Adaptive Wavelet Collocation for Elasticity Lu s Manuel Castro - PDF document

Adaptive Wavelet Collocation for Elasticity Lu´ ıs Manuel Castro Silvia Bertoluzza Instituto Superior T´ ecnico, Lisbon Istituto di Analisi Numerica del CNR, Pavia luis@civil.ist.utl.pt http://www.civil.ist.utl.pt/˜luis wavelet@dragon.ian.pv.cnr.it http://dragon.ian.pv.cnr.it/˜aivlis Wavelets in Numerical Analysis and Simulation Funchal, 11-12 March 2002 DECivil N´ ucleo de An´ alise de Estruturas - ICIST

2 Outline 1. Motivation 2. Adaptive collocation techniques 3. Plane elasticity problems 4. Numerical applications 5. Reissner-Mindlin plate bending problems 6. Numerical applications 7. Conclusions and further developments

3 Motivation Numerical Simulation of structural engineering problems

4 Adaptive Collocation techniques • Bertoluzza, S., “An Adaptive Collocation Method based on Interpolating Wavelets”, in Multi- scale Wavelet Methods for Partial Differen- tial Equations , edited by Dahmen, Kurdila and Oswald, Academic Press, 1997. • Bertoluzza, S. and Naldi, G., “A wavelet col- location method for the numerical solution of partial differential equations”, ACHA, 3, 1996. • Bertoluzza, S., “Adaptive wavelet collocation method for the solution of Burgers equation”, Transport Theory and Stat. Phys. , 25, 1996. Interpolating wavelets • Deslaurier, G. and Dubuc, S., “Symmetric it- erative interpolation processes, Constructive Approximation , 5, 1989.

5 Deslaurier-Dubuc interpolating functions 1.2 1 0.8 0.6 0.4 0.2 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -0.2 � θ N ( x ) = φ L ( y ) φ L ( y − x ) dy N = 2 L + 1 Properties • suppθ = [ − N, N ] • θ is refinable � φ L ( y ) φ L ( y − n ) dy = δ n 0 • θ ( n ) = • Polynomials up to order N can be represented as a linear combination of the integer trans- lates of θ .

6 Classical Theory of Elasticity     f x t xγ     f =  t = f y t yγ      u x ( x, y )   Displacements u = u y ( x, y )     ε xx ( x, y )         Strains ε = ε yy ( x, y )   ε xy ( x, y )         σ xx ( x, y )         Stresses σ = σ yy ( x, y )   σ xy ( x, y )      

7 Classical Theory of Elasticity Definition of the stress field ∂ u x ∂ u y σ xx = e 1 ∂ x + e 2 ∂ y ∂ u x ∂ u y σ yy = e 2 ∂ x + e 1 ∂ y � ∂u x ∂y + ∂u y � σ xy = e 3 ∂x E e 1 = 1 − ν 2 ν E e 2 = 1 − ν 2 E e 3 = 2 (1 + ν )

8 Classical Theory of Elasticity Problem 1 Find u = [ u x , u y ] T such that A u = f (Ω) u = g (Γ u ) B u = t (Γ σ )  ∂ 2 ∂ 2 ∂ 2  ∂ x 2 + e 3 ( e 2 + e 3 ) e 1   ∂ y 2 ∂ x ∂ y     A =     ∂ 2 ∂ 2 ∂ 2     ( e 2 + e 3 ) e 2 ∂ x 2 + e 1   ∂ y 2 ∂ x ∂ y ∂ ∂ ∂ ∂   ∂ x + e 3 n y ∂ y + e 3 n y e 1 n x e 2 n x   ∂ y ∂ x   B =     ∂ ∂ ∂ ∂     ∂ y + e 2 n y ∂ x + e 1 n y e 3 n x e 3 n x   ∂ x ∂ y

9 Numerical Applications 4 − 4 9 . 9 9 0 1 3 3 − 0 . 8 6 2 4 2 −3.3314 −1.1118 − 0 . 5 6 0 9 8 2 2 8 7 1 0 1 . −0.36002 3.3274 1 7 1 9 0 6 5 4 − 0 . 1 5 5 . 7.7665 0 0 0 2 4 6 8 0 2 4 6 8 4 4 −0.48343 −1.523 3 3 − −1.2631 2 . 3 2 2 0 −1.783 2 8 3 4 1 1 3 8 4 . 0 − 0 0 0 2 4 6 8 0 2 4 6 8

10 Numerical Applications 4 4 Solução exacta Solução exacta Malha uniforme com Jmax=4 Malha uniforme com Jmax=4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 −10 −8 −6 −4 −2 0 2 4 6 8 10 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 4 3.5 Solução exacta 3 2.5 Solução com uma malha uniforme com Jmax=4 2 1.5 1 0.5 0 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

11 Numerical Applications Discretisation N j 0 j max n grid n dof T CP U A 4 3 3 81 162 0.04 B 4 3 4 289 578 0.72 C 4 3 5 1089 2178 52.63 Table 1: Discretisations involved in the analysis of the square cantilever −1 10 error STRAIN ENERGY ERROR Uniform grid refinement Number of degrees of freedom −2 10 2 3 4 10 10 10

