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All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Uniform asymptotic approximations for duct eigenfunctions in a thin boundary layer flow

Analytical approximations for the acoustic modes in a duct carrying a uniform core flow with a thin shear layer at the walls are developed using the Method of Matched Asymptotic Expansions. Both two-dimensional and cylindrical duct propagation are considered. Numerical results for eigenvalues calculated using the theory are presented for the two dimensional problem and compared with results from earlier analyses. It is found that the new approximations yield a significant increase in accuracy.

Myers, M. K.↗
NASA NTRS1994

Analytic Sensitivities for Shape Optimization in Equivalent Plate Structural Wing Models

Equivalent plate modeling techniques based on Ritz analysis with simple polynomials prove to be efficient tools for structural modeling of wings in the preliminary design stage. Accuracy problems are encountered, however, when these models are used to obtain finite difference behavior sensitivities with respect to planform shape. The accuracy problems are associated with the poor numerical conditioning of static and eigenvalue equations. As higher-order polynomials are being used to Improve the analysis itself, the more sensitive is the finite difference derivative to the step size used. This article describes a formulation of wing equivalent plate modeling in which it is simple to obtain analytic, explicit expressions for stiffness and mass matrix elements without the need to perform numerical integration. This formulation leads naturally to analytic expressions for the derivatives of displacements, stresses, and natural frequencies with respect to shape design variables. This article examines the accuracy of finite difference derivatives compared with the analytic derivatives, and shows that In some cases it is impossible to obtain any information of value by finite differences. Analytic sensitivities, in this case, are still sufficiently accurate for design optimization.

Livne, Eli↗
NASA NTRS1960

Comparison of Turbulent Heat-Transfer Results for Uniform Wall Heat Flux and Uniform Wall Temperature

The purpose of this note is to examine in a more precise way how the Nusselt numbers for turbulent heat transfer in both the fully developed and thermal entrance regions of a circular tube are affected by two different wall boundary conditions. The comparisons are made for: (a) Uniform wall temperature (UWT); and (b) uniform wall heat flux (UHF). Several papers which have been concerned with the turbulent thermal entrance region problem are given. 1 Although these analyses have all utilized an eigenvalue formulation for the thermal entrance region there were differences in the choices of eddy diffusivity expressions, velocity distributions, and methods for carrying out the numerical solutions. These differences were also found in the fully developed analyses. Hence when making a comparison of the analytical results for uniform wall temperature and uniform wall heat flux, it was not known if differences in the Nusselt numbers could be wholly attributed to the difference in wall boundary conditions, since all the analytical results were not obtained in a consistent way. To have results which could be directly compared, computations were carried out for the uniform wall temperature case, using the same eddy diffusivity, velocity distribution, and digital computer program employed for uniform wall heat flux. In addition, the previous work was extended to a lower Reynolds number range so that comparisons could be made over a wide range of both Reynolds and Prandtl numbers.

Siegel, R.↗
DOE OSTI2018

Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization

The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small ε-size periodicity and a finite size defect we consider the critical ε{sup 2}-scaling for the contrast. We employ two-scale homogenization for deriving asymptotically explicit limit equations for the localized modes and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V. V. Zhikov with an emergent explicit nonlinear dependence on the spectral parameter for the spectral problem at the macroscale. Using the method of asymptotic expansions supplemented by a high contrast boundary layer analysis, we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with “ε square root” error bounds. An example for circular or spherical defects in a periodic medium with isotropic homogenized properties is given.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion

We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen–Loève expansion of Gaussian random fields. We develop, validate, and discuss an algorithm based on the Galerkin method with two-dimensional Haar wavelets as basis functions. The shape functions are constructed from the orthogonal decomposition of tensor-product spaces of one-dimensional Haar functions, and a recursive algorithm is employed to compute the matrix of the discrete eigenvalue system without the explicit calculation of integrals, allowing the implementation of a fast and efficient algorithm that provides considerable reduction in CPU time, when compared with classical Galerkin methods. Numerical experiments confirm the convergence rate of the method and assess the approximation error and the sparsity of the eigenvalue system when the wavelet expansion is truncated. We illustrate the numerical solution of a diffusion problem with random input data with the present method. In this problem, accuracy was retained after dropping the coefficients below a threshold value that was numerically determined. A similar method with scaling functions rather than wavelet functions does not need a discrete wavelet transform and leads to eigenvalue systems with better conditioning but lower sparsity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Implementation of a Pressure Correction Term in the CTF Subchannel Code

CTF is a modernized and improved version of the legacy subchannel code COBRA-TF, being jointly developed and maintained by Oak Ridge National Laboratory (ORNL) and North Carolina State University. The code was adopted by ORNL for use in the Consortium for Advanced Simulation of Light Water Reactors (CASL) in 2012 to help address CASL challenge problems. Since that time, activities related to CTF have included implementing software quality assurance measures, implementing new features and models, performing validation and verification testing, establishing and supporting a CTF user group, and developing the code for use in coupled applications. CTF is applicable to single- and two-phase flows in light water reactor geometries at normal and accident operating conditions. The two-phase flow model consists of the six-equation two-phase flow model augmented by equations for the droplet field, and it also includes appropriate source terms. This work focuses on the numerical stability of the six-equation two-phase flow model and is part of a recent effort to improve CTF capabilities for simulating flows in boiling water reactors (BWRs). The six-equation two-phase flow model is not strongly hyperbolic, meaning that some of its eigenvalues are either not real or not distinct. In a Cauchy sense, and for an initial-value problem, strong hyperbolicity was found to be equivalent to well-posedness, which ensures the uniqueness of the solution in the space-time used. This concept is also of importance when designing high-order numerical methods and also when implementing boundary conditions that rely on real eigenvalues such as Riemann solvers. A tremendous amount of research is available in the literature on how to hyperbolize the six-equation two-phase flow model by adding ad hoc terms to the system of equations. Such terms are often referred to as virtual mass or interfacial pressure correction terms. From a mathematical perspective, this approach adds terms to the phasic momentum equation containing first-order partial derivatives of the unknowns to modify the Jacobian matrix, thus the imaginary part of the eigenvalues. For this work, an interfacial pressure correction (IPC) term on the same model as what was used for the CATHARE system code, is implemented in CTF. This allows for recovery of a two-pressure formulation. In the following, the six-equation two-phase flow model is recalled, along with the IPC term and its influence on the eigenvalues. Then, numerical results and convergence studies are presented for a water-faucet problem at a pressure of 10 bar. Finally, conclusions and future work are addressed.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2021

Topology and many-body localization

We discuss the problem of localization in two dimensional electron systems in the quantum Hall (single Landau level) regime. After briefly summarizing the well-studied problem of Anderson localization in the non-interacting case, we concentrate on the problem of disorder induced many-body localization (MBL) in the presence of electron–electron interactions using numerical exact diagonalization and eigenvalue spacing statistics as a function of system size. We provide evidence showing that MBL is not attainable in a single Landau level with short range (white noise) disorder in the thermodynamic limit. We then study the interplay of topology and localization, by contrasting the behavior of topological and nontopological subbands arising from a single Landau level in two models — (i) a pair of extremely flat Hofstadter bands with an optimally chosen periodic potential, and (ii) a Landau level with a split-off nontopological impurity band. Both models provide convincing evidence for the strong effect of topology on the feasibility of many-body localization as well as slow dynamics starting from a nonequilibrium state with charge imbalance.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Integration of Online Cross-Section Generation Capability with Depletion and Transient Solvers in Griffin

