Basic principles of numerical radiation hydrodynamics and the RALEF code

transcript

Basic principles of numerical radiation

hydrodynamics and the RALEF code

M. M. Basko

ELI, Prague 10 July 2014

Keldysh Institute of Applied Mathematics, Moscow, Russia

Main constituents of the RALEF-2D package

1. Hydrodynamics

2. Thermal conduction

3. Radiation transport

4. EOS and opacities

5. Laser absorption

The RALEF-2D code has been developed with the primary goal to simulate high-temperature laser plasmas. Its principal constituent blocks are

Equations of hydrodynamics

, ( , )2

uu u p

E p u Qt

uE e e e T

– energy deposition by thermal conduction (local), – energy deposition by

radiation (non-local), – eventual external heat sources.

The RALEF-2D code is based on a one-fluid, one-temperature hydrodynamics model in two spatial dimensions (either x,y, or r,z):

),(or epp

ideal hydrodynamics

Selection of principal dependent variables

qupExt

Illustration with the 1D planar example:

Divergent form: ρ, ρu, ρE Non-divergent form: ρ, u, e

Naturally leads to a conservative numerical scheme

Easier to avoid unphysical values of the internal energy e and temperature T

Selection of principal independent variables

qupExt

Divergent Eulerian form: t, x

Non-divergent Lagrangian form: t, m

Eulerian simulation: spatial mesh

in x,y,z is fixed in time.

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 x,y,z mesh –

independent of the motion of the fluid!

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

0.0 0.5 1.0 1.5 2.0

block 5

block 4

block 3

block 2

block 1

block 10

block 9

block 8

block 7

block 6

0.0 0.5 1.0 1.5 2.0

block 1

block 2

block 3

block 4

block 5

block 6

block 7

-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

block 11

block 10

block 9

block 8

block 7

block 6

block 5

block 4

block 3

block 2

block 1

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

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.

Computational mesh: block structure

Every block can be subdivided into np1np2 topologically rectangular parts;

different parts can be composed of different materials.

0.0 0.5 1.0 1.5 2.0

part 9

part 7

part 5

part 3

part 1

part 10

part 8

part 6

part 4

part 2

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4-0.4

0.4Ncyc = 1 Ncyc = 100

RALEF: the ALE technique in action

Limitations of the ALE technique

Example: laser irradiation of a Cu foil

Cu foil

t = 1.6 nsfixed-pressure boundary

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

reflective (symmetry) boundary

free-outflow 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

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

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!

0 2 4 6 8 10

v [cm3/g]

4500 K

4000 K

3000 K

CP: TC = 4868 K, p

C = 5613 bar,

C = 0.66 g/cm3 An additional conceptual difficulty arises for a van-

der-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.

This latter problem has not been resolved yet !

Implicit and explicit numerical hydrodynamics

Numerical stability of explicit schemes requires the Courant–Friedrichs–Lewy condition (CFL condition) to be fulfilled

;vector;sionalmultidimenais;:Equation nn tUUUUF

Explicit scheme Implicit scheme

Implicit schemes are numerically stable and allow large time steps, but are difficult to implement.

which for certain problems may incur prohibitively long computing times.

1|| ucxt

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

cell i

vertex i

i, ui, Ei

No artificial viscosity is needed !

Principal variables:

uu E e

Equation of state: ( , )p p e― all assigned to the cell centers !

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.

-1 0 10.0

cell i+1

cell i

← 1st order

A fast non-iterative Riemann solver by J.K.Dukowicz (1985) is used.

A “snag”: node velocities uvi !

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

Thermal conduction

(a test bed for radiation transport)

, ( , )2

uu u p

E p u Qt

uE e e e T

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):

),(or epp

Implicit versus explicit algorithms for conduction

;,;:Equation inn

ix xtTTTLx

,...,,..., 11

ni TTTL

Explicit scheme Fully implicit scheme

An explicit scheme is stable under the condition , which practically always incurs prohibitively long computing times.

212 xt

,...,,..., 11

ni TTTL

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

,...,,..., 11

ni TTTL

RALEF is based on the SSI algorithm

for both the thermal conduction and

radiation transport.

The SSI method 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 ,

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 and radiation transport into the 2D Godunov method (in order to avoid costly matrix inversion required by fully implicit methods).

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:

1 2 , 1,1 1, ,2

, 1 2 , 1,1 1, ,2

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

Ts is a problem-specific sensitivity threshold for temperature variations.

Test problems: linear steady-state solutions

0 1000 2000 3000 40001E-10

t = 0.001

t = 0.0005

t = 0.002

0 = 0.2,

1 = 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 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).

( , )T x y x

Piecewise linear solutions with -jumps are reproduced exactly only with harmonic mean f,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

0.76 0.77 0.78 0.79

exact arithmetic

harmonic f, f0=0.01T

Parameters:

Initial condition:

(0, ) 0

( ,0) 1

Boundary condition:

Solution:0

( , ) ( ), ,/ 2

xT t x

Test results:

This solution cannot be simulated with the

harmonic-mean f , whereas excellent

results are obtained with the arithmetic-

mean f : the front position is reproduced

with an error of 0.1% for .

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

Square grid

Kershaw grid

Wavy grid

Random grid

Only the Kershaw grid

is strongly distorted in

the sense that some of

the coefficients (1)(1) 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), Shashkov et al. (1996), Breil & Maire (2007).

