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Shock and Vibration
Volume 2015, Article ID 735219, 19 pages
http://dx.doi.org/10.1155/2015/735219
Review Article

Nondestructive Damage Assessment of Composite Structures Based on Wavelet Analysis of Modal Curvatures: State-of-the-Art Review and Description of Wavelet-Based Damage Assessment Benchmark

Institute of Fundamentals of Machinery Design, Silesian University of Technology, 18A Konarskiego Street, 44-100 Gliwice, Poland

Received 13 February 2015; Revised 6 June 2015; Accepted 11 June 2015

Academic Editor: Roger Serra

Copyright © 2015 Andrzej Katunin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. Montalvão, N. M. M. Maia, and A. M. R. Ribeiro, “A review of vibration-based structural health monitoring with special emphasis on composite materials,” Shock and Vibration Digest, vol. 38, no. 4, pp. 295–324, 2006. View at Google Scholar · View at Scopus
  2. A. K. Pandey, M. Biswas, and M. M. Samman, “Damage detection from changes in curvature mode shapes,” Journal of Sound and Vibration, vol. 145, no. 2, pp. 321–332, 1991. View at Publisher · View at Google Scholar · View at Scopus
  3. Y. K. Ho and D. J. Ewins, “On the structural damage identification with mode shapes,” in Proceedings of the European Conference on System Identification and Structural Health Monitoring, pp. 677–686, Madrid, Spain, 2000.
  4. C. S. Hamey, W. Lestari, P. Qiao, and G. Song, “Experimental damage identification of carbon/epoxy composite beams using curvature mode shapes,” Structural Health Monitoring, vol. 3, no. 4, pp. 333–353, 2004. View at Publisher · View at Google Scholar · View at Scopus
  5. D. Wu and S. S. Law, “Damage localization in plate structures from uniform load surface curvature,” Journal of Sound and Vibration, vol. 276, no. 1-2, pp. 227–244, 2004. View at Publisher · View at Google Scholar · View at Scopus
  6. Z. Ismail, H. Abdul Razak, and A. G. Abdul Rahman, “Determination of damage location in RC beams using mode shape derivatives,” Engineering Structures, vol. 28, no. 11, pp. 1566–1573, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. J. F. Gauthier, T. M. Whalen, and J. Liu, “Experimental validation of the higher-order derivative discontinuity method for damage identification,” Structural Control and Health Monitoring, vol. 15, no. 2, pp. 143–161, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. P. Moreno-García, J. V. Araújo dos Santos, and H. Lopes, “A new technique to optimize the use of mode shape derivatives to localize damage in laminated composite plates,” Composite Structures, vol. 108, no. 1, pp. 548–554, 2013. View at Publisher · View at Google Scholar · View at Scopus
  9. M. S. Cao, M. Radzieński, W. Xu, and W. Ostachowicz, “Identification of multiple damage in beams based on robust curvature mode shapes,” Mechanical Systems and Signal Processing, vol. 46, no. 2, pp. 468–480, 2014. View at Publisher · View at Google Scholar · View at Scopus
  10. C. P. Ratcliffe, “Damage detection using a modified laplacian operator on mode shape data,” Journal of Sound and Vibration, vol. 204, no. 3, pp. 505–517, 1997. View at Publisher · View at Google Scholar · View at Scopus
