An optimal, fully implicit, fully conservative, and ...

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An optimal, fully implicit, fully conservative, and equilibrium preserving Vlasov-Fokker-Planck

code for spherical ICF capsule implosions

William T. Taitano (T5/XCP6) Luis Chacón (T5 )

Andrei N. Simakov (XCP6 ) Brett Keenan (XCP6 ) T5: Applied Mathematics and Plasma Physics XCP6: Plasma Theory and Applications Group

March 2, 2017

CSE 2017, Atlanta, GA

Metropolis Postdoctoral Fellowship, Thermonuclear Burn Initiative, LDRD

LA-UR 17-21529

Got converted 2 days ago!!!

NOTE: This is the lab

color palette. Outline of talk

3/1/17 | 2 Los Alamos National Laboratory

March 2, 2017

10:50AM-11:10AM

Grand Ballroom 2nd floor MS222

•  Brief physics motivation •  Vlasov-Fokker-Planck

– Numerical challenges •  Length and time scales •  Discrete conservation and equilibrium

preserving properties – Our solution

•  Adaptive grid and implicit solver •  Discrete nonlinear constraints

•  Verification studies and preliminary simulation of imploding Omega capsule

•  Conclusion


color palette.


Progress since last CSE (2015)


color palette. Progress since last CSE (2015)


•  At last CSE meeting: – 0D2V grid adaptivity – Coarse grained asymptotics for disparate vth ratio – Conservation (mass, momentum, energy) for collision operator and Vlasov pieces

•  Updates: – Fluid electrons and electric fields – Spherical geometry and adaptivity in space – New equilibrium preserving discretization for the Rosenbluth-Fokker-Planck form – New null space preserving discretization for the geometrical inertial term – Suite of verification studies – Physics simulations

We have many updates. In fact so many that we cannot possibly cover all!


color palette. Main take away from this talk


•  Project began exactly 3 years ago (very high paced R&D) •  iFP is a first of a kind multi-scale simulation capability

– Fully implicit, scalable (both algorithmic and parallel) – Optimal grid adaptivity – Analytical equilibrium preserving property and other discrete null space preserving

properties – Strict conservation enforced – Strict verification campaign against hydro limit and other codes

•  Began ICF physics campaign simulation First capsule implosion simulation with hydro boundary conditions


color palette. How does ICF (indirect driver) work at NIF?


https://www.youtube.com/watch?v=Wg8R1lrAiM4


color palette.

Recent ICF experiments have highlighted wide gaps in our understanding and predictive simulation capabilities


•  Contrary to radhydro simulation predictions, NIF’s NIC and consecutive campaigns have failed to achieve ignition

•  Recent OMEGA campaigns have highlighted serious deficiencies in our

ability to – Predict capsule compression and yield (both are over-predicted) – Predict time-dependent core mix (especially when hydro instabilities are not

expected to play a role)


color palette.

Hydrodynamics breaks down in certain regimes of ICF capsule implosion and a kinetic model is required

3/1/17 | 8

Hydro Kinetic

YO

C⌘

Yie

ldexperim

ental

Yie

ldhydro,no

mix

Figure from M.J. Rosenberg, PRL (2014)

Discrepancies btw experiment and simulations consistently increase with Knudsen number. Radiation-hydrodynamics is not valid in many key stages in ICF implosion!

Los Alamos National Laboratory


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We need high fidelity Vlasov-Fokker-Planck code to understand impact of missing physics in hydro


•  Kinetic ions: Vlasov-Fokker-Planck is considered a first principles model for weakly coupled plasmas

DfiDt

⌘ @fi@t

+ ~v ·rfi + ~ai ·rvfi =X

j

Cij (fi, fj)

•  Fluid model for electrons

•  Quasi-neutrality and ambipolarity to close system

ne = �q�1e

NsX

↵

q↵n↵

r · ~J = r ·"qene~ue +

NsX

↵

q↵n↵u↵

#= 0

~a =q↵m↵

~E =q↵m↵

rPe +PNs

↵~Fe↵

qene

3

2@t (neTe) +

5

2r · (~ueneTe) + ~ue ·rPe +r · ~Qe =

NsX

↵

hWe↵ + ~ue · ~Fe↵

i


color palette.

