Hybrid method for modelling anti-plane vibrations of layered waveguides with bonded composite joints
DOI:
https://doi.org/10.7242/1999-6691/2023.16.1.8Keywords:
hybrid method, composite structures, anti-plane vibrations, spectral element method, semi-analytical method, non-destructive evaluation, elastic wavesAbstract
The use of mesh-based methods for modeling elongated composite structures with inhomogeneities leads to an increase in computational costs when discretizing the waveguide part of greater linear dimensions, while the semi-analytical numerical methods do not allow one to directly describe the structures with local inhomogeneities of an arbitrary shape. To compensate for the shortcomings of these two classes of numerical methods, we propose a hybrid scheme based on the spectral element method (SEM) and the semi-analytical finite element method (SAFEM) for studying the anti-plane vibrations of a composite structure in the frequency domain. Thus, for a waveguide, this scheme makes it possible to represent the solution via the SAFEM as a sum of modes or guided waves, and the adjacent regions are discretized using the SEM. The displacement and stress continuity conditions are imposed on the common boundary of two domains. To couple the solutions, we introduce an auxiliary function of displacement, which is approximated by applying the same basis functions as those used in SEM and SAFEM (Lagrange interpolation polynomials on the Gauss–Legendre–Lobatto nodal points). The unknown expansion coefficients of this function are determined by the Galerkin and collocation methods. It has been established that both methods provide the same accuracy. The results obtained by the hybrid approach employing the Galerkin and collocation methods are compared with the results calculated in the standard finite element software. It is shown that they are in good agreement. The presented hybrid approach can be straightforwardly extended to the case of in-plane motion, but it requires significant refinement for a three-dimensional case.
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Kaufmann M., Zenkert D., Wennhage P. Integrated cost/weight optimization of aircraft structures. Struct. Multidisc. Optim., 2010, vol. 41, pp. 325-334. https://doi.org/10.1007/s00158-009-0413-1
Rubino F., Nisticò A., Tucci F., Carlone P. Marine application of fiber reinforced composites: A review. J. Mar. Sci. Eng., 2020, vol. 8, 26. https://doi.org/10.3390/jmse8010026
Kupski J., de Freitas S.T. Design of adhesively bonded lap joints with laminated CFRP adherends: Review, challenges and new opportunities for aerospace structures. Compos. Struct., 2021, vol. 268, 113923. https://doi.org/10.1016/j.compstruct.2021.113923
Wang W., de Freitas S.T., Poulis J.A., Zarouchas D. A review of experimental and theoretical fracture characterization of bi-material bonded joints. Compos. B Eng., 2021, vol. 206, 108537. https://doi.org/10.1016/j.compositesb.2020.108537
Philibert M., Soutis C., Gresil M., Yao K. Damage detection in a composite T-joint using guided Lamb waves. Aerospace, 2018, vol. 5, 40. https://doi.org/10.3390/aerospace5020040
Mitra M., Gopalakrishnan S. Guided wave based structural health monitoring: A review. Smart Mater. Struct., 2016, vol. 25, 53001. https://doi.org/10.1088/0964-1726/25/5/053001
Zhuang Y., Kopsaftopoulos F., Dugnani R., Chang F.-K. Integrity monitoring of adhesively bonded joints via an electromechanical impedance-based approach. Struct. Health Monit., 2018, vol. 17, pp. 1031-1045. https://doi.org/10.1177/1475921717732331
Mueller I., Memmolo V., Tschöke K., Moix-Bonet M., Möllenhoff K., Golub M.V., Venkat R.S., Lugovtsova Ye., Eremin A., Moll J. Performance assessment for a guided wave-based SHM system applied to a stiffened composite structure. Sensors, 2022, vol. 22, 7529. https://doi.org/10.3390/s22197529
Burago N.G., Nikitin I.S., Yakushev V.L. Hybrid numerical method with adaptive overlapping meshes for solving nonstationary problems in continuum mechanics. Comput. Math. and Math. Phys., 2016, vol. 56, pp. 1065-1074. https://doi.org/10.1134/S0965542516060105
Lisitsa V., Tcheverda V., Botter C. Combination of the discontinuous Galerkin method with finite differences for simulation of seismic wave propagation. J. Comput. Phys., 2016, vol. 311, pp. 142-157. https://doi.org/10.1016/j.jcp.2016.02.005
Lu J.-F., Liu Y., Feng Q.-S. Wavenumber domain finite element model for the dynamic analysis of the layered soil with embedded structures. Eur. J. Mech. Solid., 2022, vol. 96, 104696. https://doi.org/10.1016/j.euromechsol.2022.104696
Komatitsch D., Vilotte J.-P., Vai R., Castillo-Covarrubias J.M., Sánchez-Sesma F.J. The spectral element method for elastic wave equations – application to 2-D and 3-D seismic problems. Int. J. Numer. Meth. Eng., 1999, vol. 45, pp. 1139-1164. https://doi.org/10.1002/(sici)1097-0207(19990730)45:9<1139::aid-nme617>3.0.co;2-t
Bazhenov V.G., Igumnov L.A. Metody granichnykh integral′nykh uravneniy i granichnykh elementov v reshenii zadach trekhmernoy dinamicheskoy teorii uprugosti s sopryazhennymi polyami [The boundary integral equation method and the boundary element method for three-dimensional elastodynamic problems with conjugate fields]. Moscow, Fizmatlit, 2008. 352 p.
