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NASA NTRS1989

On an efficient and accurate method to integrate restricted three-body orbits

This work is a quantitative analysis of the advantages of the Bulirsch-Stoer (1966) method, demonstrating that this method is certainly worth considering when working with small N dynamical systems. The results, qualitatively suspected by many users, are quantitatively confirmed as follows: (1) the Bulirsch-Stoer extrapolation method is very fast and moderately accurate; (2) regularization of the equations of motion stabilizes the error behavior of the method and is, of course, essential during close approaches; and (3) when applicable, a manifold-correction algorithm reduces numerical errors to the limits of machine accuracy. In addition, for the specific case of the restricted three-body problem, even a small eccentricity for the orbit of the primaries drastically affects the accuracy of integrations, whether regularized or not; the circular restricted problem integrates much more accurately.

Murison, Marc A.↗

NASA NTRS1992

On the reliability of gravitational N-body integrations

In a self-gravitating system of point particles such as a spherical star cluster, small disturbances to an orbit grow exponentially on a time-scale comparable with the crossing time. The results of N-body integrations are therefore extremely sensitive to numerical errors: in practice it is almost impossible to follow orbits of individual particles accurately for more than a few crossing times. We demonstrate that numerical orbits in the gravitational N-body problem are often shadowed by true orbits for many crossing times. This result enhances our confidence in the use of N-body integrations to study the evolution of stellar systems.

Quinlan, Gerald D.↗

NASA NTRS1992

Numerical methods for the simulation of complex multi-body flows with applications for the integrated Space Shuttle vehicle

This project forms part of the long term computational effort to simulate the time dependent flow over the integrated Space Shuttle vehicle (orbiter, solid rocket boosters (SRB's), external tank (ET), and attach hardware) during its ascent mode for various nominal and abort flight conditions. Due to the limitations of experimental data such as wind tunnel wall effects and the difficulty of safely obtaining valid flight data, numerical simulations are undertaken to supplement the existing data base. This data can then be used to predict the aerodynamic behavior over a wide range of flight conditions. Existing computational results show relatively good overall comparison with experiments but further refinement is required to reduce numerical errors and to obtain finer agreements over a larger parameter space. One of the important goals of this project is to obtain better comparisons between numerical simulations and experiments. In the simulations performed so far, the geometry has been simplified in various ways to reduce the complexity so that useful results can be obtained in a reasonable time frame due to limitations in computer resources. In this project, the finer details of the major components of the Space Shuttle are modeled better by including more complexity in the geometry definition. Smaller components not included in early Space Shuttle simulations will now be modeled and gridded.

Chan, William M.↗

NASA NTRS1998

Annual Research Briefs, 1998

The topics contained in this progress report are direct numerical simulation of turbulent non-premixed combustion with realistic chemistry; LES of non-premixed turbulent reacting flows with conditional source term estimation; measurements of the three-dimensional scalar dissipation rate in gas-phase planar turbulent jets; direct simulation of a jet diffusion flame; on the use of interpolating wavelets in the direct numerical simulation of combustion; on the use of a dynamically adaptive wavelet collocation algorithm in DNS (direct numerical simulation) of non-premixed turbulent combustion; 2D simulations of Hall thrusters; computation of trailing-edge noise at low mach number using LES and acoustic analogy; weakly nonlinear modeling of the early stages of bypass transition; interactions between freestream turbulence and boundary layers; interfaces at the outer boundaries of turbulent motions; largest scales of turbulent wall flows; the instability of streaks in near-wall turbulence; an implementation of the v(sup 2) - f model with application to transonic flows; heat transfer predictions in cavities; a structure-based model with stropholysis effects; modeling a confined swirling coaxial jet; subgrid-scale models based on incremental unknowns for large eddy simulations; subgrid scale modeling taking the numerical error into consideration; towards a near-wall model for LES of a separated diffuser flow; on the feasibility of merging LES with RANS (Reynolds Averaging Numerical simulation) for the near-wall region of attached turbulent flows; large-eddy simulation of a separated boundary layer; numerical study of a channel flow with variable properties; on the construction of high order finite difference schemes on non-uniform meshes with good conservation properties; development of immersed boundary methods for complex geometries; and particle methods for micro and macroscale flow simulations.

