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.
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Lagrangian Shock Hydrodynamics Introduction and Definitions Hydrocode: “An algorithm for the transient dynamic analysis of compressible flows (and, by abuse of terminology, solids), typically in Lagrangian coordinates” Applications of Hydrocodes: Impact/blast scenarios - Weapon/armor design, - Blast/explosion survivability analysis, - High energy density physics, astrophysics, etc. - Crash worthiness analysis: Most likely, your car has been designed using hydrocodes! Algorithmic Features: Lagrangian or ALE/Lagrangian hydrocodes are preferred to Eulerian methods when - fluid/solid or fluid/fluid interface tracking is crucial - complex equations of state are used
LS-DYNA
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Motivation I State of the art for hydrocodes Traditional hydrocodes are based on hexahedral Q1/Q0 finite elements (continuous linear kinematic variables and discontinuous piece-wise constant thermodynamic variables) Explicit time integration usually preferred Spurious hourglass modes may occur (element divergence-free modes) Limited to labor-intensive hexahedral meshing Adaptivity is not robust under shock conditions due to hanging nodes in hexahedrons
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Motivation II Tetrahedra: A disruptive change Automatic mesh generation is mature technology for tetrahedra (Delaunay, etc.): Meshing times/costs are a fraction of what they are for hexahedra Error-driven automatic mesh adaptation is mature on tetrahedra Multi-material remap/interface reconstruction is easier on tetrahedra Overall, design/analysis/optimization/UQ cycle can be made faster and more user-friendly Recent work, focused on solid dynamics, of: - Lew-Radovitzky-Ortiz at CALTECH - Dorman-Key-Bochev-Heinstein at SANDIA Noteworthy work on compressible fluids: - On 2D general polyhedrons: Shaskov, Loubère, Maire - On 2D triangles: Loubère, Déspres, Campbell
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
A New Finite Element Paradigm Variational multi-scale (VMS) hydrodynamics Collaborators, at various stages of the framework development: - Brian Carnes, Dave Hensinger, Ed Love, Bill Rider, John Shadid (Sandia National Laboratories) - Alejandro Lopez-Ortega (Galcit, Caltech) - Thomas J. R. Hughes (U. Texas at Austin) Works equally well on hexahedra and tetrahedra Nodal-based: Iso-parametric P1 or Q1 finite elements Software implementation very similar to traditional hydrocodes Constant gradient tetrahedra yield finite-difference-like assembly: fast methods Globally conservative for mass, linear/angular momentum, and total energy Second order in space and time (with predictor-corrector time integrator) Von Neumann-Richtmyer artificial viscosities for shock computations Related publications at www.cs.sandia.gov/~gscovaz/
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Lagrangian Equations of Motion Lagrangian reference frame
Displacement equations Mass conservation equation Momentum equations Energy equation Caloric E.O.S.
“ Displacement BCs
” Traction BCs
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
VMS Analysis & Stabilization Scale decomposition Given we decompose the solution trial space as
, i. e.
Coarse (discrete) scales Fine (subgrid) scales
and, analogously, we decompose the test space as Isoparametric linear finite element spaces for the coarse scales:
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
VMS Analysis & Stabilization Coarse-scale equations
Fine-scale equations:
A simple approximation
Residuals
Green’s function scaling
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
VMS Analysis & Stabilization Conservation properties
Total energy is conserved
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mid-point time integration Discrete equations: Momentum
Mass lumping
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mid-point time integration Discrete equations: Energy
Mass lumping
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mid-point time integration Conservation of total energy: Momentum equation Zero Neumann BCs. Energy equation
Total energy is conserved (also with mass lumping)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mid-point time integration Discrete equations: Mass
Co-volume diagonal matrix
Discrete equations: Displacement
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Algorithm and discrete energy conservation
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
von Neumann-Richtmyer artificial viscosity Artificial stress and heat flux: Sketches of element length scales
Brick-type elements:
Triangular elements:
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Predictor/multi-corrector time integrator Conservative and second order in space/time:
=2 Spectral radius
=2 Dispersion error
Stability limit (1d case):
— First iterate — Second iterate — Third iterate — Fourth iterate — Implicit
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Saltzmann test Initial meshes (e0=1.0E-14, ρ0=1.0)
(I) Piston velocity (II) Piston velocity (III) Piston velocity
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Saltzmann test Quadrilaterals (I)
Triangles (II)
Campbell-Shashkov (JCP Vol. 172, 2001)
Shock location
Triangles (III)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Saltzmann test Quadrilaterals (I)
Triangles (II)
Triangles (III)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Noh tests on unit quadrant Velocity initial condition Quadrilaterals (I) 2500 square elements
Triangles (II) 5000 square triangles
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Noh tests on unit quadrant Mesh deformation & density contours T=0.6 (CFL=0.8) Quadrilaterals (I) 2500 square elements
Triangles (II) 5000 square triangles
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Noh tests on unit quadrant Numerical solution (radial plots) Quadrilaterals (I)
Triangles (II)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Noh tests on unit quadrant Numerical solution (radial plots) Quadrilaterals (I)
