Fourier analysis by the method of selected ordinates
Selected ordinates for Fourier analysis on periodic curve
Hallock, J. N.↗
SEARCH · Engineering Papers
Results for “Ordinate method”
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.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
High-energy /less than 400-Mev/ neutron transport using the method of discrete ordinates.
High energy neutron transport calculated using one dimensional discrete ordinates code with anisotropic scattering, comparing results with nucleon transport code calculations
Alsmiller, R. G., Jr.↗
Low-Energy Electron Transport with the Method of Discrete Ordinates
One-dimensional code adaptation for electron transport
Bartine, D. E.↗
Discrete-ordinates finite-element method for atmospheric radiative transfer and remote sensing
The principal features of the discrete-ordinates finite-element method are reviewed, and the applicability of general-purpose discrete-ordinates codes to atmospheric radiative transfer and remote sensing problems is demonstrated. In particular, numerical results for typical problems arising in meteorology, climatology, and remote sensing are shown to be in good agreement with results from other methods and measurements. A sample two-dimensional calculation demonstrates that specific capabilities available in the discrete-ordinates code TWOTRAN can produce new results that are valuable in the characterization of atmospheric effects on remote sensing (e.g., the adjacency effect). The intrinsic limitations of the method are also considered, and it is concluded that the strengths of the discrete-ordinates finite-element method outweigh its weaknesses.
Gerstl, S. A. W.↗
Radiative transfer of visible radiation in turbid atmosphere
Methods are presented for solving radiative transfer problems; they include the doubling method and the closely related matrix method, iterative method, Chandrasekhar's method of discrete ordinates, and Monte Carlo method. To consider radiation transport through turbid atmosphere, an atmospheric model was developed characterizing aerosols by parameters. Intensity and polarization of radiation in turbid atmospheres is discussed, as well as lower atmospheric heating due to solar radiation absorption by aerosols.
Yamamoto, G.↗
Improvements to the Griffin Transport Solvers
Griffin is a Multiphysics Object-Oriented Simulation Environment (MOOSE) based reactor multiphysics analysis application jointly developed by Idaho National Laboratory and Argonne National Laboratory. The code includes a variety of steady-state solvers for fixed-source, k-eigenvalue, adjoint, and subcritical multiplication, as well as transient solvers for point-kinetics, improved quasi-static, and spatial dynamics. This document summarizes the transport solver development efforts pursued during Fiscal Year 2022. We added the multiphysics transient capability for the coarse-mesh finite difference accelerated Richardson iteration for discontinuous finite element method discrete ordinates (DFEM-SN) scheme to support high-order heterogeneous transport simulations. HFEM (hybrid finite element method) - PN (spherical harmonics expansion) was completed and red-black iteration was added for solving the HFEM-PN system with both preconditioned Jacobian-free Newton Krylov and Richardson iteration solvers. The HFEM-PN scheme, as one of the low-order transport schemes, is expected for supporting routine design simulations. Pin power reconstruction capability was also designed and implemented with the Griffin ISOXML module to enhance all the low-order transport solvers for more accurate multiphysics simulations. Numerical results are presented for demonstrating the capabilities and verifying their performance, and future works are discussed.
97 MATHEMATICS AND COMPUTING↗
First-collision source treatment for ray effect mitigation in discrete-ordinate radiation transport solutions
Deterministic transport codes play a fundamental role in the modelling and simulation of neutron transport. One of the most common deterministic methods is the method of discrete ordinates, also known as the S method. While offering significant advantages over other deterministic methods or stochastic methods like Monte Carlo, the method of discrete ordinates suffers from non-physical artifacts in its local solution due to its discretization of angle. These artifacts, referred to as ray effects because of their ray-like appearance, tend to be worse in problems with small sources in areas with little scattering. Significant effort has gone into developing methods to mitigate ray effects, such as the first-collision source treatment, which separates the angular flux into the uncollided and collided fluxes and solving them using non-traditional techniques such as ray tracing. One such code capable of doing this is Lawrence Livermore National Laboratory's deterministic transport code ARDRA. Current ray tracing methods typically trace to a set of points inside a zone to compute an overall flux. However, this approach has significant drawbacks, such as a low order of convergence and not being conservative. Therefore, a new method has been developed that traces instead to a set of points on each of a zone's surfaces and computing the currents, before using these to obtain the flux. A comparison between these two ray tracing methods showed significant advantages to the new surface method, including inherent conservation, a higher convergence rate, and an increase in calculable information like leakage. This work performed under the auspices of the U.S. Department of Energy by Lawrence Liver- more National Laboratory under Contract DE-AC52-07NA27344. (authors)
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scattering
Here we present an error analysis for the discontinuous Galerkin (DG) method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation with isotropic scattering. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior and boundary layer errors. Some numerical tests are also provided in this reduced setting.