Spinks, Debra↗

DOE OSTI2022

Verification of MOOSE/Bison's Heat Conduction Solver Using Combined Spatiotemporal Convergence Analysis

Bison is a computational physics code that uses the finite element method to model the thermo-mechanical response of nuclear fuel. Since Bison is used to inform high-consequence decisions, it is important that its computational results are reliable and predictive. One important step in assessing the reliability and predictive capabilities of a simulation tool is the verification process, which quantifies numerical errors in a discrete solution relative to the exact solution of the mathematical model. One step in the verification process—called code verification—ensures that the implemented numerical algorithm is a faithful representation of the underlying mathematical model, including partial differential or integral equations, initial and boundary conditions, and auxiliary relationships. In this paper, the code verification process is applied to spatiotemporal heat conduction problems in Bison. Simultaneous refinement of the discretization in space and time is employed to reveal any potential mistakes in the numerical algorithms for the interactions between the spatial and temporal components of the solution. For each verification problem, the correct spatial and temporal order of accuracy is demonstrated for both first- and second-order accurate finite elements and a variety of time-integration schemes. Furthermore, these results provide strong evidence that the Bison numerical algorithm for solving spatiotemporal problems reliably represents the underlying mathematical model in MOOSE. The selected test problems can also be used in other simulation tools that numerically solve for conduction or diffusion.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

NASA NTRS1981

Pitfalls and guidelines for the numerical evaluation of moderate-order system frequency response

The design and evaluation of a feedback control system via frequency response methods relies heavily upon numerical methods. In application, one can usually develop low order simulation models which for the most part are devoid of numerical problems. However, when complex feedback interactions, for example, between instrument control systems and their flexible mounting structure, must be evaluated, simulation models become moderate to large order and numerical problems become common. A large body of relevant numerical error analysis literature is summarized in a large language understandable to nonspecialists. The intent is to provide engineers using simulation models with an engineering feel for potential numerical problems without getting intertwined in the complexities of the associated mathematical theory. Guidelines are also provided by suggesting alternate state of the art methods which have good numerical evaluation characteristics.

Frisch, H. P.↗

NASA NTRS2017

Automatic Estimation of Verified Floating-Point Round-Off Errors via Static Analysis

This paper introduces a static analysis technique for computing formally verified round-off error bounds of floating-point functional expressions. The technique is based on a denotational semantics that computes a symbolic estimation of floating-point round-o errors along with a proof certificate that ensures its correctness. The symbolic estimation can be evaluated on concrete inputs using rigorous enclosure methods to produce formally verified numerical error bounds. The proposed technique is implemented in the prototype research tool PRECiSA (Program Round-o Error Certifier via Static Analysis) and used in the verification of floating-point programs of interest to NASA.

Moscato, Mariano↗

DOE OSTI2013

Quantifying the Impact of Single Bit Flips on Floating Point Arithmetic

In high-end computing, the collective surface area, smaller fabrication sizes, and increasing density of components have led to an increase in the number of observed bit flips. If mechanisms are not in place to detect them, such flips produce silent errors, i.e. the code returns a result that deviates from the desired solution by more than the allowed tolerance and the discrepancy cannot be distinguished from the standard numerical error associated with the algorithm. These phenomena are believed to occur more frequently in DRAM, but logic gates, arithmetic units, and other circuits are also susceptible to bit flips. Previous work has focused on algorithmic techniques for detecting and correcting bit flips in specific data structures, however, they suffer from lack of generality and often times cannot be implemented in heterogeneous computing environment. Our work takes a novel approach to this problem. We focus on quantifying the impact of a single bit flip on specific floating-point operations. We analyze the error induced by flipping specific bits in the most widely used IEEE floating-point representation in an architecture-agnostic manner, i.e., without requiring proprietary information such as bit flip rates and the vendor-specific circuit designs. We initially study dot products of vectors and demonstrate that not all bit flips create a large error and, more importantly, expected value of the relative magnitude of the error is very sensitive on the bit pattern of the binary representation of the exponent, which strongly depends on scaling. Our results are derived analytically and then verified experimentally with Monte Carlo sampling of random vectors. Furthermore, we consider the natural resilience properties of solvers based on the fixed point iteration and we demonstrate how the resilience of the Jacobi method for linear equations can be significantly improved by rescaling the associated matrix.