0.01 0.11E-6

square mesh wavy mesh random mesh Kershaw mesh CHIC - wavy CHIC - random CHIC - Kershaw

Radiation transport

, ( , )2

uu u p

E p u Qt

uE e e e T

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):

),(or epp

Radiation transport

Transfer equation for radiation intensity I in the quasi-static approximation (the

limit of c → ):

rQ d I d d k I B d

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 !

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:

, , , ,I k B I II

c tI t x B B T

In the present version, the absorption coefficient k and the source function B = B(T) are calculated in the LTE approximation.

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

( , ) .rQ t x k c B d

c t c c k c

E E E E

for the spectral radiation energy density ; the coupling term to the fluid is

( , , )t x

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 finite-

difference equivalent of the Gauss theorem is fulfilled,

k I B dV I n dS

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.

To the best of our knowledge, no such scheme is known in the open literature.

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.

Calculation of Iν: the method of short characteristics

, , , , ,L LI k B I I I t x

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 , the radiation field I is found by solving the transfer equation

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.

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):

In our example on the left we have

I n I dl

1 2 2 3 3 4

2 2 21

out in

H I I l I I l I I l

H I I l H

Conclusion: a numerical scheme based on nodal collocation points is generally non-conservative !

1 2 , 1,1 1, ,2r

ij ij ij i j i jW H H H H

Conservativeness can be restored by assigning the intensity I to cell faces (R.Ramis, 1992) ― equivalent to using the fluxes H as independent variables

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

In this limit we can apply the asymptotic expansion:

4div .

3rR i j k

I B B Ok k

BQ d k I B d B O

k x x x

34divdiv

Q B l Tk

In the quasi-static LTE case the diffusion limit is equivalent to the approximation of radiative thermal conduction:

k >> | ln T|, | ln |, | ln k |

Heating by two opposite beamlets in the diffusion limit

Consider two opposite beamlets propagating along an infinitely thin column h:

, ,dI dI

B I B I k ds k d xd d

( ) ( ) ... ,

( ) ( ) ...

dB d BI B

In the diffusion limit we have

Having combined the two opposite beams, we get

2 3 24

d B d B d IQ k I B d k I B k I B k O k

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.

Our numerical scheme

is not conservative (violates the finite-difference version of the Gauss theorem), and

has no strict convergence in the diffusion limit.

Failures

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

0.5 1.0 1.5 2.0 2.5 3.00.0

y=0.05

y=0.0125

x=0.1915

y=0.6940

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 0 1 0

, ( ) sin , 0 1;

2 ' ' 2 ;r

Ik B I B y y y

Q y k k B y E k y y dy B y

Exact solution:

0 1 2 3 16 17 18 19 200.0

vertical symmetry axis

Computational mesh:

Typical accuracy in problems with optically thin mesh cells is 12%.

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

exact S

, nx=n

k , random mesh

Diffusion limit in a slab

0.0 0.2 0.4 0.6 0.8 1.0-0.3

square grid random grid, 2010 random grid, 2013

Qr i /

, nx=n

y=20, = 104

= 8% -> 1.9%

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

T T Tr

t r r c k r

which admits an analytical self-similar solution with the front position

1/10 1

0 00 0

t Tr t r k k

0.0 0.5 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.

0.0 0.5 1.0 1.50.0

exact RALEF, n=50T

Thermal conduction: cylinder, xy-geometry

Final state: the conduction solution

By t = 0.002102926 the exact position of the front must be at 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.

0 1 2 3 40

block 1

block 2

0 10 20 30 40 50 60 70 80 90

(degrees)

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

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-Fock-Slater atomic modeling).

0.1 1 10

THERMOS data 8 -groups 32 -groups

Photon energy h (keV)

W: T=0.25 keV, =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 in RALEF-2D

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.

Present model:

Illustrative problems

Importance of radiation transport for laser plasmas

Peak absorbed laser flux: F00 = 1.41010 W/cm2

Sn foil: Δzfoil = 0.5 µm, Rfoil = 75 μm, Rmesh = 120 μm.

Temporal profile (normalized)Ring-Gaussian spatial profile

(normalized)

CO2 laser

0 20 40 60 80 100 1200.0

p t(t)

t (ns)

-0.10 -0.05 0.00 0.05 0.100.0

p s(r)

r (mm)

Sn foil

No radiative transfer:thermal conduction only

Radiation transport:

S8 with 8 spectral groups

Irradiation by a short (10-20ps) YAG laser pulse

Laser energy: Elas-absorbed = 1 mJ

(reflection ignored)

Sn microsphere: RSn = 15 μm, Rmesh = 150 μm.

Gaussian temporal profile (normalized) Gaussian spatial profile (normalized)

0 10 20 30 40 50 600.0

p t(t)

t (ps)

-0.05 0.00 0.050.0

p s(r)

r (mm)

80 m at ps=e-2

Sn microsphere irradiated by CO2-laser ( = 10.6 μm)

Sn microsphere: RSn = 15 μm, Rmesh = 150 μm.

Gaussian temporal profile and Gaussian focal spot .

Peak absorbed laser flux: Imax = 6109 W/cm2

Spectral Sn opacities

0.01 0.1 11E-3

THERMOS 24 -groups

h (keV)

Sn: T = 20 eV,

= 10-4 g/ccIn band: 2 -groups

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(Maxwell construction in the two-phase region)

Metastable EOS(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.

Basic principles of numerical radiation hydrodynamics and the RALEF code

Documents