Basic principles of numerical radiation
hydrodynamics and the RALEF code
M. M. Basko
Keldysh Institute of Applied Mathematics, Moscow, Russia
ELI, Prague 10 July 2014
Main constituents of the RALEF-2D package
The RALEF-2D code has been developed with the primary goal to simulate
high-temperature laser plasmas. Its principal constituent blocks are
1.
Hydrodynamics
2.
Thermal conduction
3.
Radiation transport
4.
EOS and opacities
5.
Laser absorption
Equations of hydrodynamics
The RALEF-2D code is based on a one-fluid, one-temperature hydrodynamics model
in two spatial dimensions (either x,y, or r,z):

    u  0,
t
 
  
 u
t
ideal hydrodynamics
  u  u   p  0,
E
     E  p  u      T   Qr  Qdep ,
t
u2
E  e  , e  e(  , T )

or p  p(  , e)
2
  T  – energy deposition by thermal conduction (local), Qr – energy deposition by
radiation (non-local),
Qdep
– eventual external heat sources.
Selection of principal dependent variables
Illustration with the 1D planar example:
Divergent form: ρ, ρu, ρE
 
 u   0,
t x
 ( u ) 
  p  u 2   0,
t
x
 E  
 E  p u   q;
t
x

Naturally leads to a conservative
numerical scheme
RALEF
Non-divergent form: ρ, u, e
 
 u   0,
t x
u
u 1 p
u

 0,
t
x  x
e
e p u
u 
 q;
t
x  x

Easier to avoid unphysical values
of the internal energy e and
temperature T
Selection of principal independent variables
Divergent Eulerian form: t, x
 
 u   0,
t x
 ( u ) 
  p  u 2   0,
t
x
 E  
 E  p u   q;
t
x
Eulerian simulation: spatial mesh
in x,y,z is fixed in time.
Non-divergent Lagrangian form: t, m
x
 u,
t
u p

