Jiří Plešek, Radek Kolman, Miloslav Okrouhĺık

Transcription

1 DISPERSNÍ ANALÝZA VLNOVÉHO ŘEŠENÍ V METODĚ KONEČNÝCH PRVKŮ Jiří Plešek, Radek Kolman, Miloslav Okrouhĺık Ústav termomechaniky Akademie věd České republiky Praha

2 Contents Dispersion diagrams (overview) Quadratic finite elements spatial discretization error time discretization error mass lumping for explicit schemes Numerical experiments Outlook

3 Dispersion curves After Newton, Kelvin, Born... mü i = k(u i 1 2u i + u i+1 ) solution form u i = û sin K(x i ct) wave number K = 2π Λ = ω c solvability condition c = function(ω)

4 Propagation of wave packets Definition of group speeds is essential for higher order elements. phase velocity c = ω K group velocity c = dω dk

5 Three dimensional lattices Brillouin, L.: Wave Propagation in Periodic Structures. Dover Publications, Inc., New York c-ω plot c-h plot ω-k plot 1.2 longitudinal longitudinal exact c/c l transverse ω H π c l c / c l longitudinal transverse ω H 3 c l longitudinal transverse transverse exact ω H / c l H / λ H / λ (Pictures shown from tri-linear FE analysis.)

6 Effect of propagation angle Characteristic length of element L = f(h, θ) defined. definition of L ω-h plot ω-l plot The worst case θ = when L = H is treated further on.

7 Finite element method Belytschko, T., Mullen, R.: On dispersive properties of finite element solutions, In: Modern Problems in Elastic Wave Propagation. Wiley Abboud, N.N., Pinsky, P.M.: Finite element dispersion analysis for the threedimensional second-order scalar wave equation. Int. J. Num. Meth. Engrg., 35, pp , 1992.

8 Linear versus quadratic elements serendipity linear longitudinal 1 c cl c / cl transverse H/λ H/λ.4.5 Accuracy of quadratic finite elements is by far better. There are, however, four spurious branches called the optical modes. The optical modes are not eigenvectors so that they do not affect numerical stability.

9 Group velocity Lamb, H.: On group-velocity. Proc. Lond. Math. Soc., ser. 2, 1, pp , 194. Mandel shtam, L.I.: Group velocity in a crystal lattice. Zhurn. Eksp. Teor. Fiz., 15, pp , 1945 (in Russian) ωh cl 1 7 longitudinal exact transverse exact 3 cg cl spatial resolution limit H/λ H/ λ Group velocities cg = dω/dk are finite! Negative speed observed!.4.5

10 Optical modes and band filters c-ω spectrum Optical mode 3

11 Effect of time integration Dimensionless Courant number Co=(c l t)/h Explicit (CDF) Implicit (Newmark) c l -dispersion analysis now includes spatial and time discretization.

12 Mass matrix diagonalization Hinton-Rock-Zienkiewicz lumping scheme used for serendipity elements. c-h plot cg -H plot Favourable properties of serendipity elements with consistent mass matrix were spoiled. Transversal wave can overtake the longitudinal wave!

13 Comparing element types Row sum and HRZ used. quadratic linear consistent mass matrix 1.4 c cl lumped mass matrix.1.2 H/λ Similar performance advantage lost.

14 Numerical experiments Infinite plate under plane stress conditions. Point source loading at 45. Mesh size L/Λ =.1 (fine) L/Λ =.5 (critical) Material properties E = 1 Pa ρ = 1 kg/m 3 ν =.3 Newmark method t

15 Test 1: transversal wave (fine mesh) Longitudinal and transversal wave excited. No optical mode. longitudinal λ + 2G c g c = ρ transversal G c g c = ρ

16 Test 2: longitudinal wave (fine mesh) Longitudinal and transversal wave excited. No optical mode. longitudinal λ + 2G c g c = ρ transversal G c g c = ρ

17 Test 3: optical T-mode (fine mesh) Four optical modes excited. No acousto-elastic wave. x-y polarisation c g c 1 15

18 Test 4: optical L-mode (fine mesh) Optical mode doublet excited. No acousto-elastic wave. trans. x-y polarisation long. c g c 1 2 optical speed cg /cos ( π/4)

19 Test 5: transversal wave (coarse mesh) Longitudinal and transversal wave excited. Transversal strongly damped. trans. longitudinal longitudinal λ + 2G c g c = ρ transversal G c g c = ρ

20 Test 6: acousto-elastic damping Longitudinal and transversal waves damped plus two optical modes excited. longitudinal λ + 2G c g c = ρ transversal G c g c = ρ

21 Test 7: high frequency cut-off locking Everything damped out and the mesh has locked!

22 Conclusions Dispersion of quadratic serendipity elements : theoretical curves (as if t ) effects of explicit and implicit time integration error maps for Courant-element wave numbers effects of HRZ mass lumping Kolman, R., Plešek, J., Okrouhĺık, M.: Numerical dispersion of finite elements in elastodynamics: Part I: Analysis of spatial discretization error. Part II: Time integration and stability analysis. (IJNME, to appear) Future work: better lumping scheme proposal dynamic patch test proposal