Elliott, James J.↗

DOE OSTI2024

When should PIC simulations be applied to atmospheric pressure plasmas? Impact of correlation heating

Abstract Molecular dynamics simulations are used to test when the particle-in-cell (PIC) method applies to atmospheric pressure plasmas. It is found that PIC applies only when the plasma density and macroparticle weight are sufficiently small because of two effects associated with correlation heating. The first is the physical effect of disorder-induced heating (DIH). This occurs if the plasma density is large enough that a species (typically ions) is strongly correlated in the sense that the Coulomb coupling parameter exceeds one. In this situation, DIH causes ions to rapidly heat following ionization. PIC is not well suited to capture DIH because doing so requires using a macroparticle weight of one and a grid that well resolves the physical interparticle spacing. These criteria render PIC intractable for macroscale domains. The second effect is a numerical error due to artificial correlation heating (ACH). ACH is like DIH in that it is caused by the Coulomb repulsion between particles, but differs in that it is a numerical effect caused by a macroparticle weight larger than one. Like DIH, it is associated with strong correlations. However, here the macroparticle coupling strength is found to scale as Γ w 2 / 3 , where Γ is the physical coupling strength and w is the macroparticle weight. So even if the physical coupling strength of a species is small, as is expected for electrons in atmospheric pressure plasmas, a sufficiently large macroparticle weight can cause the macroparticles to be strongly coupled and therefore heat due to ACH. Furthermore, it is shown that simulations in reduced dimensions exacerbate these issues.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2012

A Fast and Efficient Version of the TwO-Moment Aerosol Sectional (TOMAS) Global Aerosol Microphysics Model

This study develops more computationally efficient versions of the TwO-Moment Aerosol Sectional (TOMAS) microphysics algorithms, collectively called Fast TOMAS. Several methods for speeding up the algorithm were attempted, but only reducing the number of size sections was adopted. Fast TOMAS models, coupled to the GISS GCM II-prime, require a new coagulation algorithm with less restrictive size resolution assumptions but only minor changes in other processes. Fast TOMAS models have been evaluated in a box model against analytical solutions of coagulation and condensation and in a 3-D model against the original TOMAS (TOMAS-30) model. Condensation and coagulation in the Fast TOMAS models agree well with the analytical solution but show slightly more bias than the TOMAS-30 box model. In the 3-D model, errors resulting from decreased size resolution in each process (i.e., emissions, cloud processing wet deposition, microphysics) are quantified in a series of model sensitivity simulations. Errors resulting from lower size resolution in condensation and coagulation, defined as the microphysics error, affect number and mass concentrations by only a few percent. The microphysics error in CN70CN100 (number concentrations of particles larger than 70100 nm diameter), proxies for cloud condensation nuclei, range from 5 to 5 in most regions. The largest errors are associated with decreasing the size resolution in the cloud processing wet deposition calculations, defined as cloud-processing error, and range from 20 to 15 in most regions for CN70CN100 concentrations. Overall, the Fast TOMAS models increase the computational speed by 2 to 3 times with only small numerical errors stemming from condensation and coagulation calculations when compared to TOMAS-30. The faster versions of the TOMAS model allow for the longer, multi-year simulations required to assess aerosol effects on cloud lifetime and precipitation.

Lee, Yunha↗

DOE OSTI2015

Enabling Predictive Simulation and UQ of Complex Multiphysics PDE Systems by the Development of Goal-Oriented Variational Sensitivity Analysis and a-Posteriori Error Estimation Methods

This project addressed the challenge of predictive computational analysis of strongly coupled, highly nonlinear multiphysics systems characterized by multiple physical phenomena that span a large range of length- and time-scales. Specifically, the project was focused on computational estimation of numerical error and sensitivity analysis of computational solutions with respect to variations in parameters and data. In addition, the project investigated the use of accurate computational estimates to guide efficient adaptive discretization. The project developed, analyzed and evaluated new variational adjoint-based techniques for integration, model, and data error estimation/control and sensitivity analysis, in evolutionary multiphysics multiscale simulations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Fast computation of synthetic seismograms within a medium containing remote localized perturbations: a numerical solution to the scattering problem

