High Order Curvilinear Finite Elements for Lagrangian Hydrodynamics Part II: Numerical Results Veselin Dobrev, Tzanio Kolev and Robert Rieben Lawrence Livermore National Laboratory March 4, 2011
LLNL-PRES-465874 This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344
Review of Our High Order FEM Hydro Algorithm The computational kernel of our method is the evaluation of the Generalized Corner Force matrix Z (F)ij = (σ : ∇~ wi ) φj Ω(t)
Semi-discrete finite element method Momentum Conservation:
Mv
dv = −F · 1 dt
Energy Conservation:
Me
de = FT · v dt
Equation of Motion:
dx =v dt
F is assembled locally from zonal corner force matrices Fz . Locally FLOP-intensive evaluation of Fz requires high order quadrature Density is computed at each quadrature point using strong mass conservation Pressure is computed at each quadrature point through the EOS (“sub-zonal pressure”). Rieben et al. (LLNL)
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Our Research Codes BLAST: C++ High order curvilinear FEM Lagrangian hydrocode: webpage: www.llnl.gov/CASC/blast Object-oriented implementation of our general FEM framework in 2Dxy, 2Drz and 3D Supports curvilinear tri/quad/tet/hex meshes with many finite element options Open Source Software: MFEM: Modular C++ finite element library mfem.googlecode.com GLVis: OpenGL visualization tool glvis.googlecode.com
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Taylor-Green Vortex: High Order Convergence for Smooth Problems 32x32 structured grid – 2nd order
Q2−Q1−RK2Avg
−1
10
unstructured grid – 3rd order
L , α=2.021
−2
2
Velocity error
10
L∞, α=1.930 Velocity error
−2
10
Q3−Q2−RK4 Unstructured
−1
10
L1, α=2.044
−3
10
L1, α=2.993 L , α=2.723 2
L∞, α=2.241
−3
10
−4
10
−4
10
−5
10
−5
10 −3 10
−6
−2
10 h
−1
10
2D Taylor–Green Vortex, t = 0.75 Rieben et al. (LLNL)
10 −3 10
−2
10 h
−1
10
Manufactured solution. No viscosity
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2D Sedov Blast Wave on Cartesian Grid
Exact Deformation
Density and Curvilinear Mesh, 20x20 zones 1.2
1.0
0.8
0.6
0.4
0.2
0.0 0.0
Symmetry is preserved
Rieben et al. (LLNL)
0.2
0.4
0.6
0.8
1.0
1.2
Curved mesh gives better approximation
High Order Curvilinear FEM for Lag. Hydro, Part II
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2D Sedov Blast Wave on Cartesian Grid Density Scatter Plots
Evaluated at 9 points/zone Rieben et al. (LLNL)
Converges to exact solution High Order Curvilinear FEM for Lag. Hydro, Part II
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2D Sedov Blast Wave on Cartesian Grid Density Scatter Plots
Evaluated at zone centers Rieben et al. (LLNL)
No undershoots and overshoots High Order Curvilinear FEM for Lag. Hydro, Part II
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2D Noh Implosion on Cartesian Grid Density
Speed
32x32 Cartesian grid
Symmetry is preserved
Small oscillations in mesh (due to ε(v ))
Shock is resolved in a single zone!
