LUTh, Paris-Meudon, May 31, 2012
2D simulations of radiation2D simulations of radiation--dominated dominated plasmasplasmas
M. M. Basko1
J. Maruhn2, Anna Tauschwitz2, V.G. Novikov3, A.S. Grushin3
in collaboration with
1 Institute for Theoretical and Experimental Physics, Moscow, Russia2 Frankfurt University, Frankfurt am Main, Germany3 Keldysh Institute of Applied Mathematics, Moscow, Russia
Equations of hydrodynamics
( )( ) ( )( ) ( ) ( )
2
0,
0,
,
, ( , )2
depr
ut
uu u p
tE
E p u Qt
uE e e e T
QT
ρ ρ
ρρ
ρρ
ρ
κ
∂+ ∇ ⋅ =
∂
∂+ ∇ ⋅ ⊗ + ∇ =
∂∂
+ ∇ ⋅ + = + +⎡ ⎤⎣ ⎦∂
= + =
∇ ⋅ ∇
– energy deposition by thermal conduction (local), – energy deposition by
radiation (non-local), – eventual external heat sources.( )Tκ∇ ⋅ ∇ rQ
depQ
The newly developed RALEF-2D code is based on the one-fluid, one-temperaturehydrodynamics model in two spatial dimensions (either x,y, or r,z):
Radiation transport
Transfer equation for radiation intensity Iν in the quasi-static approximation:
( ) ( )4
rQ d I d d k I B dν ν ν νπ
ν ν= −∇ ⋅ Ω Ω = Ω −∫ ∫ ∫ ∫
Quasi-static approximation: radiation transports energy infinitely fast (compared to the fluid motion) ⇒ the energy residing in the radiation field at any given time is infinitely small !
Radiation transport adds 3 extra dimensions (two angles and the photon frequency) ⇒the 2D hydrodynamics becomes a 5D radiation hydrodynamics !
Coupling with the fluid energy equation:
( ) ( ) ( ), ,1 , , , ,I k B I IIc t
I t x B B Tν ν ν ν ν ν νν
νν ν∂+
∂Ω⋅∇ = − = Ω =
In the present version, the absorption coefficient kν and the source function Bν = Bν(T)are calculated in the LTE approximation.
New quality due to radiation transport
Pure hydrodynamics (with or without thermal conductivity) is local.
Radiation hydrodynamics is non-local !
⇒ poses serious difficulties for the development of adequate numerical algorithms in 2 and 3 dimensions;
⇒ RALEF-2D is based on a newly developed original algorithm for radiation transport (not published yet).
Main constituents of the RALEF-2D package
1. Hydrodynamics
2. Thermal conduction
3. Radiation transport
4. EOS and opacities
5. Laser absorption
Hydrodynamics
Numerical scheme for hydrodynamics
The numerical scheme for the 2D hydrodynamics is built upon the CAVEAT-2D (LANL, 1990) hydrodynamics package and has the following properties:
it uses cell-centered principal variables on a multi-block structured quadrilateral mesh (either in the x-y or r-z geometry);
is fully conservative and belongs to the class of the second-order Godunovschemes (no artificial viscosity is needed);
the mesh is adapted to the hydrodynamic flow by applying the ALE (arbitrary Lagrangian-Eulerian) technique;
the numerical method is based on a fast non-iterative Riemann solver (J.K.Dukowicz, 1985), easily applied to an arbitrary equation of state.
Computational mesh: general topology
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0block 5
block 4
block 3
block 2
x
y
block 1
21
21
block 10
block 9
block 8
block 7
block 6
21
21
21 1
2
12
12
12
12
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
12
12
12
2
1
2
1
2
1
2
1
x
block 1
block 2
block 3
block 4
block 5
block 6
block 7
y
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
block 11
block 10
2
21
1
11
2
2
2
2
2 1
1
1
121
block 9
block 8
block 7
block 6
block 5
block 4
block 3
block 2
y
x
block 1
2
2 1
12
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
y
x
Computational mesh consists of blocks with common borders. Each block is topologically equivalent to a rectangle. Common block faces have equal number of cells.
• Cartesian (x,y)• cylindrical (r,z)
Mesh geometry:
Mesh library:Different mesh options, distinguished by the value of variable igeom, are combined into a mesh library, which is being continuously expanded.
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4Ncyc = 1 Ncyc = 100
ALE techniqueIn the arbitrary Lagrangian-Eulerian (ALE) technique, the Lagrangian phase of every hydrocycle is followed by the rezoning (mesh movement) and remapping phases. Material interfaces are traced as Lagrangian surfaces.
