Some nonconforming finite elements for the plate bending problem

We consider various non-conforming finite element methods to solve the plate bending problem. We show thaï all the éléments pass the ''Patch test", a pratical condition for convergence. Using gênerai results, we dérive in each case the error bounds for both strains and displacements. 1 INTRODUCTION We shall study the convergence and accuracy of the approximate solutions of the plate bending problem obtained with finite element methods using some nonconforming éléments. These éléments are the quadratic triangular element of Morley [15], two cubic triangular éléments recently introduced by Fraeijs de Veubeke [9], the rectangular element of Adini [1] and the triangular element of Zienkiewicz [3], To obtain the corresponding error estimâtes, the keystone is the patch test of Irons [10]. The first three éléments pass the patch test for polynomials of degree less than or equal to 2 whatever the mesh geometry. Zienkiewicz's triangle passes it only if the mesh is generated by three sets of parallel lines. Adini's rectangle passes it for polynomials with degree less than or equal to 2, whatever are the dimensions of the rectangles and it passes a "super" patch test — so called by Strang [19] when ail rectangles are equal. (1) Commissariat à l'énergie atomique Centre d'Etudes de Limeil Villeneuve-Saint-Georges — France Revue Française d'Automatique, Informatique et Recherche Opérationnelle n° avril 1975, R-l. 10 P. LASCAUX ; P. LESAINT The main results are the following : Let u be the exact solution, let un be the approximate solution that one gets on a mesh of order h (the supremum of the element side length), then : \\U-Uh\\ <ch ( | t t | 3 + * | M | 4 ) II tt l i j || < ( I « I 3 + * I « I 4 ) for the first three triangular éléments, and : II u u . || <ch n 2 | UU, || <Ch | « | for Zienkiewicz's triangular element and Adini's element, and finally : II M « A II <ch | « | 4 for Adini's element, when ail rectangles are equal. In these inequalities, c is a constant independent of h and || • ||^ (resp, I * i^) dénotes the discrete equivalent of the SobolevH* norm (resp. semi norm). We have assumed that the exact and approximated domains are similar, which is an important restriction if one wants to use Zienkiewicz' or Adini's element (see the last section). The proofs for the error bounds are first given in the case of the clamped plate, and we show (in the last section) that they are still valid with some other boundary conditions, such as, for example, in the case of a simply supported plate. An outline of the paper is as follows : In section II, we introducé some notations and the plate équations, and we dérive some gênerai estimâtes which are used subsequently. In section III, we study the éléments of Morley and Fraeijs de Veubeke, which seem to have similar properties. Section IV is devoted to the element of Adini, and section V to the element of Zienkiewicz. In each of these sections, we first describe the éléments, we check that the discrete energy is positive definite (for o < 1), we show that the patch test is satisfied, we give an interior error estimate, we dérive an error estimate for the terms arising at the interfaces between the éléments, and finally, using techniques similar to these developped by Ciarlet et Raviart [6], we prove the gênerai error estimâtes. Revue Française d'Automatique, Informatique et Recherche Opérationnelle NONCONFORMING ELEMENTS FOR THE PLATE PROBLEM 11 In the last section, we consider the remarks already done above, and we also show that if one sets the Poisson ratio o equal to the (irrealistic) value one in order to simplify the energy formula, which is valid in the continuous case for the clamped plate, the fust three éléments should no longer be used since the discrete energy is no longer a positive definite form. For the last two éléments, the energy is still a positive definite form. However, it is notknown whetherthis positive definitiveness is uniform with respect to h. II EQUATIONS OF THE PLATE BENDING PROBLEM AND GENERAL ESTIMATES The problem of the plate bending can be written as follows [13]. Let^Z be a bounded domain of the plane (x,y), with boundary F. We shall dénote by s a curvilinear abscissa along F and — -̂ the derivative along the outer normal on F and — the tangential derivative along F. We define the inner product (f,g) by ds (f>g) = | ƒ (x,y) g (x,y) dx dy and the corresponding norrn by : Given an integer m > 0, we consider the usual Sobolev space : # " ( « ) {v,vez, (n) , bveL (n), | a | < m } , with the norm and semi norm || \\m ^ and | \m ^ defined by : ""Un =C I liav||0>n) , I ot | = m where a is a multiindex such that a ~ (at , a 2 ) , â . > 0 ,