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Results for “Mathematical methods”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRSDate not supplied

Define Minimum Safe Operational Volume for Aerial Vehicles in Upper Class E Airspace

The variety of vehicle performance in upper Class E airspace requires a method that can efficiently compute the minimum safe operational boundary between aircraft. This work presents a mathematical method to define the minimum safe operational boundary needed for aerial vehicles operating in upper Class E airspace. This method focuses on the extra separation required by vehicle maneuverability, communication delay, and control/operator response time. A sensitivity study is then performed to provide a general understanding of the impact of these factors on the extra separation needed. Experiments with pairwise encounters are conducted to verify the results generated by the proposed methods.

Separation standard↗
NASA NTRSDate not supplied

Define Minimum Safe Operational Volume for Aerial Vehicles in Upper Class E Airspace

The variety of vehicle performance in upper Class E airspace requires a method that can efficiently compute the minimum safe operational boundary between aircraft. This work presents a mathematical method to define the minimum safe operational boundary needed for aerial vehicles operating in upper Class E airspace. This method focuses on the extra separation required by vehicle maneuverability, communication delay, and control/operator response time. A sensitivity study is then performed to provide a general understanding of the impact of these factors on the extra separation needed. Experiments with pairwise encounters are conducted to verify the results generated by the proposed methods.

Separation↗
DOE OSTI2024

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Advanced stationary and nonstationary kernel designs for domain-aware Gaussian processes

Gaussian process regression is a widely-applied method for function approximation and uncertainty quantification. The technique has gained popularity recently in the machine learning community due to its robustness and interpretability. The mathematical methods we discuss in this paper are an extension of the Gaussian-process framework. We are proposing advanced kernel designs that only allow for functions with certain desirable characteristics to be elements of the reproducing kernel Hilbert space (RKHS) that underlies all kernel methods and serves as the sample space for Gaussian process regression. These desirable characteristics reflect the underlying physics; two obvious examples are symmetry and periodicity constraints. In addition, non-stationary kernel designs can be defined in the same framework to yield flexible multi-task Gaussian processes. We will show the impact of advanced kernel designs on Gaussian processes using several synthetic and two scientific data sets. The results of our research show that including domain knowledge, communicated through advanced kernel designs, has a significant impact on the accuracy and relevance of the function approximation.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1985

Statistical methods for astronomical data with upper limits. I - Univariate distributions

The statistical treatment of univariate censored data is discussed. A heuristic derivation of the Kaplan-Meier maximum-likelihood estimator from first principles is presented which results in an expression amenable to analytic error analysis. Methods for comparing two or more censored samples are given along with simple computational examples, stressing the fact that most astronomical problems involve upper limits while the standard mathematical methods require lower limits. The application of univariate survival analysis to six data sets in the recent astrophysical literature is described, and various aspects of the use of survival analysis in astronomy, such as the limitations of various two-sample tests and the role of parametric modelling, are discussed.

Feigelson, E. D.↗
NASA NTRS1968

Interior ballistics of powder charges

The development of very large size solid rocket engines cannot be accomplished, as in the case of small rocket engines, by empirical methods requiring a great number of full-scale firings. It is therefore necessary to predict rocket operation from data obtained on charges of different size and geometry. This paper presents a mathematical method for determining analytically the rocket engine thrust and chamber pressure development curves from the charge and engine geometry, the heat of explosion the kinetics characteristics and any other parameters affecting engine operation. It is shown that it is possible to compute empirical values for burning rate and erosion effects from experimental firings. Following a brief survey of the fundamentals of this method, the procedure is illustrated by four representative examples.

J Boisson↗
DOE OSTI2021

Overview of the Tolerance Limit Calculations with Application to TSURFER

To establish confidence in the results of computerized physics models, a key regulatory requirement is to develop a scientifically defendable process. The methods employed for confidence, characterization, and consolidation, or C3, are statistically involved and are often accessible only to avid statisticians. This manuscript serves as a pedagogical presentation of the C3 process to all stakeholders—including researchers, industrial practitioners, and regulators—to impart an intuitive understanding of the key concepts and mathematical methods entailed by C3. The primary focus is on calculation of tolerance limits, which is the overall goal of the C3 process. Tolerance limits encode the confidence in the calculation results as communicated to the regulator. Understanding the C3 process is especially critical today, as the nuclear industry is considering more innovative ways to assess new technologies, including new reactor and fuel concepts, via an integrated approach that optimally combines modeling and simulation and minimal targeted validation experiments. This manuscript employs intuitive, analytical, numerical, and visual representations to explain how tolerance limits may be calculated for a wide range of configurations, and it also describes how their values may be interpreted. Various verification tests have been developed to test the calculated tolerance limits and to help delineate their values. The manuscript demonstrates the calculation of tolerance limits for TSURFER, a computer code developed by the Oak Ridge National Laboratory for criticality safety applications. The goal is to evaluate the tolerance limit for TSURFER-determined criticality biases to support the determination of upper, subcritical limits for regulatory purposes.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRSDate not supplied

On The Incorporation of Both Function-Driven Requirements and Topology Optimization in The Development of Lunar Launch & Landing Pads

In the design of additively manufactured structures, there can be a constant battle between focusing on function-driven design and topology optimization in an attempt to improve a products ability to meet all requirements. Function-driven optimization is defined as the process of identifying the best design parameters that satisfy project functionality requirements. Function-driven optimization is typically implemented by a designer(s) who, instead of computational limitations, has human bias for the input or inputs determined. On the other hand, topology optimization is defined as a mathematical method that optimizes material layout within a given design space, for a given set of loads, boundary conditions, and constraints with the goal of maximizing the performance of the system. In topology optimization, the work is defined by the limited mathematical inputs and computationally available goals/restraints. If not evaluated simultaneously, these two approaches can result in fundamentally different design concepts meant to address the same requirements. NASA’s Artemis program includes the development of a base camp near the lunar South Pole. After site selection is made, one of the first infrastructure elements needed will be a launch and landing pad (LLP). The construction of such a pad will prevent damage to nearby structures from plume-surface interaction (PSI) while also providing a stable platform for the lander. These benefits reduce the required distance between the established landing zones and other settlement infrastructure. In this paper, the authors evaluate the requirements associated with lunar LLP development, application of both functional and topological optimization techniques, methodologies to combine the two techniques, and potential impacts on final design of such a pad.

