ABHELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI A posteriori error analysis for Kirchhoff plate elements Jarkko Niiranen Laboratory of Structural Mechanics, Institute of Mathematics TKK Helsinki University of Technology, Finland Lourenço Beirão da Veiga, University of Milan, Italy Rolf Stenberg, TKK, Finland
Outline Kirchhoff plate bending model Finite element formulations Morley element Stabilized C 0 -element A posteriori error estimates Numerical results Conclusions and references 2
Kirchhoff plate bending model Displacement formulation. Find the deflection w such that, in the domain Ω R 2, it holds 1 6(1 ν) 2 w = f. Mixed formulation. Find the deflection w, rotation β and the shear stress q such that it holds div q = f, div m(β) + q = 0, with m(β) = 1 6 {ε(β) + ν div βi}, 1 ν w β = 0. Furthermore, the boundary conditions on the clamped, simply supported and free boundaries Γ C, Γ S and Γ F are imposed. 3
FE formulations Morley element We define the discrete space for the deflection as follows: W h = { v M 2,h v } = 0 E E h, n E E where E represents an edge of a triangle K in a triangulation T h, and M 2,h denotes the space of the second order piecewise polynomial functions on T h which are continuous at the vertices of all the internal triangles and zero at all the triangle vertices on the clamped boundary. Finite element method. Find w h W h such that K T h (Eε( w h ), ε( v)) K = (f, v) v W h. 4
Stabilized C 0 -element Given an integer k 1, we define the discrete spaces for the deflection and the rotation, respectively, as W h = {v W v K P k+1 (K) K T h }, V h = {η V η K [P k (K)] 2 K T h }, where P k (K) denotes the polynomial space of degree k on K. Finite element method. Find (w h, β h ) W h V h such that A h (w h, β h ; v, η) = (f, v) (v, η) W h V h, where the bilinear form A h we split as A h (z, φ; v, η) = B h (z, φ; v, η) + D h (z, φ; v, η), 5
with the stabilized (α) bending part (R M with the limit t 0) B h (z, φ; v, η) = (m(φ), ε(η)) K T h αh 2 K(Lφ, Lη) K + 1 αh 2 ( z φ αh 2 KLφ, v η αh 2 KLη) K K T h K and the stabilized (γ) free boundary part D h (z, φ; v, η) = m ns (φ), [ v η] s ΓF + [ z φ] s, m ns (η) ΓF + γ [ z φ] s, [ v η] s E h E E F h for all (z, φ) W h V h, (v, η) W h V h, where F h represents the collection of all the boundary edges on the free boundary Γ F, and the twisting moment is m ns = s mn. The first term in D h is for consistency, the second one for symmetry and the last one for stability. 6
A posteriori error estimates We use the following notation: for jumps, h E and h K for the edge length and the element diameter. Interior error indicators For the local error indicator η K we define: for all the elements K in the mesh T h, and for all the internal edges E I h, (Morley) η 2 K := h 4 K f 2 0,K, (Stabil.) η 2 K := h 4 K f + div q h 2 0,K + h 2 K w h β h 2 0,K, (Morley) η 2 E := h 3 E w h 2 0,E + h 1 E w h n E 2 0,E, (Stabil.) η 2 E := h 3 E q h n 2 0,E + h E m(β h )n 2 0,E. 7
Boundary error indicators Let the boundary Ω of the plate be divided into the parts of the different boundary conditions: clamped, simply supported and free, i.e., Ω = Γ C Γ S Γ F. For the Morley element, we assume that Ω = Γ C and for the edges on the clamped boundary Γ C (Morley) η 2 E,C = h 3 E w h 2 0,E + h 1 E w h n E 2 0,E. For the stabilized C 0 -element, for the edges on the simply supported boundary Γ S (Stabil.) η 2 E,S := h E m nn (β h ) 2 0,E, and for the edges on the free boundary Γ F (Stabil.) η 2 E,F := h E m nn (β h ) 2 0,E+h 3 E s m ns(β h ) q h n 2 0,E. 8