Importance of the 2-nd order + ALE
RALEF: 1-st order RALEF: 2-nd order
Non-linear stage of the Rayleigh-Taylor instability of a laser-irradiated thin carbon foil
Hydrodynamics: comparison with CHIC (CELIA)
CHIC code (CELIA, Bordeaux):
A more complex Riemann solver by P.-H. Maire:
1-st order, acoustic approximation non-iterative (2006) ⇒
⇒ 2-nd order, iterative (2009) [slower than RALEF?] – to be yet benchmarked !
CHIC: succeeds on the Saltzman test (pure Lagrangian mode);
RALEF: fails on the Saltzman test in the Lagrangian mode, runs in the ALE mode.
Hydrodynamics: comparison with MULTI-2D
unstructured triangular mesh,
an original version of the Lagrangian finite-difference hydrodynamics with artificial viscosity,
no ALE.
Thermal conduction
Numerical scheme for thermal conduction
The numerical scheme for thermal conduction (M.Basko, J.Maruhn & A.Tauschwitz, J.Com.Phys., 228, 2175, 2009) has the following features:
it uses cell-centered temperatures from the FVD (finite volume discretization) hydrodynamics on distorted quadrilateral grids,
is fully conservative (based on intercellular fluxes with an SSI energy correction for the next time step),
(almost) unconditionally stable,
space second-order accurate on all grids for smooth κ,
symmetric on a local 9-point stencil,
computationally efficient.
The key ingredient to the RALEF-2D code is the SSI (symmetric semi-implicit)method of E.Livne & A.Glasner (1985), used to incorporate thermal conduction andradiation transport into the 2D Godunov method (in order to avoid costly matrix inversion required by fully implicit methods).
Time discretization with the SSI method
Radiation coupling to the fluid is combined with thermal conduction within the unified SSI approach.
( ) ( )( ) ( ) ( ),
,
; ;
T r dep pij ij ijij ij ij ij
rT r r ijdep r
ij ijij ij ijV ij ij ij ij ij ijij
M E E W W W t W t
Wc M T T W W W t W W T T
T
− = + + Δ + Δ
∂− = + + Δ = + −
∂
Partially implicit discretization of the fluid energy equation
yields
( ) ( )( ),
,
,
, .
T r dep T rT r ij ij ij ij ijdepij ij ij V ij ij T r
V ij ij ij ij
T rij ijT r
ij ijij ij
W W W tW W W t c M
c M D D t
W WD D
T T
δ δ+ + Δ + ++ + Δ =
+ + Δ
∂ ∂= − = −
∂ ∂
In the SSI method we have to calculate the explicit cell heating power and its
temperature derivative .
rijW
/r rij ij ijD W T= −∂ ∂
The SSI method for thermal conduction
( )( )
1 2 , 1,1 1, ,2
, 1 2 , 1,1 1, ,2
, 1
1,
, 0,
, 1,0.
, 2,
ij ij i j i j ij ij ij ijmij ij ij ijm
ijV ij ij ij ij i j i j
i j ijmijm ijm ij ijm ijm
i j backwd
H H H H q M t HT T a
Tc M a a b b t
m Ht a t b b
m T
δτ
τδ τ
τ
+ +
+ +
−
−
+ − − + Δ + ∂≡ − = = − >
∂+ + + + Δ
= ∂⎧= Δ ⋅ + Δ ⋅ ⋅ = + >⎨ = ∂⎩
Time step control in the SSI method
Constraints on Δt are based on usual approximation-accuracy considerations, but here we need two separate criteria with two control parameters:
( )( ) ( )( )
( )
1 2 , 1,1 1, ,20 1
, 1 2 , 1,1 1, ,2
1,
,
.
rij ij i j i j ij ij ij
ij srV ij ij ij ij i j i j ij
ijij s
V ij ij
H H H H W q M tT T
c M a a b b D t
T Tc M
ε ε
δε
+ +
+ +
+ − − + + Δ≤ − +
+ + + + + Δ
≤ +
Ts is a problem-specific sensitivity threshold for temperature variations.
Test problems: linear steady-state solutions
0 1000 2000 3000 40001E-10
1E-8
1E-6
1E-4
0.01
1
δTL2
Ncyc
Δt = 0.001
Δt = 0.0005
Δt = 0.002
ε0 = 0.2, ε1 = 0.04
Time convergence to the steady stateThe linear solution
is reproduced exactly on all grids, unless special constraints to ensure positiveness are imposed on stronglydistorted grids.
For sufficiently large time steps Δt, the SSI method does not converge to the steady-state solution − an indication of its conditional stability (zero for large Δt).
