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Mathematical Problems in Engineering
Volume 2015, Article ID 260641, 8 pages
http://dx.doi.org/10.1155/2015/260641
Research Article

High-Order Spectral Finite Elements in Analysis of Collinear Wave Mixing

1School of Civil Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
2Key Laboratory of Highway Engineering of Sichuan Province, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
3College of Aerospace Engineering, Chongqing University, Chongqing 400044, China
4Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China

Received 8 October 2015; Revised 25 November 2015; Accepted 26 November 2015

Academic Editor: Roman Lewandowski

Copyright © 2015 Changfa Ai et al. 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. A. J. Croxford, P. D. Wilcox, B. W. Drinkwater, and P. B. Nagy, “The use of non-collinear mixing for nonlinear ultrasonic detection of plasticity and fatigue,” The Journal of the Acoustical Society of America, vol. 126, no. 5, pp. EL117–EL122, 2009. View at Publisher · View at Google Scholar · View at Scopus
  2. A. Demčenko, L. Mainini, and V. A. Korneev, “A study of the noncollinear ultrasonic-wave-mixing technique under imperfect resonance conditions,” Ultrasonics, vol. 57, pp. 179–189, 2015. View at Publisher · View at Google Scholar · View at Scopus
  3. M. Liu, G. Tang, L. J. Jacobs, and J. Qu, “Measuring acoustic nonlinearity parameter using collinear wave mixing,” Journal of Applied Physics, vol. 112, no. 2, Article ID 024908, 2012. View at Publisher · View at Google Scholar · View at Scopus
  4. G. Tang, M. Liu, L. J. Jacobs, and J. Qu, “Detecting plastic strain distribution by a nonlinear wave mixing method,” in Review of Progress in Quantitative Nondestructive Evaluation, vol. 32, American Institute of Physics, College Park, Md, USA, 2013. View at Google Scholar
  5. G. Tang, Wave Propagation in Nonlinear Media and Its Applications in Nondestructive Damage Assessment of Metallic Materials, Northwestern University, 2013.
  6. Z. Chen, G. Tang, Y. Zhao, L. J. Jacobs, and J. Qu, “Mixing of collinear plane wave pulses in elastic solids with quadratic nonlinearity,” Journal of the Acoustical Society of America, vol. 136, no. 5, pp. 2389–2404, 2014. View at Publisher · View at Google Scholar · View at Scopus
  7. Y. Zhao, Z. Chen, P. Cao, and Y. Qiu, “Experiment and FEM study of one-way mixing of elastic waves with quadratic nonlinearity,” NDT & E International, vol. 72, pp. 33–40, 2015. View at Publisher · View at Google Scholar
  8. T. Liszka and J. Orkisz, “The finite difference method at arbitrary irregular grids and its application in applied mechanics,” Computers and Structures, vol. 11, no. 1-2, pp. 83–95, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. W. Hackbusch, Multi-Grid Methods and Applications, vol. 4, Springer, Berlin, Germany, 1985.
  10. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method: Solid Mechanics, vol. 2, Butterworth-Heinemann, 2000.
  11. J. N. Reddy, An Introduction to the Finite Element Method, vol. 2, McGraw-Hill, New York, NY, USA, 1993.
  12. Y. Cho and J. L. Rose, “A boundary element solution for a mode conversion study on the edge reflection of Lamb waves,” Journal of the Acoustical Society of America, vol. 99, no. 4, pp. 2097–2109, 1996. View at Publisher · View at Google Scholar · View at Scopus
  13. P. K. Banerjee and R. Butterfield, Boundary Element Methods in Engineering Science, vol. 17, McGraw-Hill, London, UK, 1981. View at MathSciNet
  14. A. Bergamini and F. Biondini, “Finite strip modeling for optimal design of prestressed folded plate structures,” Engineering Structures, vol. 26, no. 8, pp. 1043–1054, 2004. View at Publisher · View at Google Scholar · View at Scopus
  15. Y. K. Cheung and L. Tham, The Finite Strip Method, vol. 17, CRC Press, 1997.
  16. Y. Sohn and S. Krishnaswamy, “Mass spring lattice modeling of the scanning laser source technique,” Ultrasonics, vol. 39, no. 8, pp. 543–551, 2002. View at Publisher · View at Google Scholar · View at Scopus
  17. H. Yim and Y. Sohn, “Numerical simulation and visualization of elastic waves using mass-spring lattice model,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 47, no. 3, pp. 549–558, 2000. View at Publisher · View at Google Scholar · View at Scopus
  18. P. P. Delsanto, T. Whitcombe, H. H. Chaskelis, and R. B. Mignogna, “Connection machine simulation of ultrasonic wave propagation in materials. I: the one-dimensional case,” Wave Motion, vol. 16, no. 1, pp. 65–80, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. B. C. Lee and W. J. Staszewski, “Modelling of Lamb waves for damage detection in metallic structures. Part I. Wave propagation,” Smart Materials and Structures, vol. 12, no. 5, pp. 804–814, 2003. View at Publisher · View at Google Scholar · View at Scopus
  20. S. Ham and K.-J. Bathe, “A finite element method enriched for wave propagation problems,” Computers & Structures, vol. 94-95, pp. 1–12, 2012. View at Publisher · View at Google Scholar · View at Scopus
