Convergence analysis of the spectral element method on the example of the Lamb problem in comparison with the analytical solution
https://doi.org/10.22405/2226-8383-2026-27-1-153-165
Abstract
The convergence analysis of the spectral element method (one of the modern modifications of the finite element method) for the dynamic problem of elasticity theory is performed by comparing the numerical solution with the analytical solution of the Lamb problem — the
problem of dynamic action on the boundary of a half-plane or half-space by a concentrated or distributed load changing according to some time law. The article considers the effect on the boundary of a load changing according to the Berlage time law. The calculations are
performed using the domestic strength software package “Fidesys”. Stress distribution graphs for the material under study are given. The dependence of the error of the numerical solution on the order of elements for a fixed number of points per Rayleigh wavelength is investigated.
About the Authors
Vladimir Anatol’evich Levin
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Anatoliy Victorovich Vershinin
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Konstantin Moiseevich Zingerman
Tver State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Evgeny Mikhailovich Ukhanov
Lomonosov Moscow State University
Russian Federation
Russian Federation
student
References
1. Lamb, H. 1904, “On the propagation of Tremors over the Surface of an Elastic Solid”, Philosophical Transactions of the Royal Society of London. Series A, vol. 203, pp. 1–42.
2. Lamb, H. 1916, “On Waves due to a Travelling Disturbance, with an Application to Waves in Super-posed Fluids”, Philosophical Magazine, vol. 31, pp. 386–399, 539–548.
3. Sedov, L.I. 1970, Continuum Mechanics, Vol. 2, Nauka, Moscow, pp. 404-409.
4. Aki, K. & Richards, P.G. 1983, Quantitative Seismology: Theory and Methods, Vol. 1, Mir Publ., Moscow, 520 p.
5. Peregudov, D.V. 2000, “The two-dimensional Lamb problem. Kanyar’s method”, Computational Seismology, no. 31, pp. 120–137.
6. Poruchikov, V.B. 1986, Metody dinamicheskoy teorii uprugosti [Methods of the Dynamic Theory of Elasticity], Nauka Publ., Moscow.
7. Bratov, V.A., Kuznetsov, S.V. & Morozov, N.F. 2022, “Lamb’s problems and related problems of dynamics”, Applied Mathematics and Mechanics, vol. 86, no. 4, pp. 451–469. doi: 10.31857/S003282352204004X.
8. Kausel, E. 2012, “Lamb’s Problem at Its Simplest”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 469, no. 2149, doi: 10.1098/rspa.2012.0462.
9. Strutt, J.W. (Lord Rayleigh) 1885, “On wave propagating along the plane surface of an elastic solid”, Proceedings of the London Mathematical Society, vol. 17, pp. 4–11.
10. Gulizzi, V. & Saye, R. 2022, “Modeling wave propagation in elastic solids via high-order accurate implicit-mesh discontinuous Galerkin methods”, Computer Methods in Applied Mechanics and Engineering, vol. 395, art. id. 114971, doi: 10.1016/j.cma.2022.114971.
11. Levin, V.A. & Vershinin, A.V. (eds.) 2015, Nonlinear Computational Strength Mechanics. Vol. 2. Numerical Methods. Parallel computing [Nelinejnaya vy‘chislitel‘naya mekhanika prochnosti. Tom 2. Chislenny‘e metody‘. Parallel‘ny‘e vy‘chisleniya na E‘VM.], Fizmatlit, Moscow, 544 p.
12. Lee, E.H. 1969, “Elastic-Plastic Deformation at Finite Strains”, Journal of Applied Mechanics, vol. 36, no. 1, pp. 1–6.
13. Levin, V.A., Zingerman, K.M. & Krapivin, K.Y. 2023, “Numerical solution of stress concentration problems in elastic-plastic bodies under the superposition of finite deformations”, Advanced Structured Materials, vol. 198, pp. 305–323, doi: 10.1007/978-3-031-43210-1_18.
14. 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, vol. 88, no. 2, pp. 368–392, doi: 10.1785/BSSA0880020368.
15. Konovalov, D., Vershinin, A., Zingerman, K. & Levin, V. 2017, “The implementation of spectral element method in a CAE system for the solution of elasticity problems on hybrid curvilinear meshes”, Modeling and Simulation in Engineering, vol. 2017, art. id. 1797561, doi: 10.1155/2017/1797561.
Review
For citations:
Levin V.A., Vershinin A.V., Zingerman K.M., Ukhanov E.M. Convergence analysis of the spectral element method on the example of the Lamb problem in comparison with the analytical solution. Chebyshevskii Sbornik. 2026;27(1):153-165. (In Russ.) https://doi.org/10.22405/2226-8383-2026-27-1-153-165
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