( , )T x y x=
Piecewise linear solutions with κ-jumps are reproduced exactly only with harmonic mean κf,ij , and only on rectangular grids.
Test problems: non-linear wave into a cold wall
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
0.76 0.77 0.78 0.79
0.48
0.49
0.50
0.51
0.52
exact arithmetic κf
harmonic κf, δκf0=0.01T
x
301,
(0, ) 0
( ,0) 1
Vc T
T x
T t
ρ κ κ= =
=
=
Parameters:
Initial condition:
Boundary condition:
Solution:0
2 4
2
( , ) ( ), ,/ 2
0
xT t xt
d dd d
τ ξ ξκ
τ τξξ ξ
= =
+ =
This solution cannot be simulated with the harmonic-mean κf , whereas excellent results are obtained with the arithmetic-
Test results:
mean κf : the front position is reproduced with an error of 0.1% for ε0=0.2, ε1=0.02.
Square grid 100x100
Temperature in all cells
Standard grids for test problems
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Square grid
x
y
x
y
Kershaw grid
x
y
Wavy grid
x
y
Random grid
Only the Kershaw grid is strongly distorted in the sense that some of the coefficients (1±ξ)(1±η)become negative.
Test problems: non-linear steady-state solution
Here we test against a steady-state solution of the form
with the source term .
4( , ) ,T x y a bx cx= + +2( , )Q Q x y x= =
Our scheme clearly demonstrates the 2nd
order convergence rate on all grids, and is no less accurate than the best published schemes of Morel et al. (1992), Shashkovet al. (1996), Breil & Maire (2007).
0.01 0.11E-6
1E-5
1E-4
1E-3
0.01
square mesh wavy mesh random mesh Kershaw mesh CHIC - wavy CHIC - random CHIC - Kershaw
|δTl2|
h
Radiation transportThe CHIC code at CELIA does not solve the equation of radiation transfer: it treats radiative energy transport in the spectral-multi-group diffusion approximation.
Numerical scheme for radiation transport
Our numerical scheme for radiation transport (not published yet) has the following basic properties:
radiation coupling to the fluid is combined with the thermal conduction within the unified SSI approach;
the angular dependence of the radiation field is treated by applying the classical Sn method with n(n+2) fixed photon propagation directions over the 4π solid angle;
the scheme is non-conservative in the sense that the energy deposition by radiation Qr is calculated not via elementary fluxes,
the algorithm has the important property of correct transition to the diffusion limit when the mean free path of photons becomes shorter than the characteristic length scale.
Spatial discretization: general outline
The numerical algorithm for radiation transport splits into two parts:
In part 1 of the algorithm the exact solution of the transfer equation across a quadrilateral cell with a fixed absorption coefficient kν = const is used; the interpolation scheme for the source function Bν is of primary importance.
Even if the radiation field is calculated with a sufficient accuracy, it is no guarantee for adequate coupling with the fluid in the diffusion limit!
part 1: calculation of the intensity field ;
part 2: calculation of the cell heating power
and its temperature derivative .
( ),ijI ν Ω
0 4
4ij
rij
V
W dV I d B dνπ
π ν∞ ⎛ ⎞
= Ω −⎜ ⎟⎝ ⎠
∫ ∫ ∫/r r
ij ij ijD W T= −∂ ∂
Part 1: the method of short characteristics
( ) ( ), , , , ,L LI k B I I I t xν ν νΩ ⋅∇ = − = Ω
with the method of short characteristics (A.Dedner, P.Vollmöller, JCP, 178, 263, 2002). Mesh nodes are chosen as collocation points for the radiation intensity I .
angular directions are discretized by using the Sn method with n(n+2) fixed photon propagation directions over the 4π solid angle;
for each angular direction and frequency ν, the radiation field I is found by solving the transfer equation
LΩ
Advantages: • even on strongly distorted meshes, it is guaranteed
that light rays pass through every mesh cell;• the algorithm is generally computationally more
efficient than that of long characteristics.
Disadvantages:
• a significant amount of numerical diffusion in space.
S6
Flux conservation in vacuum
Ω
4
32
1
An important shortcoming of the vertex collocation points for radiation intensities is violation of the flux conservation in vacuum, where the transfer equation requires
In our example on the left we have
( ) ( )0 0L
I n I dl∇ ⋅ Ω = Ω⋅⇒ =∫
( ) ( ) ( )
( )
1 2 2 3 3 4
1 4
1 1 12 2 212
in
out in
H I I l I I l I I l
H I I l H
= + + + + +
= + ≠
Conclusion: we cannot adopt the same fluxing algorithm that was used for thermal conduction to calculate the radiativeenergy deposition
1 2 , 1,1 1, ,2r
ij ij ij i j i jW H H H H+ += + − −
Our numerical scheme for radiation energy transport inevitably becomes non-conservative!
