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. 1 1 2 2 M m m m n n m d c x w x Qx dx / 1 , , 1 M x n m m m m n n w x e / m x exact m x x g e x 1 , 1 , cell J m m kj kj j x b x x k 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 1/2 1/2 m k m k x f x ,1 1 1 k k k k J J mk m m x x L M L M 1 1 1 , 1 J J mk J mk i Ji i f L M L M 2 0 1 2 2 0 1 2 1 1 lim a a a ff f f a a a 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: , m kj m kj az ,, 1 ,,1 , 1 1 mkJ mk kJ k g z gz 1 , 1 , 1, , 1. kj kj k kJ b x j b x b x gb x j 1 M m m m w a 1 1 1 1 1 2 M M m mk mk n n m n c z w z wa I L M L M 1 1 1 k k z z x L M LM m m m a 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 cel ls 5 cel ls 4 cel ls 10 cel ls x = 0.200 cm x = 1.500 cm x = 1.333 cm x = 0.564 cm Same Material Properties Total ScalarFlux 0 5 10 15 20 25 30 0 5 10 15 20 Position (cm ) ScalarFlux (Particles/cm 2 -s) Diam ond Quadratic H igh Level Exact inc = 40 (normal) inc = 20 (grazing ) x = 0.667 cm t = 3.0 cm 2 c = 0.9 x = 2 cm t = 2.0 cm 2 c = 0.9999 15 cel ls 5 cel ls Total ScalarFlux 0 5 10 15 20 25 30 0 5 10 15 20 Position (cm ) ScalarFlux (Particles/cm 2 -s) Diam ond Quadratic H igh Level Exact ' ' ' 1 l l M m m l ll l l m m m w C ' ' ' ' 2 ' ' ' N m n m m N m n n m n m N m n m n m m m n m w w w