E. E. Lewis

Monte Carlo simulation of Markov unreliability models

Abstract A Monte Carlo method is formulated for the evaluation of the unrealibility of complex systems with known component failure and repair rates. The formulation is in terms of a Markov process allowing dependencies between components to be modeled and computational efficiencies to be achieved in the Monte Carlo simulation. Two variance reduction techniques, forced transition and failure biasing, are employed to increase computational efficiency of the random walk procedure. For an example problem these result in improved computational efficiency by more than three orders of magnitudes over analog Monte Carlo. The method is generalized to treat problems with distributed failure and repair rate data, and a batching technique is introduced and shown to result in substantial increases in computational efficiency for an example problem. A method for separating the variance due to the data uncertainty from that due to the finite number of random walks is presented.

Simplified spherical harmonics in the variational nodal method

The multigroup simplified spherical harmonics equations with anisotropic scattering are derived from a variational principle that preserves nodal balance. The resulting equations are discretized using a Ritz procedure with spatial trial functions that are complete polynomials within the nodes and on the interfaces. The resulting equations are case in a response matrix form and incorporated as an option of the variational nodal spherical harmonics code VARIANT. Fixed source and multigroup eigenvalue calculations are performed on benchmark problems. The accuracy and computational efficiency of spherical harmonic and simplified spherical harmonic approximations are compared, and the compensating effects of spatial and angular truncation errors are examined. The results indicate that in most situations, simplified and standard spherical harmonics results of the same order are in close agreement, while the use of simplified spherical harmonics substantially reduces computing costs.

Spatial adaptivity applied to the variational nodal Pn equations

Abstract A spatial adaptive grid method is presented for the solution of two-dimensional neutron transport problems employing the spherical harmonics method within the framework of the variational nodal method. The work represents the generalization of an approach previously applied to the neutron diffusion equation. After reviewing pertinent aspects of the derivation of the variational nodal response matrices, an a posteriori estimator of the local error in the scalar flux is developed. An iterative adaptive procedure is then presented, and application is made to two-dimensional problems. Results are presented for a P5 solution of the well-known Iron-Water Benchmark Problem.

Load-capacity interference and the bathtub curve

Load-capacity (stress-strength) interference theory is used to derive a heuristic failure rate for an item subjected to repetitive loading which is Poisson distributed in time. Numerical calculations are performed using Gaussian distributions in load and capacity. Infant mortality, constant failure rate (Poisson failures), and aging are shown to be associated with capacity variability, load variability, and capacity deterioration, respectively. Bathtub-shaped failure rate curves are obtained when all three failure types are present. Changes in load or capacity distribution parameters often strongly affect the quantitative behavior of the failure-rate curves, but they do not affect the qualitative behavior of the bathtub curve. Neither is it likely that the qualitative behavior will be affected by the use of nonGaussian distributions. The numerical results, however, indicate that infant mortality and wear-out failures interact strongly with load variability. Thus bathtub curves arising from this model cannot be represented as simple superpositions of independent contributions from the three failure types. Only if the three failure types arise from independent failure mechanisms or in different components is it legitimate simply to sum the failure rate contributions. >

Variational nodal transport methods with anisotropic scattering

The variational nodal method is generalized to treat within-group and group-to-group anisotropic scattering in two- and three-dimensional eigenvalue and fixed source problems. The resulting formalism is implemented as the VARIational Anisotropic Nodal Transport code (VARIANT) within the shell of the Argonne National Laboratory production code DIF3D. The code is applied to a series of Cartesian and hexagonal geometry model problems and the accuracy of the results compared to those from TWODANT and TWOHEX and to the Monte Carlo code VIM, respectively, in two and three dimensions. VARIANT is then applied to multigroup hexagonal representations of the Experimental Breeder Reactor II, and results are obtained for three-dimensional eigenvalue and for two-dimensional neutron-gamma heating problems.

A load-capacity interference model for common-mode failures in 1-out-of-2: G systems

Load-capacity (stress-strength) interference theory is used to model the time-dependent behavior of a 1-out-of-2: G redundant system and to examine common-mode failures. For single units subjected to Poisson distributed load arrivals, the random failures, infant mortality, and aging are associated with load variability, capacity variability, and capacity deterioration, respectively. This paper extends the analysis to a redundant system by using a Markov model to treat Poisson distributed loads arriving at units simultaneously. Loss of s-independence of the unit failures is analyzed with joint PDFs of load and capacity. In the rare-event approximation, the degree of redundancy loss is characterized by expressing the coefficient in the beta-factor method in terms of the load and capacity distributions.

An adaptive approach to variational nodal diffusion problems

Abstract An adaptive grid method is presented for the solution of neutron diffusion problems in two dimensions. The primal hybrid finite elements employed in the variational nodal method are used to reduce the diffusion equation to a coupled set of elemental response matrices. An a posteriori error estimator is developed to indicate the magnitude of local errors stemming from the low-order elemental interface approximations. An iterative procedure is implemented in which p refinement is applied locally by increasing the polynomial order of the interface approximations. The automated algorithm utilizes the a posteriori estimator to achieve local error reductions until an acceptable level of accuracy is reached throughout the problem domain. Application to a series of X-Y benchmark problems indicates the reduction of computational effort achievable by replacing uniform with adaptive refinement of the spatial approximations.

An integral form of the variational nodal method

Abstract An integral form of the variational nodal method is formulated, implemented, and tested. The method combines an integral transport treatment of the even-parity flux within the spatial node with an odd-parity spherical harmonics expansion of the Lagrange multipliers at the node interfaces. The response matrices that result from this formulation are compatible with those in the VARIANT code at Argonne National Laboratory. Spatial discretization within each node allows for accurate treatment of homogeneous or heterogeneous node geometries. The integral method is implemented in Cartesian x-y geometry and applied to three benchmark problems. The method’s accuracy is compared to that of the standard spherical harmonic formulation of the variational nodal method, and the CPU and memory requirements of the two approaches are compared and contrasted. In general, for calculations requiring higher-order angular approximations, the integral method yields solutions with comparable accuracy while requiring substantially less CPU time and memory than the spherical harmonics approach.

A Finite Subelement Generalization of the Variational Nodal Method

Abstract The variational nodal method is generalized by dividing each spatial node into a number of triangular finite elements designated as subelements. The finite subelement trial functions allow for explicit geometry representations within each node, thus eliminating the need for nodal homogenization. The method is implemented within the Argonne National Laboratory code VARIANT and applied to two-dimensional multigroup problems. Eigenvalue and pin-power results are presented for a four-assembly Organization for Economic Cooperation and Development/Nuclear Energy Agency benchmark problem containing enriched UO2 and mixed oxide fuel pins. Our seven-group model combines spherical or simplified spherical harmonic approximations in angle with isoparametric linear or quadratic subelement basis functions, thus eliminating the need for fuel-coolant homogenization. Comparisons with reference seven-group Monte Carlo solutions indicate that in the absence of pin-cell homogenization, high-order angular approximations are required to obtain accurate eigenvalues, while the results are substantially less sensitive to the refinement of the finite subelement grids.

The Application of the Finite Element Method to the Multigroup Neutron Diffusion Equation

The finite element method is applied to the multigroup neutron diffusion equations. The one-group inhomogeneous diffusion equation is first discretized using both triangular and rectangular element...