Rounding error bounds by perturbation- condition analysis
Rounding error bounds by perturbation condition analysis, and conditioning for eigenvalue problem
ERROR↗
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Neutral air waves in the thermosphere
Neutral air waves in thermosphere - eigenvalues of characteristic waves
EIGENVALUE↗
What is H sub 0/ques/
Zero order Hamiltonian in perturbation theory for eigenvalue problems of atom in magnetic field
EIGENVALUE↗
The Givens-Householder method for symmetric matrices.
Givens-Householder method for computing eigenvalues and eigenvectors of real symmetric matrix
COMPUTER METHOD↗
On ''A method for simplifying linear dynamic systems.''
Equation for linear dynamic systems simplification applicable to all cases irrespective of nature of eigenvalues and eigenvectors
EIGENVALUE↗
Bounds for the natural frequencies of a simply supported beam carrying a uniformly distributed axial load.
Bounds for natural frequency eigenvalues of simply supported uniform beam with constant end load and uniformly distributed axial load
LOAD DISTRIBUTION↗
The initial value problem for the linearized equations of water waves.
Linearized system of water waves initial value problem, obtaining eigenvalue with parameter in boundary condition
LAPLACE EQUATION↗
Investigating the Feasibility and Stability for Modeling Broadband Acoustic Wave Scattering using a Time-Domain Boundary Integral Equation
Reducing aircraft noise is a major objective in the field of computational aeroacoustics. Acoustic liners are an effective tool for reducing aircraft noise and are characterized by a frequency-dependent impedance value. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. A Broadband Impedance Model is discussed in which the liner impedance is specified along a wide range of frequencies. An impedance boundary condition is derived and coupled with a time-domain boundary integral equation to model acoustic scattering by a flat plate. It is assumed the flat plate has surfaces treated with acoustic liners. The stability of the numerical algorithm is assessed using eigenvalue analysis.
mathmatics↗
INVESTIGATING THE FEASIBILITY AND STABILITY FOR MODELING ACOUSTIC WAVE SCATTERING USING A TIME-DOMAIN BOUNDARY INTEGRAL EQUATION WITH IMPEDANCE BOUNDARY CONDITION
Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet and environmentally friendly aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for aircraft noise reduction and are characterized by a frequency-dependent impedance. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. This work considers using either an impedance or an admittance (inverse of impedance) boundary condition to allow for acoustic scattering problems to be modeled with geometries consisting of both un-lined and lined surfaces. Three acoustic liner models are discussed: the Extended Helmholtz Resonator Model, the Three-Parameter Impedance Model, and the Broadband Impedance Model. In both the Helmholtz and Three-Parameter models, liner impedance is specified at a given frequency, whereas the Broadband model allows for the investigation of multiple frequencies simultaneously. The impedance and admittance boundary conditions for acoustic liners are derived for each model and coupled with a time-domain boundary integral equation. The scattering solution is obtained iteratively using a boundary element method with constant spatial and third-order temporal basis functions. Time-domain boundary integral equations are unfortunately prone to numerical instabilities due to resonant frequencies resulting from non-trivial solutions in the interior domain. When reformulated with the Burton-Miller method, the instabilities are eliminated. Using a Burton-Miller reformulation, the stability of the boundary element method assuming a liner boundary condition is assessed using eigenvalue analysis. The stability of each liner model is discussed, and it is shown that the Three-Parameter and Broadband models are sufficient for modeling an acoustic liner on the surface of scattering bodies. The Helmholtz model demonstrates strict limitations for stability, whereas the Three-Parameter and Broadband models are stable for most cases. Also included in this work is an assessment of the spatial accuracy of the time-domain boundary element method with respect to the surface element basis functions, as well as a performance study of the numerical algorithm.
