NUMERICAL SIMULATION OF FLUID-STRUCTURE INTERACTION PROBLEMS WITH DYNAMIC FRACTURE

Transcription

1 NUMERICAL SIMULATION OF FLUID-STRUCTURE INTERACTION PROBLEMS WITH DYNAMIC FRACTURE Kevin G. Wang (1), Patrick Lea (2), and Charbel Farhat (3) (1) Department of Aerospace, California Institute of Technology (2) Department of Mechanical Engineering, Northwestern University (3) Department of Aeronautics and Astronautics, Department of Mechanical Engineering, Institute for Computational and Mathematical Engineering, Stanford University December 4, 213

2 MOTIVATION Fluid-structure interaction problems with fracture underwater implosion - structure interacts with internal and/or surrounding fluids during fracture - post-fracture dynamics is also important! pipeline explosion shock lithotripsy

3 COMPUTATIONAL CHALLENGES Constitutive models - solid material models - equation of state for water Fracture models and methods - modeling and criteria - mesh independent solution Multi-phase fluid flows - density jump across interface: >1 X Fluid-structure interaction - large structural deformation - topological change due to fracture - fluid flow through cracks ( seepage ) - strong shocks at F/S interface numerical instabilities ~ 2 mm crack growth * element erosion * courtesy of Ravi-Chandar, K., UT Austin

4 MATHEMATICAL MODELS Multi-phase compressible fluids in Eulerian formulation - Euler equations where W = and W + F W = t ρ ρv, E F =,ρv, v ρv + pi, E + p v- - immiscible fluids / free surface φ + v φ = t v L n = v R n, p L = p R - equation of state (EOS) for water p = or p = p c + αρ β γ s 1 ρe γ s p s air before fracture f-f interface water f-s interface air water after fracture (level-set equation, for interface tracking) (Tait) (stiffened gas) left fluid v R right fluid n

5 MATHEMATICAL MODELS Elastic-plastic structure in Lagrangian formulation - equation of motion ρ s u s σ s u s, u s = f ext - 2 nd -order Green s strain tensor ε s = 1 2 ( u s + u s T + u u T ) f-f interface water f-s interface air air water - J 2 plasticity, linear (or piecewise linear) hardening before fracture - maximum strain based fracture criterion - cohesive force (to dissipate fracture energy) after fracture τ c δ = kδ, δ δ max, δ > δ max s.t. δ max τ c δ dδ = G f (fracture energy) determination of crack growth (J-H Song, 28)

6 MATHEMATICAL MODELS Fluid-structure interaction - impermeable fluid-structure transmission conditions (for inviscid fluid flow) f-f interface water f-s interface water v n s = u s n s no interpenetration pn s = σ s n s equilibrium air before fracture air after fracture

7 COMPUTATIONAL FRAMEWORK A fluid-structure coupled computational framework the fluid sub-problem (Eulerian formulation) CFD solver FIVER: a FInite Volume method based on Exact two-phase Riemann problems embedded boundary method FSI tracking fluid-structure interface enforcing transmission conditions the structural sub-problem (Lagrangian formulation) CSD solver extended finite element method (XFEM) phantom node formulation

8 EXTENDED FINITE ELEMENT METHOD XFEM with the Phantom Node Formulation* - each cracked element is replaced by two elements with phantom nodes - the cracking path within each element is tracked by a local signed distance function (φ(x, t)) I c *X e φ X, t = + e Ω S - displacement (u s )in a cracked element: u s X, t = N I X H φ X u I 1 I Elem 1 where H x =, x < 1, x + N I X H φ X u I 2 I Elem 2 * J-H Song et al. (28)

9 Projection based algorithms D h INTERFACE TRACKING S E - relies on the outward (or inward) normal of the interface - not capable of open surfaces or surfaces undergoing topological change (e.g. cracking) - examples: Udaykumar et al. (21); Mittal et al. (28)

10 Collision based algorithm D h INTERFACE TRACKING real phantom S E - detect the intersections between edges in the CFD grid and simplices in the discretized embedded interface, i.e. solve x i + tx ij x n e = for collision time t; edge V ij intersects element e if and only if < t < 1. * Wang, Gretarsson, Main, Farhat, 212

