### Abstract:

Nodal methods which are used to calculate reactor transients, control rod patterns, and fuel pin powers are investigated. The 3-D nodal code, STORM, has been modified to perform these calculations. Several numerical examples lead to the following conclusions: (1) By employing a thermal leakage-to-absorption ratio11 (TLAR) approximation for the spatial shape of the thermal fluxes for the 3-D Langenbuch-Maurer- Werner (LMW) and the superpronipt critical transient problems, the convergence of the conventional two-group scheme is accelerated. The resulting computing time is reduced eight to fourteen times while maintaining computational accuracy. The TLAR acceleration scheme assumes that even if the thermal leakage rate and absorption rate varies significantly in a given node, the ratio of those two changes little during a small time step; (2) By employing the steepest-ascent hill climbing search with heuristic strategies, Optimum Control Rod Pattern Searcher (OCRPS) is developed for solving control rod positioning problem in BWRs. Using the method of approximation programming the objective function and the nuclear and thermal-hydraulic constraints are modified as heuristic functions that guide the search. The test calculations have demonstrated that, for the first cycle of the Edwin Hatch Unit #2 reactor, OCRPS shows excellent performance for finding a series of optimum control rod patterns for six burnup steps during the operating cycle. Computing costs are modest even if the initial guess patterns have extremely deteriorated core characteristics; and (3) For the modified two-dimensional EPRI-9R problem, the least square second-order polynomial flux expansion method was demonstrated to be computationally about 30 times faster than a fine-mesh finite difference calculation in order to achieve comparable accuracy for pin powers. The basic assumption of this method is that the reconstructed flux can be expressed as a product of an assembly form function and a second-order polynomial function. The assembly form function is calculated by solving an assembly criticality calculation. The polynomial function is determined by minimizing the least-squares difference between the intra-nodal quantities obtained from the global two-dimensional coarse-mesh nodal solutions and those obtained from the evaluation of the polynomial function.