Abstract

1. Numerical Experiment: Two Region Slab S4 yinc= 40 (normal) yinc= 20 (grazing) 15 cells Dx = 0.667 cm st = 3.0 cm2 c = 0.9 5 cells Dx = 2 cm st = 2.0 cm2 c = 0.9999 yinc= 40 (normal) yinc= 20 (grazing) 15 cells 10 cells Dx = 1.333 cm Dx = 0.200 cm 5 cells 4 cells Dx = 0.564 cm Dx = 1.500 cm Same Material Properties Abstract CFEM Analysis: With Scattering We seek a separable solution and SCAF, g, defined by: We modify the basis function to define it in terms of the unknown attenuation factor: Using the CFE equations we obtain: It follows that the following eigen-problem is formed: If the same procedure is done for the pure-absorber case, we can conclude that f and g are the same functions with the directional cosine replaced by the relaxation length. We can also use this equation to show: • We analyze the following problem: • One dimensional, steady state, discrete-ordinates. • Continuous Finite Element (CFE) with discontinuous weight function. • Case-mode analysis shows: • CFE methods (CFEMs) yield the correct dispersion relation. • CFEMs yield Padè approximations of the exact single-cell attenuation factor for each mode. The Padè approximations have numerator and denominator of the same polynomial order. • CFE s can not be truly robust because their single-cell attenuation factors approach +1 as cell become thicker. • We present results showing that the analysis correctly predicts the single-cell attenuation factor of each mode. Case-Mode Analysis of Finite Element Spatial DiscretizationsAlexander E. Maslowski, Dr. Marvin L. Adams Texas A&M University Analytic Transport We consider the following transport equation: The homogeneous solution is spanned by the following modes: The exact single-cell attenuation factor (SCAF) for a given mode is: We aim at deriving similar properties for the CFE approximation: We restrict our study to the CFEMs that are symmetric and use discontinuous weight functions mapped to each cell. Separating the Modes To test the theory, individual Case-modes must be isolated. We found a Case-mode orthogonality relation for SN. We use it to obtain scalar-flux modes from the individual angular fluxes: CFEM Analysis: Without Scattering We seek the CFEM SCAF, f, such that: From the CFE equations the attenuation factor is: Since our CFEMs are symmetric, they treat right and left moving particles identically. We can show: Thus, no CFEM in this family is robust – each has SCAF ±1

2. Case 1: Unresolved Boundary Layers Case-Mode Analysis of Finite Element Spatial DiscretizationsAlexander E. Maslowski,Dr. Marvin L. Adams Texas A&M University Case 2: Resolved Boundary Layers • Conclusions • We found the CFEM have the following properties: • The single-cell attenuation factor is a Pade’ rational polynomial approximation of exp(-t), where t is the traversal distance in mean-free paths for the pure absorber case, and the cell width in relaxation lengths for the scattering case. • CFEMs obtain the correct angular shape function for each mode: 1/(n-m). • The rational polynomial approximation is not robust: it approaches +1 in the thick-cell limit. • The analysis describes the behavior of all methods of the defined CFE family. It applies to all slab problems regardless of mesh spacings. • We expect future studies to focus on using this analysis, or the insight gained from it, to improve solutions of multidimensional problems.