Implementation of a hybrid numerical-analytical approach for solving the problems of SH-wave diffraction by arbitrary-shaped obstacles
UDC
539.3:534.1DOI:
https://doi.org/10.31429/vestnik-17-2-49-56Abstract
Guided waves propagation and diffraction are considered in elastic waveguides with arbitrary shaped local inhomogeneities. Both analytical and numerical approaches are widely used to calculate and analyze displacement fields. This paper presents the implementation of a hybrid numerical-analytical approach. A numerical approach is utilized in the internal domain containing local inhomogeneity. It is based on the decomposition by the set of basic solutions obtained by the finite element method. Meanwhile, the solution in the outer semi-infinite domains has an analytical form and is based on the decomposition by the normal modes. Unknown decomposition coefficients are derived from the imposed interface conditions between the domains. The numerical computation of the basis finite-element solutions can be carried out on a specialized software. Nevertheless, a more efficient implementation is proposed based on the fact that multiple basic finite element problems share a single global stiffness matrix, which can be inverted once. The algorithm is tested by comparison with the COMSOL model which utilises perfectly matched layers. The developed package allows not only to calculate the displacement fields in the local domain containing the inhomogeneity, but also to obtain analytical representations of the waves travelling to infinity, which yield a physically descriptive representation of the solution. In addition, the presented package allows a parametric analysis of the dependence of the wave field characteristics on various medium properties and inhomogeneity geometry.
Keywords:
guided waves, unbounded waveguide, arbitrary local inhomogeneities, finite element method, modal decomposition, local-global solution, wave energyAcknowledgement
References
- Bogolyubov A.N., Bogolyubov N.A., Sveshnikov A.G. Matematicheskoye modelirovaniye volnovedushchikh sistem metodom konechnykh raznostey i konechnykh elementov [Mathematical modeling of waveguide systems by the method of finite differences and finite elements]. Fizicheskiye osnovy priborostroyeniya [Physical Foundations of Instrument Making], 2013, vol. 2, no. 1. pp. 10–17. DOI: 10.25210/jfop-1301-010017 (In Russian)
- Giurgiutiu V. Structural Health Monitoring with Piezoelectric Wafer Active Sensors (2nd Edition). Elsevier Academic Press, 2014. DOI: 10.1016/C2013-0-00155-7
- Atluri S.N., Shen Sh. The meshless local Petrov–Galerkin (MLPG) method: a simple and less costly alternative to the finite element and boundary element methods. Comput Model. Eng. Sci., 2002, vol. 3, no. 1, pp. 11–51. DOI: 10.3970/cmes.2002.003.011
- Brebbia C.A., Telles J.C.F., Wrobel L.C. Boundary Element Techniques: Theory and Applications in Engineering. Berlin etc.: Springer, 1984. DOI: 10.1007/978-3-642-48860-3
- Glushkov E.V., Glushkova N.V., Eremin A.A., Mikhaskiv V.V. The layered element method in the dynamic theory of elasticity. J. Applied Mathematics and Mechanics, 2009, vol. 73, iss. 4, pp. 449–456. DOI: 10.1016/j.jappmathmech.2009.08.005
- Shen Y., Giurgiutiu V. Effective non-reflective boundary for Lamb waves: Theory, finite element implementation, and applications. Wave Motion, 2015, vol. 58, pp. 22–41. DOI: 10.1016/j.wavemoti.2015.05.009
- Berenger J.-P. A perfectly matched layer for the absorption of electromagnetic waves. J. Computational Physics, 1994, vol. 114(2), pp. 185–200. DOI: 10.1006/jcph.1994.1159
- Joly P. An elementary introduction to the construction and the analysis of perfectly matched layers for time domain wave propagation. SeMA J., 2012, vol. 57, pp. 5–48. DOI: 10.1007/BF03322599
- Glushkov E.V., Glushkova N.V., Evdokimov A.A. Gibridnaya chislenno-analiticheskaya skhema dlya rascheta difraktsii uprugikh voln v lokal'no neodnorodnykh volnovodakh [Hybrid numerical-analytical scheme for calculating the diffraction of elastic waves in locally inhomogeneous waveguides]. Akusticheskiy zhurnal [Acoustic journal], 2018, vol. 64, no. 1, pp. 3–12. DOI: 10.7868/S0320791918010082
- Glushkov E.V., Glushkova N.V., Golub M.V., Moll J., Fritzen C.-P. Wave energy trapping and localization in a plate with a delamination. Smart Materials and Structures J., 2012. vol. 21. no. 12. DOI: 10.1088/0964-1726/21/12/125001
- Glushkov E.V., Glushkova N.V., Eremin A.A., Lammering R. Trapped mode effects in notched plate-like structures Sound and Vibration J., 2015. vol. 358. pp. 142–151. DOI: 10.1016/j.jsv.2015.08.007
Downloads
Submitted
Published
How to Cite
Copyright (c) 2020 Novikov O.I., Evdokimov A.A.
This work is licensed under a Creative Commons Attribution 4.0 International License.
