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DOE OSTI2017

Multiresolution molecular mechanics: Surface effects in nanoscale materials

Surface effects have been observed to contribute significantly to the mechanical response of nanoscale structures. The newly proposed energy-based coarse-grained atomistic method Multiresolution Molecular Mechanics (MMM) (Yang, To (2015), ) is applied to capture surface effect for nanosized structures by designing a surface summation rule SR{sup S} within the framework of MMM. Combined with previously proposed bulk summation rule SR{sup B}, the MMM summation rule SR{sup MMM} is completed. SR{sup S} and SR{sup B} are consistently formed within SR{sup MMM} for general finite element shape functions. Analogous to quadrature rules in finite element method (FEM), the key idea to the good performance of SR{sup MMM} lies in that the order or distribution of energy for coarse-grained atomistic model is mathematically derived such that the number, position and weight of quadrature-type (sampling) atoms can be determined. Mathematically, the derived energy distribution of surface area is different from that of bulk region. Physically, the difference is due to the fact that surface atoms lack neighboring bonding. As such, SR{sup S} and SR{sup B} are employed for surface and bulk domains, respectively. Two- and three-dimensional numerical examples using the respective 4-node bilinear quadrilateral, 8-node quadratic quadrilateral and 8-node hexahedral meshes are employed to verify and validate the proposed approach. It is shown that MMM with SR{sup MMM} accurately captures corner, edge and surface effects with less 0.3% degrees of freedom of the original atomistic system, compared against full atomistic simulation. The effectiveness of SR{sup MMM} with respect to high order element is also demonstrated by employing the 8-node quadratic quadrilateral to solve a beam bending problem considering surface effect. In addition, the introduced sampling error with SR{sup MMM} that is analogous to numerical integration error with quadrature rule in FEM is very small. - Highlights: • Surface effect captured by Multiresolution Molecular Mechanics (MMM) is presented. • A novel surface summation rule within the framework of MMM is proposed. • Surface, corner and edges effects are accuterly captured in two and three dimension. • MMM with less 0.3% degrees of freedom of atomistics reproduces atomistic results.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2021

Toward Predictive RANS and SRS Computations of Turbulent External Flows of Practical Interest

In this work, we investigate the main challenges to prediction of turbulent external flows of practical interest with Reynolds-Averaged Navier–Stokes equations (RANS) and Scale-Resolving Simulation (SRS) models. This represents a crucial step toward further developing and establishing these formulations so they can be confidently utilized in engineering problems without reference data. The study initiates by identifying the major challenges to prediction. A literature review is performed to illustrate their effects in RANS and SRS computations. Afterward, we evaluate the impact of the challenges to prediction by analyzing representative statistically steady and unsteady flows with prominent RANS and SRS methods. These include multiple turbulent viscosity and second-moment RANS closures, and hybrid and bridging SRS models. The results demonstrate the potential of the selected SRS models to predict engineering flows. Yet, they also show the importance of considering the challenges to prediction during the setup and conduction of numerical experiments. These can suppress the advantages of using SRS formulations. The data also indicate that only SRS models can confidently predict statistically unsteady flows. In contrast, the results demonstrate that mean-flow quantities of statistically steady flows can be efficiently calculated with RANS closures, especially second-moment closures. Among the selected SRS methods, bridging models reveal better suited for prediction due to their ability to prevent commutation errors and enable the robust evaluation of numerical and modeling errors. This last property allows the use of a new validation technique that does not require reference data.

42 ENGINEERING↗

NASA NTRS2009

Stochastic Formal Correctness of Numerical Algorithms

We provide a framework to bound the probability that accumulated errors were never above a given threshold on numerical algorithms. Such algorithms are used for example in aircraft and nuclear power plants. This report contains simple formulas based on Levy's and Markov's inequalities and it presents a formal theory of random variables with a special focus on producing concrete results. We selected four very common applications that fit in our framework and cover the common practices of systems that evolve for a long time. We compute the number of bits that remain continuously significant in the first two applications with a probability of failure around one out of a billion, where worst case analysis considers that no significant bit remains. We are using PVS as such formal tools force explicit statement of all hypotheses and prevent incorrect uses of theorems.