11 Collision based algorithm D h INTERFACE TRACKING real phantom S E - determine if the intersection point (x = x i + tx ij ) is located in the real part of a cracked element φ φ x = ξ i φ i 3 i=1 - register the intersection if and only if φ >

12 Fluid-structure demarcation (with cracking) D h INTERFACE TRACKING S F S E immiscible fluid-fluid interface: S F = {x in W f(x) = }, where f t + v : r f = (level-set equation) * Wang, Lea, McGarity, Belytschko, Farhat, (in prep)

13 MULTI-PHASE FLUID SOLVER The standard finite volume spatial discretization - Euler equations W F( W ) t - integrate over a control volume (C i ) W h dw t C j nei( i) C i ij F( W h ) n ij d i Ci C ij j - evaluate one numerical flux for each facet ( C ij ) i j F ( W ) n d Roe( W, W, n, EOS) C ij h ij h h - special treatment is required near a fluid-structure or fluid-fluid interface ij

14 Interface flux FLUID-STRUCTURE INTERFACE S E fluid 1 i j fluid 2 - <fluid 1, structure> : fluid-structure Riemann problem W i v s. n x i * Wang, Rallu, Gerbeau, Farhat 211

15 F-S RIEMANN PROBLEM One-dimensional, fluid-structure Riemann problem rarefaction p*, r*, u x = x(t) contact s discontinuity t not involved W n L p L, r L, u L i M ij j w t x = n w(x,) = W L, if x u(x(t), t) = u (M ij ) n s no interpenetration transmission condition could also be a shock

16 Interface flux FLUID-STRUCTURE INTERFACE S E fluid 1 i j fluid 2 - <fluid 1, structure> : fluid-structure Riemann problem --> W i * - <fluid 2, structure> : fluid-structure Riemann problem --> W j * F (1) ij = Roe (W i, W i*, EOS (1) ) (fluid 1, structure) F (2) ij = Roe (W j, W j*, EOS (2) ) (fluid 2, structure)

17 Interface flux FLUID-FLUID INTERFACE S E i j fluid 1 fluid 2 - <fluid 1, fluid 2> : two-phase fluid-fluid Riemann problem --> W i * and W j * F (1) ij = Roe (W i, W i*, EOS (1) ) F (2) ij = Roe (W j, W j*, EOS (2) )

18 - desired structural load on F k s = CONSERVATIVE LOAD TRANSFER Load computation using a surrogate interface k [p] f k s n d - CFD grid is non body-fitted define a surrogate interface control volume facets: * reconstructed surface: * F ij f = (p (1) f u p (2) f u ) ds ij * - convert F f to F s by virtual work principle! U f T F f = U s T F s I I ij I I ji F s = K T F f i M ij u ij M ij k, j n ij * Wang, Rallu, Gerbeau, Farhat, 211 * * C

19 CONSERVATIVE LOAD TRANSFER Numerical properties - reproduces the zero force for a constant pressure field and a closed embedded surface - w/o geometric error: locally third-order accurate, and globally second-order accurate - w/ geometric error: locally second-order accurate, and globally first-order accurate convergence analysis (steady flow)

20 * Performed by S. Kyriakides et al. at University of Texas at Austin Experiment* UNDERWATER IMPLOSION - implosive collapse of submerged aluminum tube (air-backed) - increased water pressure until tube collapsed. Tube collapsed dynamically essentially under constant pressure (197. psi). pressure sensors specimen water tank after sensor signal

21 Simulation setup UNDERWATER IMPLOSION - water / thin shell / air, no fracture - modeled half of the tube length-wise (symmetry assumed) - CFD grid: 3.7M nodes, 2.1M tetrahedron elements (3 procs.) - structural model: 14K Belytschko-Tsay shell elements - stress-strain response measured by - J 2 -plasticity, piecewise linear hardening CFD domain 2D views FE structural model (shell elements) * Farhat, Wang, et al. 213

