LLNL-PRES-821466This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC.

An Overview of Numerical Radiation Transport Techniques in Asteroid Deflection Modeling

Planetary Defense Conference 2021

Nicholas A. Gentile

April 28, 2021Vienna, Austria

P. O. Box 808, Livermore, CA 94551

Lawrence Livermore National Laboratory LLNL-PRES-8214662 / 13

§ Inertial confinement fusion (ICF) and asteroid simulations share some commonalities and challenges• Both have large length, density, and opacity scales

§ ICF codes discretize in space and time— Zones have r, T, P, radiation intensity, etc.— Energy deposition done by a radiation transport method— Hydrodynamic motion and shocks handled by a hydro method— Radiation and matter are coupled by thermal emission from material and electron

and ion conduction

§ Transport dominates the simulation run time• We face tradeoffs in the radiation methods between speed and accuracy

Photon transport is necessary to accurately model deflection scenarios using x-ray deposition

• Both model round things suspended in vacuum hit by x-rays

• We can use the same code for both

Lawrence Livermore National Laboratory LLNL-PRES-8214663 / 13

The transport equation describes the motion of photons including interactions with moving matter

§ This is the Boltzmann equation written in terms of Intensity • I has units of Energy/(Length2-Time-Steradian)

§ Material motion corrections (MMC) need to be included• Emission isn’t isotropic, absorption is angle dependent• There are many O(v/c) MMC approximations; also many numerical

simplifications are employed, some inaccurate

§ Radiation exchanges energy and momentum with matter

1

c

@I(⌫,⌦)

@t+ ⌦ ·rI(⌫,⌦) =� �D�aI(⌫,⌦) + �a

B[⌫, T ]

[�D(⌦)]2

+

Z 1

0d⌫0

Z

4⇡d⌦0 ⌫

⌫0�s(⌫

0 ! ⌫,⌦0 ! ⌦)I(⌫0,⌦0)

1 +

c2I(⌫,⌦)

2h⌫3

�

�Z 1

0d⌫0

Z

4⇡d⌦0�s(⌫ ! ⌫0,⌦ ! ⌦0)I(⌫,⌦)

1 +

c2I(⌫0,⌦0)

2h⌫03

�

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D ⌘ 1� ⌦ · vc

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� ⌘✓1� v2

c2

◆� 12

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B(⌫, T ) =2h⌫3

c2

exp

✓h⌫

kT

◆� 1

��1

<latexit sha1_base64="glOAY0XrCAgtCXXMP0HsImbCG50=">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</latexit>

Thermal emissionDoppler-shifted

In-scatteringPhotons move in straight lines at speed c

Out-scattering

Stimulated scattering

Absorption (Doppler-shifted)

with and arising from material motion and the Planck function describing thermal emission in Local Thermodynamic Equilibrium

Lawrence Livermore National Laboratory LLNL-PRES-8214664 / 13

The two common numerical methods for transport simulations are IMC and SN§ Implicit Monte Carlo (IMC) simulates radiation by computational particles with

randomly selected emission positions and directions• Emit, scatter, track, and absorb “fake” photons• “implicit” refers to a numerical extrapolation in time of the matter temperature used in emission• Allows accurate simulation of scattering and Doppler shifts

— Energy-angle correlation in Compton scattering can be simulated• Use of random numbers causes statistical noise ~ Nparticles

-1/2 in the results— Reducing the slowly-declining noise leads to long simulation times— Discretization errors in thermal emission, both temperature and emission location, require small Dx and Dt— Stimulated Compton is approximated or ignored

§ SN or Discrete Ordinates represents I at fixed angles using finite element basis functions in each zone• The discrete angles are selected to enable Gauss integration of spherical harmonics• Faster than IMC (>10x in opaque problems)• Fully implicit in emission temperature; smaller spatial discretization error• Can simulate stimulated Compton• The use of discrete angles makes anisotropic scattering approximate and can lead to simulation

artifacts

Lawrence Livermore National Laboratory LLNL-PRES-8214665 / 13

Computational artifacts of IMC and SN

Simulation with an isotropic point source in an absorbing non-scattering medium

§ IMC simulation (top) shows statistical noise

§ SN simulation shows ray effects

§ IMC simulation of radiation flux in an illuminated asteroid shows statistical noise

