SEARCH · Engineering Papers

Results for “matrix 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 577 records · Page 32

All sources NASA NTRS DOE OSTI

NASA NTRS1978

Fracture and crack growth in orthotropic laminates

Goree, J. G.↗

NASA NTRS1980

Stresses in a three-dimensional unidirectional composite containing broken fibers

An approximate solution is developed for the determination of the interlaminar normal and shear stresses in the vicinity of a crack in a three dimensional composite containing unidirectional linearly elastic fibers in an infinite linearly elastic matrix. In order to reduce the complexity of the formulation, certain assumptions are made as to the physically significant stresses to be retained. These simplifications reduce the partial differential equations of elasticity to differential-difference equations which are tractable using Fourier transform techniques. This 'material modeling' approach is in contrast with solutions developed by considering each lamina as a homogeneous, orthotropic layer. The resulting solution does not contain the classical singular stress field for the fibers and the influence of broken fibers on unbroken fibers is felt by a change in stress concentration factors. The matrix stresses however, are unbounded as the fiber spacing vanishes and an equivalent fiber-matrix geometry is proposed which gives the correct singular behavior. The numerical solution is considered in detail and several specific examples are presented. The