22 UNDERWATER IMPLOSION Synchronized Output from Experiment and Simulation

23 pressure (psi) pressure (psi)

24 pressure (psi) pressure (psi)

25 pressure (psi) pressure (psi)

26 pressure (psi) pressure (psi)

27 pressure (psi) pressure (psi) not available due to limited camera frequency

28 pressure (psi) pressure (psi) not available due to limited camera frequency

29 pressure (psi) pressure (psi) not available due to limited camera frequency

30 pressure (psi) pressure (psi) not available due to limited camera frequency

31 pressure (psi) pressure (psi) not available due to limited camera frequency

32 pressure (psi) pressure (psi) not available due to limited camera frequency

33 pressure (psi) pressure (psi)

34 pressure (psi) pressure (psi)

35 pressure (psi) pressure (psi)

36 pressure (psi) pressure (psi)

37 pressure (psi) pressure (psi)

38 pressure (psi) pressure (psi)

39 pressure (psi) pressure (psi)

40 pressure (psi) pressure (psi)

41 pressure (psi) pressure (psi)

42 pressure (psi) pressure (psi)

43 UNDERWATER IMPLOSION Validation* pressure (psi) DT - p max I + I - DT + p min -1 time (ms) characteristics of the pressure pulse Sensor No. p min (psi) DT - (ms) I - (psi-ms) p max (psi) DT + (ms) I - (psi-ms) 1 exp sim exp sim exp sim exp sim exp sim * Farhat, Wang, Kyriakides, et al. (213)

44 PIPELINE EXPLOSION Experiment* detonation tube pressure gauge aluminum (661-T6) pipe (specimen) I beam detonation tube.89 mm notch pre-flawed aluminum pipe (specimen) Mylar diaphragm detonation wave 457 mm mm 916 mm clamp * performed by J. Shepherd et al. at California Institute of Technology.

45 blast pressure (kpa) PIPELINE EXPLOSION Blast pressure initial notch length initial notch length (mm)

46 Simulation setup PIPELINE EXPLOSION - air thin shell gas, with cracking - CFD grid: 2.1M nodes, 12.5M elements (on 26 processors) - air: compressible and inviscid; perfect gas (g=1.4); initial pressure: 5. MPa (inside),.1 MPa (outside) - CSD Model: 17K B-T shell elements; elasto-plastic - initial notch: 25.4mm / 38.1mm / 5.8mm / 63.5mm / 76.2mm

47 (photos from Chao, T-W. 24) PIPELINE EXPLOSION Animation

48 blast pressure (kpa) PIPELINE EXPLOSION Blast pressure initial notch length initial notch length (mm) * Wang, Lea, Belytschko, McGarity, Farhat. (in prep.)

49 - Embedded boundary method K. Wang, A. Rallu, J-F. Gerbeau, and C. Farhat, Algorithms for Interface Treatment and Load Computation in Embedded Boundary Methods for Fluid and Fluid-Structure Interaction Problems, IJNMF. 67: (211). - Interface tracking K. Wang, J. Gretarsson, A. Main, and C. Farhat, Computational Algorithms for Tracking Dynamic Fluid-Structure Interfaces in Embedded Boundary Methods, IJNMF. 7: (212) - Staggered time integrators C. Farhat, A. Rallu, K. Wang and T. Belytschko, "Robust and Provably Second-Order Explicit-Explicit and Implicit-Explicit Staggered Time-Integrators for Highly Nonlinear Fluid-Structure Interaction Problems", IJNME, 84:73-17 (21) - 17 (21). - FIVER C. Farhat, J-F. Gerbeau, and A. Rallu, FIVER: A Finite Volume Method Based on Exact Two-Phase Riemann Problems and Sparse Grids for Multi-Material Flows with Large Density Jumps, JCP, 231: (212) - Validation for underwater implosion REFERENCES C. Farhat, K. Wang, A. Main, J.S. Kyriakides, K. Ravi-Chandar, L-H. Lea, and T. Belytschko, Dynamic Implosion of Underwater Cylindrical Shells: Experiments and Computations, IJSS, 5: (213)