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All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Computing unsteady shock waves for aeroacoustic applications

The computation of unsteady shock waves, which contribute significantly to noise generation in supersonic jet flows, is investigated. The paper focuses on the difficulties of computing slowly moving shock waves. Numerical error is found to manifest itself principally as a spurious entropy wave. Calculations presented are performed using a third-order essentially nonoscillatory scheme. The effect of stencil biasing parameters and of two versions of numerical flux formulas on the magnitude of spurious entropy are investigated. The level of numerical error introduced in the calculation is quantified as a function of shock pressure ratio, shock speed, Courant number, and mesh density. The spurious entropy relative to the entropy jump across a static shock decreases with increasing shock strength and shock velocity relative to the grid, but is insensitive to Courant number. The structure of the spurious entropy wave is affected by the choice of flux formulas and algorithm biasing parameters. The effect of the spurious numerical waves on the calculation of sound amplification by a shock wave is investigated. For this class of problem, the acoustic pressure waves are relatively unaffected by the spurious numerical phenomena.

Meadows, Kristine R.↗
DOE OSTI2023

Neural networks for large eddy simulations of wall-bounded turbulence: numerical experiments and challenges

Here, we examine the application of neural network-based methods to improve the accuracy of large eddy simulations of incompressible turbulent flows. The networks are trained to learn a mapping between flow features and the subgrid scales, and applied locally and instantaneously—in the same way as traditional physics-based subgrid closures. Models that use only the local resolved strain rate are poorly correlated with the actual subgrid forces obtained from filtering direct numerical simulation data. We see that highly accurate models in a priori testing are inaccurate in forward calculations, owing to the preponderance of numerical errors in implicitly filtered large eddy simulations. A network that accounts for the discretization errors is trained and found to be unstable in a posteriori testing. We identify a number of challenges that the approach faces, including a distribution shift that affects networks that fail to account for numerical errors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2016

On the Calculation of Uncertainty Statistics with Error Bounds for CFD Calculations Containing Random Parameters and Fields

This chapter discusses the ongoing development of combined uncertainty and error bound estimates for computational fluid dynamics (CFD) calculations subject to imposed random parameters and random fields. An objective of this work is the construction of computable error bound formulas for output uncertainty statistics that guide CFD practitioners in systematically determining how accurately CFD realizations should be approximated and how accurately uncertainty statistics should be approximated for output quantities of interest. Formal error bounds formulas for moment statistics that properly account for the presence of numerical errors in CFD calculations and numerical quadrature errors in the calculation of moment statistics have been previously presented in [8]. In this past work, hierarchical node-nested dense and sparse tensor product quadratures are used to calculate moment statistics integrals. In the present work, a framework has been developed that exploits the hierarchical structure of these quadratures in order to simplify the calculation of an estimate of the quadrature error needed in error bound formulas. When signed estimates of realization error are available, this signed error may also be used to estimate output quantity of interest probability densities as a means to assess the impact of realization error on these density estimates. Numerical results are presented for CFD problems with uncertainty to demonstrate the capabilities of this framework.

Uncertainty↗
NASA NTRS2005

Multi-reference evaluation of uncertainty in earth orientation parameter measurements

Uncertainties in polar motion and length-of-day measurements are evaluated empirically using several data series from the space-geodetic techniques of the global positioning system (GPS), satellite laser ranging (SLR), and very long baseline interferometry (VLBI) during 1997-2002. In the evaluation procedure employed here, known as the three-corner hat (TCH) technique, the signal common to each series is eliminated by forming pair-wise differences between the series, thus requiring no assumed values for the 'truth' signal. From the variances of the differenced series, the uncertainty of each series can be recovered when reasonable assumptions are made about the correlations between the series. In order to form the pair-wise differences, the series data must be given at the same epoch. All measurement data sets studied here were sampled at noon (UTC); except for the VLBI series, whose data are interpolated to noon and whose UT1 values are also numerically differentiated to obtain LOD. The numerical error introduced to the VLBI values by the interpolation and differentiation is shown to be comparable in magnitude to the values determined by the TCH method for the uncertainties of the VLBI series. The TCH estimates for the VLBI series are corrupted by such numerical errors mostly as a result of the relatively large data intervals. Of the remaining data sets studied here, it is found that the IGS Final combined series has the smallest polar motion and length-of-day uncertainties.

