"An Adaptive Angular Discretization Method for Neutral-Particle Transport in Three-dimensional Geometries,"
Ph.D. Dissertation, Department of Nuclear Engineering, Texas A&M University, College Station, TX (2010).
In this dissertation, we discuss an adaptive angular
discretization scheme for the neutral-particle transport
equation in three dimensions. We mesh the direction domain by
dividing the faces of a regular octahedron into equilateral
triangles and projecting these onto "spherical triangles" on
the surface of the sphere. We choose four quadrature points
per triangle, and we deﬁne interpolating basis functions
that are linear in the direction cosines. The quadrature
point's weight is the integral of the point's linear
discontinuous ﬁnite element (LDFE) basis function over its
local triangle. Variations in the locations of the four points
produce variations in the quadrature set.
The new quadrature sets are amenable to local reﬁnement and
coarsening, and hence can be used with an adaptive algorithm.
If local reﬁnement is requested, we use the LDFE basis
functions to build an approximate angular ﬂux, Ψinterpolated,
by interpolation through the existing four points on a given
triangle. We use a transport sweep to ﬁnd the actual values,
Ψcalc, at certain test directions in the triangle and compare
against Ψinterpolated at those directions. If the results are not
within a userdeﬁned tolerance, the test directions are added to the
The performance of our uniform sets (no local reﬁnement) is
dramatically better than that of commonly used sets
(level-symmetric (LS), Gauss-Chebyshev (GC) and variants) and
comparable to that of the Abu-Shumays Quadruple Range (QR)
sets. On simple problems, the QR sets and the new sets exhibit
4th-order convergence in the scalar ﬂux as the directional
mesh is reﬁned, whereas the LS and GC sets exhibit 1.5-order and
2nd-order convergence, respectively. On diﬃcult problems
(near discontinuities in the direction domain along directions
that are not perpendicular to coordinate axes), these
convergence orders diminish and the new sets outperform the
others. We remark that the new LDFE sets have strictly positive
weights and that arbitrarily reﬁned sets can be generated
without the numerical diﬃculties that plague the generation of
high-order QR sets.
Adapted LDFE sets are more eﬃcient than uniform LDFE sets only
in diﬃcult problems. This is due partly to the high accuracy
of the uniform sets, partly to basing reﬁnement decisions on
purely local information, and partly to the diﬃculty of
mapping among diﬀerently reﬁned sets. These results are promising
and suggest interesting future work that could lead to more
accurate solutions, lower memory requirements, and faster
solutions for many transport problems.
Associated Project(s):SHIELD (Smuggled HEU Interdiction through Enhanced anaLysis and Detection): A Framework for Developing Novel Detection Systems Focused on Interdicting Shielded HEU