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Superseismic Loading

Superseismic Loading. Marco Arienti ASCI Site Review October 10 / 11, 2000. Motivations. Find a solution of a fully coupled fluid-solid problem to verify the Eulerian/Lagrangian coupling algorithm Gain understanding on curvature of shock fronts due to deformable boundaries .

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Superseismic Loading

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  1. Superseismic Loading Marco Arienti ASCI Site Review October 10 / 11, 2000

  2. Motivations • Find a solution of a fully coupled fluid-solid problem to verify the Eulerian/Lagrangian coupling algorithm • Gain understanding on curvature of shock fronts due to deformable boundaries

  3. Problem setting • Isotropic medium (0-1-2)density rLamé coefficients m, lPlane strain / small strains • Perfect gas g = 1.4 (3-4) • Dilatational front P: • Distortional front S: • Shock at angle b moving at speed Us elastic solid compressible gas

  4. Problem setting (cont’d) • Use particle velocities (in the solid material): • flow deflection (in the solid, for a frame traveling with Us): • shock polar (in the gas, given Ms= Us /cgas): to obtain a relation between the shock angle b, the deflection d=q of the boundary and the strength of the shock Ms.

  5. Results: Ms-b(rfluid=1000. kg/m3 Pfluid= 2.6x109 Pa) • Finite Elements mesh: 164,000 triangles (L1)16,000 triangles (L2) • Cartesian grid:201x201 nodes (E1)101x101 nodes (E2) • Superseismic limit for Us® cP (Ms )limit = cP/cgas @ 2.23 • Range of initial Mach numbers: Ms=2.4 to Ms=9.

  6. Self-similar solution (detail of transient close to contact point) • Initial conditions:Ms=4.5rfluid=1000. kg/m3Pfluid= 2.6x109 Pa • FE Material:r =8970. kg/m3n =0.33E=110x109 Pa • Examine simulation when a steady state is developed in the reference of the moving shock • Steady state:Ms=4.399b =87.23ºd =10.21º

  7. Dilatational & distortional fronts in the solid • Vertical sectionat x=0.1 m • Origin of the front at x=0.306 • Stress fronts are according to theory and well resolved by the FE solver

  8. Conclusions • A problem treating full fluid-solid coupling is examined. For linear elastic material this problem provides a self-similar solution in the frame of the shock. Extendible to plastic deformations for future work. • Eulerian - Lagrangian coupling algorithm makes the transient converge to the correct steady solution. • The measured inclination (from numerical simulations) of the shock angle and deflection angle converge to the correct value as the Cartesian and/or Lagrangian resolutions increase.

  9. Eulerian-Lagrangian coupling with GrACE Marco Arienti ASCI Site Review October 10 / 11, 2000

  10. Motivations • Explore issues on full E/L coupling when the Eulerian solver is equipped with Adaptive Mesh Refinement. • Examine applications of concurrent AMR for typical High Explosives simulations. • NOTE: E/L coupling scheme was extensively tested for a HE EoS in the presence of rigid boundaries within the frame of the Corner Turning Problem (cf. E. Morano presentation).

  11. GrACE (Grid Adaptive Computational Engine) [M. Parashar] • Hierarchy of scalable distributed dynamic grids (Grid Hierarchy) • Data structure and storage provided (Grid Functions) • Operations on a GF are performed in a data parallel fashion at two granularities: a) a component grid; b) the entire level of GH • Adaptive Mesh Refinement (AMR) can be supported by a GH: starting from a base coarse grid add and remove recursively finer grids as the solution evolves

  12. The Berger-Oliger Algorithm[(1984) JCP, 53, 484-512] • Suited for solving hyperbolic partial differential equations on structured computational domains • Refinement factor is the same in both space and time: nested time marching on all levels (efficient) • Leads naturally to a hierarchic tree of grids • Fine grids have a separate identity from the underlying coarse grid: relation children to parent is maintained • Uses a prolongation operator to populate the GF on a newly created grid from the underlying coarser grid • Uses a restriction operator to inject newly computed function values of a fine grid to the underlying coarser grid

  13. GrACE and the Level Set function • Level Set w is defined as a Grid Function • Over FL levels, w is reconstructed locally on each patch(levels 0, 1, … FL-1) when: - the Lagrangian solver has advanced one time step - remeshing has taken place at a level • Enforce convention to distinguish “real” (w£0) from “ghost” flow (w>0) on each patch

  14. GrACE and E/L coupling • Local truncation error (LTE) evaluation requires j : flag for refinement only when j< Dx • LTE magnified by parameter K ä 1 at the boundary, say for |j|< 0.6Dx • FE solver does not need to be synchronized with the coarsest time step Dtc: - define the scale of times: Dtc, Dtc / R , …, Dtc / RL ,… - find a GrACE level S so that the Lagrangian time step estimate Dtsolid = Dtc / RS - advance the Lagrangian Solver (by Dtsolid) only when level S advances • Modify accordingly the Berger-Oliger algorithm

  15. Procedure bno_RecursiveIntegrate(Level) If (Level==0) NoIterations = 1; else NoIterations = RefinmentFactor; Loop over NoIterations { If(RegridTime(Level)) { Evaluate LTE(Level); Cluster&Regrid levels above; }If(AdvanceLagrangianTime) { Interrogate all levels for traction boundary conditions; Advance Lagrangian Solver; Update boundary; } Populate Ghost Region at Level; March Level; If (Level+1 exists) call bno_RecursiveIntegrate(Level+1); } Increment TimeStep on current Level; If(Level+1 exists) restrict soln. from Level+1 to Level adds E/L coupling

  16. Components of the VTF+AMR prototype • Ghost fluid Eulerian - Lagrangian coupling • Finite Elements mechanics: FE solver for linear elastic isotropic material (“light” version provided by P. Hung) • High Explosive: Roe solver with Mie-Grüneisen EoS for reactants and products and chemistry modeled by heat release term (E. Morano) • GrACE calls from a c++ layer connecting the driver of the application to the High Explosive package

  17. Example: HE in highly deformable casing • Cartesian coarse spacing:Dtc= 2 mm2 levels of refinement at R=4 ® finest scale 0.125 mm • Lagrangian mesh: 26,670 quadratic triangular elements;side length ~ 0.5 mm • HE (HMX) at reference density r0=1891. Kg/m3ZND profile (reaction zone length D~10.2 mm) terminated by Taylor wave • Casing (copper, but 10 times lighter and softer):reference density rS0=897. Kg/m3;Young modulus E=110x108 Pa;Poisson’s ratio n=0.33

  18. Animation: a planar “Cylinder” Test

  19. Detail of the curved detonation front Lagrangian boundary ½ mass fraction

  20. Mesh evolution (frames #5,10, 20, 29) #5 #10 #29 #20

  21. Conclusion and path forward • A VTF prototype was integrated with the Adaptive Mesh Refinement (AMR) parallel environment of GrACE. • Proof-of-concept planar “Cylinder” Test was successful • Parallel computing capabilities of AMR-VTF prototype are now limited to rigid boundaries case (cf. Corner Turning Problem). Extension to complete coupling is required. • Substitution of “light” FE solver with Adlib is under way in order to take full advantage of the coupling with a high resolution reactive flow solver. At this point results from the Cylinder Test (runs on NPACI Blue Horizon and LANL) could be compared with actual experiments.

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