Triangles (II)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
2D Sedov test, quadrilaterals vs. triangles Mesh deformation & density contours T=1.0 (CFL=0.8) Quadrilaterals 45x45 mesh (I)
Triangles 2x45x45 mesh (II)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
2D Sedov test, quadrilaterals vs. triangles Numerical solution (radial plots)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
2D Sedov test, quadrilaterals vs. triangles Numerical solution (radial plots)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
2D Sedov test on various triangular meshes Mesh 1
Mesh 2
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
2D Sedov test on various triangular meshes Mesh 3 (T=0.0)
Mesh 3 (T=1.0)
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
2D Sedov test on various triangular meshes Numerical solution (radial plots) Mesh 1
Mesh 2
Mesh 3
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
2D Sedov test on various triangular meshes Numerical solution (radial plots) Mesh 1
Mesh 2
Mesh 3
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
3D Noh test, mesh refinement study Numerical solution (radial plots)
Deformed Cartesian mesh
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
3D Sedov Blast: A Hydrocode Killer Traditional/Mimetic hydrocode (Q1/P0 staggered discretization)
Failure due to spurious hourglass modes
New VMS-based hydrocode (Q1/Q1 collocated discretization)
No hourglass modes
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
3D Sedov Blast: VMS Algorithm Mesh refinement study: (Energy IC converges to Dirac) 223 elements
443 elements
883 elements
Density
Density
Density
Int. energy
Int. energy
Int. energy
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Testing Tetrahedral VMS-Hydro Technology 3D Saltzmann problem Note: the full solution is plotted, projected along the x-axis
No pressure oscillations VMS-Hydro is robust
Mesh A Mesh A Mesh B
Mesh B
Mesh A Mesh B
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Testing Tetrahedral VMS-Hydro Technology 3D Noh problem on a Cartesian mesh Extremely low mesh distortion Tetrahedrons deliver same accuracy as hexahedrons Tetrahedrons 438K nodes 2.4M elements Hexahedrons 705K nodes 681K el.=5.4M quad. pts. Hex/Tet solutions are overlapped Note: the full solution is plotted, projected along the radial direction
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
3D Sedov Blast: Tetrahedral Computations Numerical results Most refined computation using 438,000 nodes, 2.4M cells Extremely low mesh distortion, even on fine meshes Most traditional hydrocodes on hexahedra fail the test!
No pressure oscillations
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
ALE as Lagrangian+Remap Rationale In shock hydrodynamics, arbitrary Lagrangian Eulerian (ALE) computations always adopt a Lagrangian step followed by remap (conservative interpolation). Mesh improvement algorithms are applied at the end of Lagrangian step. Pure Lagrangian computations are preferred in terms of accuracy, if robust. History/code inertial may play a role: Lagrangian methods came before ALE.
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mesh Remap Geometric conservation law (GCL) [see also Farhat et al.]
Mesh displacements
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mesh Remap Remap equations
x
Conservation principle
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mesh Remap Space-time formulation (embedding GCL)
Computational approach Geometric conservation allows to cast remap as pure transport Density/Momentum/Total Energy are collocated at the nodes: Easier implementation w.r.t. staggered methods. Space-time formulation (embeds conservation and geometric conservation) Conserves mass, momentum, and total energy Synchronized Algebraic Flux-Corrected Transport (positive, LED scheme), using the algebraic FCT framework [Kuzmin-Möller-Turek] Compatible with the Lagrangian method: Same approximation spaces & degreesof-freedom layout
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mesh Remap Discrete remap equations
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mesh Remap Discrete remap equations
Cons. form x
GCL
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mesh Remap Remap in multiple dimensions L1-error convergence
Mesh motion
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mesh Remap Remap of a 1D Sedov-like remap test
L1-norms
Density
Velocity
Energy
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Mesh Remap Lagrangian+Remap in multiple dimensions: Sedov test Eulerian
Lagrangian
ALE
Density Lagrangian ALE Eulerian
Rezone with a very simple mass-lumping mesh projection, an iterative scheme converging to Laplacian smoothing
Exact ALE and Lagrangian virtually identical
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
Summary & Future Directions Current Status on tetrahedral hydrodynamics VMS Lagrangian method shows robustness and accuracy ALE remap has been completed, application to FSI Use of VMS for error-driven adaptivity (with Brian Carnes). The fine scales can be used as an error estimator (extending the work of G. Hauke to systems of conservation laws.) Adaptivity, with M. Shephard at RPI Entropy-based viscosities (with J.-L. Guermond)
Where to go next Building a fast & user-friendly tetrahedral hydrocode for integrated analysis/design/optimization/uncertainty quantification (with Brian Carnes and Dave Hensinger at Sandia) Application to fluid-structure interaction in complex geometry
G. Scovazzi, “ALE-Lagrangian shock hydrodynamics on tetrahedrons”
References
http://www.cs.sandia.gov/~gscovaz/ G. Scovazzi, J.N. Shadid, E. Love, and W.J. Rider, "A conservative nodal variational multiscale method for Lagrangian shock hydrodynamics", Computer Method in Applied Mechanics and Engineering, Volume 199, Issues 49-52, 15 December 2010, pp. 3059-3100. A. Lopez Ortega and G. Scovazzi, "A geometricallyconservative, synchronized, flux-corrected remap for arbitrary Lagrangian-Eulerian computations with nodal finite elements", submitted, Journal of Computational Physics, 2010.