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High Order Curvilinear FEM for Lag. Hydro, Part II
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2D Noh Implosion on Cartesian Grid
Density Scatter Plot
Evaluated at 9 points/zone
Rieben et al. (LLNL)
Speed Scatter Plot
Some oscillations and symmetry breaking
High Order Curvilinear FEM for Lag. Hydro, Part II
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2D Noh Implosion on Cartesian Grid
Density Scatter Plot
Speed Scatter Plot
Evaluated at zone centers
Symmetry is preserved
No overshoots or undershoots
Oscillations are in the element interiors
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High Order Curvilinear FEM for Lag. Hydro, Part II
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Noh and Sedov on Irregular Grids: The “Quadrant” Problem full Noh implosion
full Sedov blast
Initial Cartesian grid
Symmetry is preserved (except near origin)
Different resolutions, same viscosity
Stretched zones are not a problem
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High Order Curvilinear FEM for Lag. Hydro, Part II
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Noh and Sedov on Irregular Grids: Unstructured Curvilinear Grid Noh implosion
Sedov blast
Unstructured grid on the unit square
Symmetry is preserved
Randomized initially curved mesh
Curved zones are not a problem
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Saltzman Piston
Density and mesh for Saltzman piston problem at t = 0.7, 0.8, 0.88, 0.92, 0.94, 0.96, 0.975, 0.985, 0.987 and 0.99 for a total of 6 bounces. (We can run this further.) Each image is rescaled to an aspect ratio of 5 : 1. Rieben et al. (LLNL)
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Axisymmetric Sedov Blast Wave on Cartesian Grid 40x40 Lagrangian SGH - Density
20x20 Lagrangian FEM - Density
Symmetry is not preserved
Symmetry is preserved
Mesh distorted near the origin
Curvilinear zones match physics FE pressure treatment
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Axisymmetric Sedov Blast Wave on Cartesian Grid Coarse Mesh Scatter Plot of Density vs Radius
Fine Mesh Scatter Plot of Density vs Radius
SGH shock is too fast
SGH does not improve under refinement
FEM is good with only 20x20 zones
FEM matches exact solution very closely
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Axisymmetric Sedov Blast Wave on Cartesian Grid
Comparison of Energy Conservation
1.07 1.06
BLAST Energy Transfer
1.2
SGH Blast
1.0 0.8
1.04
Energy
Normalized Total Energy
1.05
1.03 1.02
Total Energy Kinetic Energy Internal Energy
0.6 0.4
1.01
0.2
1.00 10-7
10-6
10-5
10-4
10-3 Time
10-2
10-1
SGH gains 6% energy
100
0.0 -7 10
10-6
10-5
10-4
10-3 Time
10-2
10-1
100
BLAST converts IE to KE without loss
BLAST conserves energy to machine precision
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Axisymmetric Noh Implosion on Cartesian Grid 32x32 Lagrangian SGH
32x32 Lagrangian FEM
Symmetry is not preserved
Symmetry is preserved
Mesh distorted near the origin
Wall heating is typical for Lagrangian methods
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Axisymmetric Noh Implosion on Cartesian Grid 64x64 Arbitrary Lagrangian Eulerian SGH
32x32 Lagrangian FEM
ALE fixes mesh
Symmetry is preserved
Energy jets from wall heating
Wall heating is typical for Lagrangian methods
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High Order Curvilinear FEM for Lag. Hydro, Part II
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Axisymmetric Noh Implosion on Cartesian Grid Code Comparison
Lagrangian FEM Convergence
SGH does not preserve symmetry
FEM converges to correct solution
ALE gives good shock prediction
Wall heating is typical for Lagrangian methods
FEM preserves symmetry Rieben et al. (LLNL)
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Axisymmetric Noh Implosion on Cartesian Grid Comparison of Energy Conservation 1.005
BLAST Energy Transfer 1.8
SGH Blast
1.000
1.6 1.4 1.2
0.990
Energy
Normalized Total Energy
0.995
0.985
Total Energy Kinetic Energy Internal Energy
1.0 0.8 0.6
0.980
0.4 0.975 0.9700.0
0.2 0.1
0.2
0.3 Time
0.4
0.5
ALE SGH loses 3% energy
0.00.0
0.1
0.2
0.3 Time
0.4
0.5
0.6
BLAST converts KE to IE without loss
BLAST conserves energy to machine precision Rieben et al. (LLNL)
High Order Curvilinear FEM for Lag. Hydro, Part II
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3D Sedov Blast Wave on Cartesian Grid 16x16x16 Lagrangian FEM - Density Shrunk 4x4x4 mesh at t = 1
Same FEM framework as in XY and RZ
