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Title: Finite elements modelling of scattering problems for flexural waves in thin plates: Application to elliptic invisibility cloaks, rotators and the mirage effect

Abstract

We propose a finite elements algorithm to solve a fourth order partial differential equation governing the propagation of time-harmonic bending waves in thin elastic plates. Specially designed perfectly matched layers are implemented to deal with the infinite extent of the plates. These are deduced from a geometric transform in the biharmonic equation. To numerically illustrate the power of elastodynamic transformations, we analyze the elastic response of an elliptic invisibility cloak surrounding a clamped obstacle in the presence of a cylindrical excitation i.e. a concentrated point force. Elliptic cloaking for flexural waves involves a density and an orthotropic Young's modulus which depend on the radial and azimuthal positions, as deduced from a coordinates transformation for circular cloaks in the spirit of Pendry et al. [Science 312, 1780 (2006)], but with a further stretch of a coordinate axis. We find that a wave radiated by a concentrated point force located a couple of wavelengths away from the cloak is almost unperturbed in magnitude and in phase. However, when the point force lies within the coating, it seems to radiate from a shifted location. Finally, we emphasize the versatility of transformation elastodynamics with the design of an elliptic cloak which rotates the wavevectormore » of a flexural wave within its core.« less

Authors:
 [1]; ;  [1]
  1. Institut Fresnel-CNRS (UMR 6133), University of Aix-Marseille, 13397 Marseille cedex 20 (France)
Publication Date:
OSTI Identifier:
21499781
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 230; Journal Issue: 6; Other Information: DOI: 10.1016/j.jcp.2010.12.009; PII: S0021-9991(10)00673-X; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0021-9991
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; COORDINATES; GEOMETRY; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS; SCATTERING; TRANSFORMATIONS; YOUNG MODULUS; DIFFERENTIAL EQUATIONS; EQUATIONS; MATHEMATICAL LOGIC; MATHEMATICS; MECHANICAL PROPERTIES

Citation Formats

Farhat, M., E-mail: mohamed.farhat@fresnel.f, Guenneau, S., and Enoch, S. Finite elements modelling of scattering problems for flexural waves in thin plates: Application to elliptic invisibility cloaks, rotators and the mirage effect. United States: N. p., 2011. Web. doi:10.1016/j.jcp.2010.12.009.
Farhat, M., E-mail: mohamed.farhat@fresnel.f, Guenneau, S., & Enoch, S. Finite elements modelling of scattering problems for flexural waves in thin plates: Application to elliptic invisibility cloaks, rotators and the mirage effect. United States. doi:10.1016/j.jcp.2010.12.009.
Farhat, M., E-mail: mohamed.farhat@fresnel.f, Guenneau, S., and Enoch, S. Sun . "Finite elements modelling of scattering problems for flexural waves in thin plates: Application to elliptic invisibility cloaks, rotators and the mirage effect". United States. doi:10.1016/j.jcp.2010.12.009.
@article{osti_21499781,
title = {Finite elements modelling of scattering problems for flexural waves in thin plates: Application to elliptic invisibility cloaks, rotators and the mirage effect},
author = {Farhat, M., E-mail: mohamed.farhat@fresnel.f and Guenneau, S. and Enoch, S.},
abstractNote = {We propose a finite elements algorithm to solve a fourth order partial differential equation governing the propagation of time-harmonic bending waves in thin elastic plates. Specially designed perfectly matched layers are implemented to deal with the infinite extent of the plates. These are deduced from a geometric transform in the biharmonic equation. To numerically illustrate the power of elastodynamic transformations, we analyze the elastic response of an elliptic invisibility cloak surrounding a clamped obstacle in the presence of a cylindrical excitation i.e. a concentrated point force. Elliptic cloaking for flexural waves involves a density and an orthotropic Young's modulus which depend on the radial and azimuthal positions, as deduced from a coordinates transformation for circular cloaks in the spirit of Pendry et al. [Science 312, 1780 (2006)], but with a further stretch of a coordinate axis. We find that a wave radiated by a concentrated point force located a couple of wavelengths away from the cloak is almost unperturbed in magnitude and in phase. However, when the point force lies within the coating, it seems to radiate from a shifted location. Finally, we emphasize the versatility of transformation elastodynamics with the design of an elliptic cloak which rotates the wavevector of a flexural wave within its core.},
doi = {10.1016/j.jcp.2010.12.009},
journal = {Journal of Computational Physics},
issn = {0021-9991},
number = 6,
volume = 230,
place = {United States},
year = {2011},
month = {3}
}