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axisymmetrically loaded thin shell; singular perturbation; balanced norm; layer-adapted meshes; finite element method
We analyse a finite-element discretisation of a differential equation describing an axisymmetrically loaded thin shell. The problem is singularly perturbed when the thickness of the shell becomes small. We prove robust convergence of the method in a balanced norm that captures the layers present in the solution. Numerical results confirm our findings.
[1] Chapelle, D., Bathe, K.-J.: The Finite Element Analysis of Shells --- Fundamentals. Computational Fluid and Solid Mechanics. Springer, Berlin (2011). DOI 10.1007/978-3-642-16408-8 | MR 3234588 | Zbl 1211.74002
[2] Devloo, P. R. B., Farias, A. M., Gomes, S. M., Gonçalves, J. L.: Application of a combined continuous-discontinuous Galerkin finite element method for the solution of the Girkmann problem. Comput. Math. Appl. 65 (2013), 1786-1794. DOI 10.1016/j.camwa.2013.03.015 | MR 3055736 | Zbl 1391.74243
[3] Flügge, W.: Stresses in Shells. Springer, Berlin (1960). DOI 10.1007/978-3-642-88291-3 | MR 0116598 | Zbl 0092.41504
[4] Girkmann, K.: Flächentragwerke: Einführung in die Elastostatik der Scheiben, Platten, Schalen und Faltwerke. Springer, Vienna (1956), German. DOI 10.1007/978-3-7091-2388-1 | MR 0119567 | Zbl 0071.39404
[5] Gol'denveĭzer, A. L.: Theory of Elastic Thin Shells. International Series of Monographs in Aeronautics and Astronautics. Pergamon Press, Oxford (1961). DOI 10.1016/c2013-0-01676-3 | MR 0135763 | Zbl 0052.41901
[6] Olsson, R. Gran, Reissner, E.: A problem of buckling of elastic plates of variable thickness. J. Math. Phys., Mass. Inst. Tech. 19 (1940), 131-139. DOI 10.1002/sapm1940191131 | MR 0001717 | Zbl 0024.09002
[7] Heuer, N., Karkulik, M.: A robust DPG method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 55 (2017), 1218-1242. DOI 10.1137/15M104130 | MR 3654124 | Zbl 1362.65125
[8] Lin, R., Stynes, M.: A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 50 (2012), 2729-2743. DOI 10.1137/11083778 | MR 3022240 | Zbl 1260.65103
[9] Linß, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Lecture Notes in Mathematics 1985. Springer, Berlin (2010). DOI 10.1007/978-3-642-05134-0 | MR 2583792 | Zbl 1202.65120
[10] Miller, J. J. H., O'Riordan, E., Shishkin, G. I.: Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996). DOI 10.1142/2933 | MR 1439750 | Zbl 0915.65097
[11] Morley, L. S. D.: Analysis of developable shells with special reference to the finite element method and circular cylinders. Philos. Trans. roy. Soc. London, Ser. A 281 (1976), 113-170. DOI 10.1098/rsta.1976.0023 | Zbl 0319.73040
[12] Niemi, A. H.: Benchmark computations of stresses in a spherical dome with shell finite elements. SIAM J. Sci. Comput. 38 (2016), B440--B457. DOI 10.1137/15M1027590 | MR 3513868 | Zbl 1419.74239
[13] Niemi, A. H., Babuška, I., Pitkäranta, J., Demkowicz, L.: Finite element analysis of the Girkmann problem using the modern $hp$-version and the classical $h$-version. Engin. Comput. 28 (2012), 123-134. DOI 10.1007/s00366-011-0223-0
[14] R. E. O'Malley, Jr.: Singular perturbations, asymptotic evaluation of integrals, and computational challenges. Asymptotic Analysis and the Numerical Solution of Partial Differential Equations Lecture Notes in Pure and Applied Mathematics 130. Marcel Dekker, New York (1991), 3-16. DOI 10.1201/b16933-3 | MR 1122414 | Zbl 0736.65059
[15] Pitkäranta, J., Babuška, I., Szabó, B.: The dome and the ring: Verification of an old mathematical model for the design of a stiffened shell roof. Comput. Math. Appl. 64 (2012), 48-72. DOI 10.1016/j.camwa.2012.03.008 | MR 2927141 | Zbl 1252.74043
[16] Reissner, E.: Remark on the theory of bending of plates of variable thickness. J. Math. Phys. 16 (1937), 43-45 \99999JFM99999 63.0755.02. DOI 10.1002/sapm193716143
[17] Roos, H. G., Lin{ß}, T.: Sufficient conditions for uniform convergence on layer-adapted grids. Computing 63 (1999), 27-45. DOI 10.1007/s006070050049 | MR 1702159 | Zbl 0931.65085
[18] Schwab, C.: $p$- and $hp$-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation. Clarendon Press, Oxford (1998). MR 1695813 | Zbl 0910.73003
[19] Shishkin, G. I.: Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural Section, Ekaterinburg (1992), Russian.
[20] Tin-Loi, F., Pulmano, V. A., Thambiratnam, D.: BEF analogy for axisymmetrically loaded cylindrical shells. Comput. Struct. 34 (1990), 281-285. DOI 10.1016/0045-7949(90)90371-8 | Zbl 0713.73090
[21] Ventsel, E., Krauthammer, T.: Thin Plates and Shells: Theory, Analysis, and Applications. CRC Press, Boca Raton (2001). DOI 10.1201/9780203908723
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