Symmetry is preserved
Comparable solution to RZ
3D curvilinear zones match physics
Rieben et al. (LLNL)
High Order Curvilinear FEM for Lag. Hydro, Part II
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3D Noh Implosion on Cartesian Grid 16x16x16 Lagrangian FEM - Density Shrunk 4x4x4 mesh at t = 0.6
Same FEM framework as in XY and RZ
Symmetry is preserved
Comparable solution to RZ
3D curvilinear zones match physics
Rieben et al. (LLNL)
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2D Pressure Driven ICF-like Hydro Test Problem
Simple 2D implosion of a cylindrical shell of ideal gas using an ICF-like pressure drive 1 . The interface between the high and low density regions is subject to both Richtmyer-Meshkov (RM) and Rayleigh-Taylor (RT) instabilities. Unstructured mesh topology with random spacing in central ”box” Random angular sub-divisions Q2-Q1 spatial discretization RK2Avg temporal discretization ensuring exact total energy conservation
1 S. Galera, P-H. Maire, J. Breil, ”A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction,” J. Comput. Phys, 2010 Rieben et al. (LLNL)
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3D Velocity Driven ICF-like Hydro Test Problem Internal Energy
log(Density)
log(Density)
Axisymmetric FEM Simulation
Full 3D FEM Simulation
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3D Velocity Driven ICF-like Hydro Test Problem
log(Density)
log(Density)
Axisymmetric FEM Simulation
Full 3D FEM Simulation
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Spherical Symmetry Preservation High order methods excel at spherical symmetry preservation:
Q2Q1
Q1Q0
Q1Q0 ”SGH”
Each color scale represents a 3% variation in radius
The Q1Q0 ”SGH” version of our method uses a lumped mass matrix with piece-wise constant pressures and densities and can be incorporated into a traditional SGH code for improved 3D symmetry. Rieben et al. (LLNL)
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2D Single Material Rayleigh-Taylor Instability A purely Lagrangian simulation of the two material problem is restricted by the slip-line formed at the interface; therefore, we consider a single material version: t=6
t=7
t=8
t=9
Compressible fluids Single material with a smooth density gradient
Q1-Q0:
Calculation Failure
Heavy fluid on top of a light fluid Initially perturbed interface Constant initial pressure Small downward acceleration
High order methods allow the problem to run longer in time and resolve more of the flow features.
Rieben et al. (LLNL)
Q2-Q1:
Q3-Q2:
High Order Curvilinear FEM for Lag. Hydro, Part II
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2D Single Material Rayleigh-Taylor Instability, Vorticity Driven Suggested by A. Barlow / D. Youngs, the material interface is flat. Initial velocity is divergence free with a slip line at the interface, initial pressure is from hydrostatic equilibrium: AMR-ALE with 4 Refinement Levels
BLAST Q3Q2
The high order Lagrangian calculation is in good agreement with the AMR-ALE result. Rieben et al. (LLNL)
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Axisymmetric Single Material Rayleigh-Taylor Instability Density
Rieben et al. (LLNL)
Mesh
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Shock Passing Through Helium Bubble Density up to t = 700µs
Unstructured mesh with local refinement Shock remains straight before impact Rieben et al. (LLNL)
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Axisymmetric Helium Bubble
Unstructured mesh with local refinement Shock remains straight before impact No spurious features near the axis of symmetry Demonstrates robustness of higher order FEM
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Axisymmetric Helium Bubble: 3D Visualization
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Helium Bubble: Curved Meshes
Helium Bubble
Axisymmetric Helium Bubble
High aspect ratio curved zones Impossible to represent with straight edges
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Conclusions and Future directions
Demonstrated benefits of our high order discretization methods: Higher order convergence for smooth problems. More accurate capturing of flow features using curvilinear zones. Exact total energy conservation by construction. No need for ad-hoc hourglass filters. Substantial reduction in mesh imprinting and improved symmetry preservation We are continuing to research and develop extensions to our methods: Strength model Parallelization Extension to ALE
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