We derive a fast discrete solution to the scattering problem. This solution allows us to compute accurate synthetic seismograms or waveforms for arbitrary locations of sources and receivers within a medium containing localized perturbations. The key to efficiency is that wave propagation modelling does not need to be carried out in the entire volume that encompasses the sources and the receivers but only within the subvolume containing the perturbations or scatterers. The proposed solution has important applications, for example, it permits the imaging of remote targets located in regions where no sources or receivers are present. Our solution relies on domain decomposition: within a small volume that contains the scatterers, wave propagation is modelled numerically, while in the surrounding volume, where the medium isn't perturbed, the response is obtained through wavefield extrapolation. The originality of this work is the derivation of discrete formulas for representation theorems and Kirchhof–Helmholtz integrals that naturally adapt to the numerical scheme employed for modelling wave propagation. Our solution applies, for example, to finite difference methods or finite/spectral elements methods. The synthetic seismograms obtained with our solution can be considered ‘exact’ as the total numerical error is comparable to that of the method employed for modelling wave propagation. We detail a basic implementation of our solution in the acoustic case using the finite difference method and present numerical examples that demonstrate the accuracy of the method. We show that ignoring some terms accounting for higher order scattering effects in our solution has a limited effect on the computed seismograms and significantly reduces the computational effort. Finally, we show that our solution can be used to compute localized sensitivity kernels and we discuss applications to target oriented imaging. Extension to the elastic case is straightforward and summarized in a dedicated section.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Accelerating Multivariate Functional Approximation Computation with Domain Decomposition Techniques⋆

Modeling large datasets through Multivariate Functional Approximations (MFA) provide an elegant way to handle many visualization and scientific analysis workflows. The process necessitates scalable data partitioning methods to compute MFA representations efficiently without compromising the accuracy or continuity of the reconstructed solution. We propose a domain -decomposed method for computing the MFA with B -spline bases, which reduces the total work per task and uses a restricted Additive Schwarz (RAS) method to converge the control point data degrees -of -freedom along subdomain boundaries. We provide an in-depth analysis of the parallel approach with domain decomposition solvers, aiming to minimize local subdomain error residuals and recover high -order continuity at subdomain interfaces with appropriate choices of knot overlaps. The communication cost, determined by the overlap regions in the RAS implementation, is optimized to recover the numerical error profile of the single subdomain case. Our proposed method stands in contrast to previous methods, which typically only recover either C 0 or at best C 1 continuity for arbitrary B -spline degree expansions, or those that require post -processing to blend discontinuities in the reconstructed data. We demonstrate the effectiveness of our approach using analytical and real -world datasets in 1D, 2D, and 3D through both strong and weak scaling studies. The performance results indicate that the overall cost of computing the approximation is directly proportional to the underlying nearest -neighbor communication implementation, and is only weakly dependent on the overlap region size that determines the size of the messages. This finding underscores the efficiency and scalability of our proposed method, making it a promising solution for handling large datasets in scientific workflows.

additive Schwarz solvers↗

NASA NTRS2005

Computation of Sound Propagation by Boundary Element Method

This report documents the development of a Boundary Element Method (BEM) code for the computation of sound propagation in uniform mean flows. The basic formulation and implementation follow the standard BEM methodology; the convective wave equation and the boundary conditions on the surfaces of the bodies in the flow are formulated into an integral equation and the method of collocation is used to discretize this equation into a matrix equation to be solved numerically. New features discussed here include the formulation of the additional terms due to the effects of the mean flow and the treatment of the numerical singularities in the implementation by the method of collocation. The effects of mean flows introduce terms in the integral equation that contain the gradients of the unknown, which is undesirable if the gradients are treated as additional unknowns, greatly increasing the sizes of the matrix equation, or if numerical differentiation is used to approximate the gradients, introducing numerical error in the computation. It is shown that these terms can be reformulated in terms of the unknown itself, making the integral equation very similar to the case without mean flows and simple for numerical implementation. To avoid asymptotic analysis in the treatment of numerical singularities in the method of collocation, as is conventionally done, we perform the surface integrations in the integral equation by using sub-triangles so that the field point never coincide with the evaluation points on the surfaces. This simplifies the formulation and greatly facilitates the implementation. To validate the method and the code, three canonic problems are studied. They are respectively the sound scattering by a sphere, the sound reflection by a plate in uniform mean flows and the sound propagation over a hump of irregular shape in uniform flows. The first two have analytical solutions and the third is solved by the method of Computational Aeroacoustics (CAA), all of which are used to compare the BEM solutions. The comparisons show very good agreements and validate the accuracy of the BEM approach implemented here.

Guo, Yueping↗

NASA NTRS2001

Sources of Variability in Chlorophyll Analysis by Fluorometry and by High Performance Liquid Chromatography

The need for accurate determination of chlorophyll a (chl a) is of interest for numerous reasons. From the need for ground-truth data for remote sensing to pigment detection for laboratory experimentation, it is essential to know the accuracy of the analyses and the factors potentially contributing to variability and error. Numerous methods and instrument techniques are currently employed in the analyses of chl a. These methods range from spectrophotometric quantification, to fluorometric analysis and determination by high performance liquid chromatography. Even within the application of HPLC techniques, methods vary. Here we provide the results of a comparison among methods and provide some guidance for improving the accuracy of these analyses. These results are based on a round-robin conducted among numerous investigators, including several in the Sensor Intercomparison and Merger for Biological and Interdisciplinary Oceanic Studies (SIMBIOS) and HyCODE Programs. Our purpose here is not to present the full results of the laboratory intercalibration; those results will be presented elsewhere. Rather, here we highlight some of the major factors that may contribute to the variability observed. Specifically, we aim to assess the comparability of chl a analyses performed by fluorometry and HPLC, and we identify several factors in the analyses which may contribute disproportionately to this variability.

