Abstract
In this paper, we provide the rate of convergence for approximation of an eigenvalue problem describing vibrations of axisymmetric revolution elastic shells. The eigenvalue problem approximation is constructed by the method of finite elements. We study a system of differential equations with variable coefficients, some of which are non-integrable improperly near 0. To this aim, certain special weighted spaces are used, which were introduced earlier by the author.
Similar content being viewed by others
References
Antoci, F.: Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces. Ricerche Mat. 52, 55–71 (2003)
Arabyan, M.H.: On an eigenvalue problem. YSU Scientific Notes 3, 31–39 (2005). (in Russian)
Arabyan, M.H.: The smoothness of generalized waveforms for the problem of rotation of the shell oscillations depending on certain nonsummable coefficients. Izv. NAS of Armenia Mechanics 69, 28–40 (2016). (in Russian)
Arabyan, M.H.: The space of finite elements in the weighted spaces. Modeling of Artificial Intelligence 12, 180–186 (2016). (in Russian)
Arabyan, M.H.: On the existence of solutions of two optimization problems. J. Optim. Theory Appl. 177, 291–305 (2018)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (2010)
Ciarlet, P.h.G.: Mathematical Elasticity, Theory of Shells. Elsevier, Amsterdam (2000)
Ciarlet, P.h.G.: The Finite Element Method for Elliptic Problems. SIAM, Paris (2002)
Chapelle, D., Bathe, K.-J.: The Finite Element Analysis of Shells - Fundamentals. Springer, Berlin (2003)
Flugge, W.: Statik Und Dynamik Der Schalen. Springer, Berlin (1962)
Grigolyuk, E.I.: Nonlinear oscillations and stability of shallow rods and shells. Izv. AS of USSR, Dep. Tech. Sciences 3, 33–68 (1955). (in Russian)
Kudryavtsev, L.D., Nikol’skii, S.M.: Spaces of differentiable functions of several variables and imbedding theorems. Encycl. Math. Sci. 26, 1–140 (1990). Springer
Kufner, A.: Boundary value problems in weighted spaces. Equadiff 6, 35–48 (2006)
Larson, M.G., Bengzon, F.: The Finite Element Method: Theory, Implementation and Applications. Springer, Berlin (2013)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary-Value Problems and Applications. Springer, Berlin (1972)
Necas, J.: Sobolev Spaces and Embedding Theorems. Academia, Prague (1967)
Okereki, M., Keates, S.: Finite Element Applications. A Practical Guide to the FEM Process. Springer, New York (2018)
Sanchez Hubert, J., Sanchez Palencia, E.: Vibrations and Coupling of Continuous Systems. Springer, Berlin (1989)
Timoshenko, S.: Theory of Plates and Shells. McGraw-Hill, New York (1959)
Timoshenko, S.: Strength of Materials. Part 1: Elementary Theory and Problems. Krieger Publishing Company, Melbourne (1976)
Timoshenko, S.: Strength of Materials. Part 2: Advanced Theory and Problems. Krieger Publishing Company, Melbourne (1976)
Whiteley, J.: Finite Element Methods a Practical Guide. Springer, Berlin (2017)
Zienkiewicz, O., Taylor, R.: The Finite Element Method, 7th edn. Elsevier, Amsterdam (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Francesca Rapetti
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Arabyan, M. On the rate of convergence for approximation of an eigenvalue problem describing vibrations of axisymmetric revolution elastic shells. Adv Comput Math 46, 14 (2020). https://doi.org/10.1007/s10444-020-09775-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-020-09775-1