The discrete ordinate method development to the transport equation solving' The 3D code RADUGA5'1P a

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The discrete ordinate method developmentto the
transport equation solving.The 3D code
RADUGA-5.1(P)and multiprocessors computers

• O.V.Nikolaeva,
  L.P.Bass
• Keldysh Institute of Applied Mathematic, Moscow,
  Russia
• V.S.Kuznetsov
• Research Scientific Center "Kurchatov Institute",
  Moscow, Russia

T.A.Germogenova,
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Methods developmentto solve the transport
equationin different applications

• radiation shielding problems,
• atmosphere optic problems,
• biomedicine problems.
• All methods are included into
• the code RADUGA-5.1(P)

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Flexible algorithm

• In each subregion grids and scheme are chosen in
  dependence of solution properties in this
  subregion
• smooth solutions,
• non-smooth solutions,
• discontinuous solutions.

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Calculation of discontinuous solutionby DD and
SWDD schemes

• Step of spatial mesh 1/27 mfp.

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Phase functions representation by Legenders
polynomials
PN expansion for a regular component is
used Here is Legenders polynomial.

• Peak-forward phase function is presented by sum
  regular component and singular one

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Comparison of numerical resultsfor two methods
of phase function presentation

• Method 1 by its values at nodes of some grid,
• Method 2 by Legenders polynomials

Spatial mesh 15 ? 30 meshes Mesh of spatial
grid 0.041 mfp. Angular quadrature S50
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Solution at axis z
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Definition of geometry region

• Boundaries of materials are determined by
  surfaces of simple 2D-3D bodies
• parallelepiped
• truncated cone
• prism
• cylinder
• regular hexahenron
• sphere
• rectangle
• triangle
• circle or sector
• regular hexagon
• Each body is defined by its geometrical parameters

Circle approximation by edges of regular spatial
grid meshes
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Point source of great apertureSkyshine problem
Analytical calculation of un-scattered radiation
intensity

• Energy interval (15Mev, 0 Mev)
• 22 energetic groups
• P3 expansion for phase functions
• Spatial grid 480 ? 132 meshes
• Angular quadrature S16

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Parallel beamSolar radiation reflection by a
cloudof irregular structure.The international
Intercomparison of 3D Radiation Codes (I3RC).

• Spatial grid - 100?? 100 ?? 36
• Angular quadrature S30
• Analytical calculation of un-scattered radiation
  intensity

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Radiation reflected in zenith
Pixelss optical depth
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Isotropic conic source. Problem of radiation
safety of astronauts and equipmentin the
spacecraft with the nuclear reactor in motive
regime
Un-scattered component is calculated
semi-analytically Photons field at the
spacecraft top
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Point source of small aperture.Biomedical problem

• Un-scattered radiation intensity is defined
  analytically
• Once-scattered radiation intensity is calculated
  semi-analytically
• Multi-scattered radiation intensity is determined
  by a grid scheme
• Spatial grid - 169?100 meshes
• Angular quadrature S26

Full interior reflecting condition at the top
boundary, refraction coefficient
n1.4. Scattering is simulated by
Heney-Greenstein phase function with asymmetry
parameter g0.8
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Reflected radiation at the line AB
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Effectiveness of a parallel algorithm

• MPI standard
• Spatial decomposition of calculation region
• One region ? one processor
• Effectiveness of a parallel algorithm
• N is processor number.
• T(1) is calculation time by one processor.
• T(N) is calculation time by N processors.

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We plan to deal with the following manners

• Peak-forward phase-functions (ray therapy
  problem, atmospheric optic problems)
• Time-dependent transport equation (impulse source
  in biomedicine problem)
• Inverse problem (atmospheric optic, biomedicine)
• These plans rely on experience and intellect and
  energy and optimism of three authors of this
  report, two post-graduate students and two
  students.
• We are open to cooperate