VanHeukelem, Laurie↗

NASA NTRS2004

Nonlinear filtering and limiting in high order methods for ideal and non-ideal MHD

The various filtering mechanisms and base scheme options of the newly developed adaptive numerical dissipation control in spatially high order filter schemes for the ideal and non-ideal magnetohydrodynamics (MHD) equations are investigated. These filter schemes are applicable to complex unsteady MHD high-speed shock/shear/turbulence problems. They also provide a natural and efficient way for the minimization of Div(B) numerical error. The type of spatial base scheme to be used in conjunction with our filter idea is very general. For example, spectral, compact and non-compact spatially central finite difference schemes are possible candidates. The adaptive numerical dissipation mechanism consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and to leave the rest of the region free from numerical dissipation contamination. The numerical dissipation considered consists of high order linear dissipation for the suppression of high frequency oscillation and the nonlinear dissipative portion of high-resolution shock-capturing methods for discontinuity capturing. The applicable nonlinear dissipative portion of high-resolution shock-capturing methods is also very general. The objective of this paper is to investigate the performance of using compact and non-compact central base schemes in conjunction with three commonly used types of nonlinear numerical dissipation for both the ideal and non-ideal MHD. This extended abstract shows the performance of three nonlinear filters in conjunction with a sixth-order non-compact spatial central base scheme. In the final paper, the high order compact spatial central base scheme will be illustrated and compared with the non-compact base scheme. The reason for the investigation of the high order compact spatial central base scheme over the non-compact base scheme is to evaluate if additional accuracy can be gained in regions of fine scale turbulence that are away from shocks/shears.

Yee,H. C.↗

DOE OSTI2020

Efficient Fourier transforms for transverse momentum dependent distributions

Hadron production at low transverse momenta in semi-inclusive deep inelastic scattering can be described by transverse momentum dependent (TMD) factorization. This formalism has also been widely used to study the Drell-Yan process and back-to-back hadron pair production in $e^+e^-$ collisions. These processes are the main ones for extractions of TMD parton distribution functions and TMD fragmentation functions, which encode important information about nucleon structure and hadronization. One of the most widely used TMD factorization formalism in phenomenology formulates TMD observables in coordinate $b_\perp$-space, the conjugate space of the transverse momentum. The Fourier transform from $b_\perp$-space back into transverse momentum space is sufficiently complicated due to oscillatory integrands that it requires a careful and computationally intensive numerical treatment in order to avoid potentially large numerical errors. Within the TMD formalism, the azimuthal angular dependence is analytically integrated and the two-dimensional $b_\perp$ integration reduces to a one-dimensional integration over the magnitude $b_\perp$. In this paper we develop a fast numerical Hankel transform algorithm for such a $b_\perp$-integration that improves the numerical accuracy of TMD calculations in all standard processes. Libraries for this algorithm are implemented in Python 2.7 and 3, C++, as well as FORTRAN77. All packages are made available open source.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Relationship between the contact force strength and numerical inaccuracies in piecewise-smooth systems

This work studies the different types of behavior and inaccuracies that can occur when contact is not adequately accounted for in a dynamical system with freeplay, as the strength of the contact stiffness increases. The MATLAB® ode45 time integration solver, with the built-in Event Location capability, is first validated using past experimental data from a forced Duffing oscillator with freeplay. Next, numerical results utilizing event location are compared to results neglecting event location in order to highlight possible numerical errors and effects on multistable dynamical responses. Inaccuracies tend to occur in two different ways. First, neglecting event location can affect the boundaries between basins of attraction. Second, neglecting event location has little effect on the behaviors of the attractor solutions themselves besides merely resembling poorly converged solutions. Errors are less pronounced at the limits of soft or hard contact stiffness. Finally, this study shows the importance of accurately solving piecewise-smooth systems and the existing correlation between the strength of the contact force and possible numerical inaccuracies.

42 ENGINEERING↗