97 MATHEMATICS AND COMPUTING↗
Error analysis of numerical methods for thick diffusive neutron transport problems on Shishkin mesh
A thin layer will develop at the boundary if the incoming angular flux is anisotropic in thick diffusive neutron transport problems. Solving such singularly perturbed problems, which have non-smooth solutions with singularity near the boundary, is computationally challenging. Standard finite difference schemes on a uniform mesh cannot yield ε-uniform convergence, where ε is a small parameter, while it can be achieved on a suitable piecewise-uniform Shishkin mesh. We present a formal error analysis of the diamond difference (DD) method and step difference (SD) method for solving the S{sub N} neutron transport equation. The analysis can be extended to other finite difference methods. Numerical results are presented to confirm the error estimates and the advantages of the Shishkin mesh. (author)
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Discovering strongly lensed quasar candidates with catalogue-based methods from DESI Legacy Surveys
The Hubble tension, revealed by a ~5σ discrepancy between measurements of the Hubble-Lemaitre constant among observations of the early and local Universe, is one of the most significant problems in modern cosmology. In order to better understand the origin of this mismatch, independent techniques to measure H 0 , such as strong lensing time delays, are required. Notably, the sample size of such systems is key to minimising the statistical uncertainties and cosmic variance, which can be improved by exploring the datasets of large-scale sky surveys such as Dark Energy Spectroscopic Instrument (DESI). We identify possible strong lensing time-delay systems within DESI by selecting candidate multiply imaged lensed quasars from a catalogue of 24 440 816 candidate QSOs contained in the ninth data release of the DESI Legacy Imaging Surveys (DESI-LS). Using a friend-of-friends-like algorithm on spatial co-ordinates, our method generates an initial list of compact quasar groups. This list is subsequently filtered using a measure of the similarity of colours among a group’s members and the likelihood that they are quasars. A visual inspection finally selects candidate strong lensing systems based on the spatial configuration of the group members. We identified 620 new candidate multiply imaged lensed quasars (101 grade-A, 214 grade-B, 305 grade-C). This number excludes 53 known spectroscopically confirmed systems and existing candidate systems identified in other similar catalogues. When available, these new candidates will be further checked by combining the spectroscopic and photometric data from DESI.
79 ASTRONOMY AND ASTROPHYSICS↗
Historical review and proof-of-concept future method demonstration of adaptive mesh refinement in nuclear engineering for increased fidelity and computational efficiency
As the nuclear industry's use of computational tool increases, the need for increased fidelity and computational efficiency is well known. While most approaches to increased fidelity rely on applying a fine mesh over the problem domain, a more efficient method is to apply an adaptive mesh refinement (AMR) algorithm to the mesh definition. In the field of nuclear engineering, AMR has previously been used in conjunction with deterministic methods, including: S{sub N} transport methods, Lattice Boltzmann Methods, and COMSOL. The future of AMR in nuclear engineering is to couple it to a Monte Carlo code with the goal of reducing calculation time. A proof-of-concept example yielded positive results for using the gradient of the flux as a refinement criteria. The refinement criteria was varied from 0.01 to 0.10, which yielded a recommended range of 0.01 to 0.04, and the number of refinement iterations was varied from 0 to 7, with diminishing returns seen after 5 iterations. After the success of the proof-of-concept exercise, work began on creating a full program coupling MCNP6.2 and the AMR algorithm in the deal.II library. (authors)
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
The MP{sub N} method: a new angular discretization method based on piecewise polynomial interfaces fluxes
In transport calculations, it is well known how S{sub N} method is extremely inefficient in problems where the particle physics is dominated by streaming. The ray-effect eventually produced by the insufficient angular discretization, appears to be extremely persistent with respect to the refinement of the angular quadrature. The MP{sub N} method, that relies on continuous angular representation, offers a robust remedy to such an issue. MP{sub N} is based on the decomposition of the unit sphere into solid angles and on a piecewise continuous definition of interface fluxes, which are expanded in polynomials in each solid angle. This allows propagating more than one angular degree of freedom simultaneously while maintaining unaltered the block-diagonal pattern of the displacement plus removal operator. The method is therefore well suited for the flux resolution by means of a conventional sweep algorithm. Furthermore, unlike the S{sub N} method, MP{sub N} does not rely on discrete directions and, thus, on an angular quadrature formula, but rather constructs a set of linear equations solving for the angular moments of the flux for all discrete solid angles within the sweep. MP{sub N} shows an error convergence rate higher than S{sub N} at the expense of an increased size of the coefficient matrices, so of the computational cost. Although MP{sub N} is not free from ray-effect, the latter is effectively mitigated and less persistent with respect to the increase of the angular refinement order. (authors)