Combination↗
DOE OSTI2021

Hydro-chemo-mechanical phase field formulation for corrosion induced cracking in reinforced concrete

Steel corrosion induced cracking in reinforced concrete structures is often caused by volumetric straining at the steel-concrete interface. This volumetric strain is mainly controlled by the rate of generation of corrosion product, which depends on the diffusion of Cl{sup −} and dissolved O{sub 2} as well as the chemical reaction at the steel-concrete interface. While many numerical modeling techniques have been proposed to tackle this complex phenomenon, most have been limited to using staggered solution schemes, where the diffusion, chemical reaction, mechanical strain, and crack fields are solved for separately. Unfortunately, such approaches introduce numerical errors that can lead to unrealistic predictions. Therefore, a coupled solution, is needed to remove the formulation induced errors and understand this complex phenomenon. To provide a unifying solution framework for corrosion induced cracking in reinforced concrete, a phase field formulation is presented and implemented with FEM algorithms. The governing equations are derived from the system's free energy and Fick's law of diffusion. The proposed formulation was first validated in 1D by comparing the modeling results with the semi-analytical solution approximated via Laurent series to avoid singularities in the complex domain. 2D plane strain solutions to the embedded rebar undergoing corrosion induced cracking with pre-cracks are also presented and compared with the approximated solutions from linear elastic fracture mechanics theory (LEFM). The modeling of crack propagation enabled by the proposed formulation is also presented along with parametric studies to reveal the roles of the fracture toughness and permeability of the concrete on the corrosion induced cracking. A case study was also conducted on H-pile steel with concrete jackets to demonstrate the feasibility of modeling corrosion induced fracture in reinforced concrete structures with complex steel-concrete interfaces.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Directional finite difference method for directly solving 3D gyrokinetic field equations with enhanced accuracy

The gyrokinetic (GK) field equation is a three-dimensional (3D) elliptic equation, but it is often simplified to a set of two-dimensional (2D) equations by assuming that the field does not vary along a specific direction. However, this simplification can introduce inevitable 0th-order numerical errors, as nonlinear mode coupling in toroidal geometry can produce undesirable harmonic modes that violate the assumption. In this work, we propose a novel directional finite difference method (FDM) with a local coordinate transformation to better resolve the target field of interest. The directional FDM can accurately solve 3D GK field equations without simplifications, which can overcome the limitations of conventional methods. The accuracy and efficiency of different FDMs are analyzed in great detail for a variety of geometries, from simple 2D Cartesian coordinates to realistic 3D curvilinear coordinates. The 0th-order numerical errors of simplified 2D GK equations were found to be more problematic for low-harmonic modes and low aspect ratio geometries such as spherical tokamaks. On the other hand, the directional 3D FDM can accurately resolve a much wider range of harmonic modes aligned to the direction of interest, including the low-harmonic modes. In conclusion, we demonstrate that the directional 3D FDM is a highly effective algorithm for solving the 3D GK field equations, achieving accuracy improvements of 10 to 100 times or more, particularly for low-harmonic modes in spherical tokamaks.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS2015

Hybrid Reynolds-Averaged/Large Eddy Simulation of a Cavity Flameholder; Assessment of Modeling Sensitivities

Steady-state and scale-resolving simulations have been performed for flow in and around a model scramjet combustor flameholder. The cases simulated corresponded to those used to examine this flowfield experimentally using particle image velocimetry. A variety of turbulence models were used for the steady-state Reynolds-averaged simulations which included both linear and non-linear eddy viscosity models. The scale-resolving simulations used a hybrid Reynolds-averaged / large eddy simulation strategy that is designed to be a large eddy simulation everywhere except in the inner portion (log layer and below) of the boundary layer. Hence, this formulation can be regarded as a wall-modeled large eddy simulation. This effort was undertaken to formally assess the performance of the hybrid Reynolds-averaged / large eddy simulation modeling approach in a flowfield of interest to the scramjet research community. The numerical errors were quantified for both the steady-state and scale-resolving simulations prior to making any claims of predictive accuracy relative to the measurements. The steady-state Reynolds-averaged results showed a high degree of variability when comparing the predictions obtained from each turbulence model, with the non-linear eddy viscosity model (an explicit algebraic stress model) providing the most accurate prediction of the measured values. The hybrid Reynolds-averaged/large eddy simulation results were carefully scrutinized to ensure that even the coarsest grid had an acceptable level of resolution for large eddy simulation, and that the time-averaged statistics were acceptably accurate. The autocorrelation and its Fourier transform were the primary tools used for this assessment. The statistics extracted from the hybrid simulation strategy proved to be more accurate than the Reynolds-averaged results obtained using the linear eddy viscosity models. However, there was no predictive improvement noted over the results obtained from the explicit Reynolds stress model. Fortunately, the numerical error assessment at most of the axial stations used to compare with measurements clearly indicated that the scale-resolving simulations were improving (i.e. approaching the measured values) as the grid was refined. Hence, unlike a Reynolds-averaged simulation, the hybrid approach provides a mechanism to the end-user for reducing model-form errors.