Error indicators local and global Now, for any element K T h, let the local error indicator be η K := ( η 2 K+ 1 2 η 2 E+ η 2 E,C+ η 2 E,S+ η 2 E,F ) 1/2, E I h E K E C h E K E S h E K E F h E K with the notation I h for the collection of all the internal edges, C h, S h and F h for the collections of all the boundary edges on Γ C, Γ S and Γ F, respectively. Finally, the global error indicator is defined as η := ( K T h η 2 K ) 1/2. 9
Upper bounds Reliability With E h denoting the collection of all the triangle edges, we define the mesh dependent norms for the Morley element and for the stabilized C 0 -element, respectively, as v 2 h := v 2 2,K + h 3 E v 2 0,E + K T h E E h (v, η) 2 h := v 2 2,K + v 2 1 + h 1 E K T h E I h + K v η 2 0,K + η 2 1. K T h h 2 h 1 E E E h v n E 2 0,E Theorem. There exists positive constants C such that (Morley) w w h h Cη, (Stabil.) (w w h, β β h ) h + q q h 1, Cη. v n E 2 0,E, 10
Lower bounds Efficiency Let ω K be the collection of all the triangles in T h with a nonempty intersection with the element K. Theorem. There exists positive constants C such that (Morley) η K C ( w w h h,k + h 2 K f f h 0,K ), (Stabil.) η K C ( (w w h, β β h ) h,ωk + h 2 K f f h 0,ωK + q q h 1,,ωK ), for any element K T h. 11
Numerical results We have implemented the methods in the open-source finite element software Elmer developed by CSC the Finnish IT Center for Science. Test problems with convex rectangular domains, and with known exact solutions, we have used for investigating the effectivity index for the error estimators derived. Non-convex domains we have used for studying the adaptive performance and robustness of the methods. 12
(Morley) ι = Effectivity index η w w h h (Stabil.) ι = η (w w h, β β h ) h Effectivity Index = Error Estimator / Exact Error 10 1 10 0 10 1 10 0 10 5 Number of Elements Effectivity Index = Error Estimator / True Error 10 1 10 0 10 1 10 0 10 5 Number of Elements Figure 1: Effectivity index; Left: the Morley element (with C- boundaries); Right: the stabilized method (with C/S/F-boundaries). 13
Simply supported L-domain Starting mesh Deflection Figure 2: The stabilized method: Deflection distribution for the first mesh (constant loading). 14
Adaptively refined mesh Error estimator Figure 3: The stabilized method: Distribution of the error estimator for two adaptive steps. 15
Uniform vs. Adaptive Convergence in the norm β β h 1 + (w w h, β β h ) h 10 1 Convergence of the Error estimator 10 0 10 1 10 2 10 0 10 1 10 2 10 3 10 4 10 5 Number of elements Figure 4: The stabilized method: Convergence of the error estimator for the uniform refinements and adaptive refinements; Solid lines for global, dashed lines for maximum local ones. 16
Clamped L-domain Refinements Error estimator Figure 5: Distribution of the error estimator after adaptive refinements: Left: the Morley element; Right: the stabilized method. 17
Conclusions A posteriori error analysis has been accomplished for Kirchhoff plates: the Morley element for clamped boundaries the stabilized C 0 -continuous element for general boundary conditions efficient and reliable error estimators for both methods. Numerical benchmarks confirm the adaptive performance and robustness of the error indicators. 18
References [1] L. Beirão da Veiga, J. Niiranen, R. Stenberg: A posteriori error estimates for the Morley plate bending element; Numerische Matematik, 106, 165 179 (2007). [2] L. Beirão da Veiga, J. Niiranen, R. Stenberg: A family of C 0 finite elements for Kirchhoff plates I: Error analysis; accepted for publication in SIAM Journal on Numerical Analysis (2007). [3] L. Beirão da Veiga, J. Niiranen, R. Stenberg: A family of C 0 finite elements for Kirchhoff plates II: Numerical results; accepted for publication in Computer Methods in Applied Mechanics and Engineering (2007). 19
How do we deal with a blinking star? 20
We snow it by adaptively refined mesh flakes! 21