Griffin is a Multiphysics Object-Oriented Simulation Environment (MOOSE)-based reactor multiphysics analysis application jointly developed by Argonne and Idaho National Laboratories under the DOENE Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. In FY25, an online crosssection generation capability based on the Self-Shielding Application Programming Interface (SSAPI) was demonstrated for TRISO-fueled reactor problems under steady-state conditions. This fiscal year, that capability was extended to support depletion and transient multiphysics calculations, enabling high-fidelity analyses that generate self-shielded cross sections on the fly from the actual evolving composition and temperature states rather than from pre-tabulated libraries. For depletion, a two-way coupling was established in which SSAPI computes compact-averaged self-shielded cross sections that the depletion solver then uses to advance the Bateman equations, with the updated compositions returned to SSAPI at each step; the depletion module was refactored to support both library-based and SSAPI-based cross sections, and additional logic was added to track daughter isotopes and to exclude minor isotopes for efficiency. For transient analysis, the SSAPI multigroup library was extended with the kinetics data required for time-dependent calculations, the Improved Quasi-Static (IQS) scheme was coupled with SSAPI, and several supporting capabilities were implemented, including a self-shielding treatment that lets control rods and drums move within a self-shielded model, which had previously been impossible and had ruled out rod- and drum-movement transients with on-the-fly cross sections altogether, a new mixing scheme for delayed-neutron precursor decay constants, a checkpoint-based restart workflow, and performance improvements such as pointwise cross-section interpolation and the bypassing of unnecessary Dancoff factor calculations. The implemented capabilities were verified against Serpent Monte Carlo solutions. For depletion, a prismatic pin-cell problem based on a Next Generation Nuclear Plant (NGNP) Very High Temperature Reactor benchmark showed excellent agreement, with eigenvalue differences within 200 pcm over the entire burnup range (up to 140 MWD/kgU) and fission-product and actinide inventories agreeing to within 0.8% and 2.5%, respectively; a heat-pipe microreactor assembly problem with a much higher fuel loading confirmed the same behavior and quantified the bias introduced when the multigroup equivalence effect is neglected. For transient analysis, a pin-cell problem with a step reactivity insertion and temperature feedback reproduced the analytically expected asymptotic power and showed close agreement between the direct and IQS solutions, and a two-dimensional microreactor core problem with control-drum rotation exercised the new moving-drum self-shielding treatment and demonstrated successful coupling of the online crosssection generation with both the direct and IQS transient methods. The capability was further exercised on a full-core pebble-bed problem, in which Griffin was coupled with the System Analysis Module (SAM) to simulate load-following operation of the gPBR with the Doppler feedback resolved at the TRISO fuel kernel temperature. These developments in Griffin provide a convenient, high-fidelity approach to cross-section generation for advanced thermal reactors with geometrically complex and highly heterogeneous configurations, including TRISO-fueled prismatic and pebble-bed systems, and support steady-state, depletion, and transient multiphysics calculations. They also enable self-shielded cross sections to be evaluated directly at the actual coupled state of the system, thereby establishing a foundation for high-fidelity, fully coupled multiphysics analysis of advanced reactors

Park, H.↗
DOE OSTI2019

Comment on: “Solving many-body Schrödinger equations with kinetic energy partition method”, Ann. Phys. 388 (2018) 54–68 by Y-H. Chen and S.D. Chao

We discuss a recent test of the performance of the kinetic energy partition method (KEP) through its application to two separable quantum-mechanical models. We argue that one of the benchmark models is exceedingly simple for testing any realistic approximate method and that almost any reasonable approach yields better results. In the second example our exact benchmark eigenvalues disagree considerably with those chosen by the authors for comparison, which casts doubts on the accuracy of their KEP approach.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

A non-modal analytical method to predict turbulent properties applied to the Hasegawa-Wakatani model