Solving Vlasov-Fokker-Planck requires many algorithmic innovations


DfiDt

⌘ @fi@t

+ ~v ·rfi + ~ai ·rvfi =X

j

Cij (fi, fj)

Cij (fi, fj) = �ijrv · [ Dj ·rvfi �mi

mjAjfi ]

r2vHj (~v) = �8⇡fj (~v)

r2vGj (~v) = Hj (~v)

Dj = rvrvGj Aj = rvHj

•  A nonlinear integral-differential equation •  Supports conservation of mass, momentum, and energy and positivity (numerically, these

are constraints that must be ensured for long time accuracy and stability) •  Supports disparate length and time scales •  Supports a non-trivial null space (Maxwellian)


color palette.


Challenges of the problem


color palette. Mesh resolution challenges of VFP


v

x

Under-resolved

cold distribution

Resolved hot

distribution

vth,hot

vth,cold

• Disparate temperatures during implosion dictate velocity resolution. – vth,max determines Lv

– vth,min determines Δv

•  Shock width and capsule size dictate physical space resolution

Cold speciesessentially a deltafunction on hotspecies' mesh

Hot

Cold

Taitano et al., JCP, 318, 2016


color palette. Static mesh with explicit methods, not practical


•  Intra species vth,max /vth,min~100 •  Inter species (vth,α /vth,β)max~30 • Nv~ [10(vth,max/vth,min)x(vth,α /vth,β)]2 ~109 • Nr ~ 103-104 • N=NrNv~1012-1013 unknowns in 1D2V!

•  tsim=1 ns • Nt>=109 time steps

3 Numerical Challenges and Algorithms 3

becomes quite restrictive. In a typical shell implosion, the radius decreases from a mm to a few tensof µm, about 10-30 times. Assuming the lower limit, a static mesh would have to provide su�cientresolution at the final point of compression, say 50-100 points. This, in turn, would require aninitial uniform static radial mesh of:

Nr

⇠ 103.

Temporal resolution. Challenges in the temporal integration of VFP stem from the two mainphenomena at play: streaming and collisions. These have to be compared against the total simu-lation time, i.e., the implosion time. Assuming an initial radius R of 1mm, and a typical implosionvelocity v

i

of 20 cm/µs, we find:

⌧i

⇡ R

vi

⇠ 5 ns.

The temporal stability limit for integrating collisions with an explicit time integration scheme isgiven by the corresponding Courant condition:

�tcoll

exp

⇠ �v2

D⇠ �v2

v2

th

⌫coll

.

We take typical conditions at the end of the compression phase for density and temperature,

ni

⇡ 200 g/cm3 ⇡ 5⇥ 1025 cm�3 , Ti

⇡ 2.5 keV.

For these conditions, we have vth

⇡ 10vmin

th

, and the ion collision frequency is:

⌫coll

= 4.8⇥ 10�8

✓

mp

mi

◆

1/2 ni

(cc�1) ln⇤T

i

(eV )3/2

s�1 ⇡ 105 ns�1,

where we have assumed ln ⇤ ⇠ 10. It follows that:

�tcoll

exp

⇠ 110

✓

�v

vmin

th

◆

2

⌫�1

coll

⇠ 10�9 ns,

where we have used Eq. 1. Time steps for the early compression phase will be orders of magnitudemore restrictive even though the plasma is less dense (n

i

⇠ 0.225 g/cm3), because it is significantlycolder (T

i

⇠ 1 eV). This result implies that an explicit scheme, for the conditions and velocity-spaceresolution stated, will require > 109 � 1010 time steps to reach the implosion time scale.

Regarding streaming, the time step explicit stability limits of advection in physical and velocityspace are comparable, and are given by the corresponding CFL constraint for the maximum velocityunder consideration in our discretization, v

max

⇠ 10vmax

th

:

�tstrexp

⇠ �r

vmax

⇠ R

Nr

10vmax

th

.

For Tmax

⇡ 100 keV , we have vmax

th

⇠ 3 ⇥ 108 cm/s. Therefore, for R ⇡ 0.1 cm and Nr

⇠ 103, wefind:

�tstrexp

⇠ 10�4 � 10�5 ns.

