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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 NTRSDate not supplied

Sliding Mode Transient Scaling Controller for Gas Turbine Engine Emulation on an Electric Machine

The testing of hybrid-electric aircraft propulsion system components and control strategies has dictated the need for a low-cost gas turbine engine (GTE) emulation testbed. Within that testbed, electric machines (EMs) are used to emulate the dynamics of a GTE. GTE mathematical models are used to provide control torque commands to EMs. Most GTEs operate at much higher power levels than commercially available EMs. Therefore, the transient shaft dynamics of the GTE must be scaled appropriately. The sliding mode controller being presented here aims to deliver a control torque to an electric machine with a fixed inertia that will allow it to mimic the response of a GTE with a different inertia. This solution provides an easily scalable and tunable transient emulation method without having to scale the GTE mathematical model and/or add inertia to the physical EM shaft. This controller has been proven in simulation to produce nearly identical transient rotational speed response to a reference GTE shaft model which has a large difference in rotational inertia compared to the motor. Also, the inherent robustness of the sliding mode control scheme results in negligible changes in performance for large amounts of parameter uncertainty while minimizing high frequency fluctuations in the control signal.

sliding mode↗
NASA NTRS1944

Some Notes on the Determination of the Stick-Free Neutral Point from Wind-Tunnel Data

Two graphical methods are presented for determining the stick-free neutral point, and they are extensions of the methods commonly used to determine the stick-free neutral point. A mathematical formula for computing the stick-free neutral point is also given. These methods may be applied to determine approximately the increase in tail size necessary to shift the neutral point (stick fixed or free) to any desired location on an airplane having inadequate longitudinal stability.

STABILITY CALCULATIONS↗
NASA NTRSDate not supplied

Comparison and Application of Battery Modeling Methods for Conceptual Electric Aircraft

Growing interest in design and optimization of electrified aircraft propulsion concepts prompts the need for accurate, flexible, and efficient methods to model battery systems. Presented in this paper are three battery modeling methods that have been used at NASA’s Glenn Research Center, each representing different mathematical or electrical approaches. Thévenin equivalent circuit, normalization, and curve-fitting methods are compared against battery cell test data for the X-57 Maxwell electric aircraft technology demonstrator. The methods are then applied in a simple multidisciplinary optimization context using NASA’s Six-Passenger Electric Quadrotor concept to determine their applicability and performance. The normalization method achieves the highest accuracy for steady and unsteady discharge rates with a voltage mean error percentage of 0.423% and 1.186%, respectively. Optimal quadrotor mission range between the models varies up to 0.5 nmi, identifying current battery modeling methods as a potentially significant contributor to mission analysis error. A set of relevant tools and techniques for conceptual battery modeling are identified in this paper, with conclusions made on the utility of each modeling approach for various design challenges.

power↗
NASA NTRS1947

Determination of elastic stresses in gas-turbine disks

A method is presented for the calculation of elastic stresses in symmetrical disks typical of those of a high-temperature gas turbine. The method is essentially a finite-difference solution of the equilibrium and compatibility equations for elastic stresses in a symmetrical disk. Account can be taken of point-to-point variations in disk thickness, in temperature, in elastic modulus, in coefficient of thermal expansion, in material density, and in Poisson's ratio. No numerical integration or trial-and-error procedures are involved and the computations can be performed in rapid and routine fashion by nontechnical computers with little engineering supervision. Checks on problems for which exact mathematical solutions are known indicate that the method yields results of high accuracy. Illustrative examples are presented to show the manner of treating solid disks, disks with central holes, and disks constructed either of a single material or two or more welded materials. The effect of shrink fitting is taken into account by a very simple device.

Manson, S S↗
DOE OSTI2025

Reliable and Efficient Machine Learning (Final Technical Report)

Modern scientific experiments generate massive amounts of data at a pace much faster than humans can manually analyze. While machine learning has revolutionized commercial data analysis (such as recommending movies or recognizing faces), applying these tools to complex scientific discovery is challenging because scientific answers must be precise, interpretable, and adhere to physical laws. The research under this project aims to develop new mathematical tools and computer algorithms specifically designed for scientific applications. Major progress has been made in automatically cleaning and deconstructing messy experimental data, analyzing the visual information of physical phenomena, determining the underlying physical variables, and providing rig orous mathematical analysis of interesting algorithms and concepts widely used in machine learning. This project addressed the critical gap between our ability to generate massive scientific data and our ability to extract interpretable information from it. We established mathematical foundations for Scientific Machine Learning (SciML) aimed at effective data analytics and automated discovery. Our work focused on three core objectives: (1) developing reliable feature extraction methods for dynamic high-dimensional data, (2) establishing mathematical foundations for discovering dynamics via neural networks, and (3) creating rigorous optimization techniques for these models. Key outcomes come from two fronts. On the practical side, they include the development of algorithms that significantly enhance the extraction of signals from field data, as well as the capability to handle situations that exhibit smooth variations or physical stretching due to temperature changes. They also include the creation of an automated framework for discovering fundamental state variables from raw experimental data, demonstrating the ability to identify intrinsic physical dimensions without prior knowledge of the governing laws. On the theoretical front, the research results in theoretical advances in Optimal Transport, a widely used notion in SciML, specifically regarding functions with fixed-size nodal sets, provide sharp bounds relevant to uncertainty quantification. Meanwhile, the outcomes also include the establishment of convergence theories for nonlocal gradient descent methods, enabling robust optimization with noisy data in high-dimensional settings commonly encountered in scientific modeling. The project also helps creating opportunities to train the next generation of researchers, equipping them with the necessary technical skills for today’s workplace and preparing them for future advances.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1975

A new structural analysis/synthesis capability - ACCESS

The creation of an efficient automated capability for minimum weight design of structures is reported. The ACCESS 1 computer program combines finite element analysis techniques and mathematical programming algorithms using an innovative collection of approximation concepts. Design variable linking, constraint deletion techniques and approximate analysis methods are used to generate a sequence of small explicit mathematical programming problems which retain the essential features of the design problem. Organization of the finite element analysis is carefully matched to the design optimization task. The efficiency of the ACCESS 1 program is demonstrated by giving results for several example problems.