computational mathematics↗
Analysis of SCALE Criticality and Sensitivity Calculations for Reflected HEU Cylinders [Abstract]
The SCALE code package offers several nuclear data libraries to support Monte Carlo (MC) transport, as well as MC-based derivation of $\kappa$ eff sensitivity and uncertainty (S/U) data. The CSAS sequence using the KENO MC code can utilize continuous-energy (CE) cross sections, or pre-generated multigroup (MG) cross section libraries. The use of MG libraries introduces bias into calculations in exchange for faster transport solutions. The TSUNAMI-3D sequence also utilizes KENO MC calculations. TSUNAMI-3D has two CE calculational methods: the Iterated Fission Probability (IFP) method, and the Contribution-Linked eigenvalue sensitivity/Uncertainty estimation via Tracklength importance CHaracterization (CLUTCH) method. Previous work has shown poor agreement between CLUTCH and confirmatory direct perturbation calculations in specific applications, e.g., fissionable and polyethylene reflectors
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Pin-by-Pin Core Calculation with an NEM-Based Two-Level Hybrid CMFD Algorithm
In the near future, systematic pin-by-pin core calculations are expected to provide a high level of accuracy. These are being developed in pursuit of high-fidelity tools for analyzing the reactor configurations. One such tool that has recently gained significant attention is the pin-by-pin core calculation via local-global iteration using nonlinear acceleration scheme, such as the One-Node CMFD (Coarse-Mesh Finite Difference) nonlinear algorithm. The One-Node CMFD method uses two correction factors to preserve the interface current, in contrast to the one correction term employed by conventional nonlinear iterative methods such as that proposed by Kord. S. Smith. These correction factors are updated with surface flux and net current which are obtained from a high order local problem. With two correction factors, One-Node CMFD can reduce computing time of local FMFD (Fine-Mesh Finite Difference) as it nonlinearly couples the CMFD and FMFD methods in global-local iterations. Nonetheless, very small mesh size is still required and it consequently requires long computing time for high accuracy. As such, a more flexible approach is needed to address this challenge. This paper presents one such possible approach named the hCMFD (Hybrid CMFD) method. In this method, One-Node CMFD is used to solve the global problem, while Two-Node NEM (Nodal Expansion Method) CMFD replaces FMFD as the solver for the local problem. In this paper, two kinds of CMFD methods has been simultaneously implemented for a global-local nonlinear iteration in order to do pin-by-pin heterogeneous reactor calculations. The global problem is solved by a one-node CMFD which uses two correction factors, while the local fixed-source problem is solved by the conventional two-node NEM CMFD which use one correction factor on a surface. From solution of the global eigenvalue problem, incoming partial current is obtained and it is used as boundary condition of the local fixed-source problems. It has been shown that over 99% of the whole calculations can be done in parallel with the new CMFD method for 2-D problems and the algorithm can be easily parallelized by using the OpenMP parallel scheme. The hCMFD method will enable an efficient pin-by-pin reactor analysis since a very efficient parallel calculation is possible with the hCMFD method. In future works, the hCMFD method will be extended to three-dimensional reactor problem. (authors)
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Validation of KENO Thermal Moderator Doppler Broadening Method in SCALE 6.2 Beta5 Using Continuous-Energy B-VII.1 Library
KENO, the Monte Carlo criticality safety program within the SCALE code package, has until recently used continuous-energy (CE) cross sections corresponding to the set of evaluated nuclear data closest to the temperature of interest. This method, which is used for both 1-D cross sections and thermal scattering probability data, does not fully represent the phenomenon of Doppler broadening (DB) and can result in significant differences in eigenvalues when the material temperature lies between the evaluated data sets. A method for creating temperature-dependent nuclear data has been implemented in KENO for CE calculations. This work summarizes the results of incorporating this method into two critical experiments using low-enriched uranium (LEU) in light water moderated pin lattices. The LEUCOMP-THERM-046 (LCT-046) experiment is a benchmark documented in the International Handbook of Evaluated Criticality Safety Benchmark Experiments (ICSBEP Handbook); it consists of 22 critical configurations or cases. The KRITZ-2:1 experiment, a part of the International Reactor Physics Experiment Evaluation Project (IRPhEP), consists of two cases which span a significant range of moderator temperatures. The results of this work give a good indication of the value of the newly incorporated cross section processing. However, more diverse evaluations should be performed to provide a broader validation of this method. The results suggest that when dealing with systems similar to LCT-046, there is a significant improvement made when the moderator scattering data are properly broadened and sampled. The DB of the hydrogen (in water) 2-D cross sections resulted in a consistent C/E value across the temperature range, whereas the DB of the 1-D only cross sections did not exhibit this improvement. With respect to criticality safety, this consistency adds precision to the calculational method, which could potentially impact subcriticality limits. After comparing the KRITZ and LCT-046 results, the most plausible conclusion to make as to why the added thermal moderator treatment improved the results for LCT-046 and not for the KRITZ experiment is that the two systems were dissimilar with respect to their temperatures. That is, the relative temperature differences among the cases in each experiment are more significant for LCT-046 than for KRITZ. The cross section changes as a function of temperature are also smaller at higher temperatures. This difference resulted in a lesser impact of broadening in general for the KRITZ hot case.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Neutron Number Probability Distributions in a Subcritical System Using the Forward Master Equation
