DoubleGauss Quadrature for Discrete Ordinate Transport Equations in Cylindrical Geometry

1
Double-Gauss Quadrature for Discrete Ordinate
Transport Equations in Cylindrical Geometry
The 19th International Conference on Transport
Theory Budapest Hungary

• Zhu Ruidong

The Institute of Applied Physics and
Computational Mathematics Beijing China
2
Introduction
Exact analytic solutions of transport
equations can be obtained only for the extreme
simple problems, but for the general practical
ones, approximate numerical solutions must be
adopted. The merit of the discrete ordinate
method (DOM), which is adopted extensively in
neutron transport and radiative transport lies on
its direct discretization for all the variables
and the simple process of solving.
3
Outline

• Introduction
• Design principles of quadrature schemes
• Two examples
• Cnclusions

4
Introduction
On the work of how to determinate quadrature,
numerous significant contributions have been
made. However, most of the known quadratures are
more suit for three-dimensional problems than for
two-dimensional problems, additionally,
quadrature schemes pay more attention to the
rotational invariance of the three-dimensional
problems, and the symmetries in two-dimensional
problems have not been fully utilized.
5
Transport Equation in Two-dimensional Cylindrical
Geometry
6
Design Principles of Quadrature Schemes
The basic requirement to be satisfied by any
quadrature scheme is that the integral over the
unit sphere of any quantity can be
expressed by a weighted sum taken at the
directions of the discrete ordinates
7
Design Principles of Quadrature Schemes

• Despite the lack of a sound mathematical
  theory, general design principles for DOM
  quadrature schemes can be given
• All discrete ordinates must
  be placed on the unit sphere, i.e.
• The weight factors wm must all be positive. This
  requirement ensures that error of the quadrature
  schemes will be minimized.
• The quadrature must be invariant to any rotation
  of the arrangement of the discrete directions
  around the center of the unit sphere.

8
In numerical computation, the following typical
integrals appear frequently.
Here, the integral is split into a product of two
integrals. Next, we will discuss the
discretization of the azimuth angle and polar
angle.
9
The azimuth angle discretization
Chebyshev Gauss quadrature is optimal.
10
The polar angle discretization
Gauss quadrature is optimal.
Since the singularity of the angular flux in the
direction that u is equal to zero.So double Gauss
quadrature is optimal.
11
We denote

• SN(M)
• M is the discretized number in four octants.
• For Lee quadrature LeeN(M)
• Gauss SgN(M) and
• double Gauss DgN(M)

12
Arrangement of the Sg8(72) Gauss quadrature
13
Two examples

• Source problem
• Critical problem

14
Source problem
The model is a column in two-dimensional
cylindrical geometry, which the half height is
2cm and the radius is 2cm. It is a constant
source region (S1) in the center. Furthermore,
the total cross section is and
scattering cross section is optional. . The
spatial region is subdivided into equally spaced
mesh intervals , we
calculate the net neutron fluxes at the centers
of the boundary mesh cells.
15
(a). Lee8(48)
quadrature (b). Lee8(48)
quadrature
The ray effects are indeed more pronounced the
lower the scattering cross sections, as is
evident from Figs. a) and (b) by Lee8(48)
quadrature. Furthermore, the dash line oscillates
more severely than the solid line that indicates
that the results at the cylinder boundary mesh
cells (r2) are worse than those at the top
boundary mesh cells (z2). In the following, only
the scattering section is adopted. As compared,
the results are obtained used of DOT program, as
is shown in Figs. (b). The results of two program
have a little difference, but they are neither
good.
16
(c).Lee4(16), Lee12(96) quadrature

• The results of Figs. (c) are obtained by the
  quadratures Lee4(16) and Lee12(96). By comparison
  of the results of the quadratures Lee4(16),
  Lee8(48) and Lee12(96), it is found that, in
  order to ensure an high precision and smooth
  spatial net flux, the number of discretized
  directions are not less than the quadrature
  Lee8(48).

17
(d).Sg8(72) quadrature
(e).Dg8(72) quadrature
The results of Figs. (d) are obtained by the
quadrature Sg8(72), which are close with those of
Figs. (e) obtained by the quadrature Dg8(72).
18
(f).Dg12(84) quadrature

• The results of the quadrature Dg12(84) are
  considered the best, which are composed of
  double-Gauss quadrature (N12) in the polar angle
  and equational Chebyshev-Gauss quadrature (
  ) in the azimuth angle. Meanwhile, the results
  by the quadrature Dg12(84) used of the program
  DOT are also shown Fig. b.

19
Critical problem
The model is a uniform cylinder whose half height
and radius are all A(cm). When reflecting
boundary is at RA the in the r direction, the
model corresponds to an infinite flat reflecting
boundary is at the ZA in the z direction, it
corresponds to an infinite long cylinder. The
total cross section is , the scattering
cross section is and the fission source is
.
20
As the number of discretized direction is N the
spatial region is subdivided into equally
spaced mesh intervals. For the infinite flat
plank, the infinite long cylinder and finite long
cylinder, the critical radii A(cm) are calculated
by the fixed the value of c and different
quadratures, as is shown in the table 1. The
critical half thickness of the infinite flat of
H.G.Kapar and the critical radius of the infinite
long cylinder of J.R.Thomas. However, for the
finite long cylinder, the results of the
quadrature Sg24(600) are recognized as the
standard.
21
Table 1 The results of Critical Problems A (cm)
22
Results

• For the critical half thickness of the infinite
  flat plank, it may have a good precision by
  double-Gauss quadrature in the polar angle,
  meanwhile, Lee quadrature and single-Gauss
  quadrature have a poor precision, which the
  critical thicknesses are all a little great.
• For the critical radius of a finite cylinder, the
  results of single-Gauss quadrature are all a
  little small, the results of Lee8(48) quadrature
  accord with exact values. As the numbers of
  discretized directions increase, computing may
  show not convergent.
• For a finite long cylinder that is closer to the
  practice, the precisions of the three quadratures
  are very nearly. Furthermore, the precisions of
  Lee quadrature and single-Gauss quadrature are a
  little higher than that of double-Gauss
  quadrature.

23
Conclusions
Lee quadrature is generally adopted to calculate
the critical size of a sphere or a cylinder that
its half height is nearly equal to its radius,
also Lee quadrature is used to check
two-dimensional program and may gain good
results. To a long elliptical sphere or a short
elliptical sphere Lee quadrature can not be
satisfied. Based on the analyses above, it
indicates that double-gauss quadrature in the
polar direction, Chebyshev-Gauss quadrature in
the azimuth direction and less than 20
discretized directions in one octant can gain a
high precision and good symmetry results to this
example. The polar direction and azimuth
direction could have different discrete numbers.
24

• Thanks