 0,
t m
e
u
p
 q;
t
m
Lagrangian simulation: Lagrangian mesh
in m=∫ρdx is “attached” to the fluid;
typically, the simulation time is severely
limited by mesh “collapse”.
The ALE technique (adaptive mesh)
The Arbitrary Lagrangian-Eulerian approach allows free motion of the
– independent of the motion of the fluid!
x,y,z
mesh
Implementation in RALEF: every time step consists of 2 substeps:
1. a purely Lagrangian step – the mesh is “attached” to the fluid, followed by
2. a rezoning step, where a completely new x,y,z mesh is constructed, and the
principal dynamic variables are remapped to the new mesh.
At substep 2, a new mesh is constructed by solving the Dirichlet problem for a separate
Poisson-like differential equation with a user-defined weight function.
In RALEF, there are quite a number of user-accessible parameters for control over the
mesh dynamics – which is often the most tricky part of a successful simulation!
Mesh topology in RALEF
Computational mesh consists of blocks with common borders. Each block is topologically equivalent to
a rectangle. Common block faces have equal number of cells.
0.4
1.0
2
2
0.8
1
block 5
1
block 4
2
2
y
0.3
1
block 10
1
block 9
Mesh geometry:
block 3
2
block 8
2
1
1
2
2
0.2
1
block 2
1
block 1
2
1
2
1
0.1
• Cartesian (x,y)
• cylindrical (r,z)
0.6
0.4
0.2
y
0.0
block 7
-0.1
block 6
-0.2
0.0
-0.3
0.0
0.5
1.0
x
1.5
2.0
-0.4
2.0
3.0
2
2.5
2
2.0
1
1.5
block 9
y 0.5
block 3
block 11
1
2
block 2
block 7
1
1
0.5
2
block 7
1
2
2
-1.5
block 5
block 1
1
1
block 8
2
-2.0
0.2
0.3
0.4
Different mesh options,
distinguished by the value of
variable igeom, are combined
into a mesh library, which is
being continuously expanded.
2
2
1
1
1
2
2
1
block 1
1
block 4
0.1
Mesh library:
2
-1.0
1 block 4
1.0
1
-0.5
0.0
2
block 5
2
-0.1
block 6
y
1 block 2
2
0.0
1
1.5
block 6
2
-0.2
2
1.0
-0.3
x
block 10
1
2
-0.4
1
block 3
0.0
-2.5
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.0
0.5
1.0
x
1.5
2.0
Computational mesh: block structure
Every block can be subdivided into np1np2 topologically rectangular parts;
different parts can be composed of different materials.
1.0
part 9
part 10
part 7
part 8
part 5
part 6
part 3
part 4
part 1
part 2
0.8
y
0.6
0.4
0.2
0.0
0.0
0.5
x
1.0
1.5
2.0
RALEF: the ALE technique in action
0.4
0.4
Ncyc = 1
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.4
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Ncyc = 100
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Limitations of the ALE technique
Example: laser irradiation of a Cu foil
t = 1.6 ns
fixed-pressure boundary
Cu foil
laser
Simulation stops because a singularity develops at the plasma-vacuum boundary !
Boundary conditions
To obtain a particular solution of the hydrodynamics equations, we need
 to define the initial state, and
 to impose certain boundary conditions.
The initial state is conceptually simple and fully free to be set by the user: either equilibrium
or non-equilibrium.
Setting up a physically consistent and computationally efficient combination of boundary
conditions is a considerably more complex task (requires basic understanding of
hydrodynamics and some physical intuition).
Basic types of boundary conditions in RALEF:
 reflective (symmetry) boundary (fixed in space → Eulerian)
 prescribed pressure (moves in space → Lagrangian)
 prescribed velocity (moves in space → Lagrangian)
 free outflow (fixed in space → Eulerian)
 prescribed inflow (fixed in space → Eulerian)
Additional degree of freedom: the boundary conditions and the ALE mode can be
changed at will at any time during the simulation.
Laser irradiation of a thin Sn disk
free-outflow boundary
reflective (symmetry) boundary
Different materials
By finding an appropriate combination of boundary conditions and ALE options, one can
adequately simulate practically any 2D problem with a single material.
Multiple materials pose an additional challenge:
In RALEF, mixing of different materials
within a single mesh cell is not
allowed  hence, any material
Al
Sn
interface must be treated as a
Lagrangian curve, which usually tends
to get tangled: as a result, the
simulation stops.
Equation of state
For ideal hydrodynamics (without thermal conduction, viscosity, etc) we need an equation
of state in the form
p  p , e
RALEF can easily accommodate any thermodynamically stable EOS.
However, because the principal variable is E=e+u2/2 rather than e (or T), the numerical
scheme is not positive with respect to T  sporadic appearance of negative temperatures!
CP: TC = 4868 K, pC = 5613 bar,
6000
s
4000
3000
p [bar]
C = 0.66 g/cm
b
5000
s
An additional conceptual difficulty arises for a vander-Waals type of EOS in the region of liquid-vapor
phase coexistence – i.e. below the binodal, where
we have two different branches of the same EOS:
 a metastable branch, and
 an absolutely stable equilibrium branch,
obtained by means of Maxwell construction.
3
b
4500 K
2000
4000 K
1000
0
-1000
3000 K
-2000
This latter problem has not been resolved yet !
0
2
4
6
3
v [cm /g]
8
10
Implicit and explicit numerical hydrodynamics
Equation :
dU
 F U ; U is a multidimen sional vector; U n  U tn ;
dt
Implicit scheme
Explicit scheme
U n 1  U n
 F U n 1 
t
U n 1  U n
 F U n 
t
Implicit schemes are numerically stable and allow large time steps, but are difficult to
implement.
Numerical stability of explicit schemes requires the Courant–Friedrichs–Lewy
condition (CFL condition) to be fulfilled
t  x c  | u |
1
which for certain problems may incur prohibitively long computing times.
RALEF has explicit hydrodynamics, i.e. is oriented on simulating fast processes.
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 second-order Godunov
schemes (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), well adapted for handling arbitrary equations of state.
The Godunov numerical method
Principal variables:
― all assigned to the cell centers !
i2+
Fi
Equation of state:
2+1
cell i
face i2
u2
 , u, E  e 
2
Fi2
i, ui, Ei
p  p (  , e)
Lagrangian phase: the Riemann problem is solved at each
cell face  fluxes F of momentum u and total energy E
 new ui(t+dt) and Ei(t+dt) ; mass is conserved.
Fi
1+2
Fi1
pi
1.5
p
vertex i
face i1
i1+
← 1st order
pf
1.0
cell i
uf
pi+1
0.5
A fast non-iterative Riemann solver by
J.K.Dukowicz (1985) is used.
cell i+1
0.0
-1
No artificial viscosity is needed !
0
1
x
A “snag”: node velocities uvi !
Importance of the 2-nd order + ALE
Non-linear stage of the Rayleigh-Taylor instability of a laser-irradiated thin carbon foil
RALEF: 1-st order
RALEF: 2-nd order
Thermal conduction
(a test bed for radiation transport)
Equations of hydrodynamics
The newly developed RALEF-2D code is based on a one-fluid, one-temperature
hydrodynamics model in two spatial dimensions (either x,y, or r,z):

