Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley - PowerPoint PPT Presentation

AB HELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D’HELSINKI A posteriori error analysis for the Morley plate element Jarkko Niiranen Department of Structural Engineering and Building Technology TKK – Helsinki University of Technology, Finland Louren¸ co Beir˜ ao da Veiga, University of Milan, Italy Rolf Stenberg, Institute of Mathematics, TKK, Finland

Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 2

Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 3

Introduction ◮ Thin structures (shells, plates, membranes, beams) are the main building blocks in modern structural design. ◮ Beside the classical fields as civil engineering, the variety of applications have strongly increased also in many other fields as aeronautics, biomechanics, surgical medicine or microelectronics. ◮ In particular, new applications arise when thin structures are combined with functional, smart or composite materials (shape memory alloys, piezo-electric cheramics etc.). ◮ Increasing demands for accuracy and productivity have created a need for adaptive (automated, efficient, reliable) computational methods for thin structures. 4

Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 5

Kirchhoff plate bending model ◮ We consider bending of a thin planar structure occupied by P = Ω × ( − t 2 , t 2) , where Ω ⊂ R 2 denotes the midsurface of the plate and t ≪ diam(Ω) denotes the thickness of the plate. ◮ Kinematical assumptions for the dimension reduction: — Straight fibres normal to the midsurface remain straight and normal. — Fibres normal to the midsurface do not stretch. — The midsurface moves only in the vertical direction. 6

Deformations ◮ Under these assumptions, with the deflection w , the displacement field u = ( u x , u y , u z ) takes the form u x = − z ∂w ( x, y ) u y = − z ∂w ( x, y ) u z = w ( x, y ) . , , ∂x ∂y ◮ The corresponding deformation is defined by the symmetric linear strain tensor ε ( u ) = 1 ∇ u + ( ∇ u ) T � � , 2 in the component form as e xx = − z ∂ 2 w e yy = − z ∂ 2 w e zz = 0 , ∂x 2 , ∂y 2 , e xy = − z ∂ 2 w ∂x∂y , e xz = 0 , e yz = 0 . 7

Stress resultants Next, we define the stress resultants, the moments and the shear forces: ⎛ ⎞ � t/ 2 ⎝ M xx M xy M = with M ij = − i, j = x, y , z σ ij dz , ⎠ M yx M yy − t/ 2 ⎛ ⎞ � t/ 2 ⎝ Q x Q = with Q i = i = x, y , σ iz dz , ⎠ Q y − t/ 2 where the stress tensor is assumed to be symmetric: σ ij = σ ji , i, j = x, y, z. 8

Equilibrium equations and boundary conditions The principle of virtual work gives, with the load resultant F , the equilibrium equation div div M = F with div M + Q = 0 . and the boundary conditions w = 0 , ∇ w · n = 0 on Γ C , w = 0 , n · Mn = 0 on Γ S , ∂ 2 n · Mn = 0 , ∂ s 2 ( s · Mn ) + n · div M = 0 on Γ F , ( s 1 · Mn 1 )( c ) = ( s 2 · Mn 2 )( c ) ∀ c ∈ V , where the indices 1 and 2 refer to the sides of the boundary angle at a corner point c on the free boundary Γ F . 9

Constitutive assumptions ◮ The material of the plate is assumed to be — linearly elastic (defined by the generalized Hooke’s law) — homogeneous (independent of the coordinates x, y, z ) — isotropic (independent of the coordinate system). ◮ Furthermore, we assume that the transverse normal stress vanishes: σ zz = 0. 10

Variational formulation Let the deflection w belong to the Sobolev space W = { v ∈ H 2 (Ω) | v = 0 on Γ C ∪ Γ S , ∇ v · n = 0 on Γ C } , where n indicates the unit outward normal to the boundary Γ. Problem. Variational formulation: Find w ∈ W such that ( Eε ( ∇ w ) , ε ( ∇ v )) = ( f, v ) ∀ v ∈ W , with the elasticity tensor E defined as ν E � � ∀ ε ∈ R 2 × 2 , Eε = ε + 1 − ν tr( ε ) I 12(1 + ν ) with Young’s modulus E and the Poisson ratio ν . 11

Morley finite element formulation Let E denote an edge of a triangle K in a triangulation T h . We define the discrete space for the deflection as � ∀ E ∈ E i h ∪ E c � � W h = v ∈ M 2 ,h | � ∇ v · n E � = 0 , h E where 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 of Γ C ∪ Γ S . Finite element method. Morley: Find w h ∈ W h such that � ( Eε ( ∇ w h ) , ε ( ∇ v )) K = ( f, v ) ∀ v ∈ W h . K ∈T h 12

