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DOE OSTI2021

Solution Irregularity Remediation for Spatial Discretization Error Estimation for S N Transport Solutions

The discrete ordinates linear Boltzmann transport equation is typically solved in its spatially discretized form, incurring spatial discretization error. Quantification of this error for purposes such as adaptive mesh refinement or error analysis requires an a posteriori estimator, which utilizes the numerical solution to the spatially discretized equation to compute an estimate. Because the quality of the numerical solution informs the error estimate, irregularities, present in the true solution for any realistic problem configuration, tend to cause the largest deviation in the error estimate vis-a-vis the true error. In this paper, an analytical partial singular characteristic tracking (pSCT) procedure for reducing the estimator’s error is implemented within our novel residual source estimator for a zeroth-order discontinuous Galerkin scheme, at the additional cost of a single inner iteration. Here, a metric-based evaluation of the pSCT scheme versus the standard residual source estimator is performed over the parameter range of a Method of Manufactured Solutions test suite. The pSCT scheme generates near-ideal accuracy in the estimate in problems where the dominant source of the estimator’s error is the solution irregularity, namely, problems where the true solution is discontinuous and problems where the true solution’s first derivative is discontinuous and the scattering ratio is low. In problems where the scattering ratio is high and the true solution is discontinuous in the first derivative, the error in the scattering source, which is not converged by the pSCT scheme, is greater than the error incurred due to the irregularity. Ultimately, a pSCT scheme is judged to be useful for error estimation in problems where the computational cost of the scheme is justified. In the presence of many irregularities, such a scheme may be intractable for general use, but in benchmarks, as an analytical tool, or in problems that have nondissipative discontinuities, the scheme may prove invaluable.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1979

Magnetic field directional discontinuities. 1: Minimum variance errors

Errors associated with the minimum variance analysis of directional discontinuity normal components were investigated using both computer simulation of discontinuities with controlled properties and the examination of current sheets observed by the Mariner 10 spacecraft. The simulated discontinuities were created by adding fluctuations, represented by isotropic noise, to exactly known but varying (in a plane) magnetic field components. An empirical expression for the magnitude of the error in an estimated discontinuity normal component, relative to the total field across the discontinuity, was derived, as well as other relevant statistical properties. Use of the empirical relation in the analysis of 644 discontinuities observed by Mariner 10 provides a more precise, but probably conservative, estimate of an upper bound on the relative normal component value for tangential discontinuities that can be used to separate rotational from tangential discontinuities in studies using only magnetic field data from a single spacecraft, at least for the interplanetary region of space considered.

Lepping, R. P.↗

NASA NTRSDate not supplied

A Deep Neural Network for Achieving Spectrally Consistent and Seamless Infrared Radiance Measurements Across Geostationary Satellite Domains

The NASA Clouds and the Earth's Radiant Energy System (CERES) project provides the scientific community with observed top-of-atmosphere (TOA) shortwave and longwave fluxes for climate monitoring and climate model validation. To achieve this goal, CERES relies on TOA broadband fluxes derived from geostationary satellite (GEO) imagery to account for the diurnal flux variations between the CERES observation intervals. Consistent global flux derivation depends on accurate and consistent cloud retrievals. Scene-dependent spectral measurement inconsistency of the instruments that make up the contiguous ring of GEO observations (GEO-Ring), as well as limb darkening effects, can cause discontinuities in derived cloud properties and radiative fluxes at the boundaries of adjacent imager domains. Although the algorithms utilize radiative transfer models to account for instrument-band-dependent atmospheric correction and viewing zenith angle (VZA) dependency, small discontinuities may persist due to uncertainties inherent to the multiple imager-specific algorithms. Furthermore, while hyperspectral-instrument-based spectral band adjustment factors may effectively account for spectrally induced bias, they are less effective at reducing variance owed to the specific composition of the viewed scene, which is challenging to robustly characterize. As such, this article highlights the use of a deep neural network (DNN) to resolve spectral-and VZA-induced biases between GEO-Ring imagers. The DNN uses available infrared (IR) channels from the GEO instruments, along with viewing and solar illumination geometry, to estimate homogenized, VIIRS-like IR radiances for use in the GEO cloud algorithm. This approach is effective at mitigating scene-dependent spectral variance and VZA dependency, resulting in consistent radiance measurements across the GEO-Ring, thereby leading toward a more seamless global cloud assessment.

