The Implementation of a General Higher-Order Remap Algorithm

1. LA-UR 11-03531 The Implementation of a General Higher-Order Remap Algorithm Vincent P. Chiravalle MultiMat 2011 Conference Arcachon, France

2. Outline • Cercion code structure • Linked list data structures • Lagrangian solution • Calculation of stress tensor • Remap algorithm • Test Problems • Riemann shock tube • Aluminum flyer plate • Imploding cylindrical shell • Conclusions

3. The hydrodynamics is solved on a block structured mesh • Block structured mesh • Four-sided cells • Fixed connectivity among mesh blocks • Velocities at the cell vertices A simple mesh with three structured blocks (2 zones each)

4. Cercion makes extensive use of cell oriented data structures • Cell oriented data structures • C implementation • Fortran-type array indexing • Cell_t data structure • Storage for vertex quantities such as position and velocity components • Cell-centered quantities such as density, pressure and volume • Pointer to a linked list of materials in the cell (mat_t) • Pointer to a linked list of material fluxes for the top boundary (flux_t) • Pointer to a linked list of material fluxes for the right boundary (flux_t)

5. cell_t header mat_t tail bas_t str_t Each cell has a linked list to store information about all the materials in the cell Material Linked List for a Cell Containing 1 Material • A single mat_t element for each material in the cell • Ordering of the elements according to onion skin method • Two pointers for each element • Basic material properties in bas_t data structure • volume fraction, mass and energy • density, energy density and associated derivatives • parameters for interface reconstruction • Strength properties in str_t data structure • stress deviator components and derivatives • equivalent plastic strain

6. cell_t header flux_t tail sflux_t Each cell has two additional linked lists to store material fluxes through the top and right boundaries • A flux_t element for each material crossing the boundary • volume, mass and energy fluxes • pointer to sflux_t data structure for stress energy and strain fluxes Material Flux Linked List for a Cell Boundary with 1 Material Crossing the Boundary

7. A three phase approach is used to solve the hydrodynamics equations • Lagrangian Phase • Uses the algorithm from HEMP (Wilkins 1963) • Spatially staggered grid with vertex velocities • Material strength included • four stress deviator components • yield stress correction • Margolin method for calculating strain rates and flow divergence • Margolin anti-hourglass treatment • Grid Relaxation Phase • Simple finite difference mesh relaxer • nine point stencil • Remap Phase • donor cell procedure from the SALE code (Amsden et al. 1980) for cell-centered quantities • second order correction to the remapped fluxes from the donor cell method • derivatives using Barth-Jespersen slope-limited method • Mass fluxes from the density remap for the momentum remap

8. Vertex (i,j) and surrounding cells A,B,C, and D The Lagrangian equations of motion are solved using the HEMP approach Vertex quantities for the equations of motion Axial Velocity Equation Radial Velocity Equation

9. Strain rates are calculated using the Margolin method Equations for the strain rate components Cell (i,j) and its four neighboring vertices Flow divergence

10. Material fluxes are used to redistribute materials among cells during the remap phase • Mesh relaxation moves (xp,yp) to (x,y) • Total volume fluxes (Fr, Fz) for donor cell method • Mass fluxes (MFr,MFz) for each material leaving the cell based on material volume fluxes and upwind densities Donor Cell Method Update of material k mass from Lagrangian value, M, to remapped value M*

11. A higher order correction to the donor cell method is used for cells with a single material correction donor cell method

12. Riemann shock tube problem was used to test the code • An initial discontinuity between two ideal gas regions (g=1.4) • Region 1 • density of 1 kg/m3 • pressure of 105 N/m2 • Region 2 • density of 0.01 kg/m3 • pressure of 103 N/m2 • A uniform box mesh with 500 (axial) by 2 (radial) zones • Eulerian calculation • The solution is obtained at 0.01s

13. The locations of the shock and contact discontinuity are captured (a) Density (b) Pressure

14. The velocity and sound speed spatial profiles are reasonably well represented (a) Velocity (b) Sound Speed

15. Cercion calculates the density as well as other numerical methods for the Riemann problem • An initial discontinuity between two ideal gas regions (g=1.4) • Region 1 • density of 1 kg/m3 • pressure of 1 N/m2 • Region 2 • density of 0.125 kg/m3 • pressure of 0.1 N/m2 • A uniform box mesh with 100 axial zones • Eulerian calculation • The solution is obtained at 0.14s

16. Cercion compares well with other numerical methods in capturing the velocity profile

17. Flyer plate problem tests the material strength routines in the code • Cylindrical geometry with 1 radial zone • Aluminum target • 1 cm thick with 200 axial zones • Aluminum projectile • 0.2 cm thick with 40 axial zones • Gruneisen EOS • r0=2.707 • C0=0.5386 • S1=1.339 • g0=1.97 • b=0.48 • Material strength model • yield strength of 0.0004 Mbar • shear modulus of 0.271 Mbar • Lagrangian calculation • Companion FLAG calculation

18. The temporal velocity profile at the target-vacuum interface from the code and FLAG agree fairly well

19. 5.0cm 8.0cm P=0.588 Mbar r=1.84 g/cc 2.0cm steel P=0.0 Mbar r=0.001 g/cc r z An imploding steel shell is a good test of energy conservation in converging cylindrical geometry • Two steel shells each with 10 radial zones • Outer shell is 0.25 cm thick • Inner shell is 0.5 cm thick • Gruneisen EOS for steel • r0=7.9 • C0=0.457 • S1=1.49 • g0=1.93 • b=0.5 • Material strength model • yield strength of 0.05 Mbar • shear modulus of 0.895 Mbar • Ideal gas EOS for the pressure driver material (80 zones) • Companion FLAG calculation with same mesh • 5 ms run time Initial Geometry

20. Cercion and FLAG give similar kinetic energies for the inner shell when material strength is not included Internal Energy Kinetic Energy

21. When material strength is included the agreement between Cercion and FLAG does not change Internal Energy Kinetic Energy The Cercion calculation is insensitive to the use of ALE

22. FLAG and Cercion predict virtually the same total energy for the inner steel shell Without Strength With Strength

23. Conclusions • Cell-centered data structures simplify the programming of Cercion and allow for efficient memory allocation for multi-material problems • Cercion uses proven numerical methods for the solution of the Langrangian equations of motion and the calculation of material strength properties • A second-order accurate remap method is implemented for density, energy and momentum, enabling ALE calculations to be performed • Cercion shows excellent agreement with the analytic solution for the Riemann shock tube problem • In pure Langrangian mode both Cercion and FLAG give similar velocity profiles at the target-vacuum interface for the flyer plate test problem • The cylindrical implosion test problem illustrates that the Cercion solution is relatively insensitive to the use of ALE in the calculation • Cercion calculations with and without strength for the cylindrical implosion problem generally agree with the corresponding FLAG results

24. The Sedov blast wave problem tests the symmetry of the code solution in two dimensional cylindrical geometry • An ideal gas (g=1.4) • A uniform box mesh (6 cm by 3 cm) • 300 axial zones • 150 radial zones • Energy source at the origin • 85 kJ • 4 zones (2 axial by 2 radial) • Eulerian calculation • The solution is obtained at 10.0 ms • Comparison RAGE calculation

25. The code and RAGE give different pressures behind the blast wave and both have a lower density at the front than the self-similar solution Pressure Density