workshop on numerical methods for multi-material fluid flows, prague, czech republic, september...
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Workshop on Numerical Methods for Multi-material Fluid Flows,Workshop on Numerical Methods for Multi-material Fluid Flows,
Prague, Czech Republic, September 10-14, 2007Prague, 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 LaboratoriesMikhail Shashkov, Group T-7, Los Alamos National Laboratory
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Motivation and outlineQ1/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 equationsLagrangian framework and constitutive relations:
Materials with a caloric EOS
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Momentum equation
Energy equation
Lagrangian hydrodynamics equationsMid-point time integrator:
Zero traction BCs
Total energy is conserved (even with mass lumping!)
Mass equation=piecewise linear kinematic vars.
piecewise constant thermodynamic vars.
=
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Algorithm and discrete energy conservation
Every iteration:MassMomentumAngular momentumTotal energyare conserved
3D Sedov test, energy history
Scale is 10-14
Total energyrelative error
To ensure conservation
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Lagrangian hydrodynamics equationsVariational Multi-scale (VMS) Stabilization:
Pressure correction
VMS
Assumptions:1. 2. Quadratic fine-scale terms are neglected3. Fine-scale displacements are neglected4. is negligible5. Time derivatives of fine scales are neglected6. The divergence of fine-scale velocity is neglected
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Lagrangian hydrodynamics equationsVMS fine-scale problem through linerarization:
where and
Physical interpretation: The pressure residual samples the production of entropy due to the numerical approximation (Clausius-Duhem)
needs multi-point evaluation
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Lagrangian hydrodynamics equationsNumerical interpretation of VMS mechanisms:Given the decomposition
and recalling that away from shocks
Energy:
Momentum:
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Acoustic pulse computationsInitial mesh “seeded” with an hourglass pattern
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
A closer look at the artificial viscosityArtificial viscosity à la von-Neumann/Richtmyer:
Sketches of element length scales
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
VMS-controlNo hourglass control
Two-dimensional Sedov blast testMesh deformation, pressure, and density (45x45 mesh)
Mesh deformation Element density contours Num. vs exact solution
Pressure
Density
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
VMS stabilization in three dimensionsHourglass “dilemma” and its space decomposition:
Additional deviatoric hourglass viscosity
Modes with non-zero divergence
Pointwise divergence-free modes (non-homogenous shear)
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Three-dimensional tests on Cartesian meshes
3D-Noh test on cartesian mesh
(density)
Noh test, 303 mesh, density Sedov test, 203 mesh, density
Flanagan-Belytschko cannot solve both, VMS does:
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]/~gscovaz
Contact & pre-prints:[email protected]/~gscovaz
Scovazzi-Love-Shashkov, “VMS-hydrodynamics”
Tensor artificial viscosity
Two-dimensional Noh implosion testMesh distortion comparison
No spurious jets
Pressure-like artificial viscosity
Spurious jets
Radial tri-sector mesh