Outline Kirchhoff plate bending model Finite element formulations - PowerPoint PPT Presentation

AB HELSINKI 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 Louren¸ co Beir˜ ao 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 , m ( β ) = 1 ν div m ( β ) + q = 0 , with 6 { ε ( β ) + 1 − ν div βI } , ∇ 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: � � ∂v � � W h = v ∈ M 2 ,h | � = 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 � ( Eε ( ∇ w h ) , ε ( ∇ v )) K = ( f, v ) ∀ v ∈ W h . K ∈T 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 } , | η | K ∈ [ P k ( K )] 2 ∀ K ∈ T h } , V h = { η ∈ V 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 ) � αh 2 B h ( z, φ ; v, η ) = ( m ( φ ) , ε ( η )) − K ( Lφ , Lη ) K K ∈T h 1 � ( ∇ z − φ − αh 2 K Lφ , ∇ v − η − αh 2 + K Lη ) K αh 2 K K ∈T h and the stabilized ( γ ) free boundary part D h ( z, φ ; v, η ) = � m ns ( φ ) , [ ∇ v − η ] · s � Γ F + � [ ∇ z − φ ] · s , m ns ( η ) � Γ F 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 , η 2 K := h 4 K � f � 2 (Morley) ˜ 0 ,K , 0 ,K + h − 2 η 2 K := h 4 K � f + div q h � 2 K �∇ w h − β h � 2 (Stabil . ) ˜ 0 ,K , E � � ∂w h E := h − 3 0 ,E + h − 1 η 2 E � � w h � � 2 � � 2 (Morley) 0 ,E , ∂ n E η 2 E := h 3 E � � q h · n � � 2 0 ,E + h E � � m ( β h ) n � � 2 (Stabil . ) 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 E � � ∂w h E,C = h − 3 0 ,E + h − 1 η 2 E � � w h � � 2 � � 2 (Morley) 0 ,E . ∂ n E ◮ For the stabilized C 0 -element, for the edges on the simply supported boundary Γ S η 2 E,S := h E � m nn ( β h ) � 2 (Stabil . ) 0 ,E , and for the edges on the free boundary Γ F E � ∂ η 2 E,F := h E � m nn ( β h ) � 2 0 ,E + h 3 ∂sm ns ( β h ) − q h · n � 2 (Stabil . ) 0 ,E . 8

Error indicators — local and global ◮ Now, for any element K ∈ T h , let the local error indicator be K +1 � 1 / 2 � � � � � η 2 η 2 η 2 η 2 η 2 η K := ˜ E + E,C + E,S + , E,F 2 E ∈I h E ∈C h E ∈S h E ∈F h E ⊂ ∂K E ⊂ ∂K E ⊂ ∂K 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 � 1 / 2 � � η 2 η := . K K ∈T h 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 E � � ∂v � � � h − 3 h − 1 �| v �| 2 | v | 2 E � � v � � 2 � � 2 h := 2 ,K + 0 ,E + 0 ,E , ∂ n E K ∈T h E ∈E h E ∈E h E � � ∂v � � h − 1 �| ( v, η ) �| 2 | v | 2 2 ,K + � v � 2 � � 2 h := 1 + 0 ,E ∂ n E K ∈T h E ∈I h � h − 2 K �∇ v − η � 2 0 ,K + � η � 2 + 1 . K ∈T h 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 . 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 �| w − w h �| h,K + h 2 � � (Morley) η K ≤ C K � f − f h � 0 ,K , �| ( w − w h , β − β h ) �| h,ω K + h 2 � (Stabil . ) η K ≤ C K � f − f h � 0 ,ω K � + � q − q h � − 1 , ∗ ,ω K , 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

Effectivity index η η (Morley) ι = (Stabil . ) ι = �| w − w h �| h �| ( w − w h , β − β h ) �| h 1 1 Effectivity Index = Error Estimator / Exact Error 10 10 Effectivity Index = Error Estimator / True Error 0 0 10 10 −1 −1 10 10 0 5 0 5 10 10 10 10 Number of Elements 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 1 10 Convergence of the Error estimator 0 10 −1 10 −2 10 0 1 2 3 4 5 10 10 10 10 10 10 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 refine- ments: 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. Beir˜ ao da Veiga, J. Niiranen, R. Stenberg: A posteriori error estimates for the Morley plate bending element; Numerische Matematik , 106, 165–179 (2007). ao da Veiga, J. Niiranen, R. Stenberg: A family of C 0 [2] L. Beir˜ finite elements for Kirchhoff plates I: Error analysis; accepted for publication in SIAM Journal on Numerical Analysis (2007). ao da Veiga, J. Niiranen, R. Stenberg: A family of C 0 [3] L. Beir˜ 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