×

Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides. (English) Zbl 1469.74069

Summary: We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in the unbounded straight parts of the guide. Each algorithm can be related to a so-called domain decomposition method, with an overlap between the domains. Specific transmission conditions are used, so that at each step of the algorithm only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using a bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized. An original choice of transmission conditions is proposed which enhances the effect of the overlap and allows us to handle arbitrary anisotropic materials. As a by-product, we derive transparent boundary conditions for an arbitrary anisotropic waveguide. The transparency of these new boundary conditions is checked for two- and three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for two- and three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.

MSC:

74J20 Wave scattering in solid mechanics
35P25 Scattering theory for PDEs
35Q74 PDEs in connection with mechanics of deformable solids
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Lowe, M. J.S.; Cawley, P., Long range guided wave inspection usage—current commercial capabilities and research directions, Int. Report (2006), Imperial College London
[2] Lowe, M. J.S.; Alleyne, D. N.; Cawley, P., Defect detection in pipes using guided waves, Ultrasonics, 36, 147-154 (1998)
[3] Givoli, D., Numerical Methods for Problems in Infinite Domains (1992), Elsevier Science Publishers · Zbl 0788.76001
[4] Skelton, E. A.; Adams, S. D.M.; Craster, R. V., Guided elastic waves and perfectly matched layers, Wave Motion, 44, 7-8, 573-592 (2007) · Zbl 1231.74188
[5] Bonnet-Ben Dhia, A.-S.; Chambeyron, C.; Legendre, G., On the use of perfectly matched layers in the presence of long or backward guided elastic waves, Wave Motion, 51, 2, 266-283 (2014) · Zbl 1456.74078
[6] Drozdz, M.; Moreau, L.; Castaings, M.; Lowe, M. J.S.; Cawley, P., Efficient numerical modelling of absorbing regions for boundaries of guided waves problems, AIP Conf. Proc., 820, 126 (2006)
[7] Halla, M.; Nannen, L., Hardy space infinite elements for time-harmonic two-dimensional elastic waveguide problems, Wave Motion, 59, 94-110 (2015) · Zbl 1467.65109
[8] Benmeddour, F.; Treyssède, F.; Laguerre, L., Numerical modeling of guided wave interaction with non-axisymmetric cracks in elastic cylinders, Internat. J. Solids Struct., 48, 5, 764-774 (2011) · Zbl 1236.74279
[9] Treyssède, F., Mode propagation in curved waveguides and scattering by inhomogeneities: application to the elastodynamics of helical structures, J. Acoust. Soc. Am., 129, 4, 1857-1868 (2011)
[10] Mencik, J.-M.; Ichchou, M. N., Multi-mode propagation and diffusion in structures through finite elements, Eur. J. Mech. A Solids, 24, 5, 877-898 (2005) · Zbl 1125.74347
[11] Gravenkamp, H.; Prager, J.; Saputra, A. A.; Song, C., The simulation of Lamb waves in a cracked plate using the scaled boundary finite element method, J. Acoust. Soc. Am., 132, 3, 1358-1367 (2012)
[12] Baronian, V.; Bonnet-Ben Dhia, A.-S.; Lunéville, E., Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide, J. Comput. Appl. Math., 234, 6, 1945-1952 (2010) · Zbl 1405.35121
[13] Baronian, V., Couplage des méthodes modale et éléments finis pour la diffraction des ondes élastiques guidées. Application au contrôle non destructif (2009), Ecole Polytechnique, (Ph.D. Thesis)
[14] Baronian, V.; Lhémery, A.; Bonnet-Ben Dhia, A.-S., Simulation of non-destructive inspections and acoustic emission measurements involving guided waves, J. Phys.: Conf. Ser., 195 (2009)
