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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 649 records · Page 36
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

The solution of coupled Schroedinger equations using an extrapolation method

In this paper, extrapolation to the limit in a finite-difference method is applied to solve a system of coupled Schroedinger equations. This combination results in a method that only requires knowledge of the potential energy functions for the system. This numerical procedure has several distinct advantages over the more conventional methods. Namely, initial guesses for the term values are not needed; assumptions need be made about the behavior of the wavefunctions, such as the slope or magnitude in the nonclassical region; and the algorithm is easy to implement, has a firm mathematical foundation, and provides error estimates. Moreover, the method is less sensitive to round-off error than other methods since a small number of mesh points is used and it can be implemented on small computers. A comparison of the method with another numerical method shows results agreeing within 1 part in 10 exp 4.

Goorvitch, D.↗
NASA NTRS1989

Spacecraft attitude determination using the earth's magnetic field

A method is presented by which the attitude of a low-Earth orbiting spacecraft may be determined using a vector magnetometer, a digital Sun sensor, and a mathematical model of the Earth's magnetic field. The method is currently being implemented for the Solar Maximum Mission spacecraft (as a backup for the failing star trackers) as a way to determine roll gyro drift.

Simpson, David G.↗
NASA NTRS1981

Application of HCMM data to regional geologic analysis for mineral and energy resource evaluation

Experimentation with several potentially promising techniques led to the selection of a fairly simple scheme for registration of data from the HCMM using an affine transformation. A method based solely on remote sensing was developed to estimate those meteorological effects which are required for thermal inertia mapping. It assumes that the atmospheric fluxes are spatially invariant and that the solar, sky and sensible heat fluxes can be approximated by a simple mathematical form. Coefficients are determined from least squares method by fitting observational data to the thermal model.

Watson, K.↗
NASA NTRSDate not supplied

GASP Advancements With Symbolic Computations, Optimization, and Decoupled Numerical Methods

NASA is exploring advancements in hybrid-electric propulsion concepts for aircraft design. A new analysis capability has been deployed at NASA’s Ames Research Center using a new mathematical modeling framework in Python that leverages symbolic representation of mathematical expressions, optimization-based problem formulations, and standalone numerical methods. This capability, called Gascon, has been verified against NASA’s General Aviation Synthesis Program (GASP) results for several vehicles that exercise the analysis capability for a variety of aircraft classes and engine models, including both turbofans and turboprops. We believe this capability will set the stage for more rapid development of novel aircraft models with more flexible design assessments using optimization, and more traceable analysis for reproducibility. Additionally, a novel Pareto optimization-based design space exploration that shows the trade between range flown and fuel burn by manipulating the thrust split during cruise for a true parallel hybrid aircraft was developed by leveraging the unique features of Gascon.

aircraft design↗
NASA NTRS1974

Multilateration - A nondegenerate method of obtaining station coordinates and satellite ephemerides

A technique for the determination of three-dimensional station coordinates and satellite ephemerides is developed which is based on the principle of multilateration. The method makes use of a system of six ground stations which simultaneously measure the slant range between each station and one or two satellites. It is demonstrated that a minimum of six stations is required in order to yield a system which will be free of mathematical degeneracies. It will be seen that the method of multilateration is not dependent upon the position of the satellite or any other dynamical considerations in the equations used to determine the relative station coordinates. In fact, the satellite coordinates are obtained as a direct by-product of the method. Numerical results are presented which indicate that the method of multilateration can determine the relative three-dimensional station coordinates with an accuracy that is limited only by the hardware measurement system. If a highly accurate laser ranging system is used, then accuracies in the 1-cm range can be expected.

Ong, K. M.↗
NASA NTRS1994

Feature isolation and quantification of evolving datasets

Identifying and isolating features is an important part of visualization and a crucial step for the analysis and understanding of large time-dependent data sets (either from observation or simulation). In this proposal, we address these concerns, namely the investigation and implementation of basic 2D and 3D feature based methods to enhance current visualization techniques and provide the building blocks for automatic feature recognition, tracking, and correlation. These methods incorporate ideas from scientific visualization, computer vision, image processing, and mathematical morphology. Our focus is in the area of fluid dynamics, and we show the applicability of these methods to the quantification and tracking of three-dimensional vortex and turbulence bursts.

Source record↗
DOE OSTI2024

Robust Iterative Method for Symmetric Quantum Signal Processing in All Parameter Regimes

Here, this paper addresses the problem of solving nonlinear systems in the context of symmetric quantum signal processing (QSP), a powerful technique for implementing matrix functions on quantum computers. Symmetric QSP focuses on representing target polynomials as products of matrices in SU(2) that possess symmetry properties. We present a novel Newton’s method tailored for efficiently solving the nonlinear system involved in determining the phase factors within the symmetric QSP framework. Our method demonstrates rapid and robust convergence in all parameter regimes, including the challenging scenario with ill-conditioned Jacobian matrices, using standard double precision arithmetic operations. For instance, solving symmetric QSP for a highly oscillatory target function α cos(1000x) (polynomial degree ≈ 1433) takes 6 iterations to converge to machine precision when α = 0.9, and the number of iterations only increases to 18 iterations when α = 1 – 10 -9 with a highly ill-conditioned Jacobian matrix. Leveraging the matrix product state structure of symmetric QSP, the computation of the Jacobian matrix incurs a computational cost comparable to a single function evaluation. Moreover, we introduce a reformulation of symmetric QSP using real-number arithmetics, further enhancing the method’s efficiency. Extensive numerical tests validate the effectiveness and robustness of our approach, which has been implemented in the QSPPACK software package.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1986

Boundary perturbation method for free boundary problem in convectively cooled continuous casting

Novel mathematical techniques are here used to treat general boundary conditions in convectively cooled continuous ingot casting problems, applying a constant convective heat transfer coefficient at the ingot sides (so that the temperature distribution along the sides is unknown). The problem is first inverted to treat the physical coordinates as dependent variables of temperature and heat flow, mapping the ingot into a region that is nearly rectangular. A boundary perturbation method is then used to obtain an analytical solution in this near-rectangular region. Solidification interface shapes depend on two dimensionless parameters: one is a function of the casting velocity, while the other is dependent on the convective heat transfer coefficient at the cooled wall.