Baurle, R. A.↗
NASA NTRS2004

Error and Symmetry Analysis of Misner's Algorithm for Spherical Harmonic Decomposition on a Cubic Grid

In an earlier paper, Misner (2004, Class. Quant. Grav., 21, S243) presented a novel algorithm for computing the spherical harmonic components of data represented on a cubic grid. I extend Misner s original analysis by making detailed error estimates of the numerical errors accrued by the algorithm, by using symmetry arguments to suggest a more efficient implementation scheme, and by explaining how the algorithm can be applied efficiently on data with explicit reflection symmetries.

Fiske, David R.↗
DOE OSTI2021

An asymptotic-preserving 2D-2P relativistic Drift-Kinetic-Equation solver for runaway electron simulations in axisymmetric tokamaks

We propose an asymptotic-preserving (AP), uniformly convergent numerical scheme for the relativistic collisional Drift-Kinetic Equation (rDKE) to simulate runaway electrons in axisymmetric toroidal magnetic field geometries typical of tokamak devices. The approach is derived from an exact Green's function solution with numerical approximations of quantifiable impact, and results in a simple, two-step operator-split algorithm, consisting of a collisional Eulerian step, and a Lagrangian orbit-integration step with analytically prescribed kernels. The AP character of the approach is demonstrated by analysis of the dominant numerical errors, as well as by numerical experiments. We demonstrate the ability of the algorithm to provide accurate answers regardless of plasma collisionality on a circular axisymmetric tokamak geometry.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

An unstructured C-grid type variational formulation for the sea ice dynamics

Historically, B-grid formulations of sea ice dynamics have been dominant because they have matched the grid type used by ocean models. The reason for the grid match is simple – it facilitates penetration of the curl of ice-ocean stress into the deep ocean with minimal numerical diffusivity because sea ice and ocean velocity are co-located. In recent years, as ocean models have increasingly progressed to C-grids, sea ice models have followed suit on quadrilateral meshes, but the implementation of an unstructured C-grid sea ice models is new. We present an unstructured C-grid discretization of the Elastic Viscous Plastic (EVP) rheology, where the velocity unknowns are discretized at the edges of the mesh cells with n-sides, where typically n is greater than or equal to four, rather than at the vertices, as in the B-grid. Our framework of choice is the Model for Prediction Across Scales (MPAS) within E3SM, the climate model of the U.S. Department of Energy, although our approach is general and could be applied to other models as well. While MPAS-Seaice is currently defined on a B-grid, MPAS-Ocean runs on a C-grid, hence interpolation operators are heavily used when coupled simulations are performed. In this work, we describe a mathematical formulation to transition the dynamics of MPAS-Seaice to a C-grid, in order to ultimately facilitate the coupling with MPAS-Ocean and reduce numerical errors associated with this communication. Numerical results are reported to highlight the features of the method.

58 GEOSCIENCES↗
DOE OSTI2020

Sampling Low-Dimensional Markovian Dynamics for Preasymptotically Recovering Reduced Models from Data with Operator Inference