deep learning↗

NASA NTRS2000

Experiment Evaluation of Bifurcation in Sands

The basic principles of bifurcation analysis have been established by several investigators, however several issues remain unresolved, specifically how do stress level, grain size distribution, and boundary conditions affect general bifurcation phenomena in pressure sensitive and dilatant materials. General geometrical and kinematics conditions for moving surfaces of discontinuity was derived and applied to problems of instability of solids. In 1962, the theoretical framework of bifurcation by studying the acceleration waves in elasto-plastic (J2) solids were presented. Bifurcation analysis for more specific forms of constitutive behavior was examined by studying localization in pressure-sensitive, dilatant materials, however, analyses were restricted to plane deformation states only. Bifurcation analyses were presented and applied to predict shear band formations in sand under plane strain condition. The properties of discontinuous bifurcation solutions for elastic-plastic solids under axisymmetric and plane strain loading conditions were studied. The study focused on theory, but also references and comparisons to experiments were made. The current paper includes a presentation of a summary of bifurcation analyses for biaxial and triaxial (axisymmetric) loading conditions. The Coulomb model is implemented using incremental piecewise scheme to predict the constitutive relations and shear band inclination angles. Then, a comprehensive evaluation of bifurcation phenomena is presented based on data from triaxial experiments performed under microgravity conditions aboard the Space Shuttle under very low effective confining pressure (0.05 to 1.30 kPa), in which very high peak friction angles (47 to 75 degrees) and dilatancy angles (30 to 31 degrees) were measured. The evaluation will be extended to include biaxial experiments performed on the same material under low (10 kPa) and moderate (100 kPa) confining pressures. A comparison between the behavior under biaxial and triaxial loading conditions will be presented, and related issues concerning influence of confining pressure will be discussed.

Alshibi, Khalid A.↗

DOE OSTI2022

Kink solutions with power law tails

We present a comprehensive review about the various facets of kink solutions with a power law tail, which have received considerable attention during the last few years. This area of research is in its early stages; although several aspects have become clear by now, there are a number of issues which have only been partially understood or not understood at all. We first discuss the aspects which are reasonably well known and then address in some detail the issues which are only partially or not understood at all. We present a wide class of higher (than sixth) order field theory models admitting implicit kink as well as mirror kink solutions where the two tails facing each other have a power law or a power-tower type fall off, whereas the other two ends not facing each other could have either an exponential or a power law tail. The models admitting implicit kink solutions where the two ends facing each other have an exponential tail while the other two ends have a power law tail are also discussed. Moreover, we present several field theory models which admit explicit kink solutions with a power law fall off; we note that in all these polynomial models while the potential V ( ϕ ) is continuous, its derivative is discontinuous. We also discuss one of the most important and only partially understood issues of the kink–kink and the kink–antikink forces in case the tails facing each other have a power law fall off. Finally, we briefly discuss the kink–antikink collisions at finite velocity and present some open questions.

power-tower tail↗

NASA NTRS1987

Quantitative analysis of the reconstruction performance of interpolants

The analysis presented provides a quantitative measure of the reconstruction or interpolation performance of linear, shift-invariant interpolants. The performance criterion is the mean square error of the difference between the sampled and reconstructed functions. The analysis is applicable to reconstruction algorithms used in image processing and to many types of splines used in numerical analysis and computer graphics. When formulated in the frequency domain, the mean square error clearly separates the contribution of the interpolation method from the contribution of the sampled data. The equations provide a rational basis for selecting an optimal interpolant; that is, one which minimizes the mean square error. The analysis has been applied to a selection of frequently used data splines and reconstruction algorithms: parametric cubic and quintic Hermite splines, exponential and nu splines (including the special case of the cubic spline), parametric cubic convolution, Keys' fourth-order cubic, and a cubic with a discontinuous first derivative. The emphasis in this paper is on the image-dependent case in which no a priori knowledge of the frequency spectrum of the sampled function is assumed.