Thus, we find streaming to be significantly less restrictive numerically than collisions.


color palette. Conservation is also critical


No conservation With conservation



color palette. Equilibrium preservation is also critical


t0 0.05 0.1 0.15 0.2

T

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3TH,16x8TD,16x8

t0 0.05 0.1 0.15 0.2

T

1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

2.2

TH,32x16TH,64x32TH,128x64TD,32x16TD,64x32TD,128x64

With equilibrium preservation Without equilibrium preservation

Taitano et al., JCP, 2017, under review


color palette.


Our solution: Adaptive grid Implicit solver

Exact discrete conservation


color palette.

Adaptive mesh and implicit method makes problem tractable


• v-space adaptivity with vth normalization, , Nv~104-105

• Lagrangian mesh in physical space, Nr~102

• N=NvNr~106~107 (vs. 1012 with static mesh)

• Multigrid preconditioned optimal nonlinear implicit solver [Chacon et al., JCP, 157 (2000)], Δtimp=Δtstr~10-3 ns

• Nt~103-104 (vs. 1010 with explicit methods)

bv = v/vth


color palette.

vth adaptivity in velocity space allows optimal mesh resolution throughout domain


v

x

Resolved cold

distribution

Resolved hot

distribution

dvth/dt > 0

dvth/dt < 0

v

x

Under-resolved

cold distribution

Resolved hot

distribution

vth,hot

vth,cold

Static Mesh vth adaptive Mesh


color palette.

Moving mesh in physical space: Implode mesh with capsule to track shock



color palette. To have grid adaptivity, we transform the equation


1D spherical (with logical mesh); 2D cylindrical geometry in velocity space 624 CHACON ET AL.

FIG. 1. Diagram of the local cylindrical velocity coordinate system (vr , vp) considered in this work. Cylin-drical symmetry is assumed. The spherical radius vector r is included for reference.

by-product, the required numerical representation of the subordinate problems for H andG, which will be somewhat different from the traditional treatment.

3.1. Definition of the Computational Velocity Domain

So far, the discussionhas been independent of a particular geometry and/or dimensionalityin velocity space. To focus the discussion that follows, a 2D cylindrical velocity space withangular symmetry is adopted. This space is spanned by (vr , vp), where vr is the cylindricalz-axis, and vp is the cylindrical r -axis (Fig. 1), and vr ∈ [0, vlimit]; vp ∈ [0, vlimit]. Here, vlimitis typically set to several times the characteristics velocity of the problem, v0.The domain is discretized with an integer mesh and a half mesh (Fig. 2). The integer

mesh is defined using Nr (+1) nodes in the vr axis, and Np(+1) nodes in the vp axis, withthe constraints

vr,1 = 0, vr,Nr = vlimit

vp,1 = 0, vp,Np = vlimit.

Each velocity node is characterized by a pair (vr,i , vp, j ), with i = 1, . . . , Nr (+1), andj = 1, . . . , Np(+1). The additional (i = Nr + 1, j) and (i, j = Np + 1) nodes at the bound-aries will serve a double purpose: (1) they will be used to impose the far-field boundaryconditions for the Rosenbluth potentials, and (2) they will allow an accurate determina-tion of the friction and diffusion coefficients of the Fokker–Planck collision operator at the

FIG. 2. Diagram of the 9-point stencil in velocity space employed in the discretization of the Fokker–Planckcollision operator.

Coordinate transformation:

Jacobian of transformation:

Always fixed

Ẋ = dr/dt (mesh speed)

rt,0

rt,1

ξ

bv|| ⌘~v ·~brvth,↵

, bv? ⌘

qv2 � v2||

vth,↵

pgv (t, r, bv?) ⌘ v3th,↵ (t, r) r2bv?


color palette.

Adaptivity introduces inertial terms in the conservation equation


•  VRFP equation in transformed coordinates

bv|| = �

bv||2

⇣v

�2th,↵@tv

2th,↵ + J

�1r⇠

⇣bv|| � b

x

⌘v

�1th,↵@⇠v

2th,↵

⌘+

bv2?vth,↵r

+q↵E||

Jr⇠m↵vth,↵

bv? = �bv?