  11. M. M. A. Wahab and G. D. De Roeck, “Damage detection in bridges using modal curvatures: application to a real damage scenario,” Journal of Sound and Vibration, vol. 226, no. 2, pp. 217–235, 1999. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Chandrashekhar and R. Ganguli, “Structural damage detection using modal curvature and fuzzy logic,” Structural Health Monitoring, vol. 8, no. 4, pp. 267–282, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. E. Sazonov and P. Klinkhachorn, “Optimal spatial sampling interval for damage detection by curvature or strain energy mode shapes,” Journal of Sound and Vibration, vol. 285, no. 4-5, pp. 783–801, 2005. View at Publisher · View at Google Scholar · View at Scopus
  14. M. K. Yoon, D. Heider, J. W. Gillespie Jr., C. P. Ratcliffe, and R. M. Crane, “Local damage detection using the two-dimensional gapped smoothing method,” Journal of Sound and Vibration, vol. 279, no. 1-2, pp. 119–139, 2005. View at Publisher · View at Google Scholar · View at Scopus
  15. J.-H. Chou and J. Ghaboussi, “Genetic algorithm in structural damage detection,” Computers & Structures, vol. 79, no. 14, pp. 1335–1353, 2001. View at Publisher · View at Google Scholar · View at Scopus
  16. J. N. Yang, Y. Lei, S. Lin, and N. Huang, “Hilbert-Huang based approach for structural damage detection,” Journal of Engineering Mechanics, vol. 130, no. 1, pp. 85–95, 2004. View at Publisher · View at Google Scholar · View at Scopus
  17. M. Imregun and W. J. Visser, “A review of model updating techniques,” Shock & Vibration Digest, vol. 23, no. 1, pp. 9–20, 1991. View at Google Scholar
  18. J. E. Mottershead and M. I. Friswell, “Model updating in structural dynamics: a survey,” Journal of Sound and Vibration, vol. 167, no. 2, pp. 347–375, 1993. View at Publisher · View at Google Scholar · View at Scopus
  19. S. V. Modak, T. K. Kundra, and B. C. Nakra, “Comparative study of model updating methods using simulated experimental data,” Computers and Structures, vol. 80, no. 5-6, pp. 437–447, 2002. View at Publisher · View at Google Scholar · View at Scopus
  20. H. H. Khodaparast, J. E. Mottershead, and M. I. Friswell, “Perturbation methods for the estimation of parameter variability in stochastic model updating,” Mechanical Systems and Signal Processing, vol. 22, no. 8, pp. 1751–1773, 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. W. Wang, J. E. Mottershead, and C. Mares, “Mode-shape recognition and finite element model updating using the Zernike moment descriptor,” Mechanical Systems and Signal Processing, vol. 23, no. 7, pp. 2088–2112, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. W. Wang, J. E. Mottershead, C. M. Sebastian, and E. A. Patterson, “Shape features and finite element model updating from full-field strain data,” International Journal of Solids and Structures, vol. 48, no. 11-12, pp. 1644–1657, 2011. View at Publisher · View at Google Scholar · View at Scopus
  23. S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, Academic Press, Burlington, Mass, USA, 2008.
  24. S. H. Mortazavi and S. M. Shahrtash, “Comparing denoising performance of DWT, WPT, SWT and DT-CWT for partial discharge signals,” in Proceedings of the 43rd International Universities Power Engineering Conference (UPEC '08), pp. 1–6, Padova, Italy, September 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. C. Surace and R. Ruotolo, “Crack detection of a beam using the wavelet transform,” in Proceedings of the 12th International Modal Analysis Conference, pp. 1141–1147, Honolulu, Hawaii, USA, 1994.
