WAVELETS ON THE INTERVAL APPLICATION TO ELASTICITY PROBLEMS 9 12 - PDF document

WAVELETS ON THE INTERVAL APPLICATION TO ELASTICITY PROBLEMS 9 –12 April 2001 Marseille, France

• 3D o Dams (concrete)

ARCH DAMS BUTTRESS DAMS

1. Introduction to the problems we study o the domain Ve o the surface Γ divided in two complementary parts: o Dirichlet boundary Γ ue o Neumann boundary Γσ e EQUILIBRIUM COMPATIBILITY ELASTICITY CONDITIONS CONDITIONS CONDITIONS ν E E 1 σ = − ε + ε δ σ + = ε = + b j 0 ( u u ) on V e ij ij kk ij + υ + υ − υ ij , i ij i , j j , i 1 ( 1 )( 1 2 ) 2 + b = D σ 0 ε = f σ * = ε D u σ n = t u = u ij i j on Γ i i N σ = t u = u

STRETCHING PLATES BENDING PLATES APPROXIMATION CRITERIA = ⋅ = ⋅ = ⋅ σ S X u U V q u U Γ q V Γ EQUILIBRIUM IN THE EQUILIBRIUM ON THE DOMAIN BOUNDARY ( ) ( ) ∫ ∫ T T + = − = U D σ b dV 0 U N σ t d Γ 0 V Γ σ V Γ σ ( ) ∫ T ( ) ∫ = T A DS U dV = A NS U d Γ V V Γ σ Γ Γ σ T = − A T A X Q = Q X V V Γ Γ

COMPATIBILITY IN THE DOMAIN ( ) ( ) ∫ ∫ T T = − + e DS u.dV NS u.d Γ V Γ ( ) ∫ T * − = S ε D u dV 0 V = − + + e A q A q e V V Γ Γ ELASTICITY CONDITION ( ) ∫ t − = S ε f σ dV 0 V ∫ T fSdV = F S V e = FX

GOVERNING SYSTEM   − F A A     X e V   Γ     T   ⋅ ⋅ = − A q Q     V V V     T − ⋅ ⋅   − A q Q       Γ Γ  Γ 

2. Why wavelets? Localization & Adaptability • Modeling singularities in tension (cracks, damage, …) What we look for in functions… • Hierarchical • Orthogonal • Fast computation and numerical analysis 2.1 What work was previously done? • The use of only scaling functions (Daubechies orthogonal wavelets) o Elasticity o Plasticity • Manipulation of the wavelets o Problems with the boundaries o Loss of orthogonality

2.2 Options • Wavelets on the Interval • Use of other wavelet systems even if not orthogonal 2.3 What we did… • Application of Daubechies orthogonal Wavelets on the Interval based on the works of: o A. Cohen, I. Daubechies and P. Vial; o V. Perrier and P. Monasse − + N 1 N 2 C ∑ ∑ ( ) ( ) ( ) Left Left Left Left φ = φ + φ x H x h x + + j , k k . l j 1 , k k , m j 1 , m = = l 0 m N ( ) ( ) Left j / 2 Left j φ = φ x 2 2 x • j . k 0 . k ( ) ( ) φ = j / 2 φ j − x 2 2 x m • j , m • C=2 k where k=0, …, N-1 (Cohen) • C=N-1 (Perrier) • Using numerical integration

3. Some results Problems • Stretching plates o Short Cantilever o Stressed stretching plate with central crack o L plate Approximations • Type 1: Scaling functions + wavelets • Type 2: Only scaling functions

