Towards an Optimization of the Spectral Collocation Method with a New Balancing Algorithm for Plotting Dispersion Curves of Composites with Large Numbers of Layers

Document Type : Research Paper


1 Laboratory of Mechanics, Engineering and Innovation, National High School of Electricity and Mechanics, Hassan II University, Casablanca, Morocco

2 M2S2I Laboratory, Department of Mechanical engineering, ENSET-Mohammedia, Hassan II University, Mohammedia, Morocco


This article proposes a new algorithm for computing dispersion curves of ultrasonic guided waves in multi-layered composites with large number of layers. The algorithm is based on balancing the eigenvalue problem of the Spectral Collocation Method (SCM) formulation. The SCM has proven effective in analyzing single-layer and simple waveguides. However, it struggles with large multi-layer structures due to numerical instability caused by irregular and sparse matrices in the eigenvalue problem. The proposed algorithm for balancing has significantly reduced both the conditioning measure and the matrix norm in the matrix system. The optimization of spectral formulation enables accurate calculation of dispersion curves and characterization of displacement/stress profiles using Matlab software. This precise characterization and mode separation are essential for selecting ultrasonic sensors for damage detection.  A comparison was made with Dispersion Calculator (DC) software. The algorithm was first validated using a hybrid multi-layered composite [CFRP-Al-CFRP-Al], which was successful. The validated algorithm was then quantitatively evaluated using a cross-play laminate T800M913 [0/90/0/90] in terms of three parameters: Number of collocation points, wave propagation direction and thickness. The algorithm is then applied to a large number of layers using three configurations: Symmetric layup T80M913[0/90]10s, hybrid layup [CFRP-Al]50 and challenging layup T800M913[0/90]100. The study found that the balancing algorithm, when combined with the SCM, is effective for structures with a large number of layers. Finally, optimizing the SCM will enhance its competitiveness as an effective tool in ultrasonic non-destructive testing for the studied structures of great industrial interest.