Linear eigenmode analysis often fails to describe turbulence in model systems that have non-normal linear operators and thus nonorthogonal eigenmodes, which can cause fluctuations to transiently grow faster than expected from eigenmode analysis. When combined with energetically conservative nonlinear mode mixing, transient growth can lead to sustained turbulence even in the absence of eigenmode instability. Since linear operators ultimately provide the turbulent fluctuations with energy, it is useful to define a growth rate that takes into account non-modal effects, allowing for prediction of energy injection, transport levels, and possibly even turbulent onset in the subcritical regime. We define such a non-modal growth rate using a relatively simple model of the statistical effect that the nonlinearities have on cross-phases and amplitude ratios of the system state variables. In particular, we model the nonlinearities as delta-function-like, periodic forces that randomize the state variables once every eddy turnover time. Furthermore, we estimate the eddy turnover time to be the inverse of the least stable eigenmode frequency or growth rate, which allows for prediction without nonlinear numerical simulation. We test this procedure on the 2D and 3D Hasegawa-Wakatani model [A. Hasegawa and M. Wakatani, Phys. Rev. Lett. 50, 682 (1983)] and find that the non-modal growth rate is a good predictor of energy injection rates, especially in the strongly non-normal, fully developed turbulence regime.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

A solvable model of a nonlinear extension of quantum mechanics

We introduce a particular nonlinear generalization of quantum mechanics which has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics problem. Here, we hope that this simple example will elucidate some of the issues of interpreting nonlinear generalization of quantum mechanics that have been put forth to resolve questions about quantum measurement theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Geometric Interpretation of a Non-Linear Extension of Quantum Mechanics

We recently introduced a particular non-linear generalization of quantum mechanics that has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics problem. In this paper, we suggest that the two components of the wave function represent the system described by the Hamiltonian H in two different asymptotic regions of spacetime and we show that the non-linear terms can be viewed as giving rise to gravitational effects.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1981

A quadratic weight selection algorithm

A new numerical algorithm is presented which determines a positive semi-definite state weighting matrix in the linear-quadratic optimal control design problem. The algorithm chooses the weighting matrix by placing closed-loop eigenvalues and eigenvectors near desired locations using optimal feedback gains. A simplified flight control design example is used to illustrate the algorithms capabilities.

Broussard, J. R.↗
NASA NTRS1994

Benchmark problems in computational aeroacoustics

A recent directive at NASA Langley is aimed at numerically predicting principal noise sources. During my summer stay, I worked with high-order ENO code, developed by Dr. Harold Atkins, for solving the unsteady compressible Navier-Stokes equations, as it applies to computational aeroacoustics (CAA). A CAA workshop, composed of six categories of benchmark problems, has been organized to test various numerical properties of code. My task was to determine the robustness of Atkins' code for these test problems. In one category, we tested the nonlinear wave propagation of the code for the one-dimensional Euler equations, with initial pressure, density, and velocity conditions. Using freestream boundary conditions, our results were plausible. In another category, we solved the linearized two-dimensional Euler equations to test the effectiveness of radiation boundary conditions. Here we utilized MAPLE to compute eigenvalues and eigenvectors of the Jacobian given variable and flux vectors. We experienced a minor problem with inflow and outflow boundary conditions. Next, we solved the quasi one dimensional unsteady flow equations with an incoming acoustic wave of amplitude 10(exp -6). The small amplitude sound wave was incident on a convergent-divergent nozzle. After finding a steady-state solution and then marching forward, our solution indicated that after 30 periods the acoustic wave had dissipated (a period is time required for sound wave to traverse one end of nozzle to other end).

Porter-Locklear, Freda↗
DOE OSTI2023

Validation of time-dependent shift using the pulsed sphere benchmarks

The detailed behavior of neutrons in a rapidly changing time-dependent physical system is a challenging computational physics problem, particularly when using Monte Carlo methods on heterogeneous high-performance computing architectures. A small number of algorithms and code implementations have been shown to be performant for time-independent (fixed source and k-eigenvalue) Monte Carlo, and there are existing simulation tools that successfully solve the time-dependent Monte Carlo problem on smaller computing platforms. To bridge this gap, a time-dependent version of ORNL’s Shift code has been recently developed. Shift’s history-based algorithm on CPUs, and its event-based algorithm on GPUs, have both been observed to scale well to very large numbers of processors, which motivated the extension of this code to solve time-dependent problems. The validation of this new capability requires a comparison with time-dependent neutron experiments. Lawrence Livermore National Laboratory’s (LLNL) pulsed sphere benchmark experiments were simulated in Shift to validate both the time-independent as well as new time-dependent features recently incorporated into Shift. A suite of pulsed-sphere models was simulated using Shift and compared to the available experimental data and simulations with MCNP. Overall results indicate that Shift accurately simulates the pulsed sphere benchmarks, and that the new time-dependent modifications of Shift are working as intended. Validated exascale neutron transport codes are essential for a wide variety of future multiphysics applications.

