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

TL;DR

This paper develops a finite elements method for modeling bending waves in thin plates, demonstrating elastic cloaking, rotation, and mirage effects through transformation elastodynamics with numerical simulations.

Contribution

It introduces a finite elements algorithm with specialized layers for infinite plates and applies elastodynamic transformations to design and analyze elastic cloaks and wave manipulation.

Findings

Elliptic cloaking reduces wave disturbance from a point force.

Wave polarization can be rotated within the cloak.

The method effectively models wave scattering and cloaking effects.

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 analyse 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 {\bf 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 polarization of a flexural wave within its core.