18 th International Conference on Numerical Simulation of Plasmas Cape Cod, Massachusetts

The Heavy Ion Fusion Virtual National Laboratory

Asymmetric PML for the Absorption of Waves.Application to Mesh Refinement in Electromagnetic

Particle-In-Cell Plasma Simulations.

18th International Conference on Numerical Simulation of Plasmas

Cape Cod, Massachusetts

September 8, 2003

J.-L. Vay Lawrence Berkeley National Laboratory, California, USA

J.-C. Adam, A. Héron CPHT, Ecole Polytechnique, France


Motivation

• Study of laser-plasma interaction in context of fast ignition involves plasma density far greater than critical density Imposes very strict conditions on mesh size and time steps

• Following the system on experimentally realistic space and time scales implies large domains I.e. boundary conditions are sufficiently remote so that they do not

contaminate the physics inside the target• A regular grid results in a lot of wasted resources in modeling large areas of

vacuum or low density plasma Mesh refinement allows finer gridding of localized area but is challenging

for electromagnetic PIC: need efficient absorbing mechanism at patch boundary

• We present a new Perfectly Matched Layer for the absorption of waves which gives very high absorption rate

• A new mesh refinement strategy which takes advantage of this new PML is introduced and tested on a laser-plasma interaction example in the context of the fast ignitor


Asymmetric Perfectly Matched Layer (APML)

yEH

tH

xE

HtH

xHE

tE

yHE

tE

xzyy

zy

yzxx

zx

zyx

y

zxy

x

*0

*0

0

0

xyxy

zyyzy

yxyx

zxxzx

zxzx

yxy

zyzy

xyx

EyE

cc

HtH

ExE

ccH

tH

HxH

ccE

tE

HyH

cc

EtE

**

*0

**

*0

0

0

Berenger PML Asymmetric PML (APML)

yE

xE

tH

xH

tE

yH

tE

xyz

zy

zx

0

0

0

Maxwell

If with u=(x,y)

=> Z=Z0: no reflection.0

*u

0

u

If and

=> Z=Z0: no reflection.

0

*

00

*

0

,

uuuu *

uu cc

Principle of PML:Field vanishes in layer surrounding domain. Layer medium impedance Z matches vacuum’s Z0

The APML introduces some asymmetry in absorption rate.

Absorption rates strictly equals for PML and APML at infinitesimal limit.

However, absorption rates discretized algorithms differ.


Plane wave analysis PML versus APML

0.0 0.5 1.0 1.5 2.01E-10

1E-91E-81E-71E-61E-51E-41E-30.01

0.11

=4 =8 =16 =32 =64 =128 =256 =512 =1024C

oeffi

cien

t of r

efle

ctio

n

Angle of incidence (rad)

0.0 0.5 1.0 1.5 2.01E-101E-91E-81E-71E-61E-51E-41E-30.010.1

1 =4 =8 =16 =32 =64 =128 =256 =512 =1024C

oeffi

cien

t of r

efle

ctio

n


0.0 0.5 1.0 1.5 2.01E-101E-91E-81E-71E-61E-51E-41E-30.010.1

1 =4 =8 =16 =32 =64 =128 =256 =512 =1024C

oeffi

cien

t of r

efle

ctio

n


0.0 0.5 1.0 1.5 2.01E-101E-91E-81E-71E-61E-51E-41E-30.01

0.11

=4 =8 =16 =32 =64 =128 =256 =512 =1024C

oeffi

cien

t of r

efle

ctio

n


Standard PML PML-matched coefficients

APML-Hybrid ([3]) APML-LWA ([1])


Plane wave analysis PML versus APML for =2/~20x/c

• Best tested APML implementation overall better than best tested PML implementation

0.0 0.5 1.0 1.51E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1 (Th.) (Num. Exp.) PML (Th.) (Num. Exp.) PML- adjusted (Th.) (Num. Exp.) APML-hybrid (Th.) (Num. Exp.) APML-LWA

Coe

ffici

ent o

f ref

lect

ion


(for more on this, see [1])


Mesh refinement

R2

G

A

R1

• most mesh refinement rely on algorithm ‘sewing’ grids at boundary

• an algorithm is applied at the patch boundary to connect the patch and the main grid solution • several solutions have been proposed, using finite-volume, centered finite-difference with ‘jumps’ inside fine grid to get to relevant data, energy conserving schemes, apply different formula depending on direction of wave [2], …• as can be shown on simple 1-D example (see next slide), most produce reflection of waves for wavelengths below coarse grid cutoff, eventually with amplification => instability


Tests of various mesh refinement schemes in 1-D

10 1001E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

10 'jump' (n=3) finite volume (n=3) 'directional' (n=3)

R

2c/xfine grid (xcoarse grid=nxfine grid)10 100

1E-5

1E-4

1E-3

0.01

0.1

1

10 'jump' space-time (n=3) energy conserving (n=2) 'directional' (n=2)

R

2c/xfine grid (xcoarse grid=nxfine grid)

Space only Space+Time

x o x o x o x o x oj-5/2 j-2 j-3/2 j-1 j-1/2 j j+1/2 j+1 j+3/2 j+5/2

x1 x2n.x1

Boundary

grid 1 grid 2

o: E[+],E[-]

x: B[+],B[-]

o: E, x:B

(for more on this, see [2])


We propose an alternative method by substitution

R1

Absorbing BCs

R1P1

R2P2

G

Outside patch:F = F(G)

Inside patch:F = F(G)-F(P1)+F(P2)

A

• normal PIC in main grid G at resolution R1• in area A

• patch P1 at res. R1• patch P2 at res. R2both terminated by APML• linear charge deposition on P2 and propagated on P1 and G• when gathering force, force at low resolution R1 is substituted by force at higher resolution R2 on patch P2