2

⇣v

�2th,↵@tv

2th,↵ + J

�1r⇠

⇣bv|| � b

x

⌘v

�1th,↵@⇠v

2th,↵

⌘�

bv||bv?vth,↵r

Taitano et al., JCP, 318, 2016 Taitano et al., JCP, 2017, in preparation Taitano et al., JCP, 2018, in preparation

Inertial terms due to vth adaptivity and Lagrangian mesh


color palette.

vth adaptivity provides an enabling capability to simulate ICF plasmas


• D-e-α, 3 species thermalization problem

• Resolution with static grid:

• Resolution with adaptivity and asymptotics:

• Mesh savings of

Nv ⇠ 2

✓vth,e,1vth,D,0

◆2

= 140000⇥ 70000

Nv = 128⇥ 64

⇠ 106Taitano et al., JCP, 318, 2016


color palette.





color palette.

Implicit solver strategy: Preconditioned Anderson Acceleration


• We drive the nonlinear residual to zero

R = @tf + V E (f)�

⇥D ·rvf �Af

⇤= 0

fk+1 =mkX

i=0

↵(k)i G (fk�mk+i)

G (fk) = fk � P�1k Rk = fk+1

Pk = Jk

•  Consider a fixed point map of form:

•  If , Newton’s method •  Anderson updates the solution by using history

(nonlinear) of solutions to accelerate convergence via:


color palette. Two stage operator splitting in PC operator


Step 1: Velocity space operators (including collisions)

Step 2: Streaming operator

Pv = @t �+V Ev � �hD ·rv � � ~A�

i

Px

= @t

+ v@x

�Preconditioner is a convergence accelerator! No splitting error will be present in the actual solution (driven by the nonlinear residual)

P�1R = P�1x

P�1v

R


color palette. Preconditioner is effective in reducing iteration


∆v0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Feva

l [#]

100

101

102

103PC ∆t=2.4×10-2τstiffPC ∆t=2.4×100τstiffno PC ∆t=2.4×10-2τstiffno PC ∆t=2.4×100τstiff

∆t/∆texp=12200

∆t/∆texp=2860

∆t/∆texp=625

∆t/∆texp=122 ∆t/∆texp=28.6 ∆t/∆texp=6.25


color palette. Implicit solver performance is optimal


Nv104 105

CPU

[sec

]

104

106

108

ImplicitprecO(Nv)ExplicitO(N2

v)4 orders of magnitude more efficient than explicit methods

Solver CPU time versus size of unknown


color palette. The code is scalable


# procs0 500 1000 1500

CPU

tim

e×105

0

0.5

1

1.5

2

2.5

3 Weak scalability testCPU timeideal


color palette.





color palette.

Rosenbluth-FP collision operator: conservation properties results from symmetries


m↵

�⌦v2, C↵�

↵~v

= �m�

�⌦v2, C�↵

↵~v

D~v, ~J�↵,G � ~J↵�,H

E

~v= 0)

Energy

h1, C↵�i~v = 0 )Mass

~J↵�,G � ~J↵�,H��~@v

= 0

and

br2

v,↵

"

bG��

✓

vth,�vth,↵

◆

4

#

| {z }

bG↵�

= bH↵� ,

which gives:

bG↵� = bG��

✓

vth,�vth,↵

◆

4

. (2.5.8)

Introducing these terms in the collision operator, and noting that the normalized collision operator is:

bC↵� = v3th,↵C↵� , (2.5.9)

we find:

bC↵� = v3th,↵�↵�

v3th,↵

1

vth,↵brv ·

"

1

v2th,↵brvbrv

vth,�

✓

vth,↵vth,�

◆

4

bG↵�

!

· 1

vth,↵brvbf↵ � m↵

m�

bf↵brv

vth,↵

bH↵�

vth,�

✓

vth,↵vth,�

◆

2

!#

,

which gives,

bC↵� =�↵�

v3th,�brv ·

brvbrvbG↵� · brv

bf↵ � m↵

m�

bf↵ brvbH↵�

�

. (2.5.10)

The above expression is obtained with: br2

vbH↵� = �8⇡ bf�

⇣

bv� = bv↵vth,↵

vth,�

⌘

or bH↵� = bH��

⇣

vth,�

vth,↵

⌘

2

, we obtain

br2

vbG↵� = bH↵� or bG↵� = bG��

⇣

vth,�

vth,↵

⌘

4

.

2.6 Treatment of Rosenbluth Potentials between Cross SpeciesVelocity Space

2.7 Conservative Discretization

We develop a mass, momentum, and energy conserving discretization for the Fokker-Planck operator.