Lansing, Donald L.↗

NASA NTRSDate not supplied

Efficient Basis Derivatives Evaluations for High-order Discontinuous Galerkin Schemes

Computational methods of evaluating a basis of Lagrange polynomials are developed for the purpose of implementing efficient discontinuous Galerkin conservation laws solvers. Special attention is payed to the computation of higher-order partial derivatives which may be required for certain applications. Two different approaches are considered; one involves hardcoding explicit one-line formulae into the source code in order to make it as simple as possible, whereas the other uses algorithms designed to minimize the asymptotic order of growth with respect to the order of the scheme. Timing experiments show that either approach can perform well if implemented effectively, and identify advantages and disadvantages of each.

Micaiah Smith-Pierce↗

NASA NTRS2002

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

Yan, Jue↗

NASA NTRSDate not supplied

Efficient Basis Derivatives Evaluations for High-order Discontinuous Galerkin Schemes

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Micaiah Smith-Pierce↗

NASA NTRS1986

Global collocation methods for approximation and the solution of partial differential equations

Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.

Solomonoff, A.↗

NASA NTRSDate not supplied

Fusion of Hyperspectral Sounder Products Via Spectral Fingerprinting Methodology

Satellite based measurements of top-of-atmosphere (TOA) spectral radiances in the infrared (IR) region have been in existence for almost two decades and are expected to be continued in the following decades. The data from multiple hyper-spectral IR sounders can therefore be combined to build a long-term data record to further global scale climate trend research. Challenges associated with the fusion of data from different sensors come from the stability and consistency requirement on the climate record. The direct radiance observations from different sounders need to be homogenized by reconciling the differences in calibration, spectral response function (SRF), and spatial-temporal sampling. When geophysical variables derived from radiances measured by different sounders are combined to form long-term climate records, the impacts of any inconsistencies between overlapping measurements on the retrieval must be carefully assessed in order to estimate the uncertainty of the corresponding climate anomalies/trends derived. This paper presents a novel climate fingerprinting methodology and establishes a rigorously-defined inverse relationship that allows us to efficiently evaluate the change in essential climate variables from the change in spectral radiances measured in prescribed spatial and temporal averaging scales. The inverse spectral fingerprinting relationship is constructed based on a unified spectral kernel scheme, providing a direct means for quantifying the potential discontinuity in the derived climate anomalies due to inconsistencies between overlapping measurements. We show in this paper a sample application of using the spectral fingerprinting scheme to derive long-term, global-scale surface temperatures from the Climate Hyperspectral Infrared Radiance Product (CHIRP) and quantify the inter-satellite biases.

Wan Wu↗

NASA NTRS1994

An introduction to generalized functions with some applications in aerodynamics and aeroacoustics

In this paper, we start with the definition of generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support. The concept of generalization differentiation is introduced next. This is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some of the results of classical analysis, such as Leibniz rule of differentiation under the integral sign and the divergence theorem, are derived using the generalized function theory. It is shown that the divergence theorem remains valid for discontinuous vector fields provided that the derivatives are all viewed as generalized derivatives. This implies that all conservation laws of fluid mechanics are valid as they stand for discontinuous fields with all derivatives treated as generalized deriatives. Once these derivatives are written as ordinary derivatives and jumps in the field parameters across discontinuities, the jump conditions can be easily found. For example, the unsteady shock jump conditions can be derived from mass and momentum conservation laws. By using a generalized function theory, this derivative becomes trivial. Other applications of the generalized function theory in aerodynamics discussed in this paper are derivation of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of finite part of divergent integrals, derivation of Oswatiitsch integral equation of transonic flow, and analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics presented here include the derivation of the Kirchoff formula for moving surfaces,the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Farassat, F.↗

