SEARCH · Engineering Papers

Results for “Approximation”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2

All sources NASA NTRS DOE OSTI

NASA NTRS2009

Validity of the Aluminum Equivalent Approximation in Space Radiation Shielding

The origin of the aluminum equivalent shield approximation in space radiation analysis can be traced back to its roots in the early years of the NASA space programs (Mercury, Gemini and Apollo) wherein the primary radiobiological concern was the intense sources of ionizing radiation causing short term effects which was thought to jeopardize the safety of the crew and hence the mission. Herein, it is shown that the aluminum equivalent shield approximation, although reasonably well suited for that time period and to the application for which it was developed, is of questionable usefulness to the radiobiological concerns of routine space operations of the 21 st century which will include long stays onboard the International Space Station (ISS) and perhaps the moon. This is especially true for a risk based protection system, as appears imminent for deep space exploration where the long-term effects of Galactic Cosmic Ray (GCR) exposure is of primary concern. The present analysis demonstrates that sufficiently large errors in the interior particle environment of a spacecraft result from the use of the aluminum equivalent approximation, and such approximations should be avoided in future astronaut risk estimates. In this study, the aluminum equivalent approximation is evaluated as a means for estimating the particle environment within a spacecraft structure induced by the GCR radiation field. For comparison, the two extremes of the GCR environment, the 1977 solar minimum and the 2001 solar maximum, are considered. These environments are coupled to the Langley Research Center (LaRC) deterministic ionized particle transport code High charge (Z) and Energy TRaNsport (HZETRN), which propagates the GCR spectra for elements with charges (Z) in the range I <= Z <= 28 (H -- Ni) and secondary neutrons through selected target materials. The coupling of the GCR extremes to HZETRN allows for the examination of the induced environment within the interior' of an idealized spacecraft as approximated by a spherical shell shield, and the effects of the aluminum equivalent approximation for a good polymeric shield material such as genetic polyethylene (PE). The shield thickness is represented by a 25 g/cm spherical shell. Although one could imagine the progression to greater thickness, the current range will be sufficient to evaluate the qualitative usefulness of the aluminum equivalent approximation. Upon establishing the inaccuracies of the aluminum equivalent approximation through numerical simulations of the GCR radiation field attenuation for PE and aluminum equivalent PE spherical shells, we Anther present results for a limited set of commercially available, hydrogen rich, multifunctional polymeric constituents to assess the effect of the aluminum equivalent approximation on their radiation attenuation response as compared to the generic PE.

Badavi, Francis F.↗

NASA NTRS1986

Sensitivity analysis and approximation methods for general eigenvalue problems

Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought.

Murthy, D. V.↗

NASA NTRS1985

UNAERO: A package of FORTRAN subroutines for approximating unsteady aerodynamics in the time domain

This report serves as an instruction and maintenance manual for a collection of CDC CYBER FORTRAN IV subroutines for approximating the unsteady aerodynamic forces in the time domain. The result is a set of constant-coefficient first-order differential equations that approximate the dynamics of the vehicle. Provisions are included for adjusting the number of modes used for calculating the approximations so that an accurate approximation is generated. The number of data points at different values of reduced frequency can also be varied to adjust the accuracy of the approximation over the reduced-frequency range. The denominator coefficients of the approximation may be calculated by means of a gradient method or a least-squares approximation technique. Both the approximation methods use weights on the residual error. A new set of system equations, at a different dynamic pressure, can be generated without the approximations being recalculated.

Dunn, H. J.↗

NASA NTRS1988

Approximation theory for LQG (Linear-Quadratic-Gaussian) optimal control of flexible structures

An approximation theory is presented for the LQG (Linear-Quadratic-Gaussian) optimal control problem for flexible structures whose distributed models have bounded input and output operators. The main purpose of the theory is to guide the design of finite dimensional compensators that approximate closely the optimal compensator. The optimal LQG problem separates into an optimal linear-quadratic regulator problem and an optimal state estimation problem. The solution of the former problem lies in the solution to an infinite dimensional Riccati operator equation. The approximation scheme approximates the infinite dimensional LQG problem with a sequence of finite dimensional LQG problems defined for a sequence of finite dimensional, usually finite element or modal, approximations of the distributed model of the structure. Two Riccati matrix equations determine the solution to each approximating problem. The finite dimensional equations for numerical approximation are developed, including formulas for converting matrix control and estimator gains to their functional representation to allow comparison of gains based on different orders of approximation. Convergence of the approximating control and estimator gains and of the corresponding finite dimensional compensators is studied. Also, convergence and stability of the closed-loop systems produced with the finite dimensional compensators are discussed. The convergence theory is based on the convergence of the solutions of the finite dimensional Riccati equations to the solutions of the infinite dimensional Riccati equations. A numerical example with a flexible beam, a rotating rigid body, and a lumped mass is given.