In-situ↗
NASA NTRS1971

Uniform random number generators

Methods are presented for the generation of random numbers with uniform and normal distributions. Subprogram listings of Fortran generators for the Univac 1108, SDS 930, and CDC 3200 digital computers are also included. The generators are of the mixed multiplicative type, and the mathematical method employed is that of Marsaglia and Bray.

Farr, W. R.↗
NASA NTRS1996

Nonlinear ARMA models for the D(st) index and their physical interpretation

Time series models successfully reproduce or predict geomagnetic activity indices from solar wind parameters. A method is presented that converts a type of nonlinear filter, the nonlinear Autoregressive Moving Average (ARMA) model to the nonlinear damped oscillator physical model. The oscillator parameters, the growth and decay, the oscillation frequencies and the coupling strength to the input are derived from the filter coefficients. Mathematical methods are derived to obtain unique and consistent filter coefficients while keeping the prediction error low. These methods are applied to an oscillator model for the Dst geomagnetic index driven by the solar wind input. A data set is examined in two ways: the model parameters are calculated as averages over short time intervals, and a nonlinear ARMA model is calculated and the model parameters are derived as a function of the phase space.

Vassiliadis, D.↗
NASA NTRS1972

A theory of supercritical wing sections, with computer programs and examples.

Mathematical methods for the design of supercritical wings, which depend on the numerical solution of the partial differential equations of two-dimensional gas dynamics, are developed. The main contribution is a computer program for the design of shockless transonic airfoils using the hodograph transformation and analytic continuation into the complex domain. The mathematical theory is described, and a manual for users of the programs is provided. Numerical examples are given and computational results are discussed, and the computer programs themselves are listed. The analysis routine can be used to ascertain whether the profiles behave well at off-design conditions, or to smooth coordinates and obtain a desirable shape more quickly when perfectly shockless flow is not essential.

Bauer, F.↗
NASA NTRS1990

Linear Calibration of Radiographic Mineral Density Using Video-Digitizing Methods

Radiographic images can provide quantitative as well as qualitative information if they are subjected to densitometric analysis. Using modern video-digitizing techniques, such densitometry can be readily accomplished using relatively inexpensive computer systems. However, such analyses are made more difficult by the fact that the density values read from the radiograph have a complex, nonlinear relationship to bone mineral content. This article derives the relationship between these variables from the nature of the intermediate physical processes, and presents a simple mathematical method for obtaining a linear calibration function using a step wedge or other standard.

Martin, R. Bruce↗
NASA NTRS1990

Linear Calibration of Radiographic Mineral Density Using Video-Digitizing Methods

Radiographic images can provide quantitative as well as qualitative information if they are subjected to densitometric analysis. Using modem video-digitizing techniques, such densitometry can be readily accomplished using relatively inexpensive computer systems. However, such analyses are made more difficult by the fact that the density values read from the radiograph have a complex, nonlinear relationship to bone mineral content. This article derives the relationship between these variables from the nature of the intermediate physical processes, and presents a simple mathematical method for obtaining a linear calibration function using a step wedge or other standard.

Martin, R. Bruce↗
DOE OSTI2025

2025 Workshop on Envisioning Frontiers in AI and Computing for Biological Research: Position Papers

This workshop aims to identify key research directions for transforming biology using artificial intelligence (AI), machine learning (ML) and computational methods to facilitate the discovery of new behaviors, mechanisms, and designs of biological processes relevant to DOE missions, underpinning a broader U.S. bioeconomy. By developing novel AI/ML technologies to analyze and interpret complex biological data, researchers can organize and simulate biological processes at various scales as well as advance predictive understanding and manipulation of biological systems. This integration of computation, experimentation, and next-generation experimental technologies can lead to discoveries in new biological behaviors and mechanisms relevant to DOE missions. The focus is on how advanced computational and mathematical methods can impact this mission by exploring digital twins, foundation models, automated laboratory experiments, modeling of complex living systems, and data-driven approaches for the biodesign of plants and microbial systems. While data management is important, it is not the primary focus of this workshop, which will assess the current state, trends, and AI/ML challenges at the interface between biology and computational science to identify opportunities for high-impact research at their intersection. The goal is to define research needs and opportunities that align with biological sciences, computational sciences, and applied mathematics research.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS2017

A Method for Estimating Meteorite Fall Mass from Weather Radar Data

Techniques such as weather RADAR, seismometers, and all-sky cameras allow new insights concerning the physics of meteorite fall dynamics and fragmentation during "dark flight", the period of time between the end of the meteor's luminous flight and the concluding impact on the Earth's surface. Understanding dark flight dynamics enables us to rapidly analyze the characteristics of new meteorite falls. This analysis will provide essential information to meteorite hunters to optimize recovery, increasing the frequency and total mass of scientifically important freshly-fallen meteorites available to the scientific community. We have developed a mathematical method to estimate meteorite fall mass using reflectivity data as recorded by National Oceanic and Atmospheric Administration (NOAA) Next Generation RADAR (NEXRAD) stations. This study analyzed eleven official and one unofficial meteorite falls in the United States and Canada to achieve this purpose.

Laird, C.↗