    u  0,
t
 
  
 u
t
  u  u   p  0,
E
     E  p  u      T   Qr  Qdep ,
t
u2
E  e  , e  e(  , T )

or p  p(  , e)
2
  T  – energy deposition by thermal conduction (local), Qr – energy deposition by
radiation (non-local),
Qdep
– eventual external heat sources.
Implicit versus explicit algorithms for conduction
Equation :
T
 T
 
 Lx T ;
t x x
Explicit scheme
Ti n 1  Ti n
 Lx ..., Ti n1 , Ti n , Ti n1 ,...
t
An explicit scheme is stable under the condition
incurs prohibitively long computing times.
Ti n  T tn , xi ;
Fully implicit scheme
Ti n 1  Ti n
 Lx ..., Ti n11 , Ti n 1 , Ti n11 ,...
t
t  2  x 2
1
, which practically always
A fully implicit scheme is always stable but requires iterative solution – which can be
implemented for thermal conduction, but becomes absolutely unrealistic for radiation transport.
Symmetric semi-implicit (SSI) scheme
Ti n 1  Ti n
 Lx ..., Ti n1 , Ti n 1 , Ti n1 ,...
t
RALEF is based on the SSI algorithm
for both the thermal conduction and
radiation transport.
The SSI method for thermal conduction
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 and
radiation transport into the 2D Godunov method (in order to avoid costly matrix
inversion required by fully implicit methods).
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 ,
 symmetric on a local 9-point stencil,
 computationally efficient.
Time step control in the SSI method
Constraints on t are based on usual approximation-accuracy considerations, but here
we need two separate criteria with two control parameters:
H
ij1
 H ij 2  H i , j 1,1  H i 1, j ,2  Wijr  qij M ij  t
cV ,ij M ij   aij1  aij 2  bi , j 1,1  bi 1, j ,2  D  t
r
ij
 ij
cV ,ij M ij
Ts

 0  1  Tij  Ts  ,
 1 Tij  Ts  .
is a problem-specific sensitivity threshold for temperature variations.
Test problems: linear steady-state solutions
Time convergence to the steady state
The linear solution
T ( x, y )  x
is reproduced exactly on all grids, unless
special constraints to ensure
positiveness are imposed on strongly
distorted grids.
For sufficiently large time steps t, the
SSI method does not converge to the
steady-state solution  an indication of its
conditional stability (neutral for large t).
Piecewise linear solutions with -jumps
are reproduced exactly only with
harmonic mean f,ij , and only on
rectangular grids.
t = 0.002
1
0 = 0.2, 1 = 0.04
0.01
1E-4
TL2
1E-6
t = 0.0005
1E-8
t = 0.001
1E-10
0
1000
2000
Ncyc
3000
4000
Test problems: non-linear wave into a cold wall
Parameters:
 cV  1,    0T 3
Initial condition:
T (0, x)  0
Boundary condition: T (t , 0)  1
Solution: T (t , x)   ( ),
d 2 4
d


0
2
d
d
x

,
 0t / 2
Square grid 100x100
1.0
T
harmonic f, f0=0.01
0.8
0.52
0.6
0.4
0.51
0.50
0.49
Test results:
This solution cannot be simulated with the
harmonic-mean f , whereas excellent
results are obtained with the arithmeticmean f : the front position is reproduced
with an error of 0.1% for  .
exact
arithmetic f
0.2
0.48
0.76
0.0
0.0
0.77
0.2
0.78
0.79
0.4
0.6
x
Temperature in all cells
0.8
1.0
Standard grids for test problems
Square grid
Kershaw grid
1.0
1.0
0.8
0.8
0.6
0.6
y
Only the Kershaw grid
is strongly distorted in
the sense that some of
the coefficients
(1)(1)
become negative.
y
0.4
0.4
0.2
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.0
0.2
0.4
x
0.6
0.8
1.0
x
Wavy grid
Random grid
1.0
1.0
0.8
0.8
0.6
0.6
y
y
0.4
0.4
0.2
0.2
0.0
0.0
0.2
0.4
0.6
x
0.8
1.0
0.0
0.0
0.2
0.4
0.6
x
0.8
1.0
Test problems: non-linear steady-state solution
Here we test against a steady-state solution of
the form
0.01
T ( x, y)  a  bx  cx 4 ,
with the source term Q  Q( x, y)  x .
2
2nd
Our scheme clearly demonstrates the
order convergence rate on all grids, and is
no less accurate than the best published
schemes of Morel et al. (1992), Shashkov
et al. (1996), Breil & Maire (2007).
1E-3
|Tl2|1E-4
square mesh
wavy mesh
random mesh
Kershaw mesh
CHIC - wavy
CHIC - random
CHIC - Kershaw
1E-5
1E-6
0.01
h
0.1
Radiation transport
Equations of hydrodynamics
The newly developed RALEF-2D code is based on a one-fluid, one-temperature
hydrodynamics model in two spatial dimensions (either x,y, or r,z):