Conservativeness can be restored by assigning the intensity I to cell faces (R.Ramis, 1992).
Part 2: bidirectional cell integration
When calculating Wijr, propagation directions are combined in ± pairs.LΩ
The cell heating power Wijr is calculated by
exact integration of the column heating power Wh over a quadrilateral with kν = const:
( )( ) ( )
( )
4
3
3
,
1 ( ) ' ' /
, ( ) 2 ( 2) ,
.
,h
rij h
h E S S E h
x
h
h
F
W d W
I B x x x e
k h x
d x
W F F e B Bπ
ν
τ
φ
φ τ τ
τ
−+ −
± ± −= − = − + +
= Ω
= − + −
=
+
∫ ∫
After we employ a second-order (parabolic) interpolation for the source function B(τ)along the light rays, we do recover the diffusion limit for kν → ∞ , where
4div3r
Ross
Q Bkπ⎛ ⎞
= ∇⎜ ⎟⎝ ⎠
Radiation transport: test problems
Numerical diffusion: a searchlight beam in vacuum
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
y
x0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
exact Δy=0.05 Δy=0.0125
F
x
Ωx=0.1915Ωy=0.6940y=4.0
The short-characteristic method produces a significant amount of numerical diffusion for light beams with sharp edges.
For thermal radiation a certain amount of numerical diffusion may be more an advantage than a drawback.
Test problems: radiative cooling of a slab
( ) ( )
( ) ( ) ( ) ( )
0
1
0 0 1 00
, ( ) sin , 0 1;
2 ' ' 2 ;r
I k B I B y y yy
Q y k k B y E k y y dy B y
μ π
π
∂= − = ≤ ≤
∂
⎡ ⎤= − −⎢ ⎥
⎣ ⎦∫
Exact solution:
0 1 2 3 16 17 18 19 200.0
0.5
1.0
y
x
vertical symmetry axis
Computational mesh:
Typical accuracy in problems with optically thin mesh cells is 1−2%.
S2 S4 S6 S8 S12
error L2 23% 7.6% 2.3% 1.9% 1.15%
Convergence of the Sn method 0.2 0.4 0.6 0.8 1.0
-5
0
5
10
exact S6, nx=ny=10 S12, nx=ny=80
y-Q
r(y)
at
x =
0
k = τ0 = 2, random mesh
The diffusion limit
10-2 10-1 100 101 102 103 104
0.01
0.1
1
10
-Qr a
t y
= 0.
525
τ0
exact diffusion limit square grid random grid Lobatto quadrature
S6, nx=ny=20
At τ0 >> 1 one approaches the diffusion limit.
0.0 0.2 0.4 0.6 0.8 1.0
0.7
0.8
0.9
1.0
1.1
1.2
1.3
square grid random grid exact BO, random
Qr i /
Q(y
c,i)
y
S12, nx=ny=20, τ0 = 104
Many transport codes fail to reproduce the energy exchange rate Qr in the diffusion limit.
The RALEF-2D algorithm remains robust for mesh cells of arbitrary optical thickness; in the limit of τ0 → ∞ it produces a finite numerical error.
The “ray effect” in a cylindrical cavity
R1
R2
φ1
x
y
Consider radiation transport from a central “hot” rod across a vacuum cavity.
0 1 2 3 40
1
2
3
4
y
x
block 1
block 2
S6
0 10 20 30 40 50 60 70 80 90
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
(hr/π
B0)(R
2/R1)
φ (degrees)
S6
S12
S24
S48
R2/R1=4 S6 S12 S24 S48 S96
error max/min 41% 18% 6.8% 1.9% 0.47%
error L2 15% 8.5% 2.4% 0.52% 0.23%
Convergence of the Sn method
EOS and opacities
EOS options in RALEF-2D
The principal scheme for different EOS options has been inherited from CAVEAT:
one can define up to 30 different materials;for each material one can choose out of 9 different types of EOS.
