1. Workshop on Numerical Methods for Multi-material Fluid Flows, Prague, Czech Republic, September 10-14, 2007 Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. A multi-scale Q1/P0 approach to Lagrangian Shock Hydrodynamics Guglielmo Scovazzi 1431 Computational Shock- and Multi-physics Department Sandia National Laboratories, Albuquerque (NM) Research collaborators: Edward Love, 1431 Sandia National Laboratories Mikhail Shashkov, Group T-7, Los Alamos National Laboratory

2. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Motivation and outline Q1/Q1, P1/P1-Lagrangian hydrodynamics with SUPG/VMS  Promising results for Q1/Q1 and P1/P1 finite elements  Scovazzi-Christon-Hughes-Shadid, CMAME 196 (2007) pp. 923-966)  Scovazzi, CMAME 196 (2007) pp. 967-978.  Is it possible to extend some of the ideas to Q1/P0?  Is it possible to design multi-scale hourglass controls? A new approach for Q1/P0 finite elements in fluids  A pressure correction operator provides hourglass stabilization  A Clausius-Duhem equality is used to detect instabilities  The stabilization counters numerical entropy production  The approach is applicable to ALE (Lagrangian+remap) algorithms  Promising results in 2D and 3D compressible flow computations

3. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Lagrangian hydrodynamics equations Lagrangian framework and constitutive relations: Materials with a caloric EOS

4. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Momentum equation Energy equation Lagrangian hydrodynamics equations Mid-point time integrator: Zero traction BCs Total energy is conserved (even with mass lumping!) Mass equation = piecewise linear kinematic vars. piecewise constant thermodynamic vars. =

5. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Algorithm and discrete energy conservation Every iteration:  Mass  Momentum  Angular momentum  Total energy are conserved 3D Sedov test, energy history Scale is 10 -14 Total energy relative error To ensure conservation

6. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Lagrangian hydrodynamics equations Variational Multi-scale (VMS) Stabilization: Pressure correction VMS Assumptions: 1. 2.Quadratic fine-scale terms are neglected 3.Fine-scale displacements are neglected 4. is negligible 5.Time derivatives of fine scales are neglected 6.The divergence of fine-scale velocity is neglected 

7. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Lagrangian hydrodynamics equations VMS fine-scale problem through linerarization: whereand Physical interpretation: The pressure residual samples the production of entropy due to the numerical approximation (Clausius-Duhem) needs multi-point evaluation

8. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Lagrangian hydrodynamics equations Numerical interpretation of VMS mechanisms: Given the decomposition and recalling thataway from shocks Energy: Momentum:

9. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Acoustic pulse computations Initial mesh “seeded” with an hourglass pattern

10. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” A closer look at the artificial viscosity Artificial viscosity à la von-Neumann/Richtmyer: Sketches of element length scales

11. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” VMS-control No hourglass control Two-dimensional Sedov blast test Mesh deformation, pressure, and density (45x45 mesh) Mesh deformation Element density contoursNum. vs exact solution Pressure Density

12. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” VMS stabilization in three dimensions Hourglass “dilemma” and its space decomposition: Additional deviatoric hourglass viscosity Modes with non-zero divergence Pointwise divergence-free modes (non-homogenous shear)

13. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Three-dimensional tests on Cartesian meshes 3D-Noh test on cartesian mesh (density) Noh test, 30 3 mesh, density Sedov test, 20 3 mesh, density Flanagan-Belytschko cannot solve both, VMS does:

14. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Summary and future directions A new paradigm for hourglass control  Strongly based on physics  A Clausius-Duhem term detects instabilities  In 3D, discriminates between physical and numerical effects Future work  Complete investigation in 3D computations  More complex equations of state  Generalizations to solids (no need for deviatoric hourglass viscosity)  Application to ALE (Lagrangian+remap)  Artificial viscosity Contact & pre-prints: gscovaz@sandia.gov www.cs.sandia.gov/~gscovaz Contact & pre-prints: gscovaz@sandia.gov www.cs.sandia.gov/~gscovaz

15. Scovazzi-Love-Shashkov, “VMS-hydrodynamics” Tensor artificial viscosity Two-dimensional Noh implosion test Mesh distortion comparison No spurious jets Pressure-like artificial viscosity Spurious jets Radial tri-sector mesh