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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

The Eigenvector-Eigenvalue Identity and other pragmatic topics in linear algebra for physicists

Diagonalization of an Hermitian matrix is a common task in physics. All of us have diagonalized 2x2 matrices but few have diagonalized a 3x3 matrix algebraically except in special simplifying cases. In this colloquium, I will discuss the mathematics and methods for diagonalizing small, but larger than 2x2 marices, and discuss the recently rediscovered Eigenvector-Eigenvalue identity. As an explicit, pragmatic example I will use the propagation of neutrino's propagating through matter which is inherently a 3x3 problem.

Parke, Stephen [Fermilab] (ORCID:0000000320286782)↗
NASA NTRS2000

Model Refinement Using Eigensystem Assignment

IA novel approach for the refinement of finite-element-based analytical models of flexible structures is presented. The proposed approach models the possible refinements in the mass, damping, and stiffness matrices of the finite element model in the form of a constant gain feedback with acceleration, velocity, and displacement measurements, respectively. Once the free elements of the structural matrices have been defined, the problem of model refinement reduces to obtaining position, velocity, and acceleration gain matrices with appropriate sparsity that reassign a desired subset of the eigenvalues of the model, along with partial mode shapes, from their baseline values to those obtained from system identification test data. A sequential procedure is used to assign one conjugate pair of eigenvalues at each step using symmetric output feedback gain matrices, and the eigenvectors are partially assigned, while ensuring that the eigenvalues assigned in the previous steps are not disturbed. The procedure can also impose that gain matrices be dissipative to guarantee the stability of the refined model. A numerical example, involving finite element model refinement for a structural testbed at NASA Langley Research Center (Controls-Structures-Interaction Evolutionary model) is presented to demonstrate the feasibility of the proposed approach.

Maghami, Peiman G.↗
NASA NTRS1989

Using deflation in the pole assignment problem with output feedback

A direct algorithm is suggested for the computation of a linear output feedback for a multi input, multi output system such that the resultant closed-loop matrix has eigenvalues that include a specified set of eigenvalues. The algorithm uses deflation based on unitary similarity transformations. Thus researchers hope the algorithm is numerically stable; however, this has not been proven as yet.

Miminis, George↗
DOE OSTI2016

Implementation and testing of the on-the-fly thermal scattering Monte Carlo sampling method for graphite and light water in MCNP6

Here, a proper treatment of thermal neutron scattering requires accounting for chemical binding through a scattering law S(α,β,T). Monte Carlo codes sample the secondary neutron energy and angle after a thermal scattering event from probability tables generated from S(α,β,T) tables at discrete temperatures, requiring a large amount of data for multiscale and multiphysics problems with detailed temperature gradients. We have previously developed a method to handle this temperature dependence on-the-fly during the Monte Carlo random walk using polynomial expansions in 1/T to directly sample the secondary energy and angle. In this paper, the on-the-fly method is implemented into MCNP6 and tested in both graphite-moderated and light water-moderated systems. The on-the-fly method is compared with the thermal ACE libraries that come standard with MCNP6, yielding good agreement with integral reactor quantities like k-eigenvalue and differential quantities like single-scatter secondary energy and angle distributions. The simulation runtimes are comparable between the two methods (on the order of 5–15% difference for the problems tested) and the on-the-fly fit coefficients only require 5–15 MB of total data storage.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2016