The EOS model must provide2 2( , ), ( , ), ( , ), ( , ), ( , .), ( , )V V Du Dup p e c c e T T e c c e z z e a a eρ ρ ρ ρ ρ ρ= = = = = =
Analytical EOS models:
1. polytropic gas:
2. linear EOS:
1( 1) ),,, ( 1 ,2Du VVp e a c z conste c Tγ ρ γ= − = + ==
( )0 1 0 2 0 3 0 4( ) , ...( ) / ,cold Vp c c c c e T e e cρ ρ ρ ρ ρ ρ ρ= − ⎡ + − ⎤⎣ −+ +⎦ =
Tabular EOS models:
7. general logarithmic-table (GLT) EOS with different source EOS models: Basko(Z = 1―13, 18, 22,26,28,29,36,40,42, 47 54,55,74,79,82,83,92), Novikov(THERMOS code; Z=1,6,13,22,29,74,79);
8. SESAME tabular EOS.
Opacity options in RALEF-2D
Here we profit from many years of a highly qualified work at KIAM (Moscow) in the group of Nikiforov-Uvarov-Novikov (the THERMOS code based on the Hartree-Fock-Slater atomic modeling).
0.1 1 10
1
10
100 THERMOS data 8 ν-groups 32 ν-groups
Abs
orpt
ion
coef
ficie
nt k
ν (cm
-1)
Photon energy hν (keV)
W: T=0.25 keV, ρ=0.01 g/ccOpacity options:1. power law,
2. ad hoc analytical,
3. inverse bremsstrahlung(analytical),
7. GLT tables (source opacities from Novikov)
. . . . . . .
Laser absorption
Laser absorption in RALEF-2D
individual laser beams are treated by the same method of short characteristics as the thermal radiation;
no refraction, the inverse bremsstrahlung absorption coefficient, 100% absorption at the critical surface.
Present model:
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
y
x
0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
exact Δy=0.05 Δy=0.0125
F
x
Ωx=0.1915Ωy=0.6940y=4.0
Implosion of a cylindrical tungsten column in multi-
wire Z-pinches
Problem 1 (M M Basko et al., PPCF, 54, 055003, 2012)
Multi-wire Z-pinches (Sandia, Angara-5)
40-mm diameter array of 240, 7.5-μm-diam. wires.
• 11.5 MJ stored energy
• 19 MA peak load current
• 40 TW electrical power to load
• 100-250 TW x-ray power
• 1-1.8 MJ x-ray energy
Z-machine at Sandia (USA):
X-ray pulses at Z (W.A.Stygar et al., PRE 2004)
Initial MHD phase of wire implosion (J.P.Chittenden et al.)
Problem statement for RH simulation
v0
Cylindrical implosion of an initially cold tungsten plasma cloud Initial shell parameters:
• radial thickness: 2 mm;
• implosion velocity: v0 = 400 km/s;
• uniform temperature: T0 = 20 eV;
• mass: m0 = 0.3 mg/cm (A); 6.0 mg/cm (Z);
• kinetic energy: 24 kJ/cm (A); 480 kJ/cm (Z);
• far from the axis, mass is uniformly distributed over the radius;
• possible influence of magnetic field is neglected.
In its present formulation, the problem is one-dimensional.
Tungsten EOS and opacities
The equation of state and opacities (LTE) of tungsten have been provided by the V.G.Novikov et al. from KIAM (Moscow) ; calculated with the THERMOS code based on the Hartree-Fock-Slater atomic model.
0.1 1 10
1
10
100 THERMOS data 8 ν-groups 32 ν-groups
Abs
orpt
ion
coef
ficie
nt k
ν (cm
-1)
Photon energy hν (keV)
W: T=0.25 keV, ρ=0.01 g/ccHydrodynamics was simulated with either 8 or 32 spectral groups.
The output spectra were calculated in the post-processor mode by solving the radiation transfer equation with 200spectral groups.
Densityt = 3 ns
Total X-ray emission power: case A
The imploding tungsten plasma radiates away about 90% of its initial kinetic energy.
The nominal power of implosion is
2
1 1
1 /2
24 kJ cm / 5 ns 4.8 TW cm
nom im pulseW MU t
− −
= =
= =
0 1 2 3 4 5 60
1
2
3
4
5
6
X-ra
y em
issi
on p
ower
PX (T
W/c
m)
Time (ns)
nominal power DEIRA RALEF, 8 ν-groups RALEF, 32 ν-groups
Case A
Stagnation shock: case A
Here we deal with a supercriticalRD shock front (see Zel’dovich and Raizer, chapter VII):
0.00 0.05 0.10 0.15 0.20 0.250.02
0.040.060.080.1
0.2
0.40.60.8
1
2
4
0.041 0.042 0.043 0.044
0.2
0.25
0.3
0.35
Den
sity
ρ (g
/cc)
, tem
pera
ture
T (k
eV)
Radius (mm)
density matter temperature radiation temperature
Case A: t = 3 ns
4 31 0
12
T Uσ ρ>
The shocked material radiates away about 90% of its initial kinetic energy.
