Legendre and Laguerre polynomial approach for modeling of wave propagation in layered magneto-electro-elastic media

Abstract

A numerical method to compute propagation constants and mode shapes of elastic waves in layered piezoelectric-piezomagnetic composites, potentially deposited on a substrate, is described. The basic feature of the method is the expansion of particle displacement, stress fields, electric and magnetic potentials in each layer on different polynomial bases: Legendre for a layer of finite thickness and Laguerre for the semi-infinite substrate. The exponential convergence rate of the method for propagation of Love waves is numerically verified. The main advantage of the method is to directly determine complex wave numbers for a given frequency via a matricial eigenvalue problem, in a way that no transcendental equation has to be solved. Results are presented and the method is discussed.

Mapped orthogonal functions method applied to acoustic waves-based devices

“…This is counterbalanced, for the approach in Ref. 11, by a lighter preparatory mathematical work when working with more than three distinct sets of polynomials. Probably the polynomial approach in Ref.…”

“…Both use a polynomial expansion, result through an orthogonalization process in a linear eigenvalue problem, can directly determine complex wave numbers for a given frequency and can deal with complicated acoustic wave-based devices, for instance piezoelectricpiezomagnetic composite devices. 11,49 The difference resides in the treatment of the continuity and boundary conditions. For the mapped orthogonal functions method, the continuity of the generalized displacement vector (mechanical displacement components, electric and magnetic potentials) is ensured by a wise choice of the associated polynomial expansion 47 and the continuity and boundary conditions of the generalized stress vector (normal stress components, normal electric displacement and magnetic induction) are ensured thanks to the δ-functions given by the position-dependent physical constants.…”

“…It simply consists in transforming the quadratic eigenvalue problem Eq. (16b) into a linear one via a change of variable [9][10][11] with ω real as input variable and k complex as output variable. This results in an increase in the dimensionality of the matrices by a factor of four for the same number of terms in the expansions Eqs.…”

“…Alternatives to the classical solution of a characteristic function have been proposed. [9][10][11] They allow to compute real and complex wavenumbers via a classical matricial eigenvalue problem obtained from the motion equations and the continuity and boundary conditions along with the use of a polynomial expansion approach.…”

Modelling guided waves in anisotropic plates using the Legendre polynomial method

“…Lefebvre et al [13] studied wave propagation in piezoelectric plates using the Legendre polynomial method since then this approach is used extensively to investigate the guided waves propagation in various structure [14][15][16][17][18]. The key aspect of Legendre polynomial method is the expansion of the unknown displacement quantities on a set of functions.…”