Finite element computation of trapped and leaky elastic waves in open stratified waveguides

Abstract

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results.

Contribution of leaky modes in the modal analysis of unbounded problems with perfectly matched layers

“…Both trapped and leaky modes can be accurately computed, even after PML truncation, provided that the PML is properly parametrized. 4,5 However, the PML truncation transforms the continuous sum of PML modes into a discrete one. 6 At first sight, the role of this discrete sum in the convergence of solutions might be questionable owing to the fact that discrete PML modes strongly depend on the boundary conditions arbitrarily set at the end of the truncated PML.…”

“…(5) coincides with the condition found for a PML to cancel the transverse exponential growth of a leaky mode. 4,5 To summarize, the integration contour with PML is R þ C þ þC þ as shown in Fig. 3.…”

“…More particularly, they have been successfully applied to compute modes in open systems. 4,5 It has been shown that PMLs allow getting both trapped and leaky modes as well as a third type of mode sometimes called PML modes. 6 PML modes mainly resonate inside the PML region (their fields strongly depend on PML parameters).…”

Guided wave propagation in metallic and resin plates loaded with water on single surface

“…Semi-analytical finite element (SAFE) calculations have been developed as an efficient calculation technique for guided waves propagating in elongated structures such as plates and pipes [3][4][5][6][7][8][9]. Most of the SAFE calculations were formulated for guided waves in an elastic waveguide with homogeneous geometry and material constants in the longitudinal direction under the traction-free boundary conditions.…”

“…Recently, SAFE calculations considering leaky media have become feasible by introducing absorbing layers and perfectly matched layers surrounding the leaky media or by expressing the outer open domain with the boundary element method [7][8][9][10][11]. SAFE calculations for leaky Lamb waves were formulated based on the characteristic that waves leaking into fluids propagate as plane harmonic waves with the sound speed of the fluids [1].…”