J.J. Jarrell, “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, interpolated, by interpolation through the existing four points on a given triangle. We use a transport sweep to find the actual values, calc, at certain test directions in the triangle and compare against interpolated at those directions. If the results are not within a userdefined tolerance, the test directions are added to the quadrature set. 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.