Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 1 The Implementation of a General Higher-Order Remap Algorithm Vincent P. Chiravalle LA-UR MultiMat 2011 Conference Arcachon, France

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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)

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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)

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 5 Each cell has a linked list to store information about all the materials in the cell 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 cell_t headermat_ttail bas_tstr_t Material Linked List for a Cell Containing 1 Material

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 6 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 cell_t headerflux_ttail sflux_t

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 8 The Lagrangian equations of motion are solved using the HEMP approach Vertex (i,j) and surrounding cells A,B,C, and D Axial Velocity Equation Radial Velocity Equation Vertex quantities for the equations of motion

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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 Update of material k mass from Lagrangian value, M, to remapped value M* Donor Cell Method

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 11 A higher order correction to the donor cell method is used for cells with a single material donor cell method correction

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 12 Riemann shock tube problem was used to test the code An initial discontinuity between two ideal gas regions (  =1.4) Region 1 –density of 1 kg/m 3 –pressure of 10 5 N/m 2 Region 2 –density of 0.01 kg/m 3 –pressure of 10 3 N/m 2 A uniform box mesh with 500 (axial) by 2 (radial) zones Eulerian calculation The solution is obtained at 0.01s

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 13 The locations of the shock and contact discontinuity are captured (a) Density(b) Pressure

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 14 The velocity and sound speed spatial profiles are reasonably well represented (a) Velocity(b) Sound Speed

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 15 Cercion calculates the density as well as other numerical methods for the Riemann problem An initial discontinuity between two ideal gas regions (  =1.4) Region 1 –density of 1 kg/m 3 –pressure of 1 N/m 2 Region 2 –density of kg/m 3 –pressure of 0.1 N/m 2 A uniform box mesh with 100 axial zones Eulerian calculation The solution is obtained at 0.14s

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 16 Cercion compares well with other numerical methods in capturing the velocity profile

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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= –S1=1.339 –g0=1.97 –b=0.48 Material strength model –yield strength of Mbar –shear modulus of Mbar Lagrangian calculation Companion FLAG calculation

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 18 The temporal velocity profile at the target-vacuum interface from the code and FLAG agree fairly well

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 19 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 Mbar Ideal gas EOS for the pressure driver material (80 zones) Companion FLAG calculation with same mesh 5  s run time Initial Geometry 8.0cm 2.0cm P=0.0 Mbar  =0.001 g/cc P=0.588 Mbar  =1.84 g/cc steel z r 5.0cm

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 20 Cercion and FLAG give similar kinetic energies for the inner shell when material strength is not included Kinetic Energy Internal Energy

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 21 When material strength is included the agreement between Cercion and FLAG does not change Kinetic Energy Internal Energy The Cercion calculation is insensitive to the use of ALE

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 22 FLAG and Cercion predict virtually the same total energy for the inner steel shell With StrengthWithout Strength

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 24 The Sedov blast wave problem tests the symmetry of the code solution in two dimensional cylindrical geometry An ideal gas (  =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  s Comparison RAGE calculation

Operated by Los Alamos National Security, LLC for the U.S. Department of Energy’s NNSA Slide 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