A-priori and a-posteriori error estimates for a family of Reissner-Mindlin plate elements Joint work with Lourenco Beirão da Veiga (Milano), Claudia Chinosi (Alessandria) and Carlo Lovadina (Pavia) Previous work with Dominique Chapelle (INRIA), Mikko Lyly (CSC-Scientific Computing, Finland) and Jarkko Niiranen (Helsinki).
Contents The Reissner-Mindlin model Regularity structure of plates The Falk-Tu family A priori analysis A posteriori analysis 2
The Reissner-Mindlin plate The appropriately scaled equations have the form: Find (w, β) W V such that a(β, η) + t 2 ( w β, v η) = (g, v) for all admissible deflections and rotations (v, η) W V. Here ε is the small strain operator, t is the thickness of the plate, is the bending energy and is the shear energy. a(β, η) = (ε(β), ε(η)) = ( β, η) t 2 ( w β, v η) 3
RM continued The shear force is given by q = t 2 ( w β). Suppose that we have clamped and free boundary conditions. On Γ C : On Γ F : β = 0, w = 0. ε(β)n = 0, q n = 0. The continuous spaces V = { η [H 1 (Ω)] 2 η ΓC = 0 }, W = { v H 1 (Ω) v ΓC = 0 }, 4
The Kirchhoff limit and locking When we go to the limit t 0, we obtain the Kirchhoff constraint w β = 0. In addition, we loose one boundary condition. For a straightforward FEM: Find (w h, β h ) W h V h W V such that a(β h, η) + t 2 ( w h β h, v η) = (g, v) (v, η) W h V h, the K-constraint is inherited: w h β h = 0. For low-order elements this leads to the locking : w h = 0, β h = 0. 5
Quasi-optimal error estimates { } w w h 1 + β β h 1 t 1 C inf w v 1 + inf β η 1. v W h η V h This explodes as t 0. The engineering remedy is to loosen up the constraint: Find (w h, β h ) W h V h W V such that a(β h, η) + t 2 (Π h ( w h β h ), Π h ( v η)) = (g, v). There exists O(10 2 ) O(10 3 ) of papers on this. But only O(10) with a mathematical analysis. 6
The Falk-Tu family Π h is the L 2 -projection onto a discrete shear space Q h. This is equivalent to a mixed method. Find (w h, β h, q h ) W h V h Q h such that a(β h, η) + (q h, v η) = (g, v) (v, η) W h V h, t 2 (q h, r) + ( w h β h, r) = 0 r Q h. In the limit t 0 we obtain the familiar saddle point structure (for the Kirchhoff solution). The key question is what norms have to be used. For this we take a look at: 7
Regularity and shift theorems Kirchhoff. and w 2 C g 2. w 3 C g 1. Reissner Mindlin. All works prior to 1998 use the estimates w 1 + β 1 C g 1 and w 2 + β 2 C g 0. Where do we see RM K? 8
Regularity of the Reissner Mindlin model Split w = w 0 + w r, where is the limiting Kirchhoff solution. w 0 = lim t 0 w Then it holds (Chapelle, Stenberg 88, Lyly, Niiranen, Stenberg 06) w 0 2 + t 1 w r 1 + q 1 + t q 0 C ( g 2 + t g 1 ). and for a clamped plate and convex domain w 0 3 + t 1 w r 2 + q 0 + t q 1 C ( g 1 + t g 0 ). This gives the previous results both for t = 1 and t 0. 9
Finite element norms since w h H 2 (Ω) the continous norms cannot be used. FE-norms (Babuska, Osborn, Pitkäranta 1980, Pitkäranta 1988). (v, η) 2 h = η 2 1 + K C h (t 2 + h 2 K) 1 v η 2 0,K, which in the limit t 0 gives the Kirchhoff norm with v 2 2,h = η 1 + v 2,h, K C h v 2 2,K + The dual norm for the shear is r 2 h = h 1 K E T h K C h (h 2 K + t 2 ) r 2 0. 10 v n 2 0,E.
Finite element residual norms Recall. Kirchhoff: w 2 C g 2 and Reissner-Mindlin: w 0 2 + t 1 w r 1 + q 1 + t q 0 C ( ) g 2 + t g 1. For the a-posteriori analysis this indicates (K-hhoff): 2 h 2 0 and (RM): 2 + t 1 h 2 0 + th 0. 11
The Falk-Tu spaces For the degree k 1, we define W h = { v W v K P k+1 (K) K C h }, V h = { η V η K [P k (K) + B k+3 (K)] 2 K C h }, Q h = { r [L 2 (Ω)] 2 r K [P k (K)] 2 K C h }. Here we have (a quite big) bubble space B k+3 (K) = P k+3 (K) H0 1 (K). Stability follows from Brezzis conditions, Z h -ellipticity and inf-sup. 12
Z h -ellipticity Since it holds Z h = {(v, η) W h V h ( v η, r) = 0 r Q h }. W h Q h we have Z h = {(v, η) W h V h v = Π h η }. From this the Z h -ellipticity easily follows: a(η, η) C η 2 1 C (v, η) 2 h. 13
Inf-Sup Since we have included so many bubbles we can, for r Q h given, choose η V h such that η K = b K r K K C h, where b K is the cubic bubble on K. By scaling we get (η, r) η 1 β ( K C h h 2 K r 2 0 ) 1/2. 14
A-priori estimates We now have, with C independent of t, (w w h, β β h ) h + q q h h C { } (w v, β η) h + q r h for all (v, η, r) W h V h Q h. Recall (v, η) 2 h = η 2 1 + K C h (t 2 + h 2 K) 1 v η 2 0,K. This is sharp (in contrast to FT). 15
When estimating the right hand side we write w = w 0 + w r, and use the shift theorem We get w 0 3 + t 1 w r 2 + β 2 C ( g 1 + t g 0 ). (w w, β β) h (w 0 w 0, 0) h + (w r w r, 0) h + (0, β β) h h 1 w 0 w 0 1 + t 1 w r w r 1 + h 1 β β 0 C ( h w 0 3 + t 1 h w r 2 + h β β 2 ) Ch ( g 1 + t g 0 ). The final a-priori estimate for a convex region: (w w h, β β h ) h + q q h h Ch ( g 1 + t g 0 ). 16
A-posteriori estimate Claes Johnson: The two legs of error analysis. A priori estimates are based on the stability of the discrete problem. A posteriori estimates are based on the stability of the continuous problem. The problem is now the stability of the continuous problem, i.e. the norms! We are forced to rely on one discrete leg. We use a saturation assumption. Let ( w h, β h ) be the solution of the FE equations with elements of one degree higher. There exist a positive constant α < 1 such that (w w h, β β h ) h α (w w h, β β h ) h. 17
The a posteriori estimate It holds with (w w h, β β h ) h ( K C h η 2 K) 1/2, ηk 2 = h 2 K(h 2 K + t 2 ) div q h + g 2 0,K +h 2 K div ε(β h ) + q h 2 0,K +(h 2 K + t 2 ) 1 t 2 q h ( w h β h ) 2 0,K + h E ε(β h )n 2 0,E E K + E K h E (h 2 E + t 2 ) q h n 2 0,E. 18