Monte Carlo Techniques for Nuclear Systems - Theory Lectures

These are lecture notes for a Monte Carlo class given at the University of New Mexico. The following topics are covered: course information; nuclear eng. review & MC; random numbers and sampling; computational geometry; collision physics; tallies and statistics; eigenvalue calculations I; eigenvalue calculations II; eigenvalue calculations III; variance reduction; parallel Monte Carlo; parameter studies; fission matrix and higher eigenmodes; doppler broadening; Monte Carlo depletion; HTGR modeling; coupled MC and T/H calculations; fission energy deposition. Solving particle transport problems with the Monte Carlo method is simple - just simulate the particle behavior. The devil is in the details, however. These lectures provide a balanced approach to the theory and practice of Monte Carlo simulation codes. The first lectures provide an overview of Monte Carlo simulation methods, covering the transport equation, random sampling, computational geometry, collision physics, and statistics. The next lectures focus on the state-of-the-art in Monte Carlo criticality simulations, covering the theory of eigenvalue calculations, convergence analysis, dominance ratio calculations, bias in Keff and tallies, bias in uncertainties, a case study of a realistic calculation, and Wielandt acceleration techniques. The remaining lectures cover advanced topics, including HTGR modeling and stochastic geometry, temperature dependence, fission energy deposition, depletion calculations, parallel calculations, and parameter studies. This portion of the class focuses on using MCNP to perform criticality calculations for reactor physics and criticality safety applications. It is an intermediate level class, intended for those with at least some familiarity with MCNP. Class examples provide hands-on experience at running the code, plotting both geometry and results, and understanding the code output. The class includes lectures & hands-on computer use for a variety of Monte Carlo calculations. Beginning MCNP users are encouraged to review LA-UR-09-00380, "Criticality Calculations with MCNP: A Primer (3nd Edition)" (available at http:// mcnp.lanl.gov under "Reference Collection") prior to the class. No Monte Carlo class can be complete without having students write their own simple Monte Carlo routines for basic random sampling, use of the random number generator, and simplified particle transport simulation.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2018

Solving many-body Schrödinger equations with kinetic energy partition method

We present a general formulation of our previously developed kinetic energy partition (KEP) method for solving many-bodySchrödinger equations. In atomic physics, as well as in general molecular and solid state physics, solving many-electronSchrödinger equations is a very challenging task, often called Dirac’s challenge. The central problem is how to properly handle the electron–electron Coulomb repulsion interactions. Using the KEP solution scheme, in addition to dividing the kinetic energy into partial terms, the electron–electron Coulomb interaction is also separated into parts to be associated with a “negative mass” kinetic energy term. Therefore, the full Hamiltonian can be expressed as a simple sum of subsystem Hamiltonians, each representing an effective one-body problem. Using a Hartree-like product in constructing the wave-function, we achieve fast convergence in the calculations of the ground state energies. First, the model Moshinsky atoms are used to illustrate the solution procedure. We then apply this new KEP method to harmonium atoms and obtain precise energies with an error less than 5% using only two basis functions from each subsystem. It is thus very promising that this methodology, when further extended, can be useful for general many-body systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Quantum advantages for Pauli channel estimation

We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an n-qubit Pauli channel to ±ε precision. We give an estimation protocol with an n-qubit ancilla that succeeds with high probability using only O(n/ε 2 ) copies of the Pauli channel, while proving that any ancilla-free protocol (possibly with adaptive control and channel concatenation) would need at least Ω(2 n/3 ) rounds of measurement. We further study the advantages provided by a small number of ancillas. For the case that a k-qubit ancilla (k≤n) is available, we obtain a sample complexity lower bound of Ω(2 (n-k)/3 ) for any nonconcatenating protocol, and a stronger lower bound of Ω(n 2n-k ) for any nonadaptive, nonconcatenating protocol, which is shown to be tight. We also show how to apply the ancilla-assisted estimation protocol to a practical quantum benchmarking task in a noise-resilient and sample-efficient manner, given reasonable noise assumptions. Our results provide a practically interesting example for quantum advantages in learning and also bring insights for quantum benchmarking.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1973

The inference of atmospheric ozone using satellite nadir measurements in the 1042/cm band

A description and detailed analysis of a technique for inferring atmospheric ozone information from satellite nadir measurements in the 1042 cm band are presented. A method is formulated for computing the emission from the lower boundary under the satellite which circumvents the difficult analytical problems caused by the presence of atmospheric clouds and the watervapor continuum absorption. The inversion equations are expanded in terms of the eigenvectors and eigenvalues of a least-squares-solution matrix, and an analysis is performed to determine the information content of the radiance measurements. Under favorable conditions there are only two pieces of independent information available from the measurements: (1) the total ozone and (2) the altitude of the primary maximum in the ozone profile.

Russell, J. M., III↗
DOE OSTI2022

Advanced Quantum Poisson Solver in the NISQ era

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far, either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to the linear systems through the finite difference method, we adopt the Harrow-Hassidim-Lloyd (HHL) algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. We show that our algorithm not only increases the accuracy of the solutions, but also composes more practical and scalable circuits by dynamically controlling problem size in the NISQ devices. We present both simulated and experimental results, and discuss the sources of errors. Finally, we conclude that overall results on the quantum hardware are dominated by the error in the CNOT gates.

