A fast converging CMFD nonlinear iteration scheme for two-node analytic function expansion nodal methodology

The nonlinear finite difference method (FDM) iterative scheme has been widely used as an alternative way to the core-wise response matrix formalism in modern nodal methods. This scheme turned out to be very effective in minimizing memory requirement and computing time associated with higher-order nodal methods. This conventional nonlinear FDM iterative scheme uses the modified FDM current definition with a nonlinear correction factor at an interface between two nodes. Determining the nonlinear correction factor so that the interface current should preserve the value of a higher-order nodal method makes the solution of this finite difference scheme equivalent to that of the higher-order nodal method itself. For the nonlinear FDM iterative scheme with the usual higher-order nodal methods that use the transverse-integration, this is done by solving two-node problems consisting of neighboring nodes periodically after a specified number of outer iterations of the FDM routine. Using the higher-order nodal method, the two-node problem is solved for the interface current of the two nodes with currently available node-average fluxes and transverse-leakage shapes of both nodes as boundary conditions. The nonlinear correction factor at the interface is updated by equating the resultant higher-order interface current with the modified FDM current. Then, the FDM routine is continued utilizing the updated nonlinear correction factor. The entire process is repeated until convergence of the effective multiplication factor and the node average fluxes is achieved. In this study, as an acceleration means and for the convenience of its implementation into existing FDM codes, we develop a nonlinear iterative scheme for the analytic function expansion nodal (AFEN) method. Developing a nonlinear iterative scheme for the AFEN method is not straightforward, because this method needs higher-order accurate interface and corner-point fluxes as well as interface currents in solving the two-node problem, which are not carried by the FDM routine. The new nonlinear iterative scheme developed here for the AFEN method employs two nonlinear correction factors at every interface instead of one factor in the conventional scheme. The increased degree of freedom allowed by the use of two factors provides the higher-order accurate interface fluxes and currents as well as corner-point fluxes, which are then used as the boundary conditions of the two-node problem. The nonlinear iterative AFEN method was tested on three benchmark problems and a real reactor problem. The numerical results show that the converged solutions of the nonlinear AFEN and original AFEN methods are the same if the minor truncation errors are neglected. The computing times of the new method are significantly reduced in comparison with those of the original AFEN method. The results indicate that the computing time can be reduced by a factor of ∼10 for usual two-dimensional nodal problems. The extension of the nonlinear scheme to three-dimensional geometry is straightforward. Therefore, the new nonlinear AFEN method can be effectively used in practical nuclear design problems