Neutron Transport Theory

A nodal greens function method for multidimensional neutron diffusion calculations

Nodal Green's function method for multidimensional neutron diffusion calculations

Benchmarking with the multigroup diffusion high-order response matrix method

In the Response Matrix Method (RMM) the solution of a particle transport problem in a composite (large) domain, 

$$V\mathop {\mathop U\limits_{i = 1} }\limits^N {V_i},$$

is constructed from precomputed particular solutions to the local problems associated with each of the subdomains Vi (i=1,2,...N). The RMM becomes efficient if: 

1.

the subdomains Vi, hereafter called nodes, are geometrically simpler than the whole domain of interest, V, so that the particular solutions for each node are computationally inexpensive.




2.

the problem is such that many of the Vis have identical geometrical shape and material composition.

The response matrix method

The response matrix equations (RME) are analyzed from two points of view: (a) their computational feasibility and (b) their consistency with other methods used in reactor analysis. It is shown that RME can be derived directly from the weak form of the diffusion equation without the concept of partial currents, and hence, are also applicable to the description of phenomena, where partial currents have no physical meaning (for example, the conduction of heat). By splitting the high-order RME into a coupled system of single-order equations, the analysis of the convergence properties of the iterative solutions to RME could be greatly simplified. The derived explicit expressions for the convergence ratio were verified by numerical experimentation. As an illustration, the well-known International Atomic Energy Agency benchmark problem has been calculated by two two-dimensional response matrix programs at ASEA-ATOM, CIKADA, and LABAN. The relation of RME to finite difference (FD) equations has also been investigated. It was shown that for small mesh sizes, RME are computationally not feasible. For rectangular nodes, an algorithm called the ''vectorial model'' (VM) was developed, which reduces the amount of unknowns in RME by a factor of 2. This is a generalization to two- and three-dimensional nodes ofmore » the author's earlier results. An approximate reduction of VM to scalar equations (one unknown per node) has been discussed, and its relation to recent developments in nodal methods has been emphasized.« less

Some basic properties of the response matrix equations

Response matrix equations in two-dimensional geometry have been derived in the form of a set of coupled integral equations of the Fredholm type that have been solved by the moments method. The set of Legendre polynomials defined at the material interfaces has been chosen as the base for representing the partial interface currents and the response matrices.The method has been applied to the solution of the one-group diffusion equation and its convergence has been investigated in a series of numerical experiments, involving expansions of up to order 14. It turned out that the P1 approximation should be adequate for the majority of the two-dimensional problems occurring in power reactor design. Furthermore, the response method has a substantially higher computer efficiency than the finite difference method, both in processor time and in storage locations. As a by-product, the nature of the singularities around edges and corners of material interfaces has been analyzed by numerical experimentation.

High-Order Response Matrix Equations in Two-Dimensional Geometry

Nodal Equations Derived from Invariant Imbedding Theory

Application of Invariant Imbedding to Reactor Physics

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