12 Square cantilever σ xx σ yy σ xy q V 3.000E+00 8.000E-01 0.000E+00 0.000E+00 0.000E+00 -3.000E+00 -1.000E+00 -1.300E+00 = 2.50 p a / E σ xx σ yy σ xy q V 3.000E+00 8.000E-01 0.000E+00 0.000E+00 0.000E+00 -3.000E+00 -1.000E+00 -1.300E+00 = 2.50 p a / E

13 Square cantilever 1 1 "malhai.res" "malha2.res" 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 1 "malha3.res" "malhaf.res" 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 error −1 10 STRAIN ENERGY ERROR Uniform grid refinement −2 10 Adaptive refinement Number of degrees of freedom −3 10 2 3 4 10 10 10

14 Square cantilever Mesh j max n grid n dof initial 4 289 578 2 5 597 1194 3 6 868 1736 final 7 917 1834 Table 2: Adaptive non-uniform grids used in the solution of the square plate σ xx σ yy σ xy q V 3.000E+00 8.000E-01 0.000E+00 0.000E+00 0.000E+00 -3.000E+00 -1.000E+00 -1.300E+00 = 2.50 p a / E

15 Numerical Applications 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 −1.2601 0.2 0.2 0 0 0 0.5 1 0 0.5 1 1.2 1.2 1 1 0.8 0.8 0.49277 0.6 0.6 0.4 0.4 −0.025409 0.2 0.2 0 0 0 0.5 1 0 0.5 1

16 1.2 1.2 1 1 0.8 0.8 0.49277 0.6 0.6 1.0109 0.4 0.4 9 0 4 5 2 0 0.2 0.2 . 0 − 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1 1 0.8 0.8 − 0 . 0 5 0.6 0.6 5 6 5 9 −0.13177 0.4 0.4 −0.36008 0.020447 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1 1 0.8 0.8 −0.0454 0.6 0.6 −0.13834 0.4 0.4 0.2 0.2 −0.41715 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

17 1.2 1.2 1 1 0.8 0.8 −0.038184 0.6 0.6 −0.12615 0.4 0.4 −0.21412 0.2 0.2 −0.47802 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1 1 0.8 0.8 −0.048228 0.6 0.6 −0.13607 0.4 0.4 −0.22391 0.2 0.2 −0.3996 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1 1 − 0 . 0.8 0.8 0 3 5 4 3 1 0.6 0.6 −0.12469 0.4 0.4 −0.21394 0.2 0.2 − 0 . 4 8 1 7 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

18 Square plate with a central crack Mesh j max n grid n dof 1 4 289 578 8.1355 2 5 625 1250 0.6666 3 6 670 1340 0.6591 4 7 812 1624 0.6530 5 8 823 1646 0.6529 6 9 1082 2164 0.6529 1.2 1.2 1 1 0.97356 0.8 0.8 0.6 0.6 0.4 0.4 −0.05394 64782 0.2 0.2 0 0 0 0.5 1 0 0.5 1 1.2 1.2 1 1 − 0 . 0 3 5 4 3 1 0.8 0.8 0.6 0.6 − 0 0.4 0.4 . 1 −0.21394 2 4 6 0.2 0.2 9 0 0 0 0.5 1 0 0.5 1

19 Non-rectangular elements

20 Gravity dam σ xx σ yy σ xy q V 1.000E+00 4.000E+00 5.000E+00 0.000E+00 0.000E+00 -8.000E+00 -1.000E+00 -5.000E-01 = 16.6667 p a / E 3 "mesh.res" 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5

21 Gravity dam σ xx σ yy σ xy q V 0.000E+00 0.000E+00 1.500E+01 0.000E+00 -2.000E+01 -7.000E+01 -1.500E+01 = 83.3333 p a / E 3 "mesh.res" 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5

22 Gravity dam σ xx σ yy σ xy q V 0.000E+00 0.000E+00 1.500E+01 0.000E+00 -2.000E+01 -7.000E+01 -1.500E+01 = 83.3333 p a / E 3 "mesh.res" 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5

23 Reissner-Mindlin plate bending theory     0 m xγ                 f = 0 t = m yγ      q   q γ            θ x ( x, y )         Displacements u = θ y ( x, y )   w ( x, y )         χ xx ( x, y )         χ yy ( x, y )             Strains ε = χ xy ( x, y )   γ x ( x, y )              γ y ( x, y )        m xx ( x, y )         m yy ( x, y )             Stress resultants σ = m xy ( x, y )   v x ( x, y )             v y ( x, y )      

24 Reissner-Mindlin plate bending theory Definition of the stress resultant fields � ∂ θ x ∂ x + ν ∂ θ y � m xx = D f ∂ y ν ∂ θ x ∂ x + ∂ θ y � � m yy = D f ∂ y � ∂θ x ∂y + ∂θ y � m xy = D 1 ∂x θ x + ∂w � � v x = D 2 ∂x θ y + ∂w � � v y = D 2 ∂y E h 3 D f = 12(1 − ν 2 ) D 1 = G h 3 12 D 2 = 5 6 G h