The shock front is marked by an extremely narrow peak of matter temperature, which defines the hard tail of the emitted spectrum.
The density jumps by a very large factor of ~60.
Spatially integrated emission spectra: case A
High-resolution emission spectra can be obtained in the post-processor regime even when hydrodynamics is simulated with a relatively small number (8, 32) of spectral groups.The hard component contains ~16% of the total X-ray pulse energy.
0.01 0.1 10.0
0.5
1.0
Spec
tral f
lux
F ν (TW
cm
-1 s
r-1 k
eV-1)
Photon energy hν (keV)
RALEF 8 ν-groups RALEF 32 ν-groups Planckian,
T = 0.11 keV
Case A: t = 3 ns
0 1 2 3 4 5 6 71E-5
1E-4
1E-3
0.01
0.1
1
Spec
tral f
lux
F ν (TW
cm
-1 s
r-1 k
eV-1)
Photon energy hν (keV)
RALEF, 8 ν-groups RALEF, 32 ν-groups Planckian, T = 0.34 keV
Case A: t = 3 ns
Optical thickness of the imploding plasma: case A
Depending on a selected frequency, the shock front lies at an optical depth of τ = 0.5–10.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.060.080.1
0.2
0.40.60.8
1
2
468
10
20
Rad
ial o
ptic
al d
epth
Radius (mm)
hν = 40 eV hν = 210 eV hν = 330 eV hν = 2.0 keV hν = 5.1 keV
Case A: t = 3 ns
⇒ a large portion of the X-rays emitted by the shock front is reprocessed in the tenuous infalling plasma.
X-ray images of the imploding pinch: case ASpectral X-ray images provide information on the internal structure of the imploding pinch.
-1.0 -0.5 0.0 0.5 1.00
5
10
15
20
25
30 Case A: t=3 nsR
adia
tion
inte
nsity
I ν (TW
cm
-2 s
ter-1
keV
-1)
Distance along observation slit (mm)
hν = 0.205 keV hν = 0.27 keV hν = 1.81 keV
Comparison with the Sandia experimental data(M.E.Foord et al., PRL 93, 055002, 2004)
RALEF-2D
Empty spherical hohlraum used in
experiments at GSI
Problem 2 (M M Basko et al., GSI report 2011-03)
Rescaling between 3D and 2D configurations
For 2D simulations of intrinsically 3D hohlraums we have to rescale the energy (total power) of the laser pulse !
When approximating a 3D configuration with a 2D one, we want to preserve
• the hydrodynamic flow pattern,• the temporal history of the radiation and matter temperatures.
The latter means that we preserve
• the geometric configuration and dimensions of the hohlraum,• the temporal shape of the driving laser power.
⇒ the total energy E (power) of the laser pulse must be rescaled to E !
Total energy balancein the hohlraum:
walls walls holes holes walls wh holes
walls walls holes holes walls wh holes
E F S F S S q SE EE F S F S S q S
= +⎧ +=⎨ =⎩
⇒+ +
1 11
holeswh
walls w
FqF α
≡ = ≥−
The unknown flux ratio (αw < 1 is the wall albedo) is found numerically by
performing 1 – 2 iterations.
GSI hohlraum and beam parameters
Spherical hohlraum is replaced by an “equivalent” cylindrical configuration:
• dimensions are preserved
• total power (energy) is rescaled
0.3 mmsph cylR R R= = =
(0)2 ;
2 4 2cyl sph sph
cyl
W W WW
R R Rπ π⇒= =
14 2Gaussian beam, FWHM=0.1 mm:130 J 0.14 TW; 2.7 10 W/cm , 60μm;0.9 ns
cylsph las
WW F σ
π σ= = = = × =
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
y (m
m)
x (mm)
300 μm
150 μm
Au
- zero approximation:
- first approximation:
(1)
(1)
0.6 ,
0.6 260 J/mm
sphcyl
sphcyl
WW
RE
ER
=
= =
EOS and opacityThe current version of RALEF-2D is based on LTE opacities and emissivities.In all the simulations presented below the Novikov-Grushin (KIAM) EOS and opacities of Au were used.
0.1 1 10
1
10
100
THERMOS data 7 ν-groups 200 ν-groupsA
bsor
ptio
n co
effic
ient
kν (c
m-1)
Photon energy hν (keV)
Au: T = 100 eV, ρ = 0.01 g/cc
Eulerian simulation
Run parameters: • hydro: Eulerian, Godunov 1st order
• initial fill: Au vapor, ρ0 = 10-5 g/cc
• S12, nθ×nr = 1440×160 (~373 000 cells)• 7 frequency groups
• thermal conduction: flux limit finh = 0.1
Mesh construction for Eulerian simulations
block 7
block 3bloc
k 9
bloc
k 5
block 8
block 6
block 4
block 2
block 1
We use a quasi-polar 9-block mesh with a free-float center. Each block can be divided into several parts with different cell sizes and materials.