  26. K. M. Liew and Q. Wang, “Application of wavelet theory for crack identification in structures,” Journal of Engineering Mechanics, vol. 124, no. 2, pp. 152–157, 1998. View at Publisher · View at Google Scholar · View at Scopus
  27. Q. Wang and X. Deng, “Damage detection with spatial wavelets,” International Journal of Solids and Structures, vol. 36, no. 23, pp. 3443–3468, 1999. View at Publisher · View at Google Scholar · View at Scopus
  28. C.-C. Chang and L.-W. Chen, “Vibration damage detection of a Timoshenko beam by spatial wavelet based approach,” Applied Acoustics, vol. 64, no. 12, pp. 1217–1240, 2003. View at Publisher · View at Google Scholar · View at Scopus
  29. C.-C. Chang and L.-W. Chen, “Damage detection of a rectangular plate by spatial wavelet based approach,” Applied Acoustics, vol. 65, no. 8, pp. 819–832, 2004. View at Publisher · View at Google Scholar · View at Scopus
  30. C.-C. Chang and L.-W. Chen, “Detection of the location and size of cracks in the multiple cracked beam by spatial wavelet based approach,” Mechanical Systems and Signal Processing, vol. 19, no. 1, pp. 139–155, 2005. View at Publisher · View at Google Scholar · View at Scopus
  31. J.-C. Hong, Y. Y. Kim, H. C. Lee, and Y. W. Lee, “Damage detection using the Lipschitz exponent estimated by the wavelet transform: applications to vibration modes of a beam,” International Journal of Solids and Structures, vol. 39, no. 7, pp. 1803–1816, 2002. View at Publisher · View at Google Scholar · View at Scopus
  32. E. Douka, S. Loutridis, and A. Trochidis, “Crack identification in beams using wavelet analysis,” International Journal of Solids and Structures, vol. 40, no. 13-14, pp. 3557–3569, 2003. View at Publisher · View at Google Scholar · View at Scopus
  33. E. Douka, S. Loutridis, and A. Trochidis, “Crack identification in plates using wavelet analysis,” Journal of Sound and Vibration, vol. 270, no. 1-2, pp. 279–295, 2004. View at Publisher · View at Google Scholar · View at Scopus
  34. S. Loutridis, E. Douka, L. J. Hadjileontiadis, and A. Trochidis, “A two-dimensional wavelet transform for detection of cracks in plates,” Engineering Structures, vol. 27, no. 9, pp. 1327–1338, 2005. View at Publisher · View at Google Scholar · View at Scopus
  35. A. Gentile and A. Messina, “On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams,” International Journal of Solids and Structures, vol. 40, no. 2, pp. 295–315, 2003. View at Publisher · View at Google Scholar · View at Scopus
  36. M. Rucka and K. Wilde, “Damage location in beam and plate structures by wavelet analysis of experimentally determined mode shapes,” Key Engineering Materials, vol. 293-294, pp. 313–320, 2005. View at Publisher · View at Google Scholar · View at Scopus
  37. M. Rucka and K. Wilde, “Crack identification using wavelets on experimental static deflection profiles,” Engineering Structures, vol. 28, no. 2, pp. 279–288, 2006. View at Publisher · View at Google Scholar · View at Scopus
  38. M. Rucka and K. Wilde, “Application of continuous wavelet transform in vibration based damage detection method for beams and plates,” Journal of Sound and Vibration, vol. 297, no. 3–5, pp. 536–550, 2006. View at Publisher · View at Google Scholar · View at Scopus
  39. W. Fan and P. Qiao, “A 2-D continuous wavelet transform of mode shape data for damage detection of plate structures,” International Journal of Solids and Structures, vol. 46, no. 25-26, pp. 4379–4395, 2009. View at Publisher · View at Google Scholar · View at Scopus
  40. S. Zhong and S. O. Oyadiji, “Crack detection in simply supported beams using stationary wavelet transform of modal data,” Structural Control and Health Monitoring, vol. 18, no. 2, pp. 169–190, 2011. View at Publisher · View at Google Scholar · View at Scopus
  41. D.-U. Sung, C.-G. Kim, and C.-S. Hong, “Monitoring of impact damages in composite laminates using wavelet transform,” Composites Part B: Engineering, vol. 33, no. 1, pp. 35–43, 2002. View at Publisher · View at Google Scholar · View at Scopus
  42. H. Gökdağ and O. Kopmaz, “A new damage detection approach for beam-type structures based on the combination of continuous and discrete wavelet transforms,” Journal of Sound and Vibration, vol. 324, no. 3–5, pp. 1158–1180, 2009. View at Publisher · View at Google Scholar · View at Scopus
  43. A. Katunin, “Identification of multiple cracks in composite beams using discrete wavelet transform,” Scientific Problems of Machines Operation and Maintenance, vol. 45, no. 2, pp. 41–52, 2010. View at Google Scholar