• Short Cantilever ( E = 1.0, ν ν ν = 0.3) ν 3 4 1,0 1 1 2 1,0 1,0 Mesh A Mesh B α v β ndf Problem Model nele N jo jx jv jg nnz spar 944 C T 2 Cdis 1 4 - 4 3 3 768 176 48196 0.8927 3072 608 3680 C T 2 Cdis1 1 4 - 5 4 4 186096 0.9727 3072 608 3680 C T 2 Cdis4 1 5 - 5 4 4 286054 0.9579 3072 608 3680 C T 2 Cdis7 1 6 - 5 4 4 415740 0.9388 3072 608 3680 C T 1 Cdis 1 4 3 4 3 3 38016 0.9946 3072 672 3744 C T 2 Cdis10 4 4 - 4 3 3 194176 0.9725 4 12288 2240 14528 246272 0.9977 C T 1 Cdis1 1 4 3 5 4 4 12288 2368 14656 741232 0.9931 C T 2 Cdis11 4 4 - 5 4 4 12288 8384 20672 783360 0.9963 C T 1 Cdis3 1 4 3 5 5 5 49152 8832 57984 2812224 0.9983 C T 2 Cdis12 4 4 - 6 5

N = 4 JX=4, JV=3, JG=3 JX=5, JV=4, JG=4 JX=6, JV=5, JG=5 T2_CDIS 1 T2_CDIS 2 T2_CDIS 1 ELE NNZ=48 196, NNZ=186 096, NNZ=710 084, NDF= 944 NDF=3 680 NDF= 14 528 T2_CDIS 10 T2_CDIS 11 T2_CDIS 12 4 ELE NNZ=194 176, NNZ=741 232, NNZ=2 812 224, NDF= 3 744 NDF=14 656 NDF=57 984

N = 4 N = 5 N = 6 JX=5, JV=4, JG=4 JX=5, JV=4, JG=4 JX=5, JV=4, JG=4 T2_CDIS 1 T2_CDIS 4 T2_CDIS 7 1 ELE NNZ=186 096, NNZ=286 054, NNZ=415 740, NDF=3 680 NDF=3 680 NDF=3 680

N = 4 JX=4, JV=3, JG=3 JX=5, JV=4, JG=4 JX=6, JV=5, JG=5 T2_CDIS 1 T2_CDIS 2 T2_CDIS T 2 NNZ=48 196, NNZ=186 096, NNZ=710 084, NDF= 944 NDF=3 680 NDF= 14 528 JX=4, JV=3, JG=3 JX=5, JV=4, JG=4 T1_CDIS 1 T1_CDIS T 1 J0=3 NNZ=38 016, NNZ=246 272, NDF=13 680 NDF=14 528

T1_CDIS σ xx σ yy σ xy T1_CDIS - MULTIRESOLUTION

T1_CDIS3 σ xx σ yy σ xy T1_CDIS3 - MULTIRESOLUTION

• Stressed stretching plate with central crack 3 4 3 4 1,25 1 2 0,2 1 2 0 , 2 1,0 α v β ndf Problem Model N jo jx jv jg nnz spar 14672 Crack T 2 T 4 - 5 4 4 12288 2384 740596 0.9931 14672 Crack T 1 T 4 3 4 3 3 12288 2384 153024 0.9986 58016 Crack T 1 T1 4 3 5 4 4 49152 8864 987648 0.9994 4 49152 33440 82592 3136000 0.9990 Crack T 1 T 2 4 3 5 5

T2 T σ xx σ yy σ xy

T1 T σ xx σ yy σ xy T1 T1 σ xx σ yy σ xy

T1 T2 σ xx σ yy σ xy

• L – stretching plate 2 3 a/ 2 2 3 1 a a/ 2 1 a α v β v β γ ndf Problem Model N jo jx jv jg nnz Spar 4 3 4 3 3 9216 1536 288 11040 114812 0.9981 L T 1 L L 2 4 3 5 5 4 36864 24576 576 62016 2352128 0.9987 L T 1

T1_L T1_L2 DEFORMATION

4. Future work … • Wavelet related o Implementation of analytical integrations based on works of Beylkin, Dahmen and Michelli, and Perrier o Comparison between Cohen’s and Perrier’s Wavelets on the Interval o Study of other wavelet systems on the interval ! Orthogonal (Interacting boundary wavelet) ! Non orthogonal (bi-orthogonal, …) o Implementation of adaptive schemes o Physical non-linear analysis (Elastoplasticity, fracture and damage mechanics) o 3D models