Gibson, J. S.↗

NASA NTRS1990

Differential equation based method for accurate approximations in optimization

A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.

Pritchard, Jocelyn I.↗

NASA NTRS1990

Differential equation based method for accurate approximations in optimization

This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.

Pritchard, Jocelyn I.↗

NASA NTRS1992

New literal approximations for the longitudinal dynamic characteristics of flexible flight vehicles

The goal of the literal approximation method is to obtain simple literal (analytical) approximations for key dynamic characteristics of flexible flight vehicles. A basic question regarding the method is its usefulness as an additional design tool for existing design and simulation procedures. Two aspects of this question are: (1) ease of derivation and use of the literal approximations, and (2) the suitability of one set of literal approximations to describe the dynamics of a large set of significantly different vehicles. These issues are addressed by incorporating symbolic manipulation software into the literal approximation method for the analysis of a fifth order model of the longitudinal dynamics of a flexible flight vehicle. The automated literal approximation generated in this fashion reduces the manual derivation time by an approximate factor of four. A single set of literal approximations is shown to provide adequate approximations for the dynamics of significantly different flight vehicles configurations, such as an aircraft, a missile, and a hypersonic vehicle.

Livneh, Rafael↗

NASA NTRS1993

Producing approximate answers to database queries

We have designed and implemented a query processor, called APPROXIMATE, that makes approximate answers available if part of the database is unavailable or if there is not enough time to produce an exact answer. The accuracy of the approximate answers produced improves monotonically with the amount of data retrieved to produce the result. The exact answer is produced if all of the needed data are available and query processing is allowed to continue until completion. The monotone query processing algorithm of APPROXIMATE works within the standard relational algebra framework and can be implemented on a relational database system with little change to the relational architecture. We describe here the approximation semantics of APPROXIMATE that serves as the basis for meaningful approximations of both set-valued and single-valued queries. We show how APPROXIMATE is implemented to make effective use of semantic information, provided by an object-oriented view of the database, and describe the additional overhead required by APPROXIMATE.

Vrbsky, Susan V.↗

NASA NTRS2004

A Subsonic Aircraft Design Optimization With Neural Network and Regression Approximators

The Flight-Optimization-System (FLOPS) code encountered difficulty in analyzing a subsonic aircraft. The limitation made the design optimization problematic. The deficiencies have been alleviated through use of neural network and regression approximations. The insight gained from using the approximators is discussed in this paper. The FLOPS code is reviewed. Analysis models are developed and validated for each approximator. The regression method appears to hug the data points, while the neural network approximation follows a mean path. For an analysis cycle, the approximate model required milliseconds of central processing unit (CPU) time versus seconds by the FLOPS code. Performance of the approximators was satisfactory for aircraft analysis. A design optimization capability has been created by coupling the derived analyzers to the optimization test bed CometBoards. The approximators were efficient reanalysis tools in the aircraft design optimization. Instability encountered in the FLOPS analyzer was eliminated. The convergence characteristics were improved for the design optimization. The CPU time required to calculate the optimum solution, measured in hours with the FLOPS code was reduced to minutes with the neural network approximation and to seconds with the regression method. Generation of the approximators required the manipulation of a very large quantity of data. Design sensitivity with respect to the bounds of aircraft constraints is easily generated.

Patnaik, Surya N.↗

NASA NTRS1976

Test particle propagation in magnetostatic turbulence. 2: The local approximation method

An approximation method for statistical mechanics is presented and applied to a class of problems which contains a test particle propagation problem. All of the available basic equations used in statistical mechanics are cast in the form of a single equation which is integrodifferential in time and which is then used as the starting point for the construction of the local approximation method. Simplification of the integrodifferential equation is achieved through approximation to the Laplace transform of its kernel. The approximation is valid near the origin in the Laplace space and is based on the assumption of small Laplace variable. No other small parameter is necessary for the construction of this approximation method. The n'th level of approximation is constructed formally, and the first five levels of approximation are calculated explicitly. It is shown that each level of approximation is governed by an inhomogeneous partial differential equation in time with time independent operator coefficients. The order in time of these partial differential equations is found to increase as n does. At n = 0 the most local first order partial differential equation which governs the Markovian limit is regained.

Klimas, A. J.↗

NASA NTRS1982

Difference equation state approximations for nonlinear hereditary control problems

Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.

