WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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O. Sigmund and J. S. Jensen
4. Conclusion
We have demonstrated that topology optimization is an efficient tool for the design
of materials and structures with band gaps. We have formulated two optimization
problems: one for the maximization of band-gap sizes for infinitely periodic structures
and the other for the design of finite structures with wave-stopping or waveguiding
behaviours.
The proposed optimization formulations were used to design periodic band-gap
materials with maximum relative band-gap size, and it was found that, at least for
the scalar (out-of-plane) case, the optimum inclusion shape depends on the contrast
in material properties between the material phases. For the structural problem we
have demonstrated that the optimum band-gap materials for the finite case are close
to periodic, although the inclusions close to the boundaries should be touching the
rim in order to prevent surface modes from propagating along the boundaries. We
have also shown an example of the design of a waveguide that concentrates a wide
incoming wave to a narrow and magnified output wave. The size of problems solved
was limited by MATLAB storage requirements. For future work we are rewriting
the code to FORTRAN 90, which allows for larger problems. In MATLAB, typical
computing times for an 80 x 80 element out-of-plane problem (such as figure 9) is
ca. 5-6 h for 500 iterations. This time is expected to be significantly reduced by using
the FORTRAN 90 code.
The issue of multiple local minima was not addressed in this paper, but, in the
structural case, several optimized designs can be obtained by varying the initial value
of the design variables. These designs will have different periodicities corresponding
to different band gaps. However, in our experience, topologically different solutions
obtained from different starting guesses usually have very similar objective function
values.
The analysis was restricted to deal with the uncoupled problem of in-plane and
out-of-plane waves. In the case where this decoupling cannot be performed, e.g. due to
inclusions with out-of-plane misalignments or waves not initially propagating strictly
in the plane, fully three-dimensional modelling is required. The effect of out-of-plane
scattering and the design of three-dimensional structures are subjects for further
work. In future work we will additionally consider improved objective functions and
more advanced waveguiding problems.
This work received support from Denmark's Technical Research Council through the Tal-
ent/THOR Program 'Design of microelectromechanical systems (MEMS)' and the Research
Project 'Phononic bandgap materials: analysis and optimization of wave transmission in peri-
odic materials'. The authors express their gratitude to Professors Martin P. Bends0e and Jon
J. Thomsen, Technical University of Denmark, for valuable discussions and inspiration.
References
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Bends0e, M. P. & Sigmund, 0. 2003 Topology optimization: theory, methods and applications.
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Brillouin, L. 1953 Wave propagation in periodic structures, 2nd edn. New York: Dover.
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