§ The electron and ion conduction flux also shows noise, seeded by the IMC through its effect on the electron temperature

⇥10

8erg

cm2s

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Lawrence Livermore National Laboratory LLNL-PRES-8214666 / 13

Flux-limited Diffusion is a quick but very approximate transport simulation technique§ Averaging the transport equation over angle plus an ansatz for the flux

results in diffusion equation

Here is the radiation energy density (Energy/Length3) and

§ Diffusion can’t model angular information – no shadows§ Diffusion is accurate when radiation is isotropic AND gradients in E are

small• Ad hoc flux limiter L in [0,1] needed to suppress superluminal energy flow (F > c DE)

when s is small

§ For heat conduction in electrons and ions, which typically have small flux, a similar diffusion approximation is accurate

@E

@t+r · [�LF ] +

4

3r · (Ev) +

1

3v ·rE = c�aT 4 � c�aE

<latexit sha1_base64="eDnEq3vnkx2bbRbRH87xiOf0CRk=">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</latexit>

Material motion correction terms

is the radiation flux (Energy/Length2-Time) • This expression for F is an approximation

F = � c

3(�a + �s)rE

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E = �1

c

Z

4⇡d⌦I

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Lawrence Livermore National Laboratory LLNL-PRES-8214667 / 13

The Multigroup approximation is used to express frequency dependence of s and I (or E)§ We pick O(10)-(100) fixed values of n; each range is called a “group”

• Group bounds are constant in time and space in a simulation• We solve one transport or diffusion equation per group• Scattering and absorption-reemission couple the groups and the per-group

equations— This requires iteration in SN and FLD

• Opacities are constant in each group during a time step

• Recalculated in each group at the beginning of the time step to account for changes in rand T

Lines can be represented with enough groups

Compton scattering > absorption at high n

Lawrence Livermore National Laboratory LLNL-PRES-8214668 / 13

We are using IMC in our deflection calculations because they contain vacuum and point sources

§ Diffusion has poor accuracy in vacuum• It also can’t simulate the directionality of a point source

§ SN suffers from ray effects in vacuum• Can’t accurately model strongly peaked scattering like

Compton

§ IMC can simulate point and ray sources• We have to incur and mitigate the drawbacks: — Statistical noise— Long runtimes— Use lots of zones and time steps

We must use lots of particles and processors

Lawrence Livermore National Laboratory LLNL-PRES-8214669 / 13

1D simulations simulate surface absorption, reemission, and momentum transfer

• 1 sq. cm chunk ~ 60 cm deep• Source equivalent to 1 kiloton 85 m away

• Spectrum = 1 keV Planckian• 200 groups in [3 x 10-3, 1000] keV log-spaced• Run to ~ 1e-4 sec

• Dt in [10-16, 10-9] sec• 2000 zones with Dx in [10-5,.4] cm • 106 computational photons• Materials = SiO2, Fe, H2O, Fosterite• Simulations take ~ 1 Day on 144 2.1 MHz procs• Hydrodynamics is Lagrangian

• Mesh moves with the materialIngoing shock

Ejected material

Energy escaping through asteroid surface Escaping spectrum

1 large vacuum zoneFine

ly z

oned

ast

eroi

d m

ater

ial Computational photons

are created moving parallel on vacuum face

r v

Te

Lawrence Livermore National Laboratory LLNL-PRES-82146610 / 13

2D simulations provide more realistic exploration of deposition as a function of angle

• ½ of 35 m asteroid on an axisymmetric mesh• Source is 1 kiloton, 85 m from surface

• Spectrum = 1 keV Planckian• 200 groups in [3 x 10-3, 1000] keV log-spaced• 20719 zones; sizes in [10-6, 100] cm• 108 computational photons• Materials = SiO2, Fe, H2O, Forsterite• Simulations take ~ 1 Week on 144 2.1 MHz procs• Hydrodynamics is Lagrangian

• Mesh moves with the material

Tr showing reradiated photons

Te of matter in asteroid surface

Reradiated photonsFrom asteroid

Source photons- The computational photons have exact positions on a spherical shell- The jaggedness is an artifact of the coarse vacuum zoning

Lawrence Livermore National Laboratory LLNL-PRES-82146611 / 13

Radiation hydrodynamics simulations using IMC will contribute to asteroid deflection modeling