The subject of this paper is the determination of the equilibrium probability distribution function (PDF) of the neutron population within a subcritical multiplying system in the presence of an intrinsic random source as a result of spontaneous fission (SF). Such systems are strongly stochastic and low order moments such as the mean and variance of the neutron number are not sufficient to characterize the true state of the neutron population. We base our work on the forward Master equation in a lumped model formulation for P{sub n} (t), n = 0, 1, 2,..., the probability of there existing η neutrons at time t and obtain solutions for the steady state distributions. The key parameters in this model, neutron lifetime and fission probability, are rigorously obtained by numerically solving the k-eigenvalue problem in a one-dimensional spherical system and using standard definitions of lumped parameters in terms of production and loss rates in finite systems. Previous work has examined single-chain PDFs (a single fission and the resulting progeny) for neutrons, uncoupled neutron-photon single-chain PDFs, both using a lumped backward Master equation formulation, as well as neutron PDFs in supercritical systems. In this paper, we analyze strictly subcritical systems, for which we compare analytically derived solutions with numerical computations for different multiplicity distribution models. For subcritical systems where the neutron population remains finite at steady state, we show that the forward Master equation can be very efficiently solved numerically to obtain the equilibrium distributions. Finally, this work is motivated by potential application to passive interrogation for detection of diverted nuclear materials as well to criticality safety. The equilibrium neutron number probability distribution in subcritical systems has been numerically investigated for several models of the induced and spontaneous fission multiplicity distribution. A well known approximate but closed form solution, namely the quadratic model, was shown to be unacceptably accurate for subcritical media. The matrix inversion and the recursion methods agree well and were benchmarked with the binary fission model, but the recursion formulation proves far more computationally efficient. A qualitative change in the shape of the neutron PDF is observed for increasing source strength, as expected, and a transition from a stochastic to a deterministic system is expected for a sufficiently strong source. We are currently extending this approach to obtain the equilibrium joint distribution of neutrons within the system and those that have leaked, to explicitly represent the neutron PDF as observed by an external detector. (authors)
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Prompt Neutron Decay Constants in a Highly Enriched Uranium-Lead Copper Reflected System
Rossi-{alpha} measurements were performed on highly enriched uranium (HEU)-lead copper reflected systems to determine the prompt neutron decay constants. These constants represent an eigenvalue characteristic of these particular critical configurations, which can be experimentally measured by the Rossi-a or pulse neutron source techniques and calculated by a deterministic or Monte Carlo method. In the measurements presented in this summary, highly enriched uranium interleaved with lead plates and reflected with copper were assembled to a high multiplication and the prompt neutron decay constants were determined by the Rossi-{alpha} technique. This research intended to benchmark the lead neutron cross sections; it was a collaboration between the Japanese Atomic Energy Agency (JAEA) Fast Critical Assembly Facility (FCA) in Tokai, Japan and the Los Alamos National Laboratory (LANL) in the U.S. (authors)
07 ISOTOPE AND RADIATION SOURCES↗
Application of NEM-based Pin-by-Pin HCMFD Algorithm to 3-D Core Analysis
A pin-wise reactor analysis is expected to provide a more accurate solution than the conventional nodal methods. However, a pin-wise approach is still computationally very costly though the computational performances have been enormously improved for decades. To overcome such limitation of a pin-by-pin core analysis, the HCMFD (Hybrid Course-Mesh Finite Difference) algorithm has been recently introduced. By adopting the HCMFD algorithm, a parallel computing can be applied very effectively. In the HCMFD algorithm, an iterative local-global strategy is adopted with a combination of two CMFD acceleration methods for the pin-wise core analysis. The global eigenvalue problem is solved by one-node CMFD, while the local fixed-source problems are solved by two-node CMFD based on the NEM (nodal expansion method). In the HCMFD computational scheme, a parallel computing can effectively be applied since most of the computational efforts are consumed to solve the local problems, and they can be solved in parallel. Previously, only a 2-D HCMFD algorithm was developed and its numerical efficiency was evaluated. In this paper, a 3D HCMFD scheme is introduced and some 3-D problems were analyzed to verify its applicability to 3D pin-wise core analysis. (authors)
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Coupled Neutron and Gamma Heating Calculation Based on VARIANT Nodal Transport Method