Robson, Walter↗
NASA NTRS1984

Recent advances in methods for numerical solution of O.D.E. initial value problems

In the mathematical modeling of physical systems, it is often necessary to solve an initial value problem (IVP), consisting of a system of ordinary differential equations (ODE). A typical program produces approximate solutions at certain mesh points. Almost all existing codes try to control the local truncation error, while the user is really interested in controlling the true or global error. The present investigation provides a review of recent advances regarding the solution of the IVP, giving particular attention to stiff systems. Stiff phenomena are customarily defined in terms of the eigenvalues of the Jacobian. There are, however, some difficulties connected with this approach. It is pointed out that an estimate of the Lipschitz constant proves to be a very practical way to determine the stiffness of a problem.

Bui, T. D.↗
DOE OSTI2025

Improvement and Verification of Online Cross Section Generation Capability of Griffin for TRISO-fueled Reactors

Griffin, a MOOSE-based reactor multiphysics code jointly developed by Idaho National Laboratory and Argonne National Laboratory under the DOE Office of Nuclear Energy’s NEAMS program, has pursued the development of an online multigroup cross section generation capability for a few years to enable high-fidelity, problem-dependent neutronics analyses of advanced thermal reactors. Recent advancements in Griffin’s online multigroup cross section generation capability have significantly improved the accuracy, robustness, and efficiency of self-shielding calculations for both prismatic and pebble-bed TRISO-fueled reactor applications. Key developments include a unified fuel self-shielding method applicable to both TRISO and annular compact/spherical shell fuel zone geometries; an advanced Dancoff Category-based Equivalence Theory using a bell function for non-fuel resonance treatment, achieving more than an order-of-magnitude speedup compared to the Tone method; an on-the-fly multigroup equivalence approach to mitigate group condensation errors; and a streaming correction method for pebble-bed homogenization. A proof-of-concept demonstration of on-the-fly group condensation with consistent P0 transport correction was also achieved. The method reproduced direct fine-group solutions with excellent accuracy (eigenvalue errors within 10 pcm and pin-power differences within 0.5%), but due to performance limitations of the current fixed-source solver, improvements to solver efficiency will be addressed in future work. Verification tests were performed on graphite-moderated TRISO-fueled two-dimensional core benchmark problems representing gas-cooled microreactors, heat pipe-cooled microreactors, gas-cooled pebble-bed reactors, and fluoride salt-cooled high-temperature reactors. Across all cases, Griffin showed excellent agreement with Serpent2 continuous energy Monte Carlo solutions: eigenvalue errors within 200 pcm, pin-power root-mean-square errors within 2%, and control rod and drum worth errors less than 2%. It should be noted that, for the benchmark problem, cross section generation contributed less than 3% of the total simulation times. These results demonstrate that Griffin’s online cross section generation capability delivers accurate and efficient reactor physics solutions across a wide spectrum of TRISO-fueled advanced reactor designs. With further improvements to the fine-group fixed-source solver and planned extensions to depletion, transients, and coupled neutron–gamma transport, Griffin will be well-positioned to become a powerful and comprehensive tool for advanced reactor analysis.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2016

On numerical solutions to the QCD ’t Hooft equation in the limit of large quark mass

First we give a short informal introduction to the theory behind the ’t Hooft equation. Then we consider numerical solutions to this equation in the limit of large fermion masses. It turns out that the spectrum of eigenvalues coincides with that of the Airy differential equation. Moreover when we take the Fourier transform of eigenfunctions, they look like the corresponding Airy functions with appropriate symmetry. It is known that these functions correspond to solutions of a one dimensional Schrodinger equation for a particle in a triangular potential well. So we find the analogy between this problem and the ’t Hooft equation. We also present a simple intuition behind these results.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

The Iterated Fission Matrix Method

The fission matrix (FM) method has been successfully demonstrated for different applications. One of the interesting features of this method, which is not possible via standard criticality source Monte Carlo calculations, is the capability to calculate higher forward and adjoint Eigenmodes; however, the accuracy of the estimates for higher eigenvalues strongly depends on the spatial discretization adopted for the fission matrix. A fine mesh ensures accurate results but requires matrices of large size. This is considered a limitation to the applicability of the FM method due to the memory requirements for storage of large full matrices. The adoption of sparse matrix storage (e.g., in MCNP) partially overcomes this problem; nonetheless, the sparse matrix indexing increases the complexity of the implementation of the FM algorithm and introduces the additional task of sparsifying the fission matrix. This summary presents a new method, named the iterated fission matrix (IFM), that allows to estimate higher eigenvalues of multiplying systems on much smaller meshes, but with the same level of accuracy in terms of spatial discretization errors, as compared to the FM method. The following Sections describe this new approach and provide a demonstration using two case studies: a PWR fuel assembly, and a 2D PWR core. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2014

Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple right-hand sides

The technique that was used to build the EigCG algorithm for sparse symmetric linear systems is extended to the nonsymmetric case using the BiCG algorithm. We show that, similarly to the symmetric case, we can build an algorithm that is capable of computing a few smallest magnitude eigenvalues and their corresponding left and right eigenvectors of a nonsymmetric matrix using only a small window of the BiCG residuals while simultaneously solving a linear system with that matrix. For a system with multiple right-hand sides, we give an algorithm that computes incrementally more eigenvalues while solving the first few systems and then uses the computed eigenvectors to deflate BiCGStab for the remaining systems. Our experiments on various test problems, including Lattice QCD, show the remarkable ability of EigBiCG to compute spectral approximations with accuracy comparable to that of the unrestarted, nonsymmetric Lanczos. Furthermore, our incremental EigBiCG followed by appropriately restarted and deflated BiCGStab provides a competitive method for systems with multiple right-hand sides.

Abdel-Rehim, A M↗
DOE OSTI2020

Multilevel Convergence Analysis of Multigrid-Reduction-in-Time

This study presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time-stepping operator, analyzed directly and bounded in the norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time-stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge--Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge--Kutta schemes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Gas-Cooled High-Temperature Pebble-Bed Reactor Reference Plant Model

This report details a fully coupled neutronics thermal hydraulics reference plant model for a gas-cooled high-temperature pebble-bed reactor. The multiphysics model is developed on the Nuclear Regulatory Commission's Comprehensive Reactor Analysis Bundle (BlueCRAB) available on the Idaho National Laboratory high performance computer, which natively and seamlessly couples Griffin, Pronghorn, and the BISON MOOSE-based applications. Griffin provides the reactor physics capabilities including depletion to the equilibrium core, k-eigenvalue, adjoint, and transient solutions. Pronghorn solves the porous medium equations for the fluid regions and conduction in the solid regions. BISON solves thermal conduction problems for the average pebble and TRISO in the pebble-bed core thus providing the fuel and moderator spatial fields for each pebble burnup group. The neutronics feedback relies primarily on fuel and moderator temperatures. We present results for the coupled steady state equilibrium core and a protected loss of flow event. Although this model is to be considered prototypical with regard to the capabilities in BlueCRAB, its results are consistent with published work by the Institute of Nuclear and New Energy Technology (INET) in China and other research entities.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1995

Minimization of the Vibration Energy of Thin-Plate Structures and the Application to the Reduction of Gearbox Vibration

While the vibration analysis of gear systems has been developed, a systematic approach to the reduction of gearbox vibration has been lacking. The technique of reducing vibration by shifting natural frequencies is proposed here for gearboxes and other thin-plate structures using the theories of finite elements, modal analysis, and optimization. A triangular shell element with 18 degrees of freedom is developed for structural and dynamic analysis. To optimize, the overall vibration energy is adopted as the objective function to be minimized at the excitation frequency by varying the design variable (element thickness) under the constraint of overall constant weight. Modal analysis is used to determine the sensitivity of the vibration energy as a function of the eigenvalues and eigenvectors. The optimum design is found by the gradient projection method and a unidimensional search procedure. By applying the computer code to design problems for beams and plates, it was verified that the proposed method is effective in reducing vibration energy. The computer code is also applied to redesign the NASA Lewis gear noise rig test gearbox housing. As one example, only the shape of the top plate is varied, and the vibration energy levels of all the surfaces are reduced, yielding an overall reduction of 1/5 compared to the initial design. As a second example, the shapes of the top and two side plates are varied to yield an overall reduction in vibration energy of 1/30.

Inoue, Katsumi↗
NASA NTRS1996

Classical Trajectories and Quantum Spectra

A classical model of the Schrodinger's wave packet is considered. The problem of finding the energy levels corresponds to a classical manipulation game. It leads to an approximate but non-perturbative method of finding the eigenvalues, exploring the bifurcations of classical trajectories. The role of squeezing turns out decisive in the generation of the discrete spectra.

Mielnik, Bogdan↗