Palmer, Camille J.↗
DOE OSTI2017

Implementation of a pressure correction term in the CTF sub-channel code

CTF [1] is a modernized and improved version of the legacy subchannel code, COBRA-TF [2], that is being jointly developed and maintained by Oak Ridge National Laboratory (ORNL) and North Carolina State University. The code was adopted by ORNL for use in the Consortium for Advanced Simulation of Light Water Reactors (CASL) in 2012 for aiding in addressing CASL challenge problems. Since that time, activities related to CTF have included implementing software quality assurance measures, implementing new features and models, performing validation and verification testing, establishing and supporting a CTF User Group, and developing the code for use in coupled applications [3, 4, 5]. CTF is applicable to single- and two-phase flows in light water reactor geometries at normal and accident operating conditions. The two-phase flow model consists of the six-equation two-phase flow model [6] augmented by equations for the droplet field and also includes appropriate source terms. This work focuses on the numerical stability of the six-equation two-phase flow model and is part of a recent eort to improve CTF capabilities for simulating flows in boiling nuclear reactors (BWRs). As it is of common knowledge among the thermodynamic community, the six-equation two-phase flow model is not strongly hyperbolic, meaning some of its eigenvalues are either not real or not distinct [7]. In a Cauchy sense and for an initial-value problem, strong hyperbolicity was found to be equivalent to well-posedness which ensures uniqueness of the solution in the space-time used. This concept is also of importance when designing high-order numerical methods and also when implementing boundary conditions that rely on real eigenvalues such as Riemann solvers. A tremendous amount of research is available in the literature on how to hyperbolize the six-equation two-phase flow model by adding ad hoc terms to the system of equations often referred to as virtual mass or interfacial pressure correction term. From a mathematical prospective, the main idea driving this approach is to add terms to the phasic momentum equation containing first-order partial derivatives of the unknowns to modify the Jacobian matrix and thus canceling the imaginary part of the eigenvalues.For this work it was chosen to implement a interfacial pressure correction (IPC) term on the same model as what was done for the CATHARE system code [8], that allows to recover a two-pressure formulation. In the following, the six-equation two-phase flow model is recalled along with the IPC term and its influence on the eigenvalues. Then, numerical results and convergence studies are presented for a water-faucet problem [9] at a pressure of 10 bar. Finally, conclusions and future work are addressed.

Delchini, Marco↗
DOE OSTI2018

Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines

In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadership-class computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates ill-conditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadership-class machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. Finally, this solver set is a strong choice for very large and challenging problems.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1993

Flutter analysis using transversality theory

A new method of calculating flutter boundaries of undamped aeronautical structures is presented. The method is an application of the weak transversality theorem used in catastrophe theory. In the first instance, the flutter problem is cast in matrix form using a frequency domain method, leading to an eigenvalue matrix. The characteristic polynomial resulting from this matrix usually has a smooth dependence on the system's parameters. As these parameters change with operating conditions, certain critical values are reached at which flutter sets in. Our approach is to use the transversality theorem in locating such flutter boundaries using this criterion: at a flutter boundary, the characteristic polynomial does not intersect the axis of the abscissa transversally. Formulas for computing the flutter boundaries and flutter frequencies of structures with two degrees of freedom are presented, and extension to multi-degree of freedom systems is indicated. The formulas have obvious applications in, for instance, problems of panel flutter at supersonic Mach numbers.

Afolabi, D.↗