§ We are currently running radiation hydrodynamics calculations in 1 and 2D • These expensive calculations model absorption and

reemission, shock physics, and asteroid momentum

§ These simulations allow us to characterize energy deposition with relevant physics

§ We are investigating whether we can use that deposition in hydro-only calculations and still obtain accurate results for momentum coupling• These simulations ignore radiation transport but are much

faster

Lawrence Livermore National Laboratory LLNL-PRES-82146612 / 13

A derivation of FLD with MMCEnergy conservation in lab frame:with , and

[See [2], Eqs.(6.31)—(6.38)]

raTa0rad = g0L

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g0L = g0F + vigiF

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g0F = c�aaT4 � c�aEF

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giF = �1

c�tF

iF

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EL = EF + 2vic2

FF,i ⇡ EF<latexit sha1_base64="d8hnp4LB7oQdjB0ZDJZvG0hJvTo=">AAACGHicbVDLSsNAFJ34rPUVdelmsAiCUpMo6EYoSosLFxXsA9oaJtNJO3TyYGZSLCGf4cZfceNCEbfd+TdO2iy09cDA4ZxzuXOPEzIqpGF8awuLS8srq7m1/PrG5ta2vrNbF0HEManhgAW86SBBGPVJTVLJSDPkBHkOIw1ncJP6jSHhggb+gxyFpOOhnk9dipFUkq2flu07eAXLdgUeQ6vtcoTjoU2TGD9aScWOKyc0gW0Uhjx4SlO2XjCKxgRwnpgZKYAMVVsft7sBjjziS8yQEC3TCGUnRlxSzEiSb0eChAgPUI+0FPWRR0QnnhyWwEOldKEbcPV8CSfq74kYeUKMPEclPST7YtZLxf+8ViTdy05M/TCSxMfTRW7EoAxg2hLsUk6wZCNFEOZU/RXiPlLlSNVlXpVgzp48T+pW0TwrWvfnhdJ1VkcO7IMDcARMcAFK4BZUQQ1g8AxewTv40F60N+1T+5pGF7RsZg/8gTb+AWDFnio=</latexit>

since we will drop 1

c

@FF

@t<latexit sha1_base64="h/AKc23GZSdbwYN85VWokUlS3SI=">AAACFXicbVDLSgMxFL1TX7W+Rl26CRbBhZSZKuiyKBSXFWwtdIaSSTNtaOZBkhHKMD/hxl9x40IRt4I7/8ZMO4i2HgicnPtIzvFizqSyrC+jtLS8srpWXq9sbG5t75i7ex0ZJYLQNol4JLoelpSzkLYVU5x2Y0Fx4HF6542v8vrdPRWSReGtmsTUDfAwZD4jWGmpb544vsAktbOUZGjGkRNjoRjmqNlvZunPTWV9s2rVrCnQIrELUoUCrb756QwikgQ0VIRjKXu2FSs3zRcSTrOKk0gaYzLGQ9rTNMQBlW46dZWhI60MkB8JfUKFpurviRQHUk4CT3cGWI3kfC0X/6v1EuVfuCkL40TRkMwe8hNtMEJ5RGjABCWKTzTBRDD9V0RGWEejdJAVHYI9b3mRdOo1+7RWvzmrNi6LOMpwAIdwDDacQwOuoQVtIPAAT/ACr8aj8Wy8Ge+z1pJRzOzDHxgf39R+nzw=</latexit> Express lab frame radiation

quantities in fluid frame to O(v/c) via Lorentz transformation [2] Eq.(6.30)

FL,i = FF,i + viEF + vjPF,ij +O(v

c)

