Dirac delta functions DOM for onespeed transport equation with anisotropic scattering

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Dirac delta functions DOM for one-speed transport
equation with anisotropic scattering
19th International Conference on Transport Theory

• Li Maosheng Yang Bo
• Institute of Applied Physics and Computational
  Mathematics.
• Beijing, P. R. China.

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Outline

• Introduction
• 1-D transport equation
• A particular solution
• Dirac delta functions
• Angular flux is presented by delta functions
• Numerical results
• Conclusion

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Introduction (1)
The problem of anisotropy is one of the most
important problems of transport theory. Many
methods for computing transport equation have
been proposed, such as the spherical harmonics
method , the FN method , the discrete ordinate
method (DOM) , the Monte Carlo method, to name
but a few. In general, the inclusion of
anisotropic scattering into transport equation
leads to great mathematical complexity. Thus, the
investigation of the effects of different type of
anisotropic scattering law on the particle
distribution has been the subject of many
studies.
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Introduction (2)
In one dimension case the discrete ordinate
method is equivalent to the spherical-harmonics
method under certain restrictions on the
quadrature scheme, the conventional discrete
ordinates method, PN and FN method could
represent limited degree of angular flux
according to the number of directions. As the
angular distribution has very strongly peaked in
some direction one has to take a large number of
discretized directions into account for the
highly peaked angular flux. The lower order
normal SN and PN method are thus inadequate for
this type problem. To accurately predict the
local anisotropy higher order angular flux are
necessary.
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• In this paper we calculate the critical thickness
  for one-speed neutrons in a slab. we propose an
  easy-to-use comprehensive discrete ordinate
  algorithm for computing transport equation in
  spatial homogeneous in which the high peak
  angular flux is caused by anisotropy scattering.
  The angular flux is presented by a delta function
  plus a lower order function. The determination of
  the source term is done by integrating the delta
  function and quadrature the lower order function.

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1-D transport equation in slab
The one speed linear transport equation For a
source free, infinite slab of thickness 2a, we
take a boundary conditions of that no neutrons
enter the slab from the outside Scattering
kernel
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The discrete ordinate method
In the Discrete Ordinate Method (DOM) the
transport equation is solved for a set of
discretized directions. Each direction is
associated with a solid angle in which the
intensity is assumed to be constant. All solid
angles are non-overlapping and spanning the total
angle range of 4p. The integrals over the
direction are replaced by numerical
quadrature. In each direction the transport
equation may be solved analytically and the
angular flux could be exact in the direction. The
error arise, however, not from the inability to
calculate the angular flux in the discretized
direction exactly. The error is mainly caused by
the inability of the quadrature formula.
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Angular flux
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A particular solution
If ,we can get
Integrate µ at -1,1 and simplify it,
is a solution and satisfies the
boundary conditions.
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Dirac delta function
This is a finite width delta function. The
parameter n represents the width of delta
function.
Factor ? is an important parameter. When ? is
equal to 1 the result have good precision and the
error is zero.
Angular flux at 0.25a from the surface alfa
beta 0.425 , b10, c1.1538
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Split angular flux

• When is
  a particular solution. So we split the angular
  flux into a delta function and a remained
  function
• Assume that when
  equal to angular flux
• is related to the zero moment of in
  -1 , 0

Integrate µ at -1,0 and simplify it,
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The determination of n
So we can get n and the delta function could be
determined.
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The determination of s and h
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Evaluating the error of normal discrete ordinate
The figure indicates that for a certain
discretized number of direction, for example,
N24, as the n increases the ? increases slowly
from 1.0 to 1.05, and then return to 1.0 again at
n150, after that ? quickly decreases. So n
could be a criterion to determine the discretized
number of normal discrete ordinate.
Factor ? vary as the function of n
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Angular flux is expanded by Legendre functions.
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Result 1
Critical thickness as a function of a and b1,
left figure is our result, right one is by
spherical-harmonics method
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Result 2
Critical thickness as a function of ß and b1,
left figure is our result, right one is by
spherical-harmonics method
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Result 3
Critical thickness as a function of a ß and b1,
left figure is our result, right one is by
spherical-harmonics method
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Conclusion

• Anisotropic scattering is an important research
  area in reactor physics, astronomical physics and
  radiation transfer. For the complexity of
  transport equation, its hard to obtain an
  analytic solution except for special conditions.
  Discrete ordinate method is a simple process and
  widely applied in transport problems.
• In 1-D homogenity plane geometry, spherical
  geometry or finite cylinder geometry physical
  analysis and numerical results show that the
  distribution function of angular flux is smooth
  and continuous, with maximum in one or two
  directions.

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Conclusion

• For the anisotropic scattering problem in 1-D
  slab, using the special solution of delta
  function, the angular flux is presented by a
  finite width delta function (high order) and a
  remained function. Then the higher order function
  is integrated and the lower order function is
  summed by quadrature. Numerical results show that
  accuracy results could be got with less
  discretized directions.
• We discuss the concepts and applications delta
  function discrete ordinate method in 1-D slab
  with anisotropic scattering. Making good use of
  angular flux in different directions, structuring
  the high order part of the angular flux in a
  whole to evaluate the error caused by quadrature,
  and to improve the calculating accuracy.

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Reference

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Thanks