On modelling three-dimensional elastodynamic wave propagation with boundary spectral element method
Keywords:
boundary integral equation, wave propagation, spectral element method, Boundary element method, Lobatto element, Gauss–Legendre element, Chebyshev elementAbstract
In this paper, a boundary spectral element method (BSEM) for solving the problem of three-dimensional wave propagation is introduced. In the new formulation, elastodynamics of structures is computed by the Laplace transformed boundary element method (BEM), and boundaries of structures are discretised into high-order isoparametric spectral elements. Three types of spectral elements – Lobatto, Gauss–Legendre and Chebyshev elements – have been implemented. With a significantly higher computational efficiency than the conventional BEM, the BSEM provides a competitive alternative for modelling high-frequency wave propagation in engineering applications.
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References
Aliabadi, M. H. (2002). The boundary element method. Volume 2, applications in solids and
structures. Wiley: United Kingdom.
Aliabadi, M. H., Hall, W. S., & Hibbs, T. T. (1987). Exact singularity cancelling for the
potential kernel in the boundary element method. Communications in Applied
Numerical Methods, 3(2), 123–128.
F. ZOU AND M. H. ALIABADI
Benedetti, I., Aliabadi, M., & Davi, G. (2008). A fast 3D dual boundary element method
based on hierarchical matrices. International Journal of Solids and Structures, 45(7),
–2376.
Benedetti, I., & Aliabadi, M. H. (2010). A fast hierarchical dual boundary element method
for three-dimensional elastodynamic crack problems. International Journal for
Numerical Methods in Engineering, 84(9), 1038–1067.
Dauksher, W., & Emery, A. F. (1997). Accuracy in modeling the acoustic wave equation with
Chebyshev spectral finite elements. Finite Elements in Analysis and Design, 26(2), 115–128.
Dominguez, J. (1993). Boundary elements in dynamics: Wit Press. https://www.witpress.
com/books/978-1-85312-258-3
Durbin, F. (1974). Numerical inversion of Laplace transforms: An efficient improvement to
Dubner and Abate’s method. The Computer Journal, 17(4), 371–376.
Eringen, A. C., & Suhubi, E. S. (2013). Linear theory: Academic press. https://www.elsevier.
com/books/linear-theory/eringen/978-0-12-240602-7
Giurgiutiu, V. (2005). Tuned lamb wave excitation and detection with piezoelectric wafer
active sensors for structural health monitoring. Journal of Intelligent Material Systems
and Structures, 16(4), 291–305.
Gottlieb, D., & Orszag, S. A. (1977). Numerical analysis of spectral methods: Theory and
applications (Vol. 26). Siam. https://epubs.siam.org/doi/book/10.1137/1.9781611970425
Gottlieb, D., & Orszag, S. A. (1983). Numerical analysis of spectral methods: Theory and
applications: Society for industrial and applied mathematics. https://epubs.siam.org/doi/
book/10.1137/1.9781611970425
Ha, S., & Chang, F.-K. (2010). Optimizing a spectral element for modeling PZT-induced
lamb wave propagation in thin plates. Smart Materials and Structures, 19(1), 015015.
Haywood, J., Coverley, P., Staszewski, W. J., & Worden, K. (2004). An automatic impact
monitor for a composite panel employing smart sensor technology. Smart Materials and
Structures, 14(1), 265.
Ihn, J.-B., & Chang, F.-K. (2004). Detection and monitoring of hidden fatigue crack growth
using a built-in piezoelectric sensor/actuator network: I. diagnostics. Smart Materials
and Structures, 13(3), 609.
Kim, Y., Ha, S., & Chang, F.-K. (2008). Time-domain spectral element method for built-in
piezoelectric-actuator-induced lamb wave propagation analysis. AIAA Journal, 46(3), 591–
Komatitsch, D., Barnes, C., & Tromp, J. (2000). Simulation of anisotropic wave propagation
based upon a spectral element method. Geophysics, 65(4), 1251–1260.
Komatitsch, D., & Tromp, J. (1999). Introduction to the spectral element method for threedimensional
seismic wave propagation. Geophysical Journal International, 139(3), 806–822.