NASA NTRS1993

A variable-order laminated plate theory based on the variational-asymptotical method

The variational-asymptotical method is a mathematical technique by which the three-dimensional analysis of laminated plate deformation can be split into a linear, one-dimensional, through-the-thickness analysis and a nonlinear, two-dimensional, plate analysis. The elastic constants used in the plate analysis are obtained from the through-the-thickness analysis, along with approximate, closed-form three-dimensional distributions of displacement, strain, and stress. In this paper, a theory based on this technique is developed which is capable of approximating three-dimensional elasticity to any accuracy desired. The asymptotical method allows for the approximation of the through-the-thickness behavior in terms of the eigenfunctions of a certain Sturm-Liouville problem associated with the thickness coordinate. These eigenfunctions contain all the necessary information about the nonhomogeneities along the thickness coordinate of the plate and thus possess the appropriate discontinuities in the derivatives of displacement. The theory is presented in this paper along with numerical results for the eigenfunctions of various laminated plates.

Lee, Bok W.↗

NASA NTRS2017

The Effects of Normal Metal Stripes on TES Performance

Exploring the effects of size and geometry of normal metal features on the transition shapes and performance of transition-edge sensor microcalorimeters. The spectral resolution of transition-edge sensor (TES) microcalorimeters is very sensitive to the specific dependencies of the resistance R in the superconducting transition on the current I, magnetic field B, and temperature T. In particular, it has been shown that transitions that are very steep in (R,T) space lead to a significant noise term, in excess of conventional expectations. This so-called unexplained noise is known to be reduced by the addition of normal metal stripes across the TES perpendicular to the direction of current flow. These normal metal stripes have been shown to drastically alter the oscillatory patterns seen in measurements of the critical current as a function of magnetic field. However, there are many remaining questions about the exact impact of the stripes on current distributions within the TES, the Fraunhofer pattern and, therefore, the shape of the R(I, B, T) surface. Through measurements of the resistance under DC bias of TES devices of various sizes, with different stripe patterns and dimensions, we will discuss how these stripes can affect the R(I, B, T) surface. In addition, using measurements and analysis of the noise spectra of various devices we will present how these changes to the stripe pattern may affect the performance of the TES. In particular, we will discuss strategies to reduce the presence of localized discontinuities in the derivative of R, associated with increased noise, while maintaining the globally low levels of unexplained noise currently achieved with conventional metal stripe patterns. Implementing these strategies is a path towards producing large arrays with highly uniform transitions and high spectral resolution. These large uniform arrays will be required for future x-ray astronomy applications, such as the X-IFU on ATHENA.

TES↗

NASA NTRS1984

Monolithic microwave integrated circuit devices for active array antennas

Two different aspects of active antenna array design were investigated. The transition between monolithic microwave integrated circuits and rectangular waveguides was studied along with crosstalk in multiconductor transmission lines. The boundary value problem associated with a discontinuity in a microstrip line is formulated. This entailed, as a first step, the derivation of the propagating as well as evanescent modes of a microstrip line. The solution is derived to a simple discontinuity problem: change in width of the center strip. As for the multiconductor transmission line problem. A computer algorithm was developed for computing the crosstalk noise from the signal to the sense lines. The computation is based on the assumption that these lines are terminated in passive loads.

Mittra, R.↗

NASA NTRS2023

Assessment of the Impact of Discontinuity in Satellite Instruments and Retrievals on Global PM2.5

