Case-Mode Analysis of Finite Element Spatial Discretizations Alexander E. Maslowski, Dr. Marvin L. Adams Texas A&M University Abstract 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. 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. 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 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: To test the theory, individual Case-modes must be isolated. We found a Case-mode orthogonality relation for S N. We use it to obtain scalar-flux modes from the individual angular fluxes: Separating the Modes Numerical Experiment: Two Region Slab S 4  inc = 40 (normal)  inc = 20 (grazing) 15 cells 5 cells 4 cells 10 cells  x = cm  x = cm  x = cm  x = cm Same Material Properties  inc = 40 (normal)  inc = 20 (grazing)  x = cm  t = 3.0 cm 2 c = 0.9  x = 2 cm  t = 2.0 cm 2 c = cells 5 cells

Case-Mode Analysis of Finite Element Spatial Discretizations Alexander E. Maslowski, Dr. Marvin L. Adams Texas A&M University Conclusions We found the CFEM have the following properties: The single-cell attenuation factor is a Pade’ rational polynomial approximation of exp(-  ), where  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/( ). 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. Positive Asymptotic ModeNegative Asymptotic ModePositive Transient Mode Positive Asymptotic ModeNegative Asymptotic ModePositive Transient Mode Case 2: Resolved Boundary Layers Case 1: Unresolved Boundary Layers