Siegel, R.↗
DOE OSTI2026

Measuring Success: A Refined Methodology for Estimating Long-term Continuous Improvement

Successful resource management systems require current and detailed feedback on operational and corporate-level performance. As corporate accountability concerns intensify, the precision and reliability of these performance metrics have become crucial. Traditional savings estimation methods can be difficult to understand, particularly linear regression, and can provide varying results. This paper reviews common efficiency metrics and highlights underlying mathematical inconsistencies when estimating total and percent savings with current methods. A refined approach to calculating long-term utility savings is proposed that simplifies current methodologies utilizing ratios to define an adjusted baseline, allowing for consistent and fair aggregation of results across multiple scales from resources to corporate performance. A simplified example demonstrates how the proposed methodology improves upon existing methods, especially in intermediate years. This paper’s major contributions are the simplified approach for converting modeled utility usage into estimated savings and the consistent roll-up methodology enabling more comparable, aggregable, and actionable results across scales.

Price, Chris [ORNL] (ORCID:0000000202007906)↗
DOE OSTI2024

Impact of Increased Monte Carlo Parameters on Sensitivity Calculations with SCALE [Slides]

For both models and tests, NPG and NSK parameters have only small effects on calculated sensitivity coefficients. Outside of NPG=100, only differences in CFP affected sensitivity coefficient values. Fission reactions ( 235 U) require more NPG values than scattering ( 238 U) reactions – more particles are needed to locate fission sources in the model more accurately. This work confirms the previous results with the IFP method where the CFP parameter has the greatest impact on calculated sensitivity coefficients. While immediate work focuses on fast systems, other model specifications may require a different set of MC parameters.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Impact of representative ground motion level on seismic PSA with the boundary between overestimation and underestimation

One commonly used approach in seismic probabilistic safety assessment (PSA) is the discrete method. This method follows the standard PSA framework and can be applied to various models, such as multi-unit models, while reducing computational costs using standard software. However, due to the inability to subdivide intervals infinitely, the discrete method approximates with a finite number of subintervals. In practice, different numbers of subintervals are applied, and the representative ground motion level is selected based on expert judgment. When employing a smaller number of subintervals, it is important to take caution to prevent underestimation. This study analyzes the impact of the representative ground motion level on seismic risk. It confirms that underestimation can occur with a small number of subintervals depending on the representative ground motion level. This study also proposes a method for determining the boundary of underestimation and overestimation. The method is demonstrated through examples, providing a mathematical foundation for selecting appropriate representative ground motion levels. By avoiding underestimation, this research helps prevent the oversight of significant risk contributors and enhances the understanding of seismic risk.

99 - GENERAL AND MISCELLANEOUS↗
NASA NTRS1972

On propagation of long waves in curved ducts.

A two-dimensional detailed study of the behavior of long waves in curved ducts and in junctions between straight and curved ducts will be given. The mathematical treatment of the problem utilizes the method of separation variables. Solutions and expressions for principal mode of the wave are obtained by using the linearized equation of motion solved for its characteristic values. The unavoidable approximations in the numerical solutions of the cylindrical functions are due to use of series expansion of Bessel functions and from restrictions necessary to solve infinite matrices.

Rostafinski, W.↗
NASA NTRS1975

The NATA code; theory and analysis. Volume 3: Programmer's manual

The present, third volume of the final report is a programmer's manual for the code. It provides a listing of the FORTRAN 4 source program; a complete glossary of FORTRAN symbols; a discussion of the purpose and method of operation of each subroutine (including mathematical analyses of special algorithms); and a discussion of the operation of the code on IBM/360 and UNIVAC 1108 systems, including required control cards and the overlay structure used to accommodate the code to the limited core size of the 1108. In addition, similar information is provided to document the programming of the NOZFIT code, which is employed to set up nozzle profile curvefits for use in NATA.

Bade, W. L.↗
NASA NTRS1976

Optimum lattice arrangement developed from a rigorous analytical basis

The spanwise vortex-lattice arrangement is mathematically established by lattice solutions of the slender wing which are shown to be analogous to the chordwise vortex-lattice thin wing solution. Solutions for any number N of panels wing theory lift and induced drag and thin wing theory lift and moment are predicted exactly. As N approaches infinity, the slender wing elliptic spanwise loading and thin wing cotangent chordwise loading are predicted, which proves there is mathematical convergence of the vortex-lattice method to the exact answer. Based on this planform spanwise lattice arrangement, an A-vortex-lattice spanwise system is developed for an arbitrary aspect ratio A. This A-lattice has the optimum characteristic of predicting lift accurately for any value of N.

Deyoung, J.↗
NASA NTRS1977

A method of determining spectral dye densities in color films

A mathematical analysis technique called characteristic vector analysis, reported by Simonds (1963), is used to determine spectral dye densities in multiemulsion film such as color or color-IR imagery. The technique involves examining a number of sets of multivariate data and determining linear transformations of these data to a smaller number of parameters which contain essentially all of the information contained in the original set of data. The steps involved in the actual procedure are outlined. It is shown that integral spectral density measurements of a large number of different color samples can be accurately reconstructed from the calculated spectral dye densities.

Friederichs, G. A.↗