Exposure to fine particulate matter (PM2.5) is the leading environmental risk factor for mortality globally. Satellite-derived estimates of surface PM2.5 developed from a combination of satellites, simulations, and ground monitor data are relied upon for health impact studies. The ability to develop satellite-derived PM2.5 estimates requires the continued availability of aerosol optical depth (AOD) sources. This work examines the impact of the addition or loss of satellite AOD data sources on global PM2.5 estimation and the impact of continuing the longterm record with AOD from the Visible Infrared Imaging Radiometer Suite (VIIRS) on the Suomi-National Polar orbiting Partnership (S-NPP) satellite after the loss of the MODIS (MODerate resolution Imaging Spectroradiometer) and MISR (Multi-angle Imaging Spectroradiometer) instruments on board the Terra and Aqua satellites. We find that the addition of VIIRS S-NPP AOD products to geophysical PM2.5 estimates from satellites and simulations causes regional differences that correspond to differences in the VIIRS and MODIS Deep Blue AOD algorithms and sampling. Changes in long-term trends and timeseries due to the addition or loss of AOD data sources are generally within their uncertainties. Statistical fusion with ground monitor data partially corrects for changes due to sampling differences when introducing the VIIRS AOD products, but uncertainty remains over desert regions where ground monitor coverage is sparse. This work provides promise for the sustained development of global satellite-derived PM2.5 estimates, despite discontinuities in instruments and retrieval methods.

Remote sensing↗

NASA NTRS2000

High Order Discontinuous Gelerkin Methods for Convection Dominated Problems with Application to Aeroacoustics

This project is about the investigation of the development of the discontinuous Galerkin finite element methods, for general geometry and triangulations, for solving convection dominated problems, with applications to aeroacoustics. On the analysis side, we have studied the efficient and stable discontinuous Galerkin framework for small second derivative terms, for example in Navier-Stokes equations, and also for related equations such as the Hamilton-Jacobi equations. This is a truly local discontinuous formulation where derivatives are considered as new variables. On the applied side, we have implemented and tested the efficiency of different approaches numerically. Related issues in high order ENO and WENO finite difference methods and spectral methods have also been investigated. Jointly with Hu, we have presented a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the RungeKutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples are given to illustrate the capability of the method. Jointly with Hu, we have constructed third and fourth order WENO schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. The third order schemes are based on a combination of linear polynomials with nonlinear weights, and the fourth order schemes are based on combination of quadratic polynomials with nonlinear weights. We have addressed several difficult issues associated with high order WENO schemes on unstructured mesh, including the choice of linear and nonlinear weights, what to do with negative weights, etc. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations. Jointly with P. Montarnal, we have used a recently developed energy relaxation theory by Coquel and Perthame and high order weighted essentially non-oscillatory (WENO) schemes to simulate the Euler equations of real gas. The main idea is an energy decomposition under the form epsilon = epsilon(sub 1) + epsilon(sub 2), where epsilon(sub 1) is associated with a simpler pressure law (gamma)-law in this paper) and the nonlinear deviation epsilon(sub 2) is convected with the flow. A relaxation process is performed for each time step to ensure that the original pressure law is satisfied. The necessary characteristic decomposition for the high order WENO schemes is performed on the characteristic fields based on the epsilon(sub l) gamma-law. The algorithm only calls for the original pressure law once per grid point per time step, without the need to compute its derivatives or any Riemann solvers. Both one and two dimensional numerical examples are shown to illustrate the effectiveness of this approach.

Shu, Chi-Wang↗

NASA NTRS1983

Development of a Multilevel Optimization Approach to the Design of Modern Engineering Systems

A general algorithm is proposed which carries out the design process iteratively, starting at the top of the hierarchy and proceeding downward. Each subproblem is optimized separately for fixed controls from higher level subproblems. An optimum sensitivity analysis is then performed which determines the sensitivity of the subproblem design to changes in higher level subproblem controls. The resulting sensitivity derivatives are used to construct constraints which force the controlling subproblems into chosing their own designs so as to improve the lower levels subproblem designs while satisfying their own constraints. The applicability of the proposed algorithm is demonstrated by devising a four-level hierarchy to perform the simultaneous aerodynamic and structural design of a high-performance sailplane wing for maximum cross-country speed. Finally, the concepts discussed are applied to the two-level minimum weight structural design of the sailplane wing. The numerical experiments show that discontinuities in the sensitivity derivatives may delay convergence, but that the algorithm is robust enough to overcome these discontinuities and produce low-weight feasible designs, regardless of whether the optimization is started from the feasible space or the infeasible one.

Barthelemy, J. F. M.↗