Schmit, L. A.↗
NASA NTRS1993

Formal methods for life-critical software

The use of computer software in life-critical applications, such as for civil air transports, demands the use of rigorous formal mathematical verification procedures. This paper demonstrates how to apply formal methods to the development and verification of software by leading the reader step-by-step through requirements analysis, design, implementation, and verification of an electronic phone book application. The current maturity and limitations of formal methods tools and techniques are then discussed, and a number of examples of the successful use of formal methods by industry are cited.

Butler, Ricky W.↗
NASA NTRS1993

Formal Methods for Life-Critical Software

The use of computer software in life-critical applications, such as for civil air transports, demands the use of rigorous formal mathematical verification procedures. This paper demonstrates how to apply formal methods to the development and verification of software by leading the reader step-by-step through requirements analysis, design, implementation, and verification of an electronic phone book application. The current maturity and limitations of formal methods tools and techniques are then discussed, and a number of examples of the successful use of formal methods by industry are cited.

Butler, Ricky W.↗
NASA NTRS1971

Application of Monte Carlo techniques to optimization of high-energy beam transport in a stochastic environment

An algorithm employing a modified sequential random perturbation, or creeping random search, was applied to the problem of optimizing the parameters of a high-energy beam transport system. The stochastic solution of the mathematical model for first-order magnetic-field expansion allows the inclusion of state-variable constraints, and the inclusion of parameter constraints allowed by the method of algorithm application eliminates the possibility of infeasible solutions. The mathematical model and the algorithm were programmed for a real-time simulation facility; thus, two important features are provided to the beam designer: (1) a strong degree of man-machine communication (even to the extent of bypassing the algorithm and applying analog-matching techniques), and (2) extensive graphics for displaying information concerning both algorithm operation and transport-system behavior. Chromatic aberration was also included in the mathematical model and in the optimization process. Results presented show this method as yielding better solutions (in terms of resolutions) to the particular problem than those of a standard analog program as well as demonstrating flexibility, in terms of elements, constraints, and chromatic aberration, allowed by user interaction with both the algorithm and the stochastic model. Example of slit usage and a limited comparison of predicted results and actual results obtained with a 600 MeV cyclotron are given.

Parrish, R. V.↗
NASA NTRS1990

Accumulated approximation: A new method for structural optimization by iterative improvement

A new method for the solution of non-linear mathematical programming problems in the field of structural optimization is presented. It is an iterative scheme which for each iteration refines the approximation of objective and constraint functions by accumulating the function values of previously visited design points. The method has proven to be competitive for a number of well-known examples of which one is presented here. Furthermore because of the accumulation strategy, the method produces convergence even when the sensitivity analysis is inaccurate.

Rasmussen, John↗
DOE OSTI2024

Classifying photonic topology using the spectral localizer and numerical K -theory

Recently, the spectral localizer framework has emerged as an efficient approach for classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts such as topological solitons and topological dynamics, where a system’s occupation induces a local change in its topology due to nonlinearity. For systems embedded in radiative environments that do not possess a shared bulk spectral gap, this framework enables the identification of local topology and shows that local topological protection is preserved despite the lack of a common gap. However, as the spectral localizer framework is rooted in the mathematics of C*-algebras, and not vector bundles, understanding and using this framework requires developing intuition for a somewhat different set of underlying concepts than those that appear in traditional approaches for classifying material topology. In this tutorial, we introduce the spectral localizer framework from a ground-up perspective and provide physically motivated arguments for understanding its local topological markers and associated local measure of topological protection. In doing so, we provide numerous examples of the framework’s application to a variety of topological classes, including crystalline and higher-order topology. We then show how Maxwell’s equations can be reformulated to be compatible with the spectral localizer framework, including the possibility of radiative boundary conditions. To aid in this introduction, we also provide a physics-oriented introduction to multi-operator pseudospectral methods and numerical K-theory, two mathematical concepts that form the foundation for the spectral localizer framework. Finally, we provide some mathematically oriented comments on the C*-algebraic origins of this framework, including a discussion of real C*-algebras and graded C*-algebras that are necessary for incorporating physical symmetries. Looking forward, we hope that this tutorial will serve as an approachable starting point for learning the foundations of the spectral localizer framework.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

Dynamics of a split torque helicopter transmission

Split torque designs, proposed as alternatives to traditional planetary designs for helicopter main rotor transmissions, can save weight and be more reliable than traditional designs. This report presents the results of an analytical study of the system dynamics and performance of a split torque gearbox that uses a balance beam mechanism for load sharing. The Lagrange method was applied to develop a system of equations of motion. The mathematical model includes time-varying gear mesh stiffness, friction, and manufacturing errors. Cornell's method for calculating the stiffness of spur gear teeth was extended and applied to helical gears. The phenomenon of sidebands spaced at shaft frequencies about gear mesh fundamental frequencies was simulated by modeling total composite gear errors as sinusoid functions. Although the gearbox has symmetric geometry, the loads and motions of the two power paths differ. Friction must be considered to properly evaluate the balance beam mechanism. For the design studied, the balance beam is not an effective device for load sharing unless the coefficient of friction is less than 0.003. The complete system stiffness as represented by the stiffness matrix used in this analysis must be considered to precisely determine the optimal tooth indexing position.

Krantz, Timothy L.↗
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.↗