Main Subjects

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[1] Rose, J.L., Ed., Ultrasonic Guided Waves in Solid Media, Cambridge University Press, 2014.
[2] Lamb, H., On waves in an elastic plate, Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character, 93(648), 1997, 114–128.
[3] Thomson, W.T., Transmission of Elastic Waves through a Stratified Solid Medium, Journal of Applied Physics, 21(2), 2004, 89–93.
[4] Haskell, N.A., The dispersion of surface waves on multilayered media, Bulletin of the Seismological Society of America, 43(1), 1953, 17–34.
[5] Nayfeh, A.H., The general problem of elastic wave propagation in multilayered anisotropic media, The Journal of the Acoustical Society of America, 89(4), 1991, 1521–1531.
[6] Knopoff, L., A matrix method for elastic wave problems, Bulletin of the Seismological Society of America, 54(1),1964, 431–438.
[7] Rokhlin, S.I., Wang, L., Stable recursive algorithm for elastic wave propagation in layered anisotropic media: Stiffness matrix method, The Journal of the Acoustical Society of America, 112(3), ,2002, 822–834.
[8] Lowe, M.J.S., Matrix techniques for modeling ultrasonic waves in multilayered media, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 42(4), 1995, 525–542.
[9] Barski, M., Pajak, P., Determination of Dispersion Curves for Composite Materials with the Use of Stiffness Matrix Method, Acta Mechanica et Automatica, 11(2), 2017, 121–128.
[10] Guo, S., Rébillat, M., Liu, Y., Li, Q., Lu, C., Mechbal, N., Guided waves propagation in arbitrarily stacked composite laminates: Between-layers incompatibility issue resolution using hybrid matrix strategy, Composite Structures, 322, 2023, 117360.
[11] Sorokin, S.V., Broberg, P.H., Steffensen, M.T., Ledet, L.S., Finite element modal analysis of wave propagation in homogeneous and periodic waveguides, International Journal of Mechanical Sciences, 227, 2022, 107444.
[12] Rhimini, H., Bougaze, B., El Ouahdani, M., Sidki, M., Nassim, A., Modelling of Lamb waves propagation in plane plates by the finite element method, Physical and Chemical News, 46, 2009, 63–72.
[13] Nissabouri, S., El Allami, M., Boutyour, E., Ahmed, E., Lamb wave interaction with delamination in orthotropic plate, ACM International Conference Proceeding Series, Larache, Morocco, 1–5, 2017.
[14] Mace, B., Manconi, E., Modelling wave propagation in two-dimensional structures using finite element analysis, Journal of Sound and Vibration, 318, 2008, 884–902.
[15] Bartoli, I., Marzani, A., Lanza di Scalea, F., Viola, E., Modeling wave propagation in damped waveguides of arbitrary cross-section, Journal of Sound and Vibration, 295, 2006, 685–707.
[16] Giurgiutiu, V., Haider, M.F., Propagating, Evanescent, and Complex Wavenumber Guided Waves in High-Performance Composites, Materials, 12(2), 2019, 269.
[17] Nissabouri, S., El Allami, M., Boutyour, E., Quantitative evaluation of semi-analytical finite element method for modeling Lamb waves in orthotropic plates, Comptes Rendus Mécanique, 348(5), 2020, 335–350.
[18] Dahmen, S., Amor, M.B., Ghozlen, M.H.B., Investigation of the coupled Lamb waves propagation in viscoelastic and anisotropic multilayer composites by Legendre polynomial method, Composite Structures, 153, 2016, 557–568.
[19] Adamou, AT.I., Craster, R. V., Spectral methods for modelling guided waves in elastic media, The Journal of the Acoustical Society of America, 116(3), 2004, 1524–1535.
[20] Karpfinger, F., Valero, H., Gurevich, B., Bakulin, A., Sinha, B., Spectral-method algorithm for modeling dispersion of acoustic modes in elastic cylindrical structures, Geophysics, 75(3), 2010, H19–H27.
[21] Karpfinger, F., Gurevich, B., Bakulin, A., Modeling of wave dispersion along cylindrical structures using the spectral method, The Journal of the Acoustical Society of America, 124(2), 2008, 859–865.
[22] Hernando Quintanilla, F., Lowe, M.J.S., Craster, R.V., The symmetry and coupling properties of solutions in general anisotropic multilayer waveguides, The Journal of the Acoustical Society of America, 141(1), 2017, 406-418.
[23] Georgiades, E., Lowe, MJ. S., Craster, R.V., Identification of leaky Lamb waves for waveguides sandwiched between elastic half-spaces using the Spectral Collocation Method, The Journal of the Acoustical Society of America, 155, 2023, 629-639.
[24] Ward, R.C., Balancing the Generalized Eigenvalue Problem, SIAM Journal on Scientific and Statistical Computing, 2(2), 1981, 141–152.
[25] Dispersion Calculator Armin Huber DLR-homepage, 2020.
[26] Weideman, JA., Reddy, S.C., A MATLAB differentiation matrix suite, ACM Transactions on Mathematical Software, 26(4), 2000, 465–519.
[27] Zitouni, I., Rhimini, H., Chouaf, A., Modeling the Propagation of Ultrasonic Guided Waves in a Composite Plate by a Spectral Approximation Method, Engineering Transactions, 71(2), 2023, 213-227.
[28] Zitouni, I., Rhimini, H., Chouaf, A., Comparative Study of the Spectral Method, DISPERSE and Other ‎Classical Methods for Plotting the Dispersion Curves in ‎Anisotropic Plates, Journal of Applied and Computational Mechanics, 9(4), 2023, 955–973.
[29] Quintanilla, F.H., Fan, Z., Lowe, M.J.S., Craster, R.V., Guided waves’ dispersion curves in anisotropic viscoelastic single- and multi-layered media, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2183), 2015, 20150268.
[30] Householder, A.S., The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964, Reprinted by Dover, New York, 1975.
[31] Nayfeh, A.H. Chimenti, D.E., Free Wave Propagation in Plates of General Anisotropic Media, Journal of Applied Mechanics, 56(4), 1989, 881–886.
[32] Pavlakovic, B., Lowe, M., Alleyne, D., Cawley, P., Disperse: A General Purpose Program for Creating Dispersion Curves, Review of Progress in Quantitative Nondestructive Evaluation, 16A, 1997, 185–192.
[33] Rautela, M., Huber, A., Senthilnath, J., Gopalakrishnan, S., Inverse characterization of composites using guided waves and convolutional neural networks with dual-branch feature fusion, Mechanics of Advanced Materials and Structures, 29(27), 2022, 6595–6611.
[34] Huber, A., Numerical Modeling of Guided Waves in Anisotropic Composites with application to Air-coupled Ultrasonic Inspection, Ph. D. Thesis, Faculty of Mathematics, Natural Sciences and Technology, University of Augsburg, 2023.
[35] James, R., Langou, J., Lowery, B.R., On matrix balancing and eigenvector computation, arXiv, 2014.
[36] Huber, A., Classification of solutions for guided waves in fluid-loaded viscoelastic composites with large numbers of layers, The Journal of the Acoustical Society of America, 154(2), 2023, 1073–1094.
[37] Auld, B.A., Acoustic fields and waves in solids, Krieger Publishing Company, Malabar, FL, 1990.
[38] Kamal, A., Giurgiutiu, V., Stiffness Transfer Matrix Method (STMM) for stable dispersion curves solution in anisotropic composites, Health Monitoring of Structural and Biological Systems, 9064, 2014, 293-306.
[39] Longsine, D.E., McCormick, S.F., Simultaneous Rayleigh quotient minimization methods for Ax=λBx, Linear Algebra and its Applications, 34, 1980, 195-234.