  44. A. Katunin, “Damage identification in composite plates using two-dimensional B-spline wavelets,” Mechanical Systems and Signal Processing, vol. 25, no. 8, pp. 3153–3167, 2011. View at Publisher · View at Google Scholar · View at Scopus
  45. A. Katunin and F. Holewik, “Crack identification in composite elements with non-linear geometry using spatial wavelet transform,” Archives of Civil and Mechanical Engineering, vol. 13, no. 3, pp. 287–296, 2013. View at Publisher · View at Google Scholar · View at Scopus
  46. A. Katunin, “Vibration-based damage identification in composite circular plates using polar discrete wavelet transform,” Journal of Vibroengineering, vol. 15, no. 1, pp. 355–363, 2013. View at Google Scholar · View at Scopus
  47. A. Bagheri, G. G. Amiri, and S. A. S. Razzaghi, “Vibration-based damage identification of plate structures via curvelet transform,” Journal of Sound and Vibration, vol. 327, no. 3–5, pp. 593–603, 2009. View at Publisher · View at Google Scholar · View at Scopus
  48. A. Katunin, “Spatial damage identification in composite plates using multiwavelets,” Journal of Applied Mathematics and Computational Mechanics, vol. 12, no. 3, pp. 69–78, 2013. View at Google Scholar
  49. A. Katunin, “Identification of stiff inclusion in circular composite plate based on quaternion wavelet analysis of modal shapes,” Journal of Vibroengineering, vol. 16, no. 5, pp. 2545–2551, 2014. View at Google Scholar
  50. A. Katunin, “Stone impact damage identification in composite plates using modal data and quincunx wavelet analysis,” Archives of Civil and Mechanical Engineering, vol. 15, no. 1, pp. 251–261, 2014. View at Publisher · View at Google Scholar · View at Scopus
  51. M. Unser and T. Blu, “Fractional splines and wavelets,” SIAM Review, vol. 42, no. 1, pp. 43–67, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  52. A. Katunin, “Crack identification in composite beam using causal B-spline wavelets of fractional order,” Modeling in Engineering, vol. 15, no. 46, pp. 57–63, 2013. View at Google Scholar
  53. K. N. Chaudhury and M. Unser, “Construction of Hilbert transform pairs of wavelet bases and Gabor-like transforms,” IEEE Transactions on Signal Processing, vol. 57, no. 9, pp. 3411–3425, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  54. A. Katunin and P. Przystałka, “Damage assessment in composite plates using fractional wavelet transform of modal shapes with optimized selection of spatial wavelets,” Engineering Applications of Artificial Intelligence, vol. 30, pp. 73–85, 2014. View at Publisher · View at Google Scholar · View at Scopus
  55. A. Katunin, “Vibration-based spatial damage identification in honeycomb-core sandwich composite structures using wavelet analysis,” Composite Structures, vol. 118, pp. 385–391, 2014. View at Publisher · View at Google Scholar
  56. A. Katunin, “Damage assessment in composite structures using modal analysis and 2D undecimated wavelet transform,” Journal of Vibroengineering, vol. 16, no. 8, pp. 3939–3950, 2014. View at Google Scholar
  57. L. H. Yam, Y. J. Yan, and J. S. Jiang, “Vibration-based damage detection for composite structures using wavelet transform and neural network identification,” Composite Structures, vol. 60, no. 4, pp. 403–412, 2003. View at Publisher · View at Google Scholar · View at Scopus
  58. M. Rucka and K. Wilde, “Neuro-wavelet damage detection technique in beam, plate and shell structures with experimental validation,” Journal of Theoretical and Applied Mechanics, vol. 48, no. 3, pp. 579–604, 2010. View at Google Scholar · View at Scopus
  59. H. Hein and L. Feklistova, “Computationally efficient delamination detection in composite beams using Haar wavelets,” Mechanical Systems and Signal Processing, vol. 25, no. 6, pp. 2257–2270, 2011. View at Publisher · View at Google Scholar · View at Scopus
  60. J. Morlier, F. Bos, and P. Castéra, “Diagnosis of a portal frame using advanced signal processing of laser vibrometer data,” Journal of Sound and Vibration, vol. 297, no. 1-2, pp. 420–431, 2006. View at Publisher · View at Google Scholar · View at Scopus
  61. M. Nguyen, X. Wang, Z. Su, and L. Ye, “Damage identification for composite structures with Bayesian network,” in Proceedings of the Intelligent Sensors, Sensor Networks and Information Processing Conference, pp. 307–311, Melbourne, Australia, 2004.