Density evolution(snapshots every 100 ps)
Laser energy deposition (per unit mass)(against the background of density contours)
Plasma temperature(snapshots every 100 ps)
Radiation temperature(snapshots every 100 ps)
Average radiation temperature in the hohlraum
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60
20
40
60
80
100
120
140
160
180
1 ν-group 7 ν-groups
radi
atio
n te
mpe
ratu
re T
r (eV
)
time (ns)
Average radiation temperaturenear the hohlraum center (r < 0.1 mm)
Radiation spectra from the diagnostic hole
0.0 0.2 0.4 0.6 0.8 1.0 1.20.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
F ν (TW
mm
-1 s
ter-1
)
hν (keV)
Planckian, T=120 eV
t = 0.4 ns t = 0.7 ns t = 0.9 ns
Radiation spectra from the diagnostic hole
0 1 2 3 4 5 6 7 81E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
F ν (TW
mm
-1 s
ter-1
)
hν (keV)
Planckian, T=120 eV Planckian, T=350 eV
t = 0.4 ns t = 0.7 ns t = 0.9 ns
Long-time hohlraum cooling
0.1 0.2 0.4 0.60.8 1 2 4 6 8 10 20
50
100
150
200
250
300
350
radi
atio
n te
mpe
ratu
re T
r (eV
)
time (ns)
Average temperatures nearthe hohlraum center (r < 0.1 mm)
matter
radiation
Summary
We believe that the RALEF-2D code has good chances to be recognized as one of the most powerful in its class in terms of speed, accuracy and variety of solvable problems.
Further work is needed
1) to implement radiation transport in the axially symmetric r-z geometry,
2) to improve the laser absorption model,
3) to improve the algorithm for Lagrangian node velocities.
ADDENDUM
Hydrodynamics
Computational mesh: block structure
Every block can be subdivided into np1×np2 topologically rectangular parts; different parts can be composed of different materials.
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0part 9
part 7
part 5
part 3
x
y
part 1
part 10
part 8
part 6
part 4
part 2
The Godunov numerical method
cell i
face i1
face
i2
vertex i
i2+
i1+
Fi1
Fi2
Fi2+1
Fi1+2ρi, ui, Ei
No artificial viscosity is needed !
Principal variables: 2
, ,2uu E eρ = +
Equation of state: ( , )p p eρ=― all assigned to the cell centers !
Lagrangian phase: the Riemann problem is solved at each cell face ⇒ fluxes F of momentum ρu and total energy E ⇒ new ui(t+dt) and Ei(t+dt) ; mass is conserved.
-1 0 10.0
0.5
1.0
1.5
pi+1
pi
uf
cell i+1
p
cell ipf
x
← 1st order
A “snag”:
A fast non-iterative Riemann solver by J.K.Dukowicz (1985) is used.
node velocities uvi !
Hydrodynamics: comparison with KIAM
KIAM code RALEF-2D
1. Riemann solver: iterative for a two-term EOS (Zabrodin et al.); accurate for the ideal-gas EOS; slow and capricious for realistic EOS.
2. Only the first order in the Godunovmethod.
3. Either Lagrangian or Eulerian mesh.
1. Riemann solver: non-iterative for an arbitrary EOS (J.K.Dukowicz, LANL, 1985); fast and robust for any EOS at the cost of a certain loss of accuracy.
2. More accurate second-order Godunovmethod.
3. Fully adaptive mesh with an efficient second-order rezoning algorithm [Winslow (LLNL,1981) + Brackbill(LANL) + Basko (ITEP,2009)].
Thermal conduction
Time discretization – the SSI method
For our form of the equation of state: T = T(ρ,e) .
Explicit time discretization of the energy equation (Lagrangian form):
( ) ( )T r dep pijij ij ij ij ij ijM E E W W W t W t− = + + Δ + Δ
Partially-implicit time discretization:
( ) ( )( ) ( ),
T r dep pij ij ijij ij ij ij
T r depij ij ijV ij ij ij ij
M E E W W W t W t
c M T T W W W t
− = + + Δ + Δ
− = + + Δ
To calculate the partially-implicit cell heating powers , we use the “new” central
temperature , and the “old” side temperatures (the SSI proper).