2.7.1 Discrete Mass Conservation Scheme

First, recall the continuum mass conservation statement,

h1, C↵�iv =D

1,rv ·h

~JG,↵� + ~JH,↵�

iE

v=

h

1, ~JG,↵� + ~JH,↵�

i

@v�⇠⇠⇠⇠⇠⇠⇠⇠⇠⇠D

0, ~JG,↵� + ~JH,↵�

E

v. (2.7.1)

Here, the surface integral requires the evaluation of the collisional fluxes at the velocity boundaries to bezero. In the discrete, we do the exactly identical treatment by numerically setting the fluxes to zero at theboundary.

2.7.2 Discrete Momentum Conservation Scheme

For momentum conservation, the following relation must hold:

m↵ h~v, C↵�i~v = �m� h~v, C�↵i~v . (2.7.2)

8

)Momentum

D1, Jk

↵�,G � Jk�↵,H

E

~v= 0

C↵� = �↵�rv ·h~J↵�,G � m↵

m�

~J↵�,Hi


color palette.

2V Rosenbluth-FP collision operator: numerical conservation of energy


•  The symmetry to enforce is:

•  Due to discretization error:

•  Introduce a constraint coefficient:

D~v, ~J�↵,G � ~J↵�,H

E

~v= 0

D~v, ~J�↵,G � ~J↵�,H

E

~v= O (�v)

D~v, ��↵ ~J�↵,G � ~J↵�,H

E

~v= 0 ��↵ =

D~v, ~J↵�,H

E

~vD~v, ~J�↵,G

E

~v

= 1 +O (�v)

C↵� = �↵�rv ·�↵� ~J↵�,G � m↵

m�

~J↵�,H

�



color palette.

Single-species initial random distribution thermalizes to a Maxwellian and HOLDS IT




color palette.

Conservation properties enforced down to nonlinear convergence tolerance


time0 0.01 0.02 0.03 0.04 0.05

∆M

10-16

10-14

10-12

10-10

10-8

10-6

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 0.01 0.02 0.03 0.04 0.05

∆I

10-12

10-10

10-8

10-6

10-4

10-2

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 0.01 0.02 0.03 0.04 0.05

∆U

10-15

10-10

10-5

100

ϵr=10-2

ϵr=10-4

ϵr=10-6

time0 0.01 0.02 0.03 0.04 0.05

∆S

10-2

10-1

100

101

ϵr=10-2

ϵr=10-4

ϵr=10-6



color palette. The end product is a reliable code


•  Comparison between a reference iFP and fluid simulation for a two ion species M=1.5 (weak) shock problem


color palette.


First capsule implosion simulation


color palette. First implosion calculation


• D-He3 fill Omega capsule

simulation with hydro boundary for fuel [O. Larroche, PoP, 2012, collaborator]

• Studied to investigate Rygg effect [J. R. Rygg et al. PoP, 2006]

•  FPion was first used to investigate fuel stratification to explain reactivity drop by Rygg et al.


color palette.

Simulation observes fuel stratification. We might be able to explain experiment (current effort)



color palette.

Kn reveals that D mean free path is on order capsule size for an appreciable time post shock convergence


Kn =�mfp

Rcapsule


color palette. Main take away from this talk


•  Project began exactly 3 years ago (very high paced R&D)

•  iFP is a first of a kind multi-scale simulation capability – Fully implicit, scalable (both algorithmic and parallel) – Optimal grid adaptivity – Analytical equilibrium preserving and other discrete null space preserving properties – Strict conservation enforced – Strict verification campaign against hydro limit and other codes

•  Began ICF physics campaign simulation First capsule implosion simulation with hydro boundary conditions


color palette. Future Work


•  Capsule implosion with self consistent pusher species included •  Investigate more ICF relevant physics

– Rygg (inverse) effect – Kinetically enhanced pusher mix into fuel – Kinetic effects on fuel convergence reduction

•  Implementation of neutron and thermal radiation transport packages to investigate multi-physics aspects of ICF implosion


color palette. Acknowledgements


• Mentors : Andrei N. Simakov, Luis Chacón

• Team iFP : William T. Taitano , Luis Chacón , Andrei N. Simakov, Brett Keenan

• Sponsors : Metropolis PD Fellowship, TBI , LDRD

• Collaborators : CEA, MIT, LLNL , Kinetic Effects Group at LANL


color palette. Q & A


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