<latexit sha1_base64="Xt5JxcXBch3xQ6uvUQJc8uuoLlE=">AAACKHicbZDLSsNAFIYn9VbrrerSzWARKkpJqqAbsSgWF4IV7AXaEibTSTvt5MLMpFBCHseNr+JGRJFufRInaRZaPTDw8f/ncOb8ls+okLo+1TILi0vLK9nV3Nr6xuZWfnunIbyAY1LHHvN4y0KCMOqSuqSSkZbPCXIsRprW6Dr2m2PCBfXcRznxSddBfZfaFCOpJDN/CWHVDO+OaQQvYqrGdATh2KTwxqwqHJtDWEuMYezcFzs2RzgcRyGODnNmvqCX9KTgXzBSKIC0amb+rdPzcOAQV2KGhGgbui+7IeKSYkaiXCcQxEd4hPqkrdBFDhHdMDk0ggdK6UHb4+q5Eibqz4kQOUJMHEt1OkgOxLwXi/957UDa592Qun4giYtni+yAQenBODXYo5xgySYKEOZU/RXiAVI5SJVtHIIxf/JfaJRLxkmp/HBaqFylcWTBHtgHRWCAM1ABt6AG6gCDJ/AC3sGH9qy9ap/adNaa0dKZXfCrtK9vyROiNw==</latexit>

IF =1

4⇡(cEF + ⌦F · FF )

<latexit sha1_base64="j548qFBAcuCeJui8YzYo61KW/rs=">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</latexit>

assuming IF is weakly anisotropic

PF ij ⌘1

c

Z

4⇡IF⌦i⌦j d⌦ ⇡ 1

3EF �ij

<latexit sha1_base64="XXZbeFT/VNXeVnBRYz85eHVSIAY=">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</latexit>

1)

2)

3)

4) FF = �Lc 1

3�t

@EF

@xF,i⇡ �Lc 1

3�t

@EF

@xi<latexit sha1_base64="m/vSf89zGstthjC9NDB/rkXw6dk=">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</latexit>

Flux ansatz in fluid frame andwith the flux limiter, used in term

@

@xF,i=

@

@xi+

vic2

@

@t<latexit sha1_base64="qQ5wjCLtRvQCakdq6LxWXE+LA2Y=">AAACXHicfVFbS8MwGE3r1F2cVgVffAkOQVBGOwV9EYaC+DjBXWCtI83SLSy9kKTDUfonfduLf0XTraBu6geBk/Odk+Q7cSNGhTTNuaZvFDa3toulcmWnurtn7B90RBhzTNo4ZCHvuUgQRgPSllQy0os4Qb7LSNed3Gf97pRwQcPgWc4i4vhoFFCPYiQVNTCE7XGEEztCXFLEYPoFXwfJwwVNU3gL/xPRFJ6XYS6Zqm2CXxrp3w6ZDoyaWTcXBdeBlYMayKs1MN7sYYhjnwQSMyRE3zIj6STZgZiRtGzHgkQIT9CI9BUMkE+EkyzCSeGpYobQC7lagYQL9rsjQb4QM99VSh/JsVjtZeRvvX4svRsnoUEUSxLg5UVerAYMYZY0HFJOsGQzBRDmVL0V4jFSsUj1H2UVgrU68jroNOrWZb3xdFVr3uVxFMExOAFnwALXoAkeQQu0AQZz8KEVtZL2rhf0il5dSnUt9xyCH6UffQKW3rbq</latexit>

L 2 [0, 1]<latexit sha1_base64="rnsyu/BJ4sSzsGRWrkcuF4/1MK8=">AAAB/nicbVDLSsNAFL3xWesrKq7cDBbBhZSkCrosunHhooJ9QBLKZDpph04mYWYilFDwV9y4UMSt3+HOv3HSdqGtBwYO59zLPXPClDOlHefbWlpeWV1bL22UN7e2d3btvf2WSjJJaJMkPJGdECvKmaBNzTSnnVRSHIectsPhTeG3H6lULBEPepTSIMZ9wSJGsDZS1z70Y6wHBPP8box8JpDnnLlB1644VWcCtEjcGanADI2u/eX3EpLFVGjCsVKe66Q6yLHUjHA6LvuZoikmQ9ynnqECx1QF+ST+GJ0YpYeiRJonNJqovzdyHCs1ikMzWYRV814h/ud5mY6ugpyJNNNUkOmhKONIJ6joAvWYpETzkSGYSGayIjLAEhNtGiubEtz5Ly+SVq3qnldr9xeV+vWsjhIcwTGcgguXUIdbaEATCOTwDK/wZj1ZL9a79TEdXbJmOwfwB9bnD8VilLM=</latexit>

Steps 1-5 finally yield the standard form of the diffusion equation with MMC

Eq.(11.9) in [2], Eq.(7.18) in [6]

