, we use
ks ¼ 6ð1 þ nÞ=ð7 þ 6nÞ [14] for a circular cross-section,
where n is the Poisson ratio. Timoshenko theory is used,
as the ratio between considered wave length and cross-section
dimension is smaller than the lower bound ( 20) for
using the simpler Bernoulli-Euler theory.