Finite element, nodal and response matrix methods: A variational synthesis for neutron transport

E. E. Lewis*, I. Dilber

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


The variational principle used to derive finite element approximations to the even-parity neutron transport equation is modified to include the odd parity flux as a Lagrange multiplier along the interelement boundaries. The result is a functional that guarantees nodal balance over each element or node, regardless of the form of the space-angle trial functions that are used. Ritz procedures are employed to yield nodal methods that are analogous to those used extensively in diffusion theory, but without the need for ad-hoc node-to-node interpolation of transverse leakage terms. With spherical harmonics, nodal transport methods are obtained with consistent angular approximations within the nodes and across the node interfaces. The resulting equations are in response matrix form, and from their solution the detailed flux distribution as well as the node-averaged values can be obtained. Diffusion theory comparisons of benchmark eigenvalue problems in two-dimensions demonstrated the efficacy of the new approach relative to response matrix methods in which nodal balance is not satisfied. A two-dimensional spherical harmonics implementation of the method also allows coarse spatial grids to be employed in transport calculations. This, combined with the high degree of vectorization inherent in the red-black solution algorithms, results in order of magnitude decreases in computing times from those of spherical harmonics methods in which low-order spatial finite elements are employed.

Original languageEnglish (US)
Pages (from-to)63-74
Number of pages12
JournalProgress in Nuclear Energy
Issue number1-2
StatePublished - 1986

ASJC Scopus subject areas

  • Nuclear Energy and Engineering
  • Safety, Risk, Reliability and Quality
  • Energy Engineering and Power Technology
  • Waste Management and Disposal


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