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
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.
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
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Lagrangian hydrodynamics equations
Lagrangian framework and constitutive relations:
Materials with a caloric EOS
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Lagrangian hydrodynamics equations
Mid-point time integrator:
Mass equation
Momentum equation
=
piecewise linear
kinematic vars.
=
piecewise constant
thermodynamic vars.
Zero traction BCs
Total energy is conserved (even with mass lumping!)
Energy equation
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
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Lagrangian hydrodynamics equations
Variational Multi-scale (VMS) Stabilization:
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 
VMS
Pressure correction
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Lagrangian hydrodynamics equations
VMS fine-scale problem through linerarization:
needs multi-point evaluation
where
and
Physical interpretation: The pressure residual samples the
production of entropy due to the numerical approximation
(Clausius-Duhem)
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Lagrangian hydrodynamics equations
Numerical interpretation of VMS mechanisms:
Given the decomposition
and recalling that
Momentum:
Energy:
away from shocks
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Acoustic pulse computations
Initial mesh “seeded” with an hourglass pattern
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
A closer look at the artificial viscosity
Artificial viscosity à la von-Neumann/Richtmyer:
Sketches of element length scales
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Two-dimensional Sedov blast test
Mesh deformation, pressure, and density (45x45 mesh)
Mesh deformation
No hourglass
control
VMS-control
Element density contours
Num. vs exact solution
Pressure
Density
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
VMS stabilization in three dimensions
Hourglass “dilemma” and its space decomposition:
Modes with non-zero divergence
Pointwise divergence-free modes (non-homogenous shear)
Additional deviatoric hourglass viscosity
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Three-dimensional tests on Cartesian meshes
Flanagan-Belytschko cannot solve both, VMS does:
Noh test, 303 mesh, density
Sedov test, 203 mesh, density
3D-Noh test on
cartesian mesh
(density)
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:
[email protected]
www.cs.sandia.gov/~gscovaz
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Two-dimensional Noh implosion test
Mesh distortion comparison
Pressure-like
Radial tri-sector
artificialmesh
viscosity
Spurious jets
Tensor artificial viscosity
No spurious jets