The finite element with discontiguous support multigroup method: Theory
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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 phenomena 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 Multigroup (FEDS-MG) method, a generalization of the previously introduced Petrov-Galerkin Finite-Element Multigroup (PG-FEMG) method, itself a generalization of the MG method. Like PG-FEMG, in FEDS-MG, 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 introduced in PG-FEMG and are determined by minimizing a norm of the difference between sample spectra and our finite-element space. In this paper, we present the theory of the FEDS-MG method, including the definition of the discontiguous energy mesh, definition of the finite element space, derivation of the FEDS-MG 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. A companion paper presents results.