  62. J. Xiang and M. Liang, “A two-step approach to multi-damage detection for plate structures,” Engineering Fracture Mechanics, vol. 91, pp. 73–86, 2012. View at Publisher · View at Google Scholar · View at Scopus
  63. A. Katunin and P. Przystałka, “Detection and localization of delaminations in composite beams using fractional B-spline wavelets with optimized parameters,” Eksploatacja i Niezawodnosc—Maintenance and Reliability, vol. 15, no. 3, pp. 391–399, 2014. View at Google Scholar
  64. A. Katunin and P. Przystałka, “Meta-optimization method for wavelet-based damage identification in composite structures,” in Proceedings of the Federated Conference on Computer Science and Information Systems, pp. 429–438, Warsaw, Poland, September 2014. View at Publisher · View at Google Scholar
  65. C. Sujatha, Vibration and Acoustics: Measurement and Signal Analysis, Tata McGraw Hill Education Private, New Delhi, India, 2010.
  66. C. Smith, C. M. Akujuobi, P. Hamory, and K. Kloesel, “An approach to vibration analysis using wavelets in an application of aircraft health monitoring,” Mechanical Systems and Signal Processing, vol. 21, no. 3, pp. 1255–1272, 2007. View at Publisher · View at Google Scholar · View at Scopus
  67. D. M. Siringoringo and Y. Fujino, “Experimental study of laser Doppler vibrometer and ambient vibration for vibration-based damage detection,” Engineering Structures, vol. 28, no. 13, pp. 1803–1815, 2006. View at Publisher · View at Google Scholar · View at Scopus
  68. P. Qiao, W. Lestari, M. G. Shah, and J. Wang, “Dynamics-based damage detection of composite laminated beams using contact and noncontact measurement systems,” Journal of Composite Materials, vol. 41, no. 10, pp. 1217–1252, 2007. View at Publisher · View at Google Scholar · View at Scopus
  69. M. Radzieński, M. Krawczuk, and M. Palacz, “Improvement of damage detection methods based on experimental modal parameters,” Mechanical Systems and Signal Processing, vol. 25, no. 6, pp. 2169–2190, 2011. View at Publisher · View at Google Scholar · View at Scopus
  70. M. Rucka, “Damage detection in beams using wavelet transform on higher vibration modes,” Journal of Theoretical and Applied Mechanics, vol. 49, no. 2, pp. 399–417, 2011. View at Google Scholar · View at Scopus
  71. M. Cao, L. Cheng, Z. Su, and H. Xu, “A multi-scale pseudo-force model in wavelet domain for identification of damage in structural components,” Mechanical Systems and Signal Processing, vol. 28, pp. 638–659, 2012. View at Publisher · View at Google Scholar · View at Scopus
  72. M. Cao, M. Radzieński, W. Xu, and W. Ostachowicz, “Identification of multiple damage in beams based on robust curvature mode shapes,” Mechanical Systems and Signal Processing, vol. 46, no. 2, pp. 468–480, 2014. View at Publisher · View at Google Scholar · View at Scopus
  73. W. Xu, M. Cao, M. Radzieński et al., “Detecting multiple small-sized damage in beam-type structures by Teager energy of modal curvature shape,” Journal of Vibroengineering, vol. 17, no. 1, pp. 275–286, 2015. View at Google Scholar
  74. W. Xu, M. Cao, W. Ostachowicz, M. Radzieński, and N. Xia, “Two-dimensional curvature mode shape method based on wavelets and Teager energy for damage detection in plates,” Journal of Sound and Vibration, vol. 347, pp. 266–278, 2015. View at Publisher · View at Google Scholar