T ri iW W+
ijT , 1 1,, , ...i j i jT T+ +
equation for solved at the SSI stage
ijT←
Spatial discretization ― the explicit fluxes Hijm
The integrated flux across face (i,j,m) is given by
( ), , , , ,
;ijm f ijm f ijm f ijm v ijmH g n l R
g T
κ= − ⋅
= ∇
If we assume that we know the vertex temperatures Tv,ij , the face-centered temperature gradient can be evaluated from an obvious system of two linear equations
, , ,
, , ,
f ijm v ijm v ijm
f ijm c ijm c ijm
g l T
g l T
⎧ ⋅ = Δ⎪⎨
⋅ = Δ⎪⎩
Generalization to the case of discontinuous κ and/or normal component of is straightforward.
,f ijmg
Vertex temperatures
To close the numerical scheme, we need an interpolation formula for the vertex temperatures. An effective algorithm is based on a bilinear interpolation in “natural”coordinates on a distorted c-quadrilateral: ( ) [ ] [ ], 1, 1 1, 1ξ η ∈ − + × − +
, , 1,
, , 1 , 1, 1
1 (1 )(1 ) (1 )(1 )4
(1 )(1 ) (1 )(1 )
c ij c i j
c i j c i j
x x x
x x
ξ η ξ η
ξ η ξ η
−
− − −
⎡= + + + − + +⎣
⎤+ + − + − − ⎦
, , 1,
, , 1 , 1, 1
1( ) (1 )(1 ) (1 )(1 )4
(1 )(1 ) (1 )(1 )
c ij c i j
c i j c i j
T x T T
T T
ξ η ξ η
ξ η ξ η
−
− − −
⎡= + + + − + +⎣
⎤+ + − + − − ⎦
Important property: this interpolation reconstructs exactly an arbitrary linear function
on an arbitrary quadrilateral grid ⇒ 2nd order in space !
( )T x a b x= + ⋅
For discontinuous κ the weights in the second formula are multiplied by κij, κi-1,j, κi,j-1, κi-1,j-1.
Strongly distorted grids
On strongly distorted grids the bilinear interpolation becomes non-positive!
To restore positiveness on a logically local stencil, we have to introduce additional constraints, which degrade the spatial convergence order on strongly distorted grids.
Test problems: non-linear wave into a warm wall
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
0.75 0.80 0.85
1.0
1.5T
x
exact ε1 = 0.1, arithm. κf
ε1 = 0.01, arithm. κf
ε1 = 0.01, harm. κf
Random grid 40x40401,
( , ) 1
V
t
c T
T t x
ρ κ κ
→ −∞
= =
=
Parameters:
Initial condition:
Solution: 10
2 3 44 13 4
( , ) 1 ( ), ( ),
ln 4 3
T t x t xφ ξ ξ κ
ξ φ φ φ φ φ
−= + = −
= + + + +
[originally proposed by E.Livne & A.Glasner (1985)]
• For good accuracy we need “ghost” energy control, ε1 < 0.01−0.02 .
• Arithmetic-mean κf is by far more accurate than the harmonic-mean value.
Test results:
Test problems: non-linear wave into a warm wall
Parameters:
Initial condition:
401,
( , ) 1
V
t
c T
T t x
ρ κ κ
→ −∞
= =
=
Solution: 10
2 3 44 13 4
( , ) 1 ( ), ( ),
ln 4 3
T t x t xφ ξ ξ κ
ξ φ φ φ φ φ
−= + = −
= + + + +
On strongly distorted grids both the arithmetic mean and the harmonic mean κf yield about the same accuracy.
Test results:
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
T
x
exact ε1 = 0.05, arithmetic κf
ε1 = 0.05, harmonic κf
Kershaw grid 36x36
Temperature in all cells
0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.240.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1D DEIRA (exact) 2D CAVEAT-TR
cell-
cent
er te
mpe
ratu
re
cell-center radius
Thermal conduction: comparison with CHIC
CHIC: fully implicit; RALEF: SSI – symmetric semi-implicit ⇒ faster and more accurate
Basko vs Novikov source EOS
1E-3 0.01 0.1 1 10 100 10001E-3
0.01
0.1
1
10
100 Basko EOS of Au Novikov EOS of Au
P/ρ
(1012
erg
/g)
density ρ (g/cc)
T = 1 eV
T = 10 eV
T = 100 eV
In some cases the Maxwell construction is possible for the Novikov EOS.
Final remarks
Unlike in the ideal hydrodynamics, in radiation hydrodynamics there are virtually no analytical or self-similar solutions to rely upon; hence
Without numerical simulations, it is virtually impossible to achieve adequate qualitative understanding of dynamic plasma processes, where energy transport by thermal radiation is important.
