Stability Analysis of Particle Methods with Corrected Derivatives

Transcription

1 Stability Analysis of Particle Methods with Corrected Derivatives Shaoping iao and Ted Belytschko Northwestern University Supported by Office of Naval Research

2 Outline: 1. Description of discretization and stability analysis. Results of stability analysis in 1D 3. Results of stability analysis in D 4. Computational results 5. Conclusion

3 Velocity Approximations for Eulerian and Lagrangian Kernels Eulerian Kernel h v ( x, t = w( x xi vi (commonly used in SPH Eulerian(spatial coordinates Lagrangian Kernel ( Liu & Chen (1996 v h (, t = w( v ( t I I Lagrangian(material coordinates v I = nodal values w = kernel (weight function

4 MAOR MESSAGES Two sources of instabilities: 1. Distortion of material instability occurs only for Eulerian kernel. Rank-deficiency instability occurs in both Lagrangian and Eulerian Kernel 1. Eulerian kernel distorts material instability tensile instability is a manifestation of this : Stress points only stabilize rank deficiency

5 Kernel Function with Corrected Derivatives Kernel function (Shepard (1968 w( W ( I = W ( Corrected Derivatives G I (Belytschko and Krongauz (1998 w = ai ( ( ii ( I Reproducing condition G ii I = δ i I

6 Weak Form and Discrete Equations Weak Form δ δ u ui Ω iρ0u&& idω = Ω δu ρ b dω Ω P dω 0 i 0 i 0 i + Γ 0 δu t 0 i i dγ Discrete Equations m u&& I ii = f ext ii f int ii External nodal force f ext ii ρ w b dω + = t Ω0 0 I i Γ0 w t dγ I i Internal nodal force f wi = 0 ( PidΩ = wi, x σ id int ii Ω Ω Ω u& & ii m I nodal accelerations nodal masses

7 Integration Scheme Nodal Integration (Monaghan(1988, Beissel and Belytschko (1996 f int ii = V 0 wi ( P i ( EFG with nodal integration is equivalent to SPH by collocation method for model without boundaries. EFG with nodal integration has same instability as SPH. Strain smoothing to stabilize nodal integration (.S. Chen(001

8 Integration Scheme Stress Points (Dyka(1995,1997, Libersky and Randles(1996 ( ( ( ( int S i N S I 0S M i N M I 0M ii P w V P w V f S M + =

9 Stability Analysis classical linearized stability analysis: u = u + u~ homogeneous u~ = ge iωt+ iκn Perturbation of displacements 0 ω κ 0 n frequency wave number The normal direction of the wavefront

10 Stability Results for Continuum (for rate-independent material classical Hill, Mandel(1906 κ C ω = C P = C ρ SE 0 F + PF 1 norminal stress C SE tangent modulus so S& = C E SE & C effective tangent modulus Result: if C < 0, material is unstable

11 Stability Analysis in 1D for Nodal Integration 0 w ac ] sin( ( [ = κ ρ ω Eulerian kernel Lagrangian kernel x x x w ac ] sin( ( [ ' = κ ρ ω } ] sin( ( [ ] cos( [ ( { x x x w a x 1 x x w κ κ ρ σ σ +σ = T C C ' is push-forward of C

12 σ Stability Diagrams for C T > 0 Continuum Lagrangian kernel Eulerian kernel

13 One Dimensional Example σ 0 v0 σ 0 Lagrangian kernel (note scales are different Eulerian kernel

14 Stability Analysis in 1D for Stress Points = m m 0 m w C ] sin( ( [ κ ρ ω Lagrangian kernel 1 SE PF F C C + = s s s s w a ]. ( sin(. (( [ κ

15 Stability Analysis in D Nodal integration with Lagrangian kernel u ~ ( r K = A e r iκ ( cosθ + Y sinθ + iωt K K Results are shown graphically in the following

16 Lagrangian kernel Eulerian kernel Frequency-wave number relations for nodal integration

17 Lagrangian kernel Eulerian kernel Frequency-wave number relations for Stress points

18 Two Dimensional Example Lagrangian kernel Eulerian kernel deformed rubber ring by stress points

19 Two Dimensional Example Lagrangian kernel Eulerian kernel deformed rubber ring by nodal integration

20 Material Instability by Lagrangian kernel nodal integration stress points shear bands due to material instability by Lagrangian kernel

21 Computational Results Needleman & u(1994 Evolution of Crack in a plate

22 Conclusion The tensile instability is a distortion of material instability due to Eulerian kernel. For a Lagrangian kernel the instability of discrete equations replicates the material instabilities in 1D. In 1D, stress points eliminate the instability due to rank deficiency but not the tensile instability. Stress points with Lagrangian kernel gives a stable discretization which accurately reproduces material instabilities.