Song C., Wolf J.P. The scaled boundary finite-element method–alias consistent infinitesimal finite-element cell method–for elastodynamics. Comput. Meth. Appl. Mech. Eng., 1997, vol. 147, pp. 329-355. https://doi.org/10.1016/s0045-7825(97)00021-2
Babeshko V.A., Glushkov E.V., Zinchenko Zh.F. Dinamika neodnorodnykh lineyno-uprugikh sred [Dynamics of inhomogeneous linearly elastic media]. Moscow, Nauka, 1989. 343 p.
Vatul’yan A.O. Obratnyye zadachi v mekhanike deformiruyemogo tverdogo tela [Inverse problems in mechanics of solids]. Moscow, Fizmatlit, 2007. 223 p.
Manolis G.D., Dineva P.S., Rangelov T.V., Wuttke F. State-of-the-Art for the BIEM. Seismic wave propagation in non-homogeneous elastic media by boundary elements. Springer, 2017. P. 9-52. https://doi.org/10.1007/978-3-319-45206-7_2
Bartoli I., Marzani A., di Scalea F.L., Viola E. Modeling wave propagation in damped waveguides of arbitrary cross-section. J. Sound Vib., 2006, vol. 295, pp. 685-707. https://doi.org/10.1016/j.jsv.2006.01.021
Vivar-Perez J.M., Duczek S., Gabbert U. Analytical and higher order finite element hybrid approach for an efficient simulation of ultrasonic guided waves I: 2D-analysis. Smart Structures and Systems, 2014, vol. 13, pp. 587-614. https://doi.org/10.12989/sss.2014.13.4.587
Zou F., Aliabadi M.H. A boundary element method for detection of damages and self-diagnosis of transducers using electro-mechanical impedance. Smart Mater. Struct., 2015, vol. 24, 095015. https://doi.org/10.1088/0964-1726/24/9/095015
Glushkov E.V., Glushkova N.V., Evdokimov A.A. Hybrid numerical-analytical scheme for calculating elastic wave diffraction in locally inhomogeneous waveguides. Acoust. Phys., 2018, vol. 64, pp. 1-9. https://doi.org/10.1134/s1063771018010086
Golub M.V., Shpak A.N. Semi-analytical hybrid approach for the simulation of layered waveguide with a partially debonded piezoelectric structure. Appl. Math. Model., 2019, vol. 65, pp. 234-255. https://doi.org/10.1016/j.apm.2018.08.019
Malik M.K., Chronopoulos D., Tanner G. Transient ultrasonic guided wave simulation in layered composite structures using a hybrid wave and finite element scheme. Compos. Struct., 2020, vol. 246, 112376. https://doi.org/10.1016/j.compstruct.2020.112376
Novikov O.I., Evdokimov A.A. Implementation of a hybrid numerical-analytical approach for solving the problems of SH-wave diffraction by arbitrary-shaped obstacles. Ekologicheskiy vestnik nauchnykh tsentrov ChEC – Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2020, vol. 17, no. 2, pp. 49-56. https://doi.org/10.31429/vestnik-17-2-49-56
Shi L., Zhou Y., Wang J.-M., Zhuang M., Liu N., Liu Q.H. Spectral element method for elastic and acoustic waves in frequency domain. J. Comput. Phys., 2016, vol. 327, pp. 19-38. https://doi.org/10.1016/j.jcp.2016.09.036
Bubenchikov A.M., Poponin V.S., Melnikova V.N. The mathematical statement and solution of spatial boundary value problems by means of spectral element method. Vest. Tom. gos. un-ta. Matematika i mekhanika – Mathematics and Mechanics, 2008, no. 3, pp. 70 76.
Golub M.V., Shpak A.N., Buethe I., Fritzen C.-P. Harmonic motion simulation and resonance frequencies determination of a piezoelectric strip-like actuator using high precision finite element method. Vychisl. mekh. splosh. sred – Computational Continuum Mechanics, 2015, vol. 8, no. 4, pp. 397-407.
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