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Research Papers

# A Benchmark Study of Modeling Lamb Wave Scattering by a Through Hole Using a Time-Domain Spectral Element Method

[+] Author and Article Information
Menglong Liu, Fangsen Cui

Institute of High Performance Computing,
A*STAR,
Singapore 138632

David Schmicker

Institute for Mechanics,
Otto-von-Guericke University of Magdeburg,
Magdeburg 39106, Germany

Zhongqing Su

Department of Mechanical Engineering,
The Hong Kong Polytechnic University,
Hong Kong 518000, China

1Corresponding authors.

Manuscript received August 25, 2017; final manuscript received November 1, 2017; published online January 16, 2018. Assoc. Editor: Paul Fromme.

ASME J Nondestructive Evaluation 1(2), 021006 (Jan 16, 2018) (8 pages) Paper No: NDE-17-1078; doi: 10.1115/1.4038722 History: Received August 25, 2017; Revised November 01, 2017

## Abstract

Ultrasonic guided waves (GWs) are being extensively investigated and applied to nondestructive evaluation and structural health monitoring. Guided waves are, under most circumstances, excited in a frequency range up to several hundred kilohertz or megahertz for detecting defect/damage effectively. In this regard, numerical simulation using finite element analysis (FEA) offers a powerful tool to study the interaction between wave and defect/damage. Nevertheless, the simulation, based on linear/quadratic interpolation, may be inaccurate to depict the complex wave mode shape. Moreover, the mass lumping technique used in FEA for diagonalizing mass matrix in the explicit time integration may also undermine the calculation accuracy. In recognition of this, a time domain spectral element method (SEM)—a high-order FEA with Gauss–Lobatto–Legendre (GLL) node distribution and Lobatto quadrature algorithm—is studied to accurately model wave propagation. To start with, a simplified two-dimensional (2D) plane strain model of Lamb wave propagation is developed using SEM. The group velocity of the fundamental antisymmetric mode ($A0$) is extracted as indicator of accuracy, where SEM exhibits a trend of quick convergence rate and high calculation accuracy (0.03% error). A benchmark study of calculation accuracy and efficiency using SEM is accomplished. To further extend SEM-based simulation to interpret wave propagation in structures of complex geometry, a three-dimensional (3D) SEM model with arbitrary in-plane geometry is developed. Three-dimensional numerical simulation is conducted in which the scattering of $A0$ mode by a through hole is interrogated, showing a good match with experimental and analytical results.