@EL

@t+

@FL,i

@xi= c�aaT

4 � c�aEF � �tvicFF,i

<latexit sha1_base64="NZILEZMpGBNCm3yUvlQyKeFOuzw=">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</latexit>

5) FF,i = �c1

3�

@E

@xi<latexit sha1_base64="snt+bgjt1IR6Of2LgGwimPz+J3s=">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</latexit>

@FL,i

@xi<latexit sha1_base64="y6GziwJz0emNZvNjEn0O3fp1qco=">AAACDnicbVDLSsNAFJ34rPUVdelmsBRcSEmqoMuiIC5cVLAPaEKYTCft0MkkzEzEEvIFbvwVNy4UcevanX/jpA2orQcuHM65d+be48eMSmVZX8bC4tLyympprby+sbm1be7stmWUCExaOGKR6PpIEkY5aSmqGOnGgqDQZ6Tjjy5yv3NHhKQRv1XjmLghGnAaUIyUljyz6gQC4dSJkVAUMXjppddHNMt+lHuPZp5ZsWrWBHCe2AWpgAJNz/x0+hFOQsIVZkjKnm3Fyk3zJzEjWdlJJIkRHqEB6WnKUUikm07OyWBVK30YREIXV3Ci/p5IUSjlOPR1Z4jUUM56ufif10tUcOamlMeJIhxPPwoSBlUE82xgnwqCFRtrgrCgeleIh0jno3SCZR2CPXvyPGnXa/ZxrX5zUmmcF3GUwD44AIfABqegAa5AE7QABg/gCbyAV+PReDbejPdp64JRzOyBPzA+vgHUZJyZ</latexit>

Flux ansatz in fluid frame without the flux limiter, used in term �t

vicFF,i

<latexit sha1_base64="Yce7AcThNBRePIAUenpOFYtUjoI=">AAACB3icbVDLSsNAFJ3UV62vqEtBBovgQkpSBV0WheKygn1AU8JkOmmHzkzCzKRQQnZu/BU3LhRx6y+482+ctllo64ELh3Pu5d57gphRpR3n2yqsrK6tbxQ3S1vbO7t79v5BS0WJxKSJIxbJToAUYVSQpqaakU4sCeIBI+1gdDv122MiFY3Eg57EpMfRQNCQYqSN5NvHnqIDjnwNvVAinI59mqU4g3U/rZ/TzLfLTsWZAS4TNydlkKPh219eP8IJJ0JjhpTquk6seymSmmJGspKXKBIjPEID0jVUIE5UL539kcFTo/RhGElTQsOZ+nsiRVypCQ9MJ0d6qBa9qfif1010eN1LqYgTTQSeLwoTBnUEp6HAPpUEazYxBGFJza0QD5HJQ5voSiYEd/HlZdKqVtyLSvX+sly7yeMogiNwAs6AC65ADdyBBmgCDB7BM3gFb9aT9WK9Wx/z1oKVzxyCP7A+fwD2n5lc</latexit>

Inconsistent !

DE

Dt� @

@xiL 1

3�t

@E

@xi+

4

3E@vi@xi

= c�aT 4 � c�aE<latexit sha1_base64="TQDTOxjiydpk5Mq/295fJcZGi6E=">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</latexit>

Lawrence Livermore National Laboratory LLNL-PRES-82146613 / 13

§ [1] § [2]§ [3]§ [4]§ [5]§ [6]

References

G.C. Pomraning, Equations of Radiation Hydrodynamics, in: D. ter Harr (Ed.), International Series of Monographs in Natural Philosophy, vol. 54, Pergamon, New York, 1973.

J.I. Castor, Radiation Hydrodynamics, Cambridge University Press, New York, 2007

J. R. Buchler, Radiation Transfer in the Fluid Frame, JQSRT 30 No.5 (1983) 395–407.

R. B. Lowrie, D. Mihalas, J. E. Morel, Comoving-frame radiation transport for nonrelativistic fluid velocities, JQSRT 69 (2001) 291–304.

Mihalas and Weibel-Mihalas, Foundations of Radiation Hydrodynamics, Oxford University Press, Oxford, 1984.

R.L. Bowers and J. R. Wilson, Numerical Modeling in Applied Physics and Astrophysics, Jones and Bartlett, Boston, 1991.