A piecewise linear finite element method for the buckling and the vibration problems of thin plates

Mora, David; Rodríguez, Rodolfo

doi:10.1090/S0025-5718-09-02228-5

A piecewise linear finite element method for the buckling and the vibration problems of thin plates
HTML articles powered by AMS MathViewer

by David Mora and Rodolfo Rodríguez PDF

Math. Comp. 78 (2009), 1891-1917 Request permission

Abstract:

The aim of this paper is to analyze a piecewise linear finite element method to approximate the buckling and the vibration problems of a thin plate. The method is based on a conforming discretization of a bending moment formulation for the Kirchhoff-Love model. The analysis restricts to simply connected polygonal clamped plates, not necessarily convex. The method is proved to converge with optimal order for both spectral problems, including an improved order for the eigenvalues. Numerical experiments are reported to assess its performance and to compare it with other low-order finite element methods.

References

Mohamed Amara, Daniela Capatina-Papaghiuc, and Amna Chatti, Bending moment mixed method for the Kirchhoff-Love plate model, SIAM J. Numer. Anal. 40 (2002), no. 5, 1632–1649. MR 1950615, DOI 10.1137/S0036142900379680
M. Amara and F. Dabaghi, An optimal $\textbf {C}^0$ finite element algorithm for the 2D biharmonic problem: theoretical analysis and numerical results, Numer. Math. 90 (2001), no. 1, 19–46. MR 1868761, DOI 10.1007/s002110100284
I. Babuška and J. Osborn, Eigenvalue problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR 1115240
I. Babuška, J. Osborn, and J. Pitkäranta, Analysis of mixed methods using mesh dependent norms, Math. Comp. 35 (1980), no. 152, 1039–1062. MR 583486, DOI 10.1090/S0025-5718-1980-0583486-7
F. Brezzi and P.-A. Raviart, Mixed finite element methods for 4th order elliptic equations, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 33–56. MR 0657975
C. Canuto, Eigenvalue approximations by mixed methods, RAIRO Anal. Numér. 12 (1978), no. 1, 27–50, iii (English, with French summary). MR 488712, DOI 10.1051/m2an/1978120100271
P. G. Ciarlet, Basic error estimates for elliptic problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 17–351. MR 1115237
P. G. Ciarlet and P.-A. Raviart, A mixed finite element method for the biharmonic equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 125–145. MR 0657977
R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753
V. Girault, J. Giroire, and A. Sequeira, A stream-function–vorticity variational formulation for the exterior Stokes problem in weighted Sobolev spaces, Math. Methods Appl. Sci. 15 (1992), no. 5, 345–363. MR 1170532, DOI 10.1002/mma.1670150506
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
P. Grisvard, Elliptic Problems in Non-Smooth Domains, Pitman, Boston, 1985.
Kazuo Ishihara, A mixed finite element method for the biharmonic eigenvalue problems of plate bending, Publ. Res. Inst. Math. Sci. 14 (1978), no. 2, 399–414. MR 509196, DOI 10.2977/prims/1195189071
Kazuo Ishihara, On the mixed finite element approximation for the buckling of plates, Numer. Math. 33 (1979), no. 2, 195–210. MR 549449, DOI 10.1007/BF01399554
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
Tetsuhiko Miyoshi, A mixed finite element method for the solution of the von Kármán equations, Numer. Math. 26 (1976), no. 3, 255–269. MR 438741, DOI 10.1007/BF01395945
B. Mercier, J. Osborn, J. Rappaz, and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp. 36 (1981), no. 154, 427–453. MR 606505, DOI 10.1090/S0025-5718-1981-0606505-9
Rolf Rannacher, Nonconforming finite element methods for eigenvalue problems in linear plate theory, Numer. Math. 33 (1979), no. 1, 23–42. MR 545740, DOI 10.1007/BF01396493
Andreas Rössle, Corner singularities and regularity of weak solutions for the two-dimensional Lamé equations on domains with angular corners, J. Elasticity 60 (2000), no. 1, 57–75. MR 1832016, DOI 10.1023/A:1007639413619
Reinhard Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. Numér. 12 (1978), no. 1, 85–90, iii (English, with French summary). MR 483557, DOI 10.1051/m2an/1978120100851