A hybrid finite element method to compute the free vibration frequencies of a clamped plate

An " assumed stresses " hybnd method is studied to provide (external) approximations to the plate vibration problem, variationaîly formulated on some space which is orthogonal to all selfequilibrating stresses Convergence results and error estimâtes are denved by a Rayleigh pnnciple Resumé — On utilise une méthode d'éléments finis « hybrides duaux » pour approcher le problème des vibrations libres d'une plaque encastrée, le problème a été formulé à Vaide d^un principe vanationnel sur un espace orthogonal aux tenseurs auto-équihbrés On démontre la convergence de Vapproximation et on déduit les estimations d'erreur par un principe du type Rayleigh 0. INTRODUCTION This paper deals with the approximation of the vibration frequencies and modes of a clamped plate, via a " dual hybrid " finite element method first introduced by Tabarrok [19] and based on a variational principle due to Toupin [20]. The corresponding method for the static case, proposed by Pian and Tong (see e.g. [15]) and analyzed by Brezzi [4] and Brezzi-Marini [5], essentially involves the tensors « selfequilibrating » over each element ; on the contrary the natural space for the dynamical case is some orthogonal complement to the tensors selfequilibrating over the whole domain, in that way we can reduce our problem to a spectral problem for a compact selfadjoint operator T, whose finite dimensional approximation is however not of inner type. (*) Received in December 1979 AMS (MOS) subject classification (1980) Pnmary 65N15, 65N25, 65N30 (**) Work partially supported by the G N I M -C N R and the L A N -C N R (Pavia) O Istituto di Matematica Apphcata, Umversita di Pavia, 27100 Pavia, Italie R.A.I.R.O. Analyse numénque/Numencal Analysis, 0399-0516/1981/101 /$ 5 00 © Bordas-Dunod