WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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Working : PMMA-Alu-11-seg-7.5cm-200302-ref : Input : FFT Analyzer
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Figure 3: Response curves showing the acceleration response at the end of the bar as a function of
the excitation frequency, left: experimental data and right: corresponding theoretical predictions.
With the objective of e.g. minimizing the structural response in a certain part of the structure we
can formulate a topology optimization problem as
min |a|
x
T L |a|
s.t. : (K + iΩC − Ω 2 M) a = f
0 ≤ xe ≤ 1, e = 1, . . . , N
where K, C, M are the stiffness-, damping-, and mass-matrix, f, a the forcing and vibration/wave
amplitude and L is a zero matrix with ones at the diagonal elements corresponding to the degrees
of freedom of the nodes, lines, or areas to be damped. A design variable xe is assigned to each of
the N finite elements and the element properties are interpolated as:
p(xe) = (1 − xe)p1 + xep2
where the subscripts 1 and 2 here correspond to two different material phases and p is a relevant
physical property, e.g. mass density and two stiffness parameters in the elastic case.
Design of optimized structures
In the following, we present some examples of two-phase material structures with optimized wavereflecting
and wave-guiding properties. In all examples the structural domain is 20×20cm, out-ofplane
waves/vibrations are considered, and the two materials are epoxy and aluminum. The figures
show the optimized topology and the frequency response to the corresponding periodic loading.
In Fig. 4(a) the structure is optimized for a minimum vibration response along each edge of the
structure when it is subjected to a periodic load at 40 kHz along the opposite edge. All boundaries
are free. The frequency response from Fig. 1 is shown for comparison. In Fig. 4(b) the structure
is optimized for maximum wave amplitude at the center of the right edge with a 40 kHz load
acting on most of the left edge. The boundaries are all wave-transmitting and for comparison the
wave-amplitudes are also shown for the structures made of the pure materials.
Further research is currently focused around design of wave-guide structures with the aim of comparing
the response of optimized designs to existing wave-guide structures.
Concluding remarks
By using multiple material phases it is possible to design structures that have specific wavereflecting/vibration-damping
and wave-guiding properties. The basic phenomenon used in the de-
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