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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-AC04-94AL 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

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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 ) Scovazzi, CMAME 196 (2007) pp 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

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

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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. =

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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 Total energy relative error To ensure conservation

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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

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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

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

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

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

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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

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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)

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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:

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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: Contact & pre-prints:

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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

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