[15] Baronian, V.; Jezzine, K.; Le Bourdais, F., Hybrid modal/FE simulation of guided waves inspections, AIP Conf. Proc., 1511, 191 (2013)
[16] Fraser, W. B., Orthogonality relation for the Rayleigh-Lamb modes of vibration of a plate, J. Acoust. Soc. Am., 59, 215-216 (1976) · Zbl 0324.73022
[17] Gregory, R. D., A note on bi-orthogonality relations for elastic cylinders of general cross section, J. Elasticity, 13, 3, 351-355 (1983) · Zbl 0522.73052
[18] Ben Belgacem, F.; Fournié, M.; Gmati, N.; Jelassi, F., On the Schwarz algorithms for elliptic exterior boundary value problems, ESAIM-M2AN, 39, 4, 693-714 (2005) · Zbl 1089.65126
[19] Gmati, N.; Zrelli, N., Numerical study of some iterative solvers for acoustics in unbounded domains, ARIMA, 4, 1-23 (2006)
[20] Ben Belgacem, F.; Jelassi, F.; Gmati, N., Convergence bounds of GMRES with Schwarz preconditioner for the scattering problem, Internat. J. Numer. Methods Engrg., 80, 2, 191-209 (2009) · Zbl 1176.76060
[21] Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018
[22] Pagneux, V.; Maurel, A., Lamb wave propagation in inhomogeneous elastic waveguides, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 458, 2024, 1913-1930 (2002) · Zbl 1056.74030
[23] Pagneux, V.; Maurel, A., Scattering matrix properties with evanescent modes for waveguides in fluids and solids, J. Acoust. Soc. Am., 116, 1913-1920 (2004)
[24] Pagneux, V.; Maurel, A., Lamb wave propagation in elastic waveguides with variable thickness, Proc. R. Soc. A Math. Phys. Eng. Sci., 462, 2068, 1315-1339 (2006) · Zbl 1149.74348
[25] Nasarov, S. A., The Mandelstam energy radiation conditions and the Umov-Poynting vector in elastic waveguides, J. Math. Sci. (2013) · Zbl 1308.35302
[26] Merkulov, L. G.; Rokhlin, S. I.; Zobnin, O. P., Calculation of the spectrum of wave numbers for Lamb waves in a plate, Sov. J. Nondestruct. Test., 6, 369-373 (1970)
[27] Schwarz, H. A., Ueber einige Abbildungsaufgaben, J. Reine Angew. Math. (1869) · JFM 02.0626.01
[29] Després, B., Méthodes de décomposition de domaine pour les problèmes de propagation d’ondes en régime harmonique (1991), Université Paris IX Dauphine, (Ph.D. Thesis) · Zbl 0849.65085
[30] Collino, F.; Ghameni, S.; Joly, P., Domain decomposition method for harmonic wave propagation: a general presentation, J. Comput. Methods Appl. Mech. Eng., 184, 2-4, 171-211 (2000) · Zbl 0965.65134
[31] Gander, M. J.; Zhang, H., Optimized Schwarz methods with overlap for the Helmholtz equation, (Domain Decomposition Methods in Sci. and Eng. XXI (2014), Springer International Publishing), 207-215 · Zbl 1382.65442
[32] Boudendir, Y., Techniques de décomposition de domaine et méthode d’équations intégrales (2002), INSA Toulouse, (Ph.D. Thesis)
[33] Nakamura, G.; Uhlmann, G.; Wang, J.-N., Unique continuation for elliptic systems and crack determination in anisotropic elasticity, Contemp. Math., 362, 321-338 (2004) · Zbl 1129.35315
[34] Tonnoir, A., Conditions transparentes pour la diffraction d’ondes en milieu élastique anisotrope (2015), Ecole Polytechnique, (Ph.D. Thesis)
[35] Pagneux, V., Revisiting the edge resonance for Lamb waves in a semi-infinite plate, J. Acoust. Soc. Am., 120, 2, 649-656 (2006)
[36] Auld, B. A., Acoustic Fields and Waves in Solids (1973), Krieger Publishing Company: Krieger Publishing Company Florida
[37] Royer, D.; Dieulesaint, E., Ondes élastiques dans les Solides, Vol. Tome 1 (1996), Masson
[38] Hayashi, T.; Rose, J. L., Guided wave simulation and visualisation by a semi-analytical finite element method, Mater. Eval., 61, 75-79 (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.