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
The Effect of the Flux Separability Approximation on Multigroup Neutron Transport
The angular dependence of flux-weighted multigroup cross sections is commonly neglected when generating multigroup libraries. The error of this flux separability approximation is typically not isolated from other error sources due to a lack of availability of library generation and corresponding solvers that cannot relax this approximation. These errors can now be isolated and quantified with the availability of a multigroup Monte Carlo transport and multigroup library-generation capability in the OpenMC Monte Carlo transport code. This work will discuss relevant details of the OpenMC implementation, provide an example case useful for detailing the type of errors one can expect from making the flux separability approximation, and end with more realistic problems which show the impact of the approximation and highlight how it can strongly arise from an energy-dependent resonance absorption effect. Since the angle-dependence is intrinsically linked to the energy group structure, these examples also show that relaxing the flux separability approximation with angle-dependent cross sections could be used to reduce either the fine-tuning required to set a multigroup energy structure for a specific reactor type or the number of energy groups required to obtain a desired level of accuracy for a given problem. This trade-off could increase the costs of generating multigroup cross sections, and has the potential to require more memory for storing the multigroup library during the transport calculations, but it can significantly reduce the computational time required since the runtime of a discrete ordinates or method of characteristics neutron transport solver scales roughly linearly with the number of groups.
42 ENGINEERING↗
Bidirectional Reflectance of a Macroscopically Flat, High-Albedo Particulate Surface: An Efficient Radiative Transfer Solution and Applications to Regoliths
Many remote sensing applications rely on accurate knowledge of the bidirectional reflection function (BRF) of surfaces composed of discrete, randomly positioned scattering particles. Theoretical computations of BRFs for plane-parallel particulate layers are usually reduced to solving the radiative transfer equation (RTE) using one of existing exact or approximate techniques. Since semi-empirical approximate approaches are notorious for their low accuracy, violation of the energy conservation law, and ability to produce unphysical results, the use of numerically exact solutions of RTE has gained justified popularity. For example, the computation of BRFs for macroscopically flat particulate surfaces in many geophysical publications is based on the adding-doubling (AD) and discrete ordinate (DO) methods. A further saving of computer resources can be achieved by using a more efficient technique to solve the plane-parallel RTE than the AD and DO methods. Since many natural particulate surfaces can be well represented by the model of an optically semi-infinite, homogeneous scattering layer, one can find the BRF directly by solving the Ambartsumian's nonlinear integral equation using a simple iterative technique. In this way, the computation of the internal radiation field is avoided and the computer code becomes highly efficient and very accurate and compact. Furthermore, the BRF thus obtained fully obeys the fundamental physical laws of energy conservation and reciprocity. In this paper, we discuss numerical aspects and the computer implementation of this technique, examine the applicability of the Henyey-Greenstein phase function and the sigma-Eddington approximation in BRF and flux calculations, and describe sample applications demonstrating the potential effect of particle shape on the bidirectional reflectance of flat regolith surfaces. Although the effects of packing density and coherent backscattering are currently neglected, they can also be incorporated. The FORTRAN implementation of the technique is available on the World Wide Web, and can be applied to a wide range of remote sensing problems. BRF computations for undulated (macroscopically rough) surfaces are more complicated and often rely on time consuming Monte Carlo procedures. This approach is especially inefficient for optically thick, weakly absorbing media (e.g., snow and desert surfaces at visible wavelengths since a photon may undergo many internal scattering events before it exists the medium or is absorbed. However, undulated surfaces can often be represented as collections of locally flat tilted facets characterized by the BRF found from the traditional plane parallel RTE. In this way the MOnte Carlo procedure could be used only to evaluate the effects of surface shadowing and multiple surface reflections, thereby bypassing the time-consuming ray tracing inside the medium and providing a great savings of CPU time.