  75. J. Shang, Y. He, D. Liu, H. Zang, and W. Chen, “Laser Doppler vibrometer for real-time speech-signal acquirement,” Chinese Optics Letters, vol. 7, no. 8, pp. 732–733, 2009. View at Publisher · View at Google Scholar · View at Scopus
  76. W. Ostachowicz, P. Kudela, P. Malinowski, and T. Wandowski, “Damage localisation in plate-like structures based on PZT sensors,” Mechanical Systems and Signal Processing, vol. 23, no. 6, pp. 1805–1829, 2009. View at Publisher · View at Google Scholar · View at Scopus
  77. Z. Su, L. Ye, and X. Bu, “A damage identification technique for CF/EP composite laminates using distributed piezoelectric transducers,” Composite Structures, vol. 57, no. 1-4, pp. 465–471, 2002. View at Publisher · View at Google Scholar · View at Scopus
  78. L. Yu and V. Giurgiutiu, “In situ 2-D piezoelectric wafer active sensors arrays for guided wave damage detection,” Ultrasonics, vol. 48, no. 2, pp. 117–134, 2008. View at Publisher · View at Google Scholar · View at Scopus
  79. Y. J. Yan and L. H. Yam, “Online detection of crack damage in composite plates using embedded piezoelectric actuators/sensors and wavelet analysis,” Composite Structures, vol. 58, no. 1, pp. 29–38, 2002. View at Publisher · View at Google Scholar · View at Scopus
  80. P. Qiao, K. Lu, W. Lestari, and J. Wang, “Curvature mode shape-based damage detection in composite laminated plates,” Composite Structures, vol. 80, no. 3, pp. 409–428, 2007. View at Publisher · View at Google Scholar · View at Scopus
  81. Y. Wang, S. Yuan, and L. Qiu, “Improved wavelet-based spatial filter of damage imaging method on composite structures,” Chinese Journal of Aeronautics, vol. 24, no. 5, pp. 665–672, 2011. View at Publisher · View at Google Scholar · View at Scopus
  82. V. La Saponara, C. Brandli, L. Arronche, and W. Lestari, “Gabor wavelet transform contours for the detection of uniaxial tensile damage in woven fiberglass/epoxy composites,” Mechanics Research Communications, vol. 62, pp. 138–145, 2014. View at Publisher · View at Google Scholar
  83. W. J. Staszewski, “Intelligent signal processing for damage detection in composite materials,” Composites Science and Technology, vol. 62, no. 7-8, pp. 941–950, 2002. View at Publisher · View at Google Scholar · View at Scopus
  84. Y. Huang, D. Meyer, and S. Nemat-Nasser, “Damage detection with spatially distributed 2D Continuous Wavelet Transform,” Mechanics of Materials, vol. 41, no. 10, pp. 1096–1107, 2009. View at Publisher · View at Google Scholar · View at Scopus
  85. A. Panopoulou, T. Loutas, D. Roulias, S. Fransen, and V. Kostopoulos, “Dynamic fiber Bragg gratings based health monitoring system of composite aerospace structures,” Acta Astronautica, vol. 69, no. 7-8, pp. 445–457, 2011. View at Publisher · View at Google Scholar · View at Scopus
  86. S. Lu, M. Jiang, Q. Sui, Y. Sai, and L. Jia, “Low velocity impact localization system of CFRP using fiber Bragg grating sensors,” Optical Fiber Technology, vol. 21, pp. 13–19, 2015. View at Publisher · View at Google Scholar
  87. B. Li, Z. Li, J. Zhou, L. Ye, and E. Li, “Damage localization in composite lattice truss core sandwich structures based on vibration characteristics,” Composite Structures, vol. 126, pp. 34–51, 2015. View at Publisher · View at Google Scholar