Daumas, Marc↗

DOE OSTI2014

Error Rate Comparison during Polymerase Chain Reaction by DNA Polymerase

As larger-scale cloning projects become more prevalent, there is an increasing need for comparisons among high fidelity DNA polymerases used for PCR amplification. All polymerases marketed for PCR applications are tested for fidelity properties (i.e., error rate determination) by vendors, and numerous literature reports have addressed PCR enzyme fidelity. Nonetheless, it is often difficult to make direct comparisons among different enzymes due to numerous methodological and analytical differences from study to study. We have measured the error rates for 6 DNA polymerases commonly used in PCR applications, including 3 polymerases typically used for cloning applications requiring high fidelity. Error rate measurement values reported here were obtained by direct sequencing of cloned PCR products. The strategy employed here allows interrogation of error rate across a very large DNA sequence space, since 94 unique DNA targets were used as templates for PCR cloning. The six enzymes included in the study, Taq polymerase, AccuPrime-Taq High Fidelity, KOD Hot Start, cloned Pfu polymerase, Phusion Hot Start, and Pwo polymerase, we find the lowest error rates with Pfu , Phusion, and Pwo polymerases. Error rates are comparable for these 3 enzymes and are >10x lower than the error rate observed with Taq polymerase. Mutation spectra are reported, with the 3 high fidelity enzymes displaying broadly similar types of mutations. For these enzymes, transition mutations predominate, with little bias observed for type of transition.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2015

Smisc - A collection of miscellaneous functions

A collection of functions for statistical computing and data manipulation. These include routines for rapidly aggregating heterogeneous matrices, manipulating file names, loading R objects, sourcing multiple R files, formatting datetimes, multi-core parallel computing, stream editing, specialized plotting, etc. Smisc-package A collection of miscellaneous functions allMissing Identifies missing rows or columns in a data frame or matrix as.numericSilent Silent wrapper for coercing a vector to numeric comboList Produces all possible combinations of a set of linear model predictors cumMax Computes the maximum of the vector up to the current index cumsumNA Computes the cummulative sum of a vector without propogating NAs d2binom Probability functions for the sum of two independent binomials dataIn A flexible way to import data into R. dbb The Beta-Binomial Distribution df2list Row-wise conversion of a data frame to a list dfplapply Parallelized single row processing of a data frame dframeEquiv Examines the equivalence of two dataframes or matrices dkbinom Probability functions for the sum of k independent binomials factor2character Converts all factor variables in a dataframe to character variables findDepMat Identify linearly dependent rows or columns in a matrix formatDT Converts date or datetime strings into alternate formats getExtension Filename manipulations: remove the extension or path, extract the extension or path getPath Filename manipulations: remove the extension or path, extract the extension or path grabLast Filename manipulations: remove the extension or path, extract the extension or path ifelse1 Non-vectorized version of ifelse integ Simple numerical integration routine interactionPlot Two-way Interaction Plot with Error Bar linearMap Linear mapping of a numerical vector or scalar list2df Convert a list to a data frame loadObject Loads and returns the object(s) in an ".Rdata" file more Display the contents of a file to the R terminal movAvg2 Calculate the moving average using a 2-sided window openDevice Opens a graphics device based on the filename extension p2binom Probability functions for the sum of two independent binomials padZero Pad a vector of numbers with zeros parseJob Parses a collection of elements into (almost) equal sized groups pbb The Beta-Binomial Distribution pcbinom A continuous version of the binomial cdf pkbinom Probability functions for the sum of k independent binomials plapply Simple parallelization of lapply plotFun Plot one or more functions on a single plot PowerData An example of power data pvar Prints the name and value of one or more objects qbb The Beta-Binomial Distribution rbb And numerous others (space limits reporting).

Landon Sego, PNNL↗

DOE OSTI2015

Smisc - A collection of miscellaneous functions

Landon Sego, PNNL↗

NASA NTRS1978

Filtering and error analysis via the UDU super T covariance factorization

Kalman filter algorithms based on the UDU super T covariance factorization are discussed, with special attention given to algorithm implementation efficiency. A U-D-factored covariance error-analysis algorithm is formulated, and its efficiency and numerical stability are demonstrated in a representative orbit determination problem. The numerical results are compared with those obtained using covariance error-analysis formulas, and the comparison highlights the numerical superiority of the present algorithm. A byproduct of the U-D analysis is a highly efficient algorithm mechanization of the arbitrary gain covariance update formula.