Particle entering and leaving patch

• Ideally, the field associated with a macroparticle entering/leaving a patch should (magically) appear/vanish

• Since this may be challenging, we have opted for an operationally simple procedure– The current of a macroparticle is deposited inside a patch as soon

as it enters it and stops being deposited when it leaves it• This implies the creation of a macroparticle of opposite sign at the

entrance location and a macroparticle of same sign at the exit location• With the substitution operation F(G)-F(P1)+F(P2) inside the patch, the

contribution due to these standing charge should cancel out• Because this cancellation is not exact (two different resolutions), a

residual spurious standing field appears. • Since it is expected that this field will vanish rapidly inside the patch,

we define a band on the border of the patch in which we do not perform the substitution


The Particle-In-Cell code used for testing: EMI2D

• PIC electromagnetic 2D, linear or cubic splines, Esirkepov current deposition scheme (similar to Vilasenor-Buneman algorithm but extend to high-order splines)

• Boundary conditions: open system– particles

- ions leave the box freely- electrons reflected until an ion exit (overall charge conserved)

– EM fields: APML absorbing layer + incoming wave


• A laser impinges on a cylindrical target which density is far greater than the critical density (context of fast ignition [4])

• The center of the plasma is artificially cooled to simulate a cold high-density core

• Two cases are tested:1. Patch boundary in

plasma2. Patch boundary

surrounds plasma

Test: laser interaction with cylindrical target

coreLaser beam

=1m,1020W.cm-2

(Posc/mec~8,83)

2=28/k0

10nc, 10keV

Patch:Case 1Case 2

•The first case is expected to be especially hard on the method since we anticipate that many electrons will cross the patch boundary.


X-Y particle-density plots for ions and electrons

Case 1 Case 2

Case 1 Case 2

Very similar.See patch boundary in case 1.

Very similar.See patch boundary in case 1.


X-Vx particle plots for ions and electrons

Case 1 Case 2

Case 1 Case 2

Very similar

Very similarBackground T° higher in case 1


Y-Vy particle plots for ions and electrons

Case 1 Case 2

Case 1 Case 2

Very similar

Very similarBackground T° higher in case 1


Vx-Vy particle plots for ions and electrons

Case 1 Case 2

Case 1 Case 2

Very similar

Very similar


Bz main grid

Case 1 Case 2

In case 2, the electrons see the Laser light from G and its plasma response from P2. They have the same frequency but different wavelengths due to different numerical dispersion on G and P2. This gives a spurious residual low amplitude wave.


Bz patch P1

Case 1 Case 2

In case 2, the zone which absorbs the laser light is in the patch. The plasma response to the laser is clearly recognizable.


Bz patch P2

Case 1 Case 2

In case 2, the zone which absorbs the laser light is in the patch. The plasma response to the laser is clearly recognizable.


Ex main grid

Case 1 Case 2

Very similar


Ex Patch P1

Case 1 Case 2

In both cases, the accumulation of charge due to macroparticles entering or leaving the effective area of the patch is evident.


Ex Patch P2

Case 1 Case 2



Ey main grid

Case 1 Case 2

In case 2, the electrons see the Laser light from G and its plasma response from P2. They have the same frequency but different wavelengths due to different numerical dispersion on G and P2. This gives a spurious residual low amplitude wave.


Ey Patch P1

Case 1 Case 2



Ey patch P2

Case 1 Case 2



Discussion

• The results from the performed test appear very promising since the main features of the physical processes were retained and no instability has been observed.

• We note, however, the presence of two spurious effects1. when the laser-plasma interaction occurs inside the refined area,

different numerical dispersion in the refined patch and the main grid accounts for a spurious, although low intensity, laser trace in the plasma, due to inexact cancellation of the incident laser and the plasma response,

2. when the patch lies inside the plasma, its boundary is visible as a low density line in the plasma density plots for both species. Several explanations for this effect are being considered: spurious field from remaining charges at boundaries, different cutoffs in plasma frequency on G and P2, a bug,…

• Despite these spurious effects, we note that the phase-space projections look very similar, indicating that the macroparticle trajectories were largely unaffected.


Conclusion

• A New Asymmetric PML was introduced and higher absorption rates were obtained compared with standard PML.

• Taking advantage of these high absorption rates, a new strategy for coupling the mesh refinement technique to electromagnetic Particle-In-Cell simulations was devised.

• A first test exhibited spurious effects which, nonetheless, did not affect significantly the main physical aspects.

• A more profound analysis of the issues will be performed in order to unequivocally identify the source of the spurious effects and remedies will be explored.

• Based on our present understanding, these may involve– use of higher-order (less dispersive) Maxwell solver,– add Gauss corrector in patch (Boris, Marder or hyperbolic) to remove

standing charges due to macroparticle entering or leaving patch,– devise more elaborate procedure for particle entrance and exit of patch

which lead to reduction in magnitude, or even inexistence, of standing charges.


References

1. J.-L. Vay, “Asymmetric Perfectly Matched Layer for the Absorption of Waves” ”, J. Comp. Physics 183, 367-399 (2002)

2. J.-L. Vay, “An extended FDTD scheme for the wave equation. Application to multiscale electromagnetic simulation”, J. Comp. Physics 167, 72-98 (2001)

3. J.-L. Vay, “A new absorbing layer boundary condition for the wave equation”, J. Comp. Physics 165, 511-521 (2000)

4. M. Tabak et al., “Ignition and high gain with ultrapowerful lasers”, P. of Plasmas, Vol. 1, Issue 5, 1626-1634 (1994)

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18 th International Conference on Numerical Simulation of Plasmas Cape Cod, Massachusetts

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