  88. S.-T. Quek, Q. Wang, L. Zhang, and K.-K. Ang, “Sensitivity analysis of crack detection in beams by wavelet technique,” International Journal of Mechanical Sciences, vol. 43, no. 12, pp. 2899–2910, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  89. A. V. Ovanesova and L. E. Suárez, “Applications of wavelet transforms to damage detection in frame structures,” Engineering Structures, vol. 26, no. 1, pp. 39–49, 2004. View at Publisher · View at Google Scholar · View at Scopus
  90. S. Zhong and S. O. Oyadiji, “Sampling interval sensitivity analysis for crack detection by stationary wavelet transform,” Structural Control and Health Monitoring, vol. 20, no. 1, pp. 45–69, 2013. View at Publisher · View at Google Scholar · View at Scopus
  91. M. Feilner, D. van de Ville, and M. Unser, “An orthogonal family of quincunx wavelets with continuously adjustable order,” IEEE Transactions on Image Processing, vol. 14, no. 4, pp. 499–510, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  92. D. Van De Ville, T. Blu, and M. Unser, “Isotropic polyharmonic B-splines: scaling functions and wavelets,” IEEE Transactions on Image Processing, vol. 14, no. 11, pp. 1798–1813, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  93. V. Strela, Multiwavelet theory and applications [Ph.D. thesis], Massachusetts Institute of Technology, Cambridge, Mass, USA, 1996.
  94. M. Solís, M. Algaba, and P. Galvín, “Continuous wavelet analysis of mode shapes differences for damage detection,” Mechanical Systems and Signal Processing, vol. 40, no. 2, pp. 645–666, 2013. View at Publisher · View at Google Scholar · View at Scopus
  95. D. Černá, V. Finĕk, M. Gottfried et al., “Boundary artifact reduction in wavelet image compression,” in Mezinárodní Konference Technical Computing Prague, Prague, Czech Republic, 2009.
  96. P. Shui and Z. Bao, “Interval interpolating wavelets with robust boundary filters,” Science in China. Series E. Technological Sciences, vol. 43, no. 3, pp. 287–296, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  97. A. Messina, “Refinements of damage detection methods based on wavelet analysis of dynamical shapes,” International Journal of Solids and Structures, vol. 45, no. 14-15, pp. 4068–4097, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  98. L. Montanari, B. Basu, A. Spagnoli, and B. M. Broderick, “A padding method to reduce edge effects for enhanced damage identification using wavelet analysis,” Mechanical Systems and Signal Processing, vol. 52-53, pp. 264–277, 2015. View at Publisher · View at Google Scholar
  99. A. Katunin, K. Dragan, and M. Dziendzikowski, “Damage identification in aircraft composite structures: a case study using various non-destructive testing techniques,” Composite Structures, vol. 127, pp. 1–9, 2015. View at Publisher · View at Google Scholar
  100. B. Grübner, W. Hufenbach, R. Gottwald, M. Lepper, and B. Zhou, “Experimental and numerical validation of an analytical calculation method for notched fibre-reinforced multilayered composites under bending and compressive loads,” in Proceedings of the 19th International Conference on Composite Materials, Montreal, Canada, 2013.
  101. A. Katunin, M. Dańczak, and P. Kostka, “Automated identification and classification of internal defects in composite structures using computed tomography and 3D wavelet analysis,” Archives of Civil and Mechanical Engineering, vol. 15, no. 2, pp. 436–448, 2015. View at Publisher · View at Google Scholar