Mishchenko, Michael I.↗
ESAS-Derived Earth Departure Stage Design for Human Mars Exploration
The Vision for Space Exploration has set the nation on a course to have humans on Mars as early as 2030. To reduce the cost and risk associated with human Mars exploration, NASA is planning for the Mars architecture to leverage the lunar architecture as fully as possible. This study takes the defined launch vehicles and system capabilities from ESAS and extends their application to DRM 3.0 to design an Earth Departure Stage suitable for the cargo and crew missions to Mars. The impact of a propellant depot in LEO was assessed and sLzed for use with the EDS. To quantitatively assess and compare the effectiveness of alternative designs, an initial baseline architecture was defined using the ESAS launch vehicles and DRM 3.0. The baseline architecture uses three NTR engines, LH2 propellant, no propellant depot in LEO, and launches on the Ares I and Ares V. The Mars transfer and surface elements from DRM 3.0 were considered to be fixed payloads in the design of the EDS. Feasible architecture alternatives were identified from previous architecture studies and anticipated capabilities and compiled in a morphological matrix. ESAS FOMs were used to determine the most critical design attributes for the effectiveness of the EDS. The ESAS-derived FOMs used in this study to assess alternative designs are effectiveness and performance, affordability, reliability, and risk. The individual FOMs were prioritized using the AHP, a method for pairwise comparison. All trades performed were evaluated with respect to the weighted FOMs, creating a Pareto frontier of equivalently ideal solutions. Additionally, each design on the frontier was evaluated based on its fulfillment of the weighted FOMs using TOPSIS, a quantitative method for ordinal ranking of the alternatives. The designs were assessed in an integrated environment using physics-based models for subsystem analysis where possible. However, for certain attributes such as engine type, historical, performance-based mass estimating relations were more easily employed. The elements from the design process were integrated into a single loop, allowing for rapid iteration of subsystem analyses and compilation of resulting designs.
Flaherty, Kevin↗
High-order diamond differencing schemes for the Boltzmann Fokker-Planck equation in 3D Cartesian geometries
The Boltzmann Fokker-Planck, an approximate form of the linear Boltzmann equation is commonly used to treat efficiently the transport of charged particles in matter. This paper introduces the application of high-order diamond differencing schemes (HODD), specifically the DD1 and DD2 schemes which are 4- and 6-order accurate respectively, to handle the spatial discretization of that equation in 3D Cartesian geometries. The energy deposition solutions for the coupled transport of electrons and photons presented in this work shows that HODD, compared to classical DD scheme, provides correction to the oscillations and a reduced propensity to yield negative fluxes. They are useful tools to minimize local error, notably in regions with abrupt variations of the flux solution. They also can be used to reduced execution time by decreasing the needed number of voxels to obtain a fixed accuracy. On the tested benchmarks, the DD1 scheme is 87%- 92%-91% more accurate than the classical DD scheme for total, mean per-voxels and maximum deviation of energy deposition values respectively. For comparison, a calculation with 8 times more voxels, requiring roughly 2.5 times more time to execute, is 92%-90%-77% more accurate. (authors)
97 MATHEMATICS AND COMPUTING↗
Fast and Accurate Hybrid Stream PCRTMSOLAR Radiative Transfer Model for Reflected Solar Spectrum Simulation in the Cloudy Atmosphere
A hybrid stream PCRTM-SOLAR model has been proposed for fast and accurate radiative transfer simulation. It calculates the reflected solar (RS) radiances with a fast coarse way and then, with the help of a pre-saved matrix, transforms the results to obtain the desired high accurate RS spectrum. The methodology has been demonstrated with the hybrid stream discrete ordinate (HSDO) radiative transfer (RT) model. The HSDO method calculates the monochromatic radiances using a 4-stream discrete ordinate method, where only a small number of monochromatic radiances are simulated with both 4-stream and a larger N-stream (N ≥ 16) discrete ordinate RT algorithm. The accuracy of the obtained channel radiance is comparable to the result from N-stream moderate resolution atmospheric transmission version 5 (MODTRAN5). The root-mean-square errors are usually less than 5x10(exp -4) mW/sq cm/sr/cm. The computational speed is three to four-orders of magnitude faster than the medium speed correlated-k option MODTRAN5. This method is very efficient to simulate thousands of RS spectra under multi-layer clouds/aerosols and solar radiation conditions for climate change study and numerical weather prediction applications.
Yang, Qiguang↗
Data-driven acceleration of thermal radiation transfer calculations with the dynamic mode decomposition and a sequential singular value decomposition
In this work, we present a method for accelerating discrete ordinates radiative transfer calculations for radiative transfer. Our method works with nonlinear positivity fixes, in contrast to most acceleration schemes. The method is based on the dynamic mode decomposition (DMD) and using a sequence of rank-one updates to compute the singular value decomposition needed for DMD. Using a sequential method allows us to automatically determine the number of solution vectors to include in the DMD acceleration. We present results for slab geometry discrete ordinates calculations with the standard temperature linearization. Compared with positive source iteration, our results demonstrate that our acceleration method reduces the number of transport sweeps required to solve the problem by a factor of about 3 on a standard diffusive Marshak wave problem, a factor of several thousand on a cooling problem where the effective scattering ratio approaches unity, and a factor of 20 improvement in a realistic, multimaterial radiating shock problem.
97 MATHEMATICS AND COMPUTING↗