Simulating Thin Shells with MPM

Transcription

1 Simulating Thin Shells with MPM Biswajit Banerjee Center for the Simulation of Accidental Fires and xplosions University of Utah March 14, 2005

2 Outline The Problem. Shell Theory. MPM Formulation. Results. Conclusions. Version: March 5, 2005; Typeset on March 13, 2005,14:38 2

3 Problem: Rocket Casing (Courtesy: Wikipedia and AHPRA) Version: March 5, 2005; Typeset on March 13, 2005,14:38 3

4 Problem: Bio Membranes (Baumgart et al., 2003, Nature, 425, ) Version: March 5, 2005; Typeset on March 13, 2005,14:38 4

5 Problem: The Goal Predict deformation and stress in thin shells/membranes interacting with fluids. Version: March 5, 2005; Typeset on March 13, 2005,14:38 5

6 Shell Theory: Notation N F Director n z β X x α h h φ 0 Mid surface φ Version: March 5, 2005; Typeset on March 13, 2005,14:38 6

7 Shell Theory: Assumptions Modified Reissner-Mindlin approach. (Hughes and Liu, 1981, Computer Meth. Appl. Mech. ngrg., 26, , ) Assumptions Director remains straight. Deformed director not normal to the mid-surface. Plane stress. No momentum balance in the director direction. No curvature at material points. Version: March 5, 2005; Typeset on March 13, 2005,14:38 7

8 Shell Theory: Kinematics Ansatz. x(α, β, z, t) = ϕ(α, β, t) + z n(α, β, t) (1) Material time derivative. ẋ(α, β, z, t) = ϕ(α, β, t) + z ṅ(α, β, t) + ż n(α, β, t) (2) Ignore velocity in director direction. w(α, β, z, t) = u(α, β, t) + z r(α, β, t) (3) Velocity gradient. w = [ s u + z s r] + rn (4) where s a := a (1 nn). (Lewis et al., 1998, Proc. ASM Pressure Vessels and Piping Conference.) Version: March 5, 2005; Typeset on March 13, 2005,14:38 8

9 Shell Theory: Momentum Balance Balance of momentum in 3D. σ = ρ ẍ ; σ = σ T (5) Mid-surface balance of momentum. where σ := 1 h h + h σ(z) dz Director balance of momentum. s σ = ρ u (6) s M n σ (1 nn) = 1 12 ρ h2 ṙ (7) where M := (1 nn) [ 1 h ] h σ(z) z dz (1 nn) h + (Lewis et al., 1998, Proc. ASM Pressure Vessels and Piping Conference.) Version: March 5, 2005; Typeset on March 13, 2005,14:38 9

10 MPM: Interpolation Functions Particle characteristic functions. χ p (x) = 1 x Ω ; F (x) = F (x p ) χ p (x) (8) p p where χ p (x) = H(x (x p l p )) H(x (x p + l p )), H is the Heaviside step function, and 2 l p is the current particle spacing. (Bardenhagen et al., 2004, Comp. Model. ng. Sci.) Grid interpolation functions. S v (x) = 1 x Ω ; G(x) = G(x v ) S v (x) (9) v v where S v (ξ, η, ζ) = 1 8 (1 + ξξ v)(1 + ηη v )(1 + ζζ v ) (isoparametric). Version: March 5, 2005; Typeset on March 13, 2005,14:38 10

11 MPM: Particle to Grid Weighting functions. S p,v (1) = 1 χ p (x) S v (x) dv ; V p Ω p Ω v S (1) p,v = 1 x p Ω p (10) where V p = material point volume, and Ω p = region of non-zero support for the material point Interpolation. m v = p m p S (1) p,v (11) The state variables that are interpolated to the grid in this step are the mass (m), momentum (mu), volume (V ), external forces (f ext ), temperature (T ), and specific volume (v). Version: March 5, 2005; Typeset on March 13, 2005,14:38 11

12 MPM: Director Rotation Rate Interpolate p p = m p r p instead of angular momentum. p v = p p S p,v (1) (12) p Recover rotation rate at the grid. Why? r v = p v m v (13) The moment of inertia contains h 2 terms which can be very small for thin shells. It is also not desirable to interpolate the shell thickness to the grid. Version: March 5, 2005; Typeset on March 13, 2005,14:38 12

13 MPM: Computing Gradients Gradient weighting function. S p,v (1) = 1 χ p (x) S v (x) dv (14) V p Ω p Ω Gradients of u and r at a particle. u p = v u v S (1) p,v ; In-surface gradients. r p = v r v S (1) p,v (15) s u p = u p [1 n n pn n p] ; s r p = r p [1 n n pn n p] (16) Version: March 5, 2005; Typeset on March 13, 2005,14:38 13

