Guided elastic waves and perfectly matched layers

Elastic waveguides support propagating modes that have two possible features, negative group velocities and long wavelengths that, for some frequencies, degrade the accuracy or otherwise poison existing numerical schemes that utilise perfectly matched layers (PMLs) to mimic infinite domains. We illustrate why negative group velocities and long waves are potentially an issue and describe how these problems are overcome. Detailed numerical simulations confirm the accuracy of the modified scheme and provide both theoretical and pragmatic estimates for the parameters within the PML model, in particular for the damping function. We also contrast and compare different implementations of the PML model using spectral and finite difference methods.

[1]  Fr'ed'eric Nataf.  A new approach to perfectly matched layers for the linearized Euler system , 2006, J. Comput. Phys..

[2]  Trapped modes in topographically varying elastic waveguides , 2007 .

[3]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[4]  Orthogonality relation for Rayleigh–Lamb modes of vibration of an arbitrarily layered elastic plate with and without fluid loading , 1994 .

[5]  R. Craster,et al.  Trapped modes in curved elastic plates , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  Dan Givoli,et al.  FINITE ELEMENT FORMULATION WITH HIGH-ORDER ABSORBING BOUNDARY CONDITIONS FOR TIME-DEPENDENT WAVES , 2006 .

[7]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[8]  F. Livingstone,et al.  Review of progress in quantitative NDE: Williamsburg, VA, USA, 21–26 June 1987 , 1988 .

[9]  J Qu,et al.  Crack characterization using guided circumferential waves. , 2001, The Journal of the Acoustical Society of America.

[10]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[11]  Gunilla Kreiss,et al.  A new absorbing layer for elastic waves , 2006, J. Comput. Phys..

[12]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[13]  Vincent Pagneux,et al.  Lamb wave propagation in inhomogeneous elastic waveguides , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[14]  N. Anders Petersson,et al.  Perfectly matched layers for Maxwell's equations in second order formulation , 2005 .

[15]  W. B. Fraser.  Orthogonality relation for the Rayleigh-Lamb modes of vibration of a plate , 1976 .

[16]  J. Achenbach.  Wave propagation in elastic solids , 1962 .

[17]  Satish C. Reddy,et al.  A MATLAB differentiation matrix suite , 2000, TOMS.

[18]  C. K. Yuen,et al.  Theory and Application of Digital Signal Processing , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[19]  Marcus J. Grote,et al.  Local nonreflecting boundary condition for Maxwell's equations , 2006 .

[20]  Lanbo Liu,et al.  Acoustic pulse propagation near a right-angle wall. , 2006, The Journal of the Acoustical Society of America.

[21]  C. Tsogka,et al.  Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media , 2001 .

[22]  M. Lowe,et al.  DISPERSE: A GENERAL PURPOSE PROGRAM FOR CREATING DISPERSION CURVES , 1997 .

[23]  L. Thompson.  A review of finite-element methods for time-harmonic acoustics , 2006 .

[24]  John B. Schneider,et al.  Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation , 1996 .

[25]  Peter Cawley,et al.  The Potential of Guided Waves for Monitoring Large Areas of Metallic Aircraft Fuselage Structure , 2001 .

[26]  Patrick Joly,et al.  Stability of perfectly matched layers, group velocities and anisotropic waves , 2003 .

[27]  William H. Press,et al.  Numerical Recipes: FORTRAN , 1988 .

[28]  Richard V. Craster,et al.  Spectral methods for modelling guided waves in elastic media , 2004 .

[29]  Julius Kaplunov,et al.  Localized vibration in elastic structures with slowly varying thickness , 2005 .

[30]  I. A. Viktorov.  Rayleigh and Lamb Waves: Physical Theory and Applications , 1967 .