Completely boundary-free minimum and maximum principles for neutron transport and their least-squares and Galerkin equivalents

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Some minimum and maximum variational principles for even-parity neutron transport are reviewed and the corresponding principles for odd-parity transport are derived by a simple method to show why the essential boundary conditions associated with these maximum principles have to be imposed. The method also shows why both the essential and some of the natural boundary conditions associated with these minimum principles have to be imposed. These imposed boundary conditions for trial functions in the variational principles limit the choice of the finite element used to represent trial functions. The reasons for the boundary conditions imposed on the principles for even- and odd-parity transport point the way to a treatment of composite neutron transport, for which completely boundary-free maximum and minimum principles are derived from a functional identity. In general a trial function is used for each parity in the composite neutron transport, but this can be reduced to one without any boundary conditions having to be imposed. An alternative derivation of the functional identity gives as a by-product Davis complementary principles for composite neutron transport, which use two trial functions satisfying essential boundary conditions. If these two trial functions are replaced by one then both natural and essential boundary conditions have to be imposed. The functional identity is used to re-establish three well-known principles directly, and it shows that the boundary-free maximum principle is equivalent to a generalized least-squares method with weights in the form of operators and no boundary conditions imposed. The least-squares principle uses two positive definite volume integrals so that the divergence therorem can be used to change awkward volume integrals into manageable surface integrals without the need to impose boundary conditions on trial functions. A geometrical interpretation of the boundary-free maximum principle is given using the projection theorem for a Hilbert space with a suitable metric. This path leads to several boundary-free Galerkin equations for both the second-order and first-order forms of the transport equation.