A priori error estimate The method is stable and convergent with respect to the following discrete norm on the space W h + H 2 : E � � ∂v � � � �| v �| 2 | v | 2 h − 3 E � � v � � 2 h − 1 � � 2 h := 2 ,K + 0 ,E + 0 ,E , ∂ n E K ∈T h E ∈E h E ∈E h Proposition. (Shi 90, Ming and Xu 06) Assuming that Γ = Γ C there exists a positive constant C such that � � �| w − w h �| h ≤ Ch | w | H 3 (Ω) + h � f � L 2 (Ω) . The numerical results indicate the same convergence rate for general boundary conditions as well. 13

Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 14

A posteriori error estimates ◮ We use the following notation: � · � for jumps (and traces), h E and h K for the edge length and the element diameter. Interior error indicators ◮ For all the elements K in the mesh T h , η 2 K := h 4 K � f � 2 ˜ 0 ,K , and for all the internal edges E ∈ I h , E � � ∂w h η 2 E := h − 3 E � � w h � � 2 0 ,E + h − 1 � � 2 0 ,E . ∂ n E 15

Boundary error indicators ◮ The boundary of the plate is divided into clamped, simply supported and free parts: Γ := ∂ Ω = Γ C ∪ Γ S ∪ Γ F . ◮ For edges on the clamped and simply supported boundaries Γ C and boundary Γ S , respectively, E � � ∂w h η 2 E,C := h − 3 E � � w h � � 2 0 ,E + h − 1 � � 2 0 ,E , ∂ n E η 2 E,S := h − 3 E � � w h � � 2 0 ,E . 16

Error indicators — local and global ◮ 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 K := ˜ E + E,C + , E,S 2 E ∈I h E ∈C h E ∈S h E ⊂ ∂K E ⊂ ∂K E ⊂ ∂K with the notation — I h for the collection of all the internal edges, — C h and S h for the collections of all the boundary edges on Γ C and Γ S , respectively. ◮ The global error indicator is defined as � 1 / 2 � � η 2 η h := . K K ∈T h 17

Upper bound — Reliability Theorem. Reliability: There exists a positive constant C such that �| w − w h �| h ≤ Cη h . Lower bound — Efficiency Theorem. Efficiency: For any element K , there exists a positive constant C K such that �| w − w h �| h,K + h 2 � � η K ≤ C K K � f − f h � 0 ,K . Efficiency is proved by standard arguments; reliability needs a new Cl´ ement-type interpolant and a new Helmholtz-type decomposition. 18

Techniques for the analysis — Helmholtz decomposition Lemma. Let σ be a second order tensor field in L 2 (Ω; R 2 × 2 ) . H 1 (Ω)] 2 such that 0 (Ω) and φ ∈ [ ˜ Then, there exist ψ ∈ W , ρ ∈ L 2 ⎛ ⎞ ⎝ 0 − ρ ⎠ . σ = Eε ( ∇ ψ ) + ρ + Curl φ , with ρ = 0 ρ � ψ � H 2 (Ω) + � ρ � L 2 (Ω) + � φ � H 1 (Ω) ≤ C � σ � L 2 (Ω) . Here ˜ H m (Ω) , m ∈ N , indicate the quotient space of H m (Ω) where the seminorm | · | H m (Ω) is null. In analysis, Lemma is applied to the tensor field Eε ( ∇ ( w − w h )). 19

Contents 1 Introduction 2 Kirchhoff plate bending model 3 Morley finite element formulation 4 A posteriori error estimates 5 Numerical results 6 Conclusions and Discussion References 20

Numerical results ◮ We have implemented the method in the open-source finite element software Elmer developed by CSC – the Finnish IT Center for Science. ◮ The software provides error balancing strategy and complete remeshing for triangular meshes. ◮ We have used test problems with convex rectangular domains – and with known exact solutions – for investigating the effectivity index for the error estimator derived. ◮ Non-convex domains we have used for studying the adaptive performance and robustness of the method. 21

η h Effectivity index ι = �| w − w h �| h Effectivity Index = Error Estimator / Exact Error Effectivity Index = Error Estimator / Exact Error 1 1 10 10 0 0 10 10 −1 −1 10 10 0 1 2 3 4 5 0 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 Number of Elements Number of Elements Figure 1: Left : uniform refinements; Right : adaptive refinements. Clamped (squares), simply supported (circles) and free (triangles) boundaries included. 22

Adaptively refined mesh — Error estimator Simply supported L-corner Figure 2: Simply supported L-shaped domain: Distribution of the error estimator for two adaptive steps. 23