This work introduces a method for learning low-dimensional models from data of high-dimensional black-box dynamical systems. The novelty is that the learned models are exactly the reduced models that are traditionally constructed with classical projection-based model reduction techniques. Thus, the proposed approach learns models that are guaranteed to have the well-studied properties of reduced models known from model reduction, without requiring full knowledge of the governing equations and without requiring the operators of the high-dimensional systems. The key ingredient is a new data sampling scheme to obtain re-projected trajectories of high-dimensional systems that correspond to Markovian dynamics in low-dimensional subspaces. The exact recovery of reduced models from these re-projected trajectories is guaranteed pre-asymptotically under certain conditions for finite amounts of data and for a large class of systems with polynomial nonlinear terms. Numerical results demonstrate that the low-dimensional models learned with the proposed approach match reduced models from traditional model reduction up to numerical errors in practice. In conclusion, the numerical results further indicate that low-dimensional models fitted to re-projected trajectories are predictive even in situations where models fitted to trajectories without re-projection are inaccurate and unstable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Shadow Lagrangian dynamics for superfluidity

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS2014

Combined Uncertainty and A-Posteriori Error Bound Estimates for CFD Calculations: Theory and Implementation

Simulation codes often utilize finite-dimensional approximation resulting in numerical error. Some examples include, numerical methods utilizing grids and finite-dimensional basis functions, particle methods using a finite number of particles. These same simulation codes also often contain sources of uncertainty, for example, uncertain parameters and fields associated with the imposition of initial and boundary data,uncertain physical model parameters such as chemical reaction rates, mixture model parameters, material property parameters, etc.

CFD↗
NASA NTRS2010

An Improved Neutron Transport Algorithm for HZETRN

Long term human presence in space requires the inclusion of radiation constraints in mission planning and the design of shielding materials, structures, and vehicles. In this paper, the numerical error associated with energy discretization in HZETRN is addressed. An inadequate numerical integration scheme in the transport algorithm is shown to produce large errors in the low energy portion of the neutron and light ion fluence spectra. It is further shown that the errors result from the narrow energy domain of the neutron elastic cross section spectral distributions, and that an extremely fine energy grid is required to resolve the problem under the current formulation. Two numerical methods are developed to provide adequate resolution in the energy domain and more accurately resolve the neutron elastic interactions. Convergence testing is completed by running the code for various environments and shielding materials with various energy grids to ensure stability of the newly implemented method.

Slaba, Tony C.↗
NASA NTRS1986

Optimal control of quaternion propagation errors in spacecraft navigation

Optimal control techniques are used to drive the numerical error (truncation, roundoff, commutation) in computing the quaternion vector to zero. The normalization of the quaternion is carried out by appropriate choice of a performance index, which can be optimized. The error equations are derived from Friedland's (1978) theoretical development, and a matrix Riccati equation results for the computation of the gain matrix. Simulation results show that a high precision of the order of 10 to the -12th can be obtained using this technique in meeting the q(T)q=1 constraint. The performance of the estimator in the presence of the feedback control that maintains the normalization, is studied.

Vathsal, S.↗
NASA NTRS2012

Uncertainty Quantification for Polynomial Systems via Bernstein Expansions

This paper presents a unifying framework to uncertainty quantification for systems having polynomial response metrics that depend on both aleatory and epistemic uncertainties. The approach proposed, which is based on the Bernstein expansions of polynomials, enables bounding the range of moments and failure probabilities of response metrics as well as finding supersets of the extreme epistemic realizations where the limits of such ranges occur. These bounds and supersets, whose analytical structure renders them free of approximation error, can be made arbitrarily tight with additional computational effort. Furthermore, this framework enables determining the importance of particular uncertain parameters according to the extent to which they affect the first two moments of response metrics and failure probabilities. This analysis enables determining the parameters that should be considered uncertain as well as those that can be assumed to be constants without incurring significant error. The analytical nature of the approach eliminates the numerical error that characterizes the sampling-based techniques commonly used to propagate aleatory uncertainties as well as the possibility of under predicting the range of the statistic of interest that may result from searching for the best- and worstcase epistemic values via nonlinear optimization or sampling.

Crespo, Luis G.↗
NASA NTRS1990

Computational enhancement of an unsymmetric block Lanczos algorithm

An unsymmetric block Lanczos algorithm has been employed for the dynamic analysis of a large system which has arbitrary damping and/or repeated (or closely spaced) eigenvalues. In the algorithm development, the right and left Lanczos vectors are all theoretically biorthogonal to each other. However, these vectors may lose the biorthogonality owing to cancellation and roundoff errors. For the unsymmetric case there can be a breakdown, even without numerical errors. This paper describes computational techniques which have led to a robust unsymmetric block Lanczos algorithm.

Kim, Hyoung M.↗