Thornton, C. L.↗

NASA NTRS1992

Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations

Recent development using compact difference schemes to solve the Navier-Stokes equations show spectral-like accuracy. A study was made of the numerical characteristics of various combinations of the Runge-Kutta (RK) methods and compact difference schemes to calculate the unsteady Euler equations. The accuracy of finite difference schemes is assessed based on the evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative errors, phase velocity and harmonic content of the numerical results are of concern. As a result of the discretization procedure, the simulated unsteady flow motions actually propagate in a dispersive numerical medium. Consequently, the dispersion characteristics of the numerical schemes which relate the phase velocity and wave number may greatly impact the numerical accuracy. The aim is to assess the numerical accuracy of the simulated results. To this end, the Fourier analysis is to provide the dispersive correlations of various numerical schemes. First, a detailed investigation of the existing RK methods is carried out. A generalized form of an N-step RK method is derived. With this generalized form, the criteria are derived for the three and four-step RK methods to be third and fourth-order time accurate for the non-linear equations, e.g., flow equations. These criteria are then applied to commonly used RK methods such as Jameson's 3-step and 4-step schemes and Wray's algorithm to identify the accuracy of the methods. For the spatial discretization, compact difference schemes are presented. The schemes are formulated in the operator-type to render themselves suitable for the Fourier analyses. The performance of the numerical methods is shown by numerical examples. These examples are detailed. described. The third case is a two-dimensional simulation of a Lamb vortex in an uniform flow. This calculation provides a realistic assessment of various finite difference schemes in terms of the conservation of the vortex strength and the harmonic content after travelling a substantial distance. The numerical implementation of Giles' non-refelctive equations coupled with the characteristic equations as the boundary condition is discussed in detail. Finally, the single vortex calculation is extended to simulate vortex pairing. For the distance between two vortices less than a threshold value, numerical results show crisp resolution of the vortex merging.

Yu, Sheng-Tao↗

NASA NTRS1987

Application of TOMS data to weather analysis models

Recent studies of the errors of analysis reflected by numerical weather analysis models are reviewed. Despite the improvements in data coverage and data ingestion methods in the past decade, it is found that significant errors of analysis persist. A case study comparison of Total Ozone Mapping Spectrometer (TOMS) data and aircraft data with the NMC analysis over the North Atlantic is used to illustrate the size of local errors encountered. The possibility of using TOMS images to locate meteorologically significant features such as troughs and ridges near the tropopause is discussed.

Nastrom, Gregory D.↗

NASA NTRS1970

A numerical study of cavity radiometer emissivities

Error analysis of apparent emissivities for Lambertian cavities used in radiometers

Sydnor, C. L.↗

DOE OSTI2023

Adaptive Time Step Control for Multirate Infinitesimal Methods

Multirate methods have been used for decades to temporally evolve initial-value problems in which different components evolve on distinct time scales, and thus use of different step sizes for these components can result in increased computational efficiency. Generally, such methods select these different step sizes based on experimentation or stability considerations. For problems that evolve on a single time scale, adaptivity approaches that strive to control local temporal error are widely used to achieve numerical results of a desired accuracy with minimal computational effort, while alleviating the need for manual experimentation with different time step sizes. However, there is a notable gap in the publication record on the development of adaptive time step controllers for multirate methods. In this paper, we extend the single-rate controller work of Gustafsson [ACM Trans. Math. Software, 20 (1994), pp. 496-517] to the multirate method setting. Specifically, we develop controllers based on polynomial approximations to the principal error functions for both the "fast" and "slow" time scales within multirate infinitesimal (MRI) methods. We additionally investigate a variety of approaches for estimating the errors arising from each time scale within MRI methods. We then numerically evaluate the proposed multirate controllers and error estimation strategies on a range of multirate test problems, comparing their performance against an estimated optimal performance. Through this work, we combine the most performant of these approaches to arrive at a set of multirate adaptive time step controllers that robustly achieve desired solution accuracy with minimal computational effort.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1993

Error modeling for GPS geodetic applications

An extensive investigation was conducted to provide realistic error models for Global Positioning System (GPS) related numerical simulation. This study considers most of the important error sources for measurement and dynamic models which are currently being used for GPS geodetic applications. These error models were evaluated by comparing with real GPS data.

Rim, Hyung Jin↗

NASA NTRS1972

Numerical stability in problems of linear algebra.

Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.