Rosen, I. G.↗

NASA NTRS1984

Difference equation state approximations for nonlinear hereditary control problems

Rosen, I. G.↗

NASA NTRS1988

Mathematical algorithms for approximate reasoning

Most state of the art expert system environments contain a single and often ad hoc strategy for approximate reasoning. Some environments provide facilities to program the approximate reasoning algorithms. However, the next generation of expert systems should have an environment which contain a choice of several mathematical algorithms for approximate reasoning. To meet the need for validatable and verifiable coding, the expert system environment must no longer depend upon ad hoc reasoning techniques but instead must include mathematically rigorous techniques for approximate reasoning. Popular approximate reasoning techniques are reviewed, including: certainty factors, belief measures, Bayesian probabilities, fuzzy logic, and Shafer-Dempster techniques for reasoning. A group of mathematically rigorous algorithms for approximate reasoning are focused on that could form the basis of a next generation expert system environment. These algorithms are based upon the axioms of set theory and probability theory. To separate these algorithms for approximate reasoning various conditions of mutual exclusivity and independence are imposed upon the assertions. Approximate reasoning algorithms presented include: reasoning with statistically independent assertions, reasoning with mutually exclusive assertions, reasoning with assertions that exhibit minimum overlay within the state space, reasoning with assertions that exhibit maximum overlay within the state space (i.e. fuzzy logic), pessimistic reasoning (i.e. worst case analysis), optimistic reasoning (i.e. best case analysis), and reasoning with assertions with absolutely no knowledge of the possible dependency among the assertions. A robust environment for expert system construction should include the two modes of inference: modus ponens and modus tollens. Modus ponens inference is based upon reasoning towards the conclusion in a statement of logical implication, whereas modus tollens inference is based upon reasoning away from the conclusion. These algorithms allow one to reason accurately with uncertain data. The above environment can replicate state-f-the-art expert system environments which provides a continuity between the current expert systems which cannot be validated or verified and future expert systems which should be both validated and verified

Murphy, John H.↗

NASA NTRS1994

Comparison of daily averaged reflection, transmission, and absorption for selected radiative flux transfer approximations

This paper compares accuracy for the daily averaged reflection, transmission, and absorption of solar flux derived from the delta-four-stream approximation and a few selected two-stream approximations. In the chosen variety of two-stream approximations, a two-stream approximation suggested by Zhu produces the most accurate reflection and transmission for conservative scattering with optical thickness tau(sub 1) less than 3. Otherwise, the delta-Eddington approximation is superior to other two-stream approximations. The delta-four stream approximation produces errors that are smaller than the delta-Eddington approximation by at least a factor of 2.

Zhu, Xun↗

NASA NTRS1997

Geochemistry of approximately 1.9 Ga sedimentary rocks from northeastern Labrador, Canada

