WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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Struct Multidisc Optim<br />
DOI 10.1007/s00158-008-0257-0<br />
RESEARCH PAPER<br />
Topological material layout in plates for vibration<br />
suppression and wave propagation control<br />
Anders A. Larsen · Bogi Laksafoss ·<br />
Jakob S. Jensen · Ole Sigmund<br />
Received: 13 August 2007 / Revised: 21 February 2008 / Accepted: 28 February 2008<br />
© Springer-Verlag 2008<br />
Abstract We propose a topological material layout<br />
method to design elastic plates with optimized properties<br />
for vibration suppression and guided transport<br />
of vibration energy. The gradient-based optimization<br />
algorithm is based on a finite element model of the plate<br />
vibrations obtained using the Mindlin plate theory coupled<br />
with analytical sensitivity analysis using the adjoint<br />
method and an iterative design update procedure based<br />
on a mathematical programming tool. We demonstrate<br />
the capability of the method by designing bi-material<br />
plates that, when subjected to harmonic excitation, either<br />
effectively suppress the overall vibration level or<br />
alternatively transport energy in predefined paths in the<br />
plates, including the realization of a ring wave device.<br />
Most of this work was performed while AAL was employed<br />
at the Department of Mechanical Engineering.<br />
A. A. Larsen (B)<br />
Department of Mathematics,<br />
Technical University of Denmark, Matematiktorvet,<br />
Building 303S, 2800 Lyngby, Denmark<br />
e-mail: a.a.larsen@mat.dtu.dk<br />
B. Laksafoss<br />
NKT Flexibles, 2605 Brøndby, Denmark<br />
e-mail: bogi.laksafoss@nktflexibles.com<br />
J. S. Jensen · O. Sigmund<br />
Department of Mechanical Engineering,<br />
Technical University of Denmark, Nils Koppels Allé,<br />
Building 404, 2800 Lyngby, Denmark<br />
J. S. Jensen<br />
e-mail: jsj@mek.dtu.dk<br />
O. Sigmund<br />
e-mail: sigmund@mek.dtu.dk<br />
Keywords Mindlin plate · Topology design ·<br />
Energy transport · Vibration response<br />
1 Introduction<br />
In this paper we use the method of topology optimization<br />
to design bi-material structures for suppressing the<br />
vibration response and for controlling the transport of<br />
vibration energy in moderately thick plates. Possible<br />
applications are for structural shielding and isolation<br />
of vibration in the audible frequency range and for the<br />
creation of devices that rely on guided transportation of<br />
vibrations e.g. for absorption or harvesting of energy.<br />
Topology optimization (Bendsøe and Kikuchi 1988)<br />
is a systematic design methodology that allows to create<br />
material distributions of one or more materials that<br />
optimize a specified objective. Aside from its original<br />
application to structural optimization problems in<br />
mechanics, many recent extensions to other physics<br />
settings have appeared, e.g. in optics, fluid mechanics,<br />
electromagnetism, etc. (see e.g. Bendsøe and Sigmund<br />
(2003) for a recent comprehensive coverage of the<br />
method and its applications).<br />
The idea of using topology optimization to design<br />
mechanical structures for passive control of the propagation<br />
of elastic waves was suggested in Sigmund and<br />
Jensen (2003), in which several examples dealing with<br />
wave shielding and wave guiding devices were given.<br />
The paper considered the material design problem,<br />
in which repetitive unit cells were designed and used<br />
for periodic wave-reflectors and also the corresponding<br />
structural optimization problem for finite structures to<br />
be used for both wave reflecting and wave guiding<br />
purposes. In the examples a 2D plane strain model of