14 MPM: Computing Stress Compute velocity gradient in each layer. w top w cen p = [ s u p + h + s r p ] + r n pn n p (17) p = s u p + r n pn n p (18) w bot p = [ s u p h s r p ] + r n pn n p (19) Compute deformation gradient in each layer. F p = t w p + 1 ; F n+1 p = F p F n p (20) Compute 3D Cauchy stress in each layer. σ = K ( J 1 ) 1 + G [ ] 1 b 2 J J 3 tr( b1) where K = bulk modulus, G = shear modulus, J = det F, and b := J 2 3F F T. (21) Version: March 5, 2005; Typeset on March 13, 2005,14:38 14

15 MPM: The Plane Stress Condition xpress F p in terms of an orthogonal basis (with e 3 aligned with n) F rot p = R F n+1 p R T (22) where R = cos θ (1 aa) + aa sin θ skew(a), a is the axis of rotation, and θ is the angle of rotation. Compute the Cauchy stress components in this basis. Apply the constraint σ 33 = 0. Use a Newton iteration to determine the deformation gradient F and the stress σ. Version: March 5, 2005; Typeset on March 13, 2005,14:38 15

16 MPM: Updates Update the thickness of the shell. h + n+1 = h F zz (+z) dz ; h n+1 = h 0 Update the deformation gradients. F n+1 p Update the stresses. Update the volume. 1 0 F zz ( z) dz (23) = R T F R (24) σ n+1 p = R T σ R (25) V n+1 p = V 0 p J n+1 p (26) Version: March 5, 2005; Typeset on March 13, 2005,14:38 16

17 MPM: Internal Force/Moment Internal force. f int v = p [ σ n+1 p S (1) p,v ] V n+1 p (27) where σ n+1 p = 1 h + n+1 h n+1 h n+1 Internal moment. m int v = p where 1 s = [ 1 n n pn n p, and M n+1 p = 1 s 1 h n+1 h + n+1 h n+1 σ n+1 p (z) dz. [( M n+1 p S (1) p,v 1 s) + (28) ( n n p σ n+1 p (z) z dz σ n+1 p ] 1 s. 1 s ) S (0) p,v ] V n+1 p Version: March 5, 2005; Typeset on March 13, 2005,14:38 17

18 MPM: quations of Motion Linear momentum of mid-surface (solved on grid). u v = 1 m v ( f ext v f int ) v (29) Angular momentum of director (solved on particles). Compute the first-order gradient term on grid. m v = ( M n+1 p S p,v (1) 1 s) (30) p Interpolate m v to the particles. Solve momentum equation on the particles. ( ) 12 V [ p ṙ p = m p h 2 m ext p m p n p σ ] p 1 s (31) p Version: March 5, 2005; Typeset on March 13, 2005,14:38 18

19 MPM: Integrate Accelerations Mid-surface acceleration. Director acceleration. Compute intermediate increment. Compute implicit correction for stiffness. (CFDLib Source Code). Compute intermediate rotation rate. u n+1 v = u n v + t u v (32) r p = t ṙ p (33) r p = (1 + β1 s ) r p (34) r n+1 p = r n p + r p (35) Version: March 5, 2005; Typeset on March 13, 2005,14:38 19

20 MPM: Update Shell Director Compute incremental rotation R using Update director. Update rate of rotation. θ = r t ; a = nn p r n+1 p n n p r n+1 p (36) n n+1 p = R n n p (37) r n+1 p = R r n+1 p (38) Version: March 5, 2005; Typeset on March 13, 2005,14:38 20

21 Simulation: Punched Plate Plane Shell Hollow Cylinder Version: March 5, 2005; Typeset on March 13, 2005,14:38 21

22 Results: Deformed Plate Version: March 5, 2005; Typeset on March 13, 2005,14:38 22

23 Simulation: Punched Plate Solid Block Cylindrical Shell Version: March 5, 2005; Typeset on March 13, 2005,14:38 23

24 Results: Deformed Cylinder Version: March 5, 2005; Typeset on March 13, 2005,14:38 24

25 Results: Inflated Sphere Version: March 5, 2005; Typeset on March 13, 2005,14:38 25

26 Conclusions/Future Work asy to incorporate 3D material models. Plane stress condition difficult to apply. Need better representation of shell kinematics. Need improved update for shell director. Alternative is geometrically exact shell. More difficult to implement material models. Difficult to interpolate between shell surface and 3D grid. Version: March 5, 2005; Typeset on March 13, 2005,14:38 26

27 Questions? O F Version: March 5, 2005; Typeset on March 13, 2005,14:38 27