Fifty-eight rock chips from fifteen samples of sedimentary rocks from the Ramah Group (approximately 1.9 Ga) in northeastern Labrador, Canada, were analyzed for major and minor elements, including C and S, to elucidate weathering processes on the Earth's surface about 1.9 Ga ago. The samples come from the Rowsell Harbour, Reddick Bight, and Nullataktok Formations. Two rock series, graywackes-gray shales of the Rowsell Harbour, Reddick Bight and Nullataktok Formations, and black shales of the Nullataktok Formation, are distinguishable on the basis of lithology, mineralogy, and major and trace element chemistry. The black shales show lower concentrations than the graywackes-gray shales in TiO2 (0.3-0.7 wt% vs. 0.7-1.8 wt%), Al2O3 (9.5-20.1 wt% vs. 13.0-25.0 wt%), and sigma Fe (<1 wt% vs. 3.8-13.9 wt% as FeO). Contents of Zr, Th, U, Nb, Ce, Y, Rb, Y, Co, and Ni are also lower in the black shales. The source rocks for the Ramah Group sediments were probably Archean gneisses with compositions similar to those in Labrador and western Greenland. The major element chemistry of source rocks for the Ramah Group sedimentary rocks was estimated from the Al2O3/TiO2 ratios of the sedimentary rocks and the relationship between the major element contents (e.g., SiO2 wt%) and Al2O3/TiO2 ratios of the Archean gneisses. This approach is justified, because the Al/Ti ratios of shales generally retain their source rock values; however, the Zr/Al, Zr/Ti, and Cr/Ni ratios fractionate during the transport of sediments. The measured SiO2 contents of shales in the Ramah Group are generally higher than the estimated SiO2 contents of source rocks by approximately 5 wt%. This correction may also have to be applied when estimating average crustal compositions from shales. Two provenances were recognized for the Ramah Group sediments. Provenance I was comprised mostly of rocks of bimodal compositions, one with SiO2 contents approximately 45 wt% and the other approximately 65 wt%, and was the source for most sedimentary rocks of the Ramah Group, except for black shales of the Nullataktok Formation. The black shales were apparently derived from Provenance II that was comprised mostly of felsic rocks with SiO2 contents approximately 65 wt%. Comparing the compositions of the Ramah Group sedimentary rocks and their source rocks, we have recognized that several major elements, especially Ca and Mg, were lost almost entirely from the source rocks during weathering and sedimentation. Sodium and potassium were also leached almost entirely during the weathering of the source rocks. However, significant amounts of Na were added to the black shales and K to all the rock types during diagenesis and/or regional metamorphism. The intensity of weathering of source rocks for the Ramah Group sediments was much higher than that of typical Phanerozoic sediments, possibly because of a higher PCO2 in the Proterozoic atmosphere. Compared to the source rock values, the Fe3+/Ti ratios of many of the graywackes and gray shales of the Ramah Group are higher, the Fe2+/Ti ratios are lower, and the sigma Fe/Ti ratios are the same. Such characteristics of the Fe geochemistry indicate that these sedimentary rocks are comprised of soils formed by weathering of source rocks under an oxygen-rich atmosphere. The atmosphere about 1.9 Ga was, therefore, oxygen rich. Typical black shales of Phanerozoic age exhibit positive correlations between the organic C contents and the concentrations of S, U, and Mo, because these elements are enriched in oxygenated seawater and are removed from seawater by organic matter in sediments. However, such correlations are not found in the Ramah Group sediments. Black shales of the Ramah Group contain 1.7-2.8 wt% organic C, but are extremely depleted in sigma Fe (<1 wt% as FeO), S (<0.3 wt%), U (approximately l ppm), Mo (<5 ppm), Ni (<2 ppm), and Co (approximately 0 ppm). This lack of correlation, however, does not imply that the approximately 1.9 Ga atmosphere-ocean system was anoxic. Depletion of these elements from the Ramah Group sediments may have occurred during diagenesis.

Non-NASA Center↗

DOE OSTI2026

Direct Nonlinear Approximation for Security Region Boundary of Integrated Energy Systems: A Polynomial Chaos Expansion Solution

The strong interdependence of electricity, gas, and heating systems can facilitate fault propagation within integrated energy systems (IESs), posing significant challenges to secure operation. This paper proposes a polynomial chaos expansion (PCE)-based approximation method to accurately characterize the IES security region boundary (IES–SRB). By integrating the Karush-Kuhn-Tucker conditions with PCE theory, the IES-SRB approximation problem is reformulated as a set of nonlinear equations concerning the approximation coefficients. Using the Galerkin projection method, these equations are further transformed into a system of projection equations that govern the polynomial approximation coefficients in the IES-SRB approximation. To reduce computational complexity while maintaining high approximation accuracy, a piecewise polynomial approximation method is proposed. Numerical studies on the E39-G20-H6 and E118-G96-H52 IES test systems demonstrate that the proposed method can accurately and effectively construct IES security regions.

Wu, Chenghao [Northeast Electric Power University]↗

NASA NTRS1988

Approximation of the dielectric properties of Maxwellian plasmas - Dispersion functions and physical constraints

The dielectric properties of Maxwellian plasmas are approximated using high Pade approximants to the dispersion function and direct approximation of the distribution. Physical constraints on permissible approximations are discussed, and it is found that some previously published results can lead to predictions of qualitatively incorrect wave properties, including unphysical negative damping. Approximate dispersion functions for Maxwellian distributions are given explicitly, and some of the effects of these approximations on the resulting dispersion are discussed. The approximations discussed here are of use both in analytical work and in accelerating large-scale numerical computations.

Robinson, P. A.↗

NASA NTRS1989

Nonlinear control via approximate input-output linearization - The ball and beam example

This paper presents an approach for the approximate input-output linearization of nonlinear systems, particularly those for which relative degree is not well defined. It is shown that there is a great deal of freedom in the selection of an approximation and that, by designing a tracking controller based on the approximating system, tracking of reasonable trajectories can be achieved with small error. The approximating system is itself a nonlinear system, with the difference that it is input-output linearizable by state feedback. Some properties of the accuracy of the approximation are demonstrated and, in the context of the ball and beam example, it is shown to be far superior to the Jacobian approximation. The results are focused on finding regular SISO systems which are close to systems which are not regular and controlling these approximate regular systems.

Hauser, John↗