Komatitsch, D., & Tromp, J. (2002). Spectral-element simulations of global seismic wave
propagation—I. Validation. Geophysical Journal International, 149(2), 390–412.
Komatitsch, D., & Vilotte, J.-P. (1998). The spectral element method: An efficient tool to
simulate the seismic response of 2D and 3D geological structures. Bulletin of the
Seismological Society of America, 88(2), 368–392.
Lachat, J., & Watson, J. (1976). Effective numerical treatment of boundary integral
equations: A formulation for three-dimensional elastostatics. International Journal for
Numerical Methods in Engineering, 10(5), 991–1005.
Mallardo, V., & Aliabadi, M. (2012). An adaptive fast multipole approach to 2D wave
propagation. Computer Modeling in Engineering and Sciences, 87(2), 77.
Mallardo, V., Aliabadi, M., & Khodaei, Z. S. (2013). Optimal sensor positioning for impact
localization in smart composite panels. Journal of Intelligent Material Systems and
Structures, 24(5), 559–573.
EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 225
Mallardo, V., Sharif Khodaei, Z., & Aliabadi, F. M. (2016). A Bayesian approach for sensor
optimisation in impact identification. Materials, 9(11), 946.
Muldowney, G., & Higdon, J. (1995). A spectral boundary element approach to threedimensional
Stokes flow. Journal of Fluid Mechanics, 298, 167–192.
Occhialini, J., Muldowney, G., & Higdon, J. (1992). Boundary integral/spectral element
approaches to the Navier–Stokes equations. International Journal for Numerical
Methods in Fluids, 15(12), 1361–1381.
Patera, A. T. (1984). A spectral element method for fluid dynamics: Laminar flow in a
channel expansion. Journal of Computational Physics, 54(3), 468–488.
Peng, H., Meng, G., & Li, F. (2009). Modeling of wave propagation in plate structures
using three-dimensional spectral element method for damage detection. Journal of
Sound and Vibration, 320(4), 942–954.
Rigby, R. (1995). Boundary element analysis of cracks in aerospace structures. PhD Thesis,
Wessex Institute of Technology, University of Portsmouth.[Links].
Sharif-Khodaei, Z., & Aliabadi, M. H. (2014). Assessment of delay-and-sum algorithms for
damage detection in aluminium and composite plates. Smart Materials and Structures, 23
(7), 075007.
Shen, L., & Liu, Y. (2007). An adaptive fast multipole boundary element method for threedimensional
acoustic wave problems based on the Burton–Miller formulation.
Computational Mechanics, 40(3), 461–472.
Wen, P. H., Aliabadi, M. H., & Rooke, D. P. (1998). Cracks in three dimensions: A
dynamic dual boundary element analysis. Computer Methods in Applied Mechanics
and Engineering, 167(1), 139–151.
Worden, K., & Staszewski, W. (2000). Impact location and quantification on a composite
panel using neural networks and a genetic algorithm. Strain, 36(2), 61–68.
Żak, A. (2009). A novel formulation of a spectral plate element for wave propagation in
isotropic structures. Finite Elements in Analysis and Design, 45(10), 650–658.
Zhao, X. (2004). An efficient approach for the numerical inversion of Laplace transform
and its application in dynamic fracture analysis of a piezoelectric laminate. International
Journal of Solids and Structures, 41(13), 3653–3674.
Zienkiewicz, O. C., & Taylor, R. L. (2000). The finite element method: Solid mechanics (Vol.
. Butterworth-heinemann. https://www.amazon.com/Finite-Element-Method-Solid-
Mechanics/dp/0470395052
Zou, F., & Aliabadi, M. (2017). On modelling three-dimensional piezoelectric smart structures
with boundary spectral element method. Smart Materials and Structures, 26(5), 055015.
Zou, F., & Aliabadi, M. H. (2015). A boundary element method for detection of damages
and self-diagnosis of transducers using electro-mechanical impedance. Smart Materials
and Structures, 24, 9.
Zou, F., Benedetti, I., & Aliabadi, M. H. (2014). A boundary element model for structural
health monitoring using piezoelectric transducers. Smart Materials and Structures, 23
(1), 015022.
