The why and how of finite elements

Abstract The ‘Why’ of the title reflects the desire in safety assessments for independent means of making realistic calculations for systems of complex shape. In principle the geometrical flexibility attainable by the deterministic finite element method being the equal of the stochastic Monte Carlo method. The development of the finite element method is traced to show how ideas from structural and fluid mechanics, the calculus of variations, functional analysis and the calculus of finite differences have been forged to provide a tool which minimizes the mismatch between the behaviour of a continuous system and that of a discrete model of the system assembled from finite elements. Geometrical flexibility of the model is achieved by the use of polygonal and curved elements. The behaviour of any point of an element is described in terms of its behaviour at discrete points or nodes of the element. In treating neutron transport the finite element method can be applied to phase-space, or as in this paper the spatial dependence can be treated by the use of finite elements in conjunction with expansions in orthogonal functions for the directional dependence. The mathematical formulation is based on a mixed parity form of the Boltzmann equation for one-group transport. The minimization of the mismatch between the system and its finite element model leads to a completely boundary-free maximum principle. This variational principle is also recast into a generalized least-squares principle. When the essential boundary conditions of the classical calculus of variations are imposed the well-known minimum and maximum principles for the even- and odd-parity second-order Boltzmann equations are obtained as special cases. The maximum principle for the second-order even-parity equation is used to demonstrate the precision and flexibility of the finite element method by solving the problems of a dog-legged duct in a shield and a cylindrical fuel element in a square lattice cell. The geometrical interpretation of the boundary-free maximum principle with the aid of a suitable Hilbert space then leads to completely boundary-free weighted residual or Galerkin schemes for both the first- and second-order forms of the Boltzmann equation. Imposing essential boundary conditions leads to classical schemes. The paper concludes with a sketch of finite element treatments of the multigroup Boltzmann equation.

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