ABHELSINKI UNIVERSITY OF TECHNOLOGY

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1 ABHELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI Computational results for the superconvergence and postprocessing of MITC plate elements Jarkko Niiranen, Institute of Mathematics, TKK, Finland in collaboration with Mikko Lyly, CSC Scientific Computing Ltd., Finland Rolf Stenberg, Institute of Mathematics, TKK, Finland

2 Outline Introduction MITC finite elements for Reissner Mindlin plates Superconvergence and postprocessing Computational results Conclusions

3 Introduction The original deflection approximation is superconvergent compared to a certain interpolant. The postprocessed deflection approximation is a polynomial of one degree higher than the original one. It is constructed by utilizing the superconvergence property, which gives accuracy of one degree higher than the original one. The postprocessing is local, which implies low computational costs. 3

4 MITC finite elements for Reissner Mindlin plates The plate is assumed to be linearly elastic isotropic, with the shear modulus G and the Poissonin ratio ν. The undeformed plate midsurface Ω R is a convex polygon. The plate thickness t << diam(ω) is constant.

5 Let the boundary conditions on Γ be clamped, simply supported or free. Then the Reissner Mindlin plate problem reads: Problem. For the loading g H 1 (Ω) find the deflection w {v H 1 (Ω) v ΓC =, v ΓSS = } and the rotation β {η [H 1 (Ω)] η ΓC =, (η τ ) ΓSS = } such that a(β, η) + 1 ( w β, v η) = (g, v) (v, η) W V, t where the bending bilinear form is, with the linear strain ε, a(φ, η) = 1 {(ε(φ), ε(η)) + ν 6 (div φ, div η)}. 1 ν 5

6 The polynomial interpolation for the MITC finite element method is for the deflection approximation w h W h of order k for the components of the rotation approximation β h V h of order k, enriched by the interior bubbles of order k + 1. Method. (Bathe, Brezzi and Fortin 1989) Find w h W h W and β h V h V such that a(β h, η)+ 1 t (R h( w h β h ), R h ( v η)) = (g, v) (v, η) W h V h, where the reduction operator R h maps the shear stress into the rotated Raviart Thomas polynomial space of order k 1. 6

7 Superconvergence and postprocessing The interpolation operator I h : H s (Ω) W h, s > 1, is defined through the conditions (v I h v)(a) = vertices a K, v I h v, p E = p P k (E) edges E K, (v I h v, p) K = p P k 3 (K). The reduction and interpolation operators are closely related: R h v = I h v v H s (Ω), s. 7

8 Superconvergence For the deflection approximation w h of order k, with the mesh size h, it holds: w w h 1 Ch k, where the exact deflection w is assumed to be smooth. For the deflection approximation w h and the interpolant I h w it holds: Theorem 1. Assuming a smooth solution, I h w w h 1 C(h + t)h k. 8

9 Postprocessing The original deflection approximation is of order k in the element K: w h K P k (K). The postprocessed deflection approximation is of order k + 1 in the element K: w h K P k+1(k) = P k (K) Ŵ (K) W (K). The new degrees of freedom of order k + 1, corresponding to the element boundaries E, space Ŵ (K), and element interior, space W (K), are added to the original deflection approximation. 9

10 The postprocessing method is based on the definition of the shear stress: q = 1 t ( w β) or w = β + t q. Postprocessing scheme. Find the local postprocessed deflection approximation w h K P k+1(k) such that I h w h = w h in the element K, w h τ E, ˆv τ E E = (β h + t q h ) τ E, ˆv τ E E ˆv Ŵ (K), ( w h, v) K = (β h + t q h, v) K v W (K). 1

11 In the postprocessing we utilize the superconvergence of the original deflection approximation, I h w w h 1 C(h + t)h k. Theorem. For the postprocessed deflection approximation w h it holds, assuming a smooth solution, w w h 1 C(h + t)h k. This is an error estimate of order h + t better than the original one, w w h 1 Ch k. According to the computational results, a corresponding accuracy improvement holds also in the L -norm. 11

12 Computational results The following semi-infinite plate is considered: the midsurface Ω = {(x, y) R y > } Poisson ratio ν =.3 shear modulus G = 1 (1+ν) thickness t =.1 loading g = 1 G cos x. For the boundary Γ = {(x, y) R y = } two different types of boundary conditions are imposed: simply supported or free. The discretized domain is D = [, π/] [, 3π/]. 1