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## References

Kundu, T. , 2014, “ Acoustic Source Localization,” Ultrasonics, 54(1), pp. 25–38. [PubMed]
Su, Z. , Ye, L. , and Lu, Y. , 2006, “ Guided Lamb Waves for Identification of Damage in Composite Structures: A Review,” J. Sound Vib., 295(3), pp. 753–780.
Raghavan, A. , and Cesnik, C. E. , 2007, “ Review of Guided-Wave Structural Health Monitoring,” Shock Vib. Dig., 39(2), pp. 91–116.
Qiu, L. , Yuan, S. , Chang, F.-K. , Bao, Q. , and Mei, H. , 2014, “ On-Line Updating Gaussian Mixture Model for Aircraft Wing Spar Damage Evaluation Under Time-Varying Boundary Condition,” Smart Mater. Struct., 23(12), p. 125001.
Qiu, L. , Yuan, S. , Zhang, X. , and Wang, Y. , 2011, “ A Time Reversal Focusing Based Impact Imaging Method and Its Evaluation on Complex Composite Structures,” Smart Mater. Struct., 20(10), p. 105014.
Zhou, C. , Su, Z. , and Cheng, L. , 2011, “ Quantitative Evaluation of Orientation-Specific Damage Using Elastic Waves and Probability-Based Diagnostic Imaging,” Mech. Syst. Signal Process., 25(6), pp. 2135–2156.
Liu, M. , Wang, Q. , Zhang, Q. , Long, R. , and Su, Z. , 2018, “ Characterizing Hypervelocity (>2.5 Km/s)-Impact-Engendered Damage in Shielding Structures Using In-Situ Acoustic Emission: Simulation and Experiment,” Int. J. Impact Eng., 111, pp. 273–284.
Qiu, L. , Liu, M. , Qing, X. , and Yuan, S. , 2013, “ A Quantitative Multidamage Monitoring Method for Large-Scale Complex Composite,” Struct. Health Monit., 12(3), pp. 183–196.
Giurgiutiu, V. , 2007, Structural Health Monitoring: With Piezoelectric Wafer Active Sensors, Academic Press, Cambridge, MA.
Cho, Y. , and Rose, J. L. , 1996, “ A Boundary Element Solution for a Mode Conversion Study on the Edge Reflection of Lamb Waves,” J. Acoust. Soc. Am., 99(4), pp. 2097–2109.
Alleyne, D. N. , and Cawley, P. , 1992, “ The Interaction of Lamb Waves With Defects,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 39(3), pp. 381–397.
Sundararaman, S. , and Adams, D. E. , 2008, “ Modeling Guided Waves for Damage Identification in Isotropic and Orthotropic Plates Using a Local Interaction Simulation Approach,” ASME J. Vib. Acoust., 130(4), p. 041009.
Shen, Y. , and Cesnik, C. E. , 2017, “ Modeling of Nonlinear Interactions Between Guided Waves and Fatigue Cracks Using Local Interaction Simulation Approach,” Ultrasonics, 74, pp. 106–123. [PubMed]
Shen, Y. , and Giurgiutiu, V. , 2016, “ Combined Analytical FEM Approach for Efficient Simulation of Lamb Wave Damage Detection,” Ultrasonics, 69, pp. 116–128. [PubMed]
Patera, A. T. , 1984, “ A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion,” J. Comput. Phys., 54(3), pp. 468–488.
Komatitsch, D. , and Vilotte, J.-P. , 1998, “ The Spectral Element Method: An Efficient Tool to Simulate the Seismic Response of 2D and 3D Geological Structures,” Bull. Seismol. Soc. Am., 88(2), pp. 368–392.
Komatitsch, D. , Erlebacher, G. , Göddeke, D. , and Michéa, D. , 2010, “ High-Order Finite-Element Seismic Wave Propagation Modeling With MPI on a Large GPU Cluster,” J. Comput. Phys., 229(20), pp. 7692–7714.
Kudela, P. , Żak, A. , Krawczuk, M. , and Ostachowicz, W. , 2007, “ Modelling of Wave Propagation in Composite Plates Using the Time Domain Spectral Element Method,” J. Sound Vib., 302(4), pp. 728–745.
Zak, A. , Krawczuk, M. , and Ostachowicz, W. , 2006, “ Propagation of In-Plane Waves in an Isotropic Panel With a Crack,” Finite Elem. Anal. Des., 42(11), pp. 929–941.
Peng, H. , Meng, G. , and Li, F. , 2009, “ Modeling of Wave Propagation in Plate Structures Using Three-Dimensional Spectral Element Method for Damage Detection,” J. Sound Vib., 320(4–5), pp. 942–954.
Żak, A. , and Krawczuk, M. , 2011, “ Certain Numerical Issues of Wave Propagation Modelling in Rods by the Spectral Finite Element Method,” Finite Elem. Anal. Des., 47(9), pp. 1036–1046.
Rose, J. L. , 2014, Ultrasonic Guided Waves in Solid Media, Cambridge University Press, New York.
Taylor, R. L. , 1972, “ On Completeness of Shape Functions for Finite Element Analysis,” Int. J. Numer. Methods Eng., 4(1), pp. 17–22.
Fromme, P. , and Sayir, M. B. , 2002, “ Measurement of the Scattering of a Lamb Wave by a Through Hole in a Plate,” J. Acoust. Soc. Am., 111(3), pp. 1165–1170. [PubMed]

## Figures

Fig. 1

GLL-based node collation in 3D SEM element with the polynomial order pξ=6,pη=5, and pζ=3 in the local coordinate ξ,η,andζ∈[−1,1]

Fig. 2

Lagrangian interpolation with polynomial order pξ=6 of GLL nodes

Fig. 3

Flowchart of development of 3D SEM

Fig. 4

Illustration of SEM element built from abaqus plane element (pξ=3,pη=3,pζ=2, the shown element is represented in local coordinate ξ, η, and ζ∈[−1,1], and in the global coordinate, the element is usually irregular in order to adapt to complex geometry)

Fig. 5

Element discretization and node collation of 2D plane strain SEM model

Fig. 6

Typical displacement response containing A0 and S0 modes (the scale of displacement and coordinate in the in-plane and out-of-plane direction is adjusted for illustration)

Fig. 7

Displacement signal and wave packet at sections A–B and C–D to extract time of arrival tA,B(a) and tC,D(a)

Fig. 8

Benchmark study of calculated error of A0 mode using SEM (nλ=1−20,pξ=1−5): (a)pη=1 and (b) pη=4

Fig. 9

Benchmark study between minimum calculation error and polynomial order in the thickness direction pη

Fig. 10

Comparison of (a) consumed memory and (b) calculation time of A0 mode under 100 kHz using SEM (nλ=1−20,pξ=1−5,pη=4)

Fig. 11

Sketch of A0 mode crossing a through hole

Fig. 12

Amplitude (normalized) on a circle around the cavity based on SEM and CPT: theta direction: ψ, radial direction: normalized amplitude; plate thickness 1 mm, hole radius r0=10mm, signal acquisition at r=13mm, frequency fc=20kHz

Fig. 13

Amplitude (normalized) on a circle around the cavity based on SEM and CPT: theta direction: ψ, radial direction: normalized amplitude; plate thickness 1 mm, hole radius r0=10mm, signal acquisition at r=13mm, frequency fc=100kHz

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