Guaranteed lower eigenvalue bounds for the biharmonic equation
The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate’s vibrations. This paper shows that the eigenvalue provided by the nonconforming Morley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms.
Mathematics Subject Classification (2000)65N25 65N30 74K20
- 2.Carstensen, C., Gedicke, J.: Guaranteed lower bounds for eigenvalues. Math. Comp. Accepted for publication (2013)Google Scholar
- 4.Ciarlet, P.G.: The finite element method for elliptic problems. Studies in Mathematics and its Applications, vol. 4. North-Holland Publishing Co., Amsterdam (1978)Google Scholar
- 5.Evans, L.C.: Partial differential equations, Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)Google Scholar
- 7.Parlett, B.N.: The symmetric eigenvalue problem, Classics in Applied Mathematics, vol. 20. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1998). Corrected reprint of the 1980 originalGoogle Scholar
- 9.Timoshenko, S., Gere, J.: Theory of elastic stability. Engineering Societies Monographs. MacGraw-Hill International, New York (1985)Google Scholar