Babuska, I.↗

NASA NTRS1984

On the remote sensing of cloud properties from satellite infrared sounder data

A method for remote sensing of cloud parameters by using infrared sounder data has been developed on the basis of the parameterized infrared transfer equation applicable to cloudy atmospheres. The method is utilized for the retrieval of the cloud height, amount, and emissivity in 11 micro m region. Numerical analyses and retrieval experiments have been carried out by utilizing the synthetic sounder data for the theoretical study. The sensitivity of the numerical procedures to the measurement and instrument errors are also examined. The retrieved results are physically discussed and numerically compared with the model atmospheres. Comparisons reveal that the recovered cloud parameters agree reasonably well with the pre-assumed values. However, for cases when relatively thin clouds and/or small cloud fractional cover within a field of view are present, the recovered cloud parameters show considerable fluctuations. Experiments on the proposed algorithm are carried out utilizing High Resolution Infrared Sounder (HIRS/2) data of NOAA 6 and TIROS-N. Results of experiments show reasonably good comparisons with the surface reports and GOES satellite images.

Yeh, H. Y. M.↗

NASA NTRS1987

Geoid undulation computations at laser tracking stations

Geoid undulation computations were performed at 29 laser stations distributed around the world using a combination of terrestrial gravity data within a cap of radius 2 deg and a potential coefficient set up to 180 deg. The traditional methods of Stokes' and Meissl's modification together with the Molodenskii method and the modified Sjoberg method were applied. Performing numerical tests based on global error assumptions regarding the terrestrial data and the geopotential set it was concluded that the modified Sjoberg method is the most accurate and promising technique for geoid undulation computations. The numerical computations for the geoid undulations using all the four methods resulted in agreement with the ellipsoidal minus orthometric value of the undulations on the order of 60 cm or better for most of the laser stations in the eastern United States, Australia, Japan, Bermuda, and Europe. A systematic discrepancy of about 2 meters for most of the western United States stations was detected and verified by using two relatively independent data sets. For oceanic laser stations in the western Atlantic and Pacific oceans that have no terrestrial data available, the adjusted GEOS-3 and SEASAT altimeter data were used for the computation of the geoid undulation in a collocation method.

Despotakis, Vasilios K.↗

NASA NTRS1985

Analysis and forecast intercomparisons using the FGGE SOP-1 data base

The First GARP Global Experiment (FGGE) data base has provided the research community with the most complete global coverage of meteorological informaiton since measurements were first taken. It is important to understand the characteristics of the FGGE data base in terms of its accuracy and representativeness compared to the true state of the atmosphere. The uncertainty associated with the initial conditions used in a numerical model plays a significant role in subsequent error growth as shown by predictability theory. The ultimate limit of forecast skill is directly related to the initial uncertainty. The accuracy of analyses using FGGE data base coupled with the errors due to modeling imperfections determines the present skill of numerical models. An estimte of initial uncertainty is determined by examining differences in several FGGE analyses. The estimate is then used to determine an ultimate limit of forecast skill from derived predictability error growth rates. Several different calculations of predictability error growth are also compared. Finally, present-day forecast skill using FGGE analyses is contrasted to the estimates of ultimate skill, and the partition of model error versus initial uncertainty is discussed.

Baumhefner, D. P.↗

NASA NTRS1981

Theoretical studies of solar-pumped lasers

Estimates of the absorption and emission characteristics of molecules required to develop materials for solar-pumped lasers are addressed. These characteristics are described in terms of the Franck Condon factors, which are calculated from the molecular wave functions. Wave functions for vibrational levels in the lower and upper electronic states of I2 and NSe are calculated numerically and methods of checking errors discussed. Errors arise when the vibrational quantum numbers are high; but, using a calculated rather than measured value of the dissociation energy, wave functions up to the fiftieth vibrational level are obtained. A numerical method of evaluating the wave functions is given, which should be more accurate in the region of electronic transitions during absorption. Franck Condon factors, plotted versus the wavelength of the absorbed photons, are shown, and a check on the Franck Condon factors is made using the vibrational sum rule.

Harries, W. L.↗

NASA NTRS2006

Space-Time Error Representation and Estimation in Navier-Stokes Calculations

The mathematical framework for a-posteriori error estimation of functionals elucidated by Eriksson et al. [7] and Becker and Rannacher [3] is revisited in a space-time context. Using these theories, a hierarchy of exact and approximate error representation formulas are presented for use in error estimation and mesh adaptivity. Numerical space-time results for simple model problems as well as compressible Navier-Stokes flow at Re = 300 over a 2D circular cylinder are then presented to demonstrate elements of the error representation theory for time-dependent problems.

Barth, Timothy J.↗