John M. Christenson

Application of preconditioned transpose-free quasi-minimal residual method for two-group reactor kinetics

Two preconditioned transpose-free quasi-minimal residual methods (TFQMR) (Freund, SIAM J. Sci. Stat. Comput. 14, 470 1993) and quasi-minimal residual variant of the biconjugate gradient stabilized algorithm (QMRCGSTAB) (Chan et al., SIAM J. Sci. Stat. Comput. IS. 338 1994) are applied to solve the non-symmetric linear systems of equations which are derived from the time dependent two-dimensional two-energy-group neutron diffusion equations by finite difference approximation. We compare the TFQMR and QMRCGSTAB methods with the other popular method such as the generalized minimal residual method (GMRES), the conjugate gradient square method (CGS), and biconjugate gradient stabilized algorithm (Bi-CGSTAB). In order to accelerate the TFQMR and QMRCGSTAB we use the preconditioning technique. Two of the preconditioners are based on pointwise incomplete factorization: the incomplete factorization (ILU) and the modified incomplete factorization (MILU). Another two based on the block tridiagnal structure of the coefficient matrix are blockwise and modified blockwise incomplete factorizations, BILU and MBILU which are suitable for the system of partial differential equations such as two-energy-group neutron diffusion equations. Finally, the last two are the alternating-direction implicit (ADI) and block successive overrelaxation (BSOR) preconditioners which are derived from the basic iterative schemes. Comparisons are made by these methods combined with different preconditioners to solve a sequence of time steps reactor transient problems. Numerical results indicate that the preconditioner significantly affects the convergent rate TFQMR and QMRCGSTAB methods in three typical reactor kinetics test problems. Numerical experiments indicate that preconditioned QMRCGSTAB with the preconditioner MBILU requires fewer iterations than other methods in the three typical reactor kinetics test problems. Moreover, numerical results indicate that a good preconditioner can significantly improve the total iteration number (i.e. rate of convergence) of these generalized conjugate gradient methods, TFQMR, QMRCGSTAB, CGS, Bi-CGSTAB and GMRES. For preconditioners and MBILU and BILU, we find that all of the eigenvalues of preconditioned matrix are more clustered about 1 than the eigenvalues of other preconditioners in a typical reactor kinetics test problem. Such eigenvalue distribution is very favorable or the rate of convergence of these generalized conjugate gradient methods.

An Investigation of the Temporal Subdomain Method for Solving the Finite Element Formulation of the Space-Time Reactor Kinetics Equations

The temporal subdomain method (TSM), based on a spatial finite element formulation, is investigated as a method for the solution of the space-time-dependent multigroup neutron dynamics equations. The spatial aspect of the problem was formulated as an array of finite elements by using a two-dimensional rectangular coordinate system subdivided into contiguous triangular elements. Within each element and within each neutron group, the flux was represented by a linear polynomial. Numerical experiments using a computer program developed during the course of the investigation demonstrated that the method is straightforward to implement and that it produces stable calculations for a wide range of time steps. The stability of the method has been tested for sinusoidal, ramp, and step-changed reactivity insertions. The results show that the TSM outperforms most alternating direction implicit methods in the sense that a similar degree of accuracy can be achieved with larger time steps using the same number of nodes. System condition number calculations as a function of node number were also carried out for a series of static eigenvalue calculations to determine the likelihood of error propagation and the difficulty of inverting the global system matrices during the time-dependent calculations.

Application of optimal control theory to a load-following pressurized water reactor

In this paper, the control characteristics of a load-following pressurized water reactor are investigated through the application of a nonlinear optimization method to a simplified plant simulator. A model describing the power level control and power distribution control is developed and used to formulate an optimal control problem. In the optimal control problem formulation, all of the safety and system operating limits are included as hard constraints, and the multiple objective functionals are combined into a single performance index. The differences in the calculated optimal load-following control strategies are investigated for the cases of steady-state T[sub avg] (coolant average temperature) program operation and variable T[sub avg] operation at both beginning-of-cycle and near end-of-cycle conditions. The results show that the amount of boron control action for the demanded load variations can be significantly reduced when variable T[sub avg] operation is incorporated into the control policy.