  21. I. Babuška, F. Ihlenburg, T. Strouboulis, and S. K. Gangaraj, “A posteriori error estimation for finite element solutions of Helmholtz' equation. Part I: the quality of local indicators and estimators,” International Journal for Numerical Methods in Engineering, vol. 40, no. 18, pp. 3443–3462, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. I. Babuška, F. Ihlenburg, T. Strouboulis, and S. K. Gangaraj, “A posteriori error estimation for finite element solutions of Helmholtz' equation. Part II. Estimation of the pollution error,” International Journal for Numerical Methods in Engineering, vol. 40, no. 21, pp. 3883–3900, 1997. View at Google Scholar · View at MathSciNet · View at Scopus
  23. F. Moser, L. J. Jacobs, and J. Qu, “Modeling elastic wave propagation in waveguides with the finite element method,” NDT & E International, vol. 32, no. 4, pp. 225–234, 1999. View at Publisher · View at Google Scholar · View at Scopus
  24. W. Ostachowicz, P. Kudela, M. Krawczuk, and A. Zak, Guided Waves in Structures for SHM: The Time-Domain Spectral Element Method, John Wiley & Sons, 2011.
  25. A. T. Patera, “A spectral element method for fluid dynamics: laminar flow in a channel expansion,” Journal of Computational Physics, vol. 54, no. 3, pp. 468–488, 1984. View at Publisher · View at Google Scholar · View at Scopus
  26. Y. C. Hu, K. Y. Sze, and Y. X. Zhou, “Stabilized plane and axisymmetric Lobatto finite element models,” Computational Mechanics, vol. 56, no. 5, pp. 879–903, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. P. Kudela, M. Krawczuk, and W. Ostachowicz, “Wave propagation modelling in 1D structures using spectral finite elements,” Journal of Sound and Vibration, vol. 300, no. 1-2, pp. 88–100, 2007. View at Publisher · View at Google Scholar · View at Scopus
  28. R. Sridhar, A. Chakraborty, and S. Gopalakrishnan, “Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method,” International Journal of Solids and Structures, vol. 43, no. 16, pp. 4997–5031, 2006. View at Publisher · View at Google Scholar · View at Scopus
  29. M. Palacz and M. Krawczuk, “Analysis of longitudinal wave propagation in a cracked rod by the spectral element method,” Computers & Structures, vol. 80, no. 24, pp. 1809–1816, 2002. View at Publisher · View at Google Scholar · View at Scopus
  30. M. Krawczuk, M. Palacz, and W. Ostachowicz, “The dynamic analysis of a cracked Timoshenko beam by the spectral element method,” Journal of Sound and Vibration, vol. 264, no. 5, pp. 1139–1153, 2003. View at Publisher · View at Google Scholar · View at Scopus
  31. D. Roy Mahapatra and S. Gopalakrishnan, “A spectral finite element model for analysis of axial-flexural-shear coupled wave propagation in laminated composite beams,” Composite Structures, vol. 59, no. 1, pp. 67–88, 2003. View at Publisher · View at Google Scholar · View at Scopus
  32. P. Kudela, A. Zak, M. Krawczuk, and W. Ostachowicz, “Modelling of wave propagation in composite plates using the time domain spectral element method,” Journal of Sound and Vibration, vol. 302, no. 4-5, pp. 728–745, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  33. D. Komatitsch, R. Martin, J. Tromp, M. A. Taylor, and B. A. Wingate, “Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles,” Journal of Computational Acoustics, vol. 9, no. 2, pp. 703–718, 2001. View at Publisher · View at Google Scholar · View at Scopus
  34. C. S. Rekatsinas, C. V. Nastos, T. C. Theodosiou, and D. A. Saravanos, “A time-domain high-order spectral finite element for the simulation of symmetric and anti-symmetric guided waves in laminated composite strips,” Wave Motion, vol. 53, pp. 1–19, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. J. M. Carcione, D. Kosloff, A. Behle, and G. Seriani, “Spectral scheme for wave propagation simulation in 3-D elastic-anisotropic media,” Geophysics, vol. 57, no. 12, pp. 1593–1607, 1992. View at Publisher · View at Google Scholar · View at Scopus
  36. G. Seriani, “3-D large-scale wave propagation modeling by spectral element method on Cray T3E multiprocessor,” Computer Methods in Applied Mechanics and Engineering, vol. 164, no. 1-2, pp. 235–247, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  37. D. Komatitsch and J. Tromp, “Introduction to the spectral element method for three-dimensional seismic wave propagation,” Geophysical Journal International, vol. 139, no. 3, pp. 806–822, 1999. View at Publisher · View at Google Scholar · View at Scopus
  38. J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral-element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 1, pp. 437–444, 2006. View at Publisher · View at Google Scholar · View at Scopus
  39. H. Peng, G. Meng, and F. Li, “Modeling of wave propagation in plate structures using three-dimensional spectral element method for damage detection,” Journal of Sound and Vibration, vol. 320, no. 4-5, pp. 942–954, 2009. View at Publisher · View at Google Scholar · View at Scopus
  40. M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, vol. 237, Academic Press, San Diego, Calif, USA, 1998.
  41. D. D. Muir, One-sided ultrasonic determination of third order elastic constants using angle-beam acoustoelasticity measurements [Ph.D. thesis], Georgia Institute of Technology, Atlanta, Ga, USA, 2009.