Finite Elements with Discontiguous Support for Energy Discretization in Particle Transport
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The standard multigroup (MG) method for energy discretization of the transport equation can be sensitive to approximations in the weighting spectrum chosen for cross-section averaging. As a result, MG often inaccurately treats important phe-nomena such as self-shielding variations across a fuel pin. From a ﬁnite-element viewpoint, MG uses a single ﬁxed basis function (the pre-selected spectrum) within each group, with no mechanism to adapt to local solution behavior. In this work, we introduce the Finite-Element-with-Discontiguous-Support (FEDS) method, an extension of the previously introduced Petrov-Galerkin Finite-Element Multigroup (PG-FEMG) method, itself a generalization of the MG method. Like PG-FEMG, in FEDS, the only approximation is that the angular ﬂux is a linear combination of basis functions. The coeﬃcients in this combination are the unknowns. A basis function is non-zero only in the discontiguous set of energy intervals associated with its energy element. Discontiguous energy elements are generalizations of bands in-troduced in PG-FEMG and are determined by minimizing a norm of the diﬀerence between sample spectra and our ﬁnite-element space. We present the theory of the FEDS method, including the deﬁnition of the discontiguous energy mesh, deﬁnition of the ﬁnite element space, derivation of the FEDS transport equation and cross sections, deﬁnition of the minimization problem, and derivation of a useable form of the minimization problem that can be solved to determine the energy mesh. FEDS generates cross sections that ordinary MG codes can use without modiﬁcation, pro-vided those codes can handle upscattering, allowing FEDS answers from existing MG codes. FEDS requires that the energy domain is divided into elements, each in general a collection of discontiguous energy intervals. FEDS solves a minimization problem to ﬁnd the optimal grouping, in a certain sense, of hyperﬁne intervals into its elements. It generates accurate, convergent discretizations without need for accurate reference solutions. We show convergence in energy as energy elements are added for several types of problems, beginning with cylindrical pincell problems. Convergence is ob-tained for a variety of basis functions ranging from simple (1/E) to more complicated (space-angle-averaged reference spectra), demonstrating robustness of the method. We investigate four sets of problems. We ﬁrst investigate the same pincell prob-lems used when testing the PG-FEMG method. We use lessons learned from these pincell calculations to inform our implementation of the FEDS method on an energy-generalized version of the C5 problem, which we call the C5G∞ problem. We then ap-ply FEDS to time-dependent neutronics problems, where correctly capturing stream-ing times in a time-of-ﬂight problem becomes important. Finally, we compare the FEDS method to continuous-energy Monte Carlo one-dimensional slab pincell prob-lem. We ﬁnd FEDS to be superior in eﬃciency and accuracy to MG with the same weighting functions and number of energy unknowns. Whereas MG requires un-known counts commensurate with the number of resonances to be convergent, we ﬁnd FEDS converges in energy even at low numbers of energy unknowns.
Finite Element (FE)
Till, Andrew (2015). Finite Elements with Discontiguous Support for Energy Discretization in Particle Transport. Doctoral dissertation, Texas A & M University. Available electronically from