    u  0,
t
 
  
 u
t
  u  u   p  0,
E
     E  p  u      T   Qr  Qdep ,
t
u2
E  e  , e  e(  , T )

or p  p(  , e)
2
  T  – energy deposition by thermal conduction (local), Qr – energy deposition by
radiation (non-local),
Qdep
– eventual external heat sources.
Radiation transport
Transfer equation for radiation intensity I in the quasi-static approximation (the
limit of c → ):
1 I
  I  k  B  I  ,
c t


I  I t , x , ,  , B  B  , T 
Quasi-static approximation: radiation transports energy infinitely fast (compared to the fluid
motion)  the energy residing in radiation field at any given time is infinitely small !
In the present version, the absorption coefficient
are calculated in the LTE approximation.
k
and the source function
B = B(T)
Coupling with the fluid energy equation:
Qr    
  d   I

d


 d   k  I  B  d
4
Radiation transport adds 3 extra dimensions (two angles and the photon frequency) 
the 2D hydrodynamics becomes a 5D radiation hydrodynamics !
Not to be mixed up with spectral multi-group diffusion
In many cases the term “radiation hydrodynamics” (RH) is applied to hydrodynamic
equations augmented with the multi-frequency diffusion equation
 1
1 E
 u 1
u 
    E   E        
E
c t
c
3
c
3
k


 
 
for the spectral radiation energy density
fluid is
Qr (t , x ) 

 B


k

E
 
 
c



E  E (t , x, ) ;
the coupling term to the
 k  cE  B  d .
Here the information about the angular dependence of the radiation field is lost; one
simply has to solve some 30 – 100 additional mutually independent diffusion equations.
Not much of a challenge for computational physics: there already exist numerically stable,
positive and conservative numerical schemes on distorted (non-rectangular) grids.
In the RALEF code we have such a scheme implemented for the thermal conduction.
Challenges for computational RH
Ideally, the numerical scheme for solving the radiation transfer equation must possess
the following important properties:
 it must be positive in the sense that non-negative boundary values of Iν
guarantee Iν ≥ 0 at all collocation points ― provided that Bν ≥ 0 ;
 it must be conservative in the sense that both locally and globally the finitedifference equivalent of the Gauss theorem is fulfilled,
 k  I  B  dV
V

 I    n  dS
S
 it must reproduce the diffusion limit on arbitrary quadrilateral (or triangular) grids
in optically thick regions ― especially when the optical thickness of individual grid
cells becomes >> 1  a high-order accuracy scheme is needed.
The difficulty of constructing such a scheme has been recognized early in the theory of neutron transport:
K.D.Lathrop, J.Comp.Phys., 4 (1969) 475.
To the best of our knowledge, no such scheme is known in the open literature.
Calculation of Iν: the method of short characteristics
 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 , the radiation field I is
L
found by solving the transfer equation
 L I  k  B  I  ,


I  I t , x , ,  L ,
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 .
Advantages:
• even on strongly distorted meshes, it is guaranteed
that light rays pass through each mesh cell;
• the algorithm is generally computationally more
efficient than that with long characteristics.
Disadvantages:
• a significant amount of numerical diffusion in space.
S6
Conservativeness: flux conservation in vacuum
Consider a scheme where the radiation intensity is assigned to mesh vertices. Consider
one cell with a negligibly small absorption (k  0):
1
4
 