Accurate determination of temperature distributions in fuel pins and duct walls is essential for fuel-pin performance and assembly bowing analyses. Currently, the multi-assembly, sub-channel analysis code SE2-ANL is used in thermalhydraulic analysis of sodium cooled fast reactors (SFR). SE2- ANL is interfaced with the DIF3D/GAMSOR/DIF3D procedure for coupled neutron and gamma heating calculations, developed at Argonne National Laboratory (ANL). In this procedure, the neutron flux distribution is first calculated using the finite-difference diffusion theory option of the DIF3D code. Using the neutron flux solution, the GAMSOR code computes the gamma source distribution for the subsequent gamma flux calculation with DIF3D. However, in the current heating calculation scheme, the neutron and gamma flux calculations are limited to six triangular meshes per hexagonal assembly. In addition, the pin power distributions are estimated by assuming that the power distribution within an assembly is separable in the radial and axial directions. In each assembly, the axial power profile is obtained by averaging the power densities of six triangular meshes at each axial mesh interval. The radial power profile is evaluated at a single axial mesh (e.g., at the core mid-plane) by approximating the neutron and gamma flux distributions by a linear or quadratic polynomial of two variables x and y, depending on the polynomial order specified by the user. The coefficients of the linear and quadratic polynomials are determined by the least squares fit of six and 24 triangular mesh fluxes, respectively. Furthermore, the total heat generation rate in an assembly duct wall is simply determined by a user-specified fraction of the total gamma heating within that assembly. With the advancement of computer technology, the nodal transport theory code VARIANT is routinely used in core design analyses of SFR instead of DIF3D. It is therefore necessary to develop a new heating calculation procedure based on VARIANT neutron and gamma transport calculations. In this study, a new, coupled neutron and gamma heating calculation scheme has been developed to determine the pin power distributions more accurately by eliminating the aforementioned limitations of the current procedure. For this, the GAMSOR code has been extended to generate the intra-nodal gamma source distribution to be consistent with the trial functions of VARIANT. In the new heating calculation procedure, the neutron flux distribution is determined first by solving an eigenvalue transport problem using VARIANT. The obtained neutron flux is then used in GAMSOR to calculate the gamma source distribution. Using this gamma source distribution, the gamma flux distribution is determined by solving a fixed-source transport problem using VARIANT again. Using the computed neutron and gamma flux distributions, the power distribution in each fuel pin and duct wall is determined using a new utility code CURVE, which has been developed to evaluate the VARIANT flux solution at each fuel pin position and along the duct mid-wall. For each pin segment contained in a VARIANT node, the axial power shape is represented as a quadratic profile. The performance of the new heating calculation scheme based on VARIANT transport solutions has been tested against the existing method of SE2-ANL, using the ABR-1000 metal core design. The gamma source distribution and the assembly, fuel pin and duct wall power distributions were calculated and compared. (authors)
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Testing of graphite thermal neutron scattering law data in support of treat neutronic analysis
The Transient Reactor Test Facility (TREAT) has historically provided the capability for transient fuel performance and safety testing in the United States. The operation of this facility was suspended in 1994. However, in recent years preparations for the restart of TREAT were initiated. TREAT is a graphite moderated and air cooled reactor. Its core is composed of a 19 x 19 array of fuel assemblies that contain UO{sub 2} (HEU) fuel particles dispersed in a graphite matrix, along with reflector assemblies filled with graphite only. Each fuel assembly is 8 feet tall with a fueled region that is 4 feet tall and upper and lower regions of graphite that are 2 feet tall. In addition, a 2 feet thick radial graphite reflector region is present. Finally, a 5 feet thick concrete biological shield surrounds the core. Previous work showed that the graphite thermal scattering cross sections can introduce variations exceeding 2000 pcm in the predicted eigenvalue of the TREAT core. Recognizing that the graphite in the assembly (and thereby the core) is a porous mixture of crystalline and amorphous carbon (with a lower density in comparison to ideal crystalline graphite), it becomes clear that TREAT specific thermal neutron scattering cross section libraries will be needed (at all temperatures of interest) for use within neutronic codes seeking to simulate TREAT. (authors)
21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
Toward Linear Stability Analysis of Upper Plenum Flows
Linear stability analysis is a tool used to help identify the sensitivity of a particular domain to perturbations and evaluate the system dynamics as a transition, if any, occurs. The development of a time invariant base flow for future use in a linear stability analysis is the focus of this investigation. A test case, with a Reynolds number of 100, was brought to a steady state using the spectral element code Nek5000 and the method of selective frequency damping. The convergence criteria for this method is the volume averaged value of the forcing function which, after approximately 80,000 time-steps, decreased to 3.6x10{sup -4}. Analysis shows that the case has yet to have reached the tolerance desired, 1x10{sup -6}. With the simulation in the process of converging a preliminary perturbation analysis was conducted. Due to the low Reynolds number of this case, 100, it was hypothesized that the flow would be linearly stable. This preliminary stability analysis only focused on the largest fundamental mode and the energy growth of that perturbation over time. Upon evaluation of the perturbation energy, which was shown to be monotonically decreasing in time, it was concluded that, despite not yet being converged, the system is in fact linearly stable. Future work includes evaluating the domain at higher Reynolds numbers, solving the eigenvalue problem created when the linearized Navier-Stokes are written in operator form and calculating the energy evolution of the optimal perturbation. (authors)
42 ENGINEERING↗