13 Accuracy for the uniform meshes Interior domain D i D b Figure 1: Uniform meshes, with N =,, 6, 8; Interior domain D i ; Boundary region D b. 13

14 Simply supported boundary Deflection y coordinate y coordinate x coordinate 1 x coordinate Figure : Uniform mesh; Deflection in the discretized domain, with N =, 8 and k =. 1

15 Simply supported H 1 - and L -errors Interior domain Relative H 1 error Relative L error Number of elements in x direction Number of elements in x direction Figure 3: Uniform mesh; Convergence in the H 1 - and L -norms, with k =, 3 (red dashed line for the original, black solid line for the postprocessed deflection). 15

16 Free boundary Deflection y coordinate y coordinate x coordinate 1 x coordinate Figure : Uniform mesh; Deflection in the discretized domain, with N =, 8 and k =. 16

17 Free H 1 - and L -errors Interior domain Relative H 1 error Relative L error Number of elements in x direction Number of elements in x direction Figure 5: Uniform mesh; Convergence in the H 1 - and L -norms, with k =, 3 (red dashed line for the original, black solid line for the postprocessed deflection). 17

18 Accuracy for the uniform meshes Boundary region D i D b Figure 6: Uniform meshes, with N =,, 6, 8; Interior domain D i ; Boundary region D b. 18

19 Simply supported H 1 - and L -errors Boundary region Relative H 1 error Relative L error Number of elements in x direction Number of elements in x direction Figure 7: Uniform mesh; Convergence in the H 1 - and L -norms, with k =, 3 (red dashed line for the original, black solid line for the postprocessed deflection). 19

20 Free H 1 - and L -errors Boundary region Relative H 1 error 1 3 Relative L error Number of elements in x direction Number of elements in x direction Figure 8: Uniform mesh; Convergence in the H 1 - and L -norms, with k =, 3 (red dashed line for the original, black solid line for the postprocessed deflection).

21 Free Pointwise errors Along the line x = π/ x 1 3 Pointwise error along the line x = π/ y coordinate Figure 9: Uniform mesh; Pointwise error on the line x = π/, with N =, k = (red dashed line for the original, black solid line for the postprocessed deflection, triangles for the vertex values). 1

22 Accuracy for the non-uniform meshes Boundary region D i D b Figure 1: Non-uniform meshes, with N =,, 6, 8; Interior domain D i ; Boundary region D b.

23 Simply supported boundary Deflection y coordinate y coordinate x coordinate 1 x coordinate Figure 11: Non-uniform mesh; Deflection in the discretized domain, with N =, 8 and k =. 3

24 Simply supported H 1 - and L -errors Boundary region Relative H 1 error Relative L error Number of elements in x direction Number of elements in x direction Figure 1: Non-uniform mesh; Convergence in the H 1 - and L - norms, with k =, 3 (red dashed line for the original, blue solid line for the postprocessed deflection).

25 Free boundary Deflection y coordinate y coordinate x coordinate 1 x coordinate Figure 13: Non-uniform mesh; Deflection in the discretized domain, with N =, 8 and k =. 5

26 Free H 1 - and L -errors Boundary region Relative H 1 error 1 3 Relative L error Number of elements in x direction Number of elements in x direction Figure 1: Non-uniform mesh; Convergence in the H 1 - and L - norms, with k =, 3 (red dashed line for the original, blue solid line for the postprocessed deflection). 6

27 Free Pointwise errors Along the line y = π/ 3 x 1 3 Pointwise error along the line y = π/ x coordinate Figure 15: Non-uniform mesh; Pointwise error on the line y = π/, with N =, k = (red dashed line for the original, blue solid line for the postprocessed deflection, triangles for the vertex values). 7

28 Conclusions A superconvergence result in the H 1 -norm holds for the original deflection approximation. Improved accuracy in the H 1 -norm holds for the postprocessed deflection approximation. The numerical computations confirm the results, for both uniform and nonuniform meshes. Furthermore, the numerical computations indicate similar results also in the L -norm and show the superaccuracy of the vertex values. 8