  I  0 
    n  I dl  0
L
2
3

In our example on the left we have
1
1
1
H in   I1  I 2  l   I 2  I 3  l   I 3  I 4  l
2
2
2
1
H out   I1  I 4   3l  H in
2
Conclusion: a numerical scheme based on nodal collocation
points is generally non-conservative !
Conservativeness can be restored by assigning the intensity I to cell
faces (R.Ramis, 1992) ― equivalent to using the fluxes H as independent
variables
Wijr  Hij1  Hij 2  Hi, j 1,1  Hi 1, j ,2
But then the high-order of finite-difference approximation, needed for the diffusion limit, is lost !
The diffusion limit
The diffusion limit is the limit of
k >> | ln T|, | ln |, | ln k |
In this limit we can apply the asymptotic expansion:
 1 
1
I  B 
B  O  2  ,
k
 k 

Qr 

 d   k  I  B 
4
 3 B
 4

d  div 
B   O 
 x x x
 3k R

 i j k

 .

In the quasi-static LTE case the diffusion limit is equivalent to the approximation of
radiative thermal conduction:
 4

Qr  div 
B 
 3kR

 div  lRT 3T 
Heating by two opposite beamlets in the diffusion limit



Consider two opposite beamlets  and   
propagating along an infinitely thin column h:
dI 
dI 

 BI ,
  B  I  ,    k ds   k d   x
d
d


In the diffusion limit we have
dB d 2 B
I ( )  B( ) 

 ... ,
d d 2
dB d 2 B

I ( )  B( ) 

 ...
d d 2

Having combined the two opposite beams, we get
 d 3B 
d 2B
d 2I
Q   k  I  B d  k  I  B  k  I  B  k
O
k
 ...
2
3 
2
d

d

d



4

r


Conclusion: our finite-difference scheme must correctly reproduce the 2nd spatial
derivatives of the unknown function
Iν
– i.e. to be of high order accuracy !
The latter is a major challenge for non-rectangular grids in 2 and 3 dimensions – and
especially so when the cell optical thickness Δτ >> 1 !
Achievements
We have developed an original numerical scheme for radiation transport which
 is strictly positive,
 from the practical point of view ― reproduces the diffusion limit with a fair degree
of accuracy on not too strongly distorted meshes,
 works robustly in both (x,y) and (r,z) geometries.
Failures
Our numerical scheme
 is not conservative (violates the finite-difference version of the Gauss theorem), and
 has no strict convergence in the diffusion limit.
Radiation transport:
test problems
Numerical diffusion: a searchlight beam in vacuum
The short-characteristic method produces a significant amount of numerical diffusion for
light beams with sharp edges.
4.0
1.0
3.5
0.8
3.0
x=0.1915
F
2.5
y=0.6940
0.6
y
y=4.0
2.0
0.4
1.5
exact
y=0.05
1.0
0.2
y=0.0125
0.5
0.0
0.0
0.5
1.0
1.5
x
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
x
For thermal radiation a certain amount of numerical diffusion may be more an advantage than a drawback.
Test problems: radiative cooling of a slab
Computational mesh:
vertical symmetry axis
1.0
y
0.5
0.0
0
1
2
3
16
17
18
19
20
x

I
 k0  B  I  ,
y
B( y )  sin  y  ,
0  y  1;


Qr  y   2 k0  k0  B  y ' E1  k0 y  y '  dy  2B  y   ;
 0

1
Convergence of the Sn method
error L2
-Qr(y) at x = 0
10
Exact solution:
k , random mesh
5
exact
S6, nx=ny=10
S12, nx=ny=80
0
0.2
S2
S4
S6
S8
S12
23%
7.6%
2.3%
1.9%
1.15%
-5
Typical accuracy in problems with optically thin mesh cells is 12%.
0.4
y
0.6
0.8
1.0
Diffusion limit in a slab
S12, nx=ny=20,  = 10
0.3
4
square grid
random grid, 2010
random grid, 2013
0.2
r
Qi / Q(yc,i)-1
0.1
0.0
-0.1
-0.2
L2 = 8% -> 1.9%
-0.3
0.0
0.2
0.4
y
0.6
0.8
1.0
No convergence, the error level depends on the level of mesh distortion.
Test “fireball” (the limit of radiative heat conduction)
Consider an initially hot unit sphere with T0 = 1 and
k = 100/T. Propagation of a radial heat wave is
governed by the conduction equation
1.5
T 1   16 T 3 T 

r

t r r  3 cv k r 
1.0
which admits an analytical self-similar solution
with the front position
1/10
rf  t 
t 
 r0  
 t0 
T 
for k  k0  
 T0 
y
1
0.5
0.0
0.0
0.5
x
1.0
1.5
Fireball expansion in xy-geometry
By t = 0.002102926 the exact position of the front must be at r = 1.5.
Final state: the conduction solution
By t = 0.002102926 the exact position of the front must be at r = 1.5.
0.5
Thermal conduction: cylinder, xy-geometry
0.4
0.3
exact
RALEF, n=50
T
0.2
0.1
0.0
0.0
0.5
1.0
r
1.5
Final state: the transport solution with k0 = 100
From practical point of view, in this particular case we observe a slow convergence
to the exact diffusion solution.
Final state: the transport solution with cell = 10
In this example we observe no convergence to the exact diffusion solution.
Numerical energy “leak” is  40% !
SN method: the “ray effect”
Consider radiation transport from a central “hot” rod
across a vacuum cavity.
4
y
S6
3
R2
y
2

x
block 2
R1
1
block 1
0
0
1
2
x
3
4
1.3
1.2
(hr/B0)(R2/R1)
1.1
Convergence of the Sn method
1.0
0.9
S6
0.8
S12
0.7
S24
S48
0.6
0
10
20
30
40
50
60
 (degrees)
70
80
90
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%
Opacities
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-FockSlater atomic modeling).
W: T=0.25 keV, =0.01 g/cc
2. ad hoc analytical,
3. inverse bremsstrahlung
(analytical),
.......
7. GLT tables (source opacities
from Novikov)
-1
1. power law,
Absorption coefficient k (cm )
Opacity options:
100
THERMOS data
8 -groups
32 -groups
10
1
0.1
1
Photon energy h (keV)
10
Laser absorption in RALEF-2D
Present model:
 An arbitrary number of monochromatic cylindrical laser
beams is allowed: for every beam the transfer equation is
solved by the same method of short characteristics as for
thermal radiation.
 No refraction, no reflection, the inverse bremsstrahlung
absorption coefficient, artificially enhanced absorption
beyond the critical surface.
Illustrative problems
Importance of radiation transport for laser plasmas
r
Sn foil: Δzfoil = 0.5 µm, Rfoil = 75 μm, Rmesh = 120 μm.
CO2 laser
Peak absorbed laser flux:
z
F00 = 1.41010 W/cm2
Sn foil
Temporal profile (normalized)
Ring-Gaussian spatial profile
(normalized)
1.0
ps(r)
pt(t)
1.0
0.5
0.0
0
20
40
60
t (ns)
80
100
120
30 m
0.5
0.0
-0.10
-0.05
0.00
r (mm)
0.05
0.10
No radiative transfer:
thermal conduction only
Radiation transport:
S8 with 8 spectral groups
Irradiation by a short (10-20ps) YAG laser pulse
Sn microsphere:
laser
RSn = 15 μm, Rmesh = 150 μm.
Laser energy: Elas-absorbed = 1 mJ
(reflection ignored)
z
Gaussian temporal profile (normalized)
Gaussian spatial profile (normalized)
1.0
ps(r)
pt(t)
1.0
20 ps
0.5
0.5
FWHM
-2
80 m at ps=e
0.0
0.0
0
10
20
30
t (ps)
40
50
60
-0.05
0.00
r (mm)
0.05
Sn microsphere irradiated by CO2-laser ( = 10.6 μm)
Sn microsphere:
laser
z
RSn = 15 μm, Rmesh = 150 μm.
Peak absorbed laser flux:
Imax = 6109 W/cm2
Gaussian temporal profile and Gaussian focal spot .
Spectral Sn opacities
100
In band: 2 -groups
Sn: T = 20 eV,
-4
 = 10 g/cc
10
-1
k (cm )
1
0.1
0.01
THERMOS
24 -groups
1E-3
0.01
0.1
h (keV)
1
Click to see the movie: Time is in nanoseconds
Evolution after the laser is turned off:
equilibrium EOS
The effect of metastable EOS
Fully equilibrium EOS
Metastable EOS
(Maxwell construction in the two-phase region)
(van der Waals loops in the two-phase region)
Summary
Our agenda for future work:
 laser deposition with refraction and reflection,
 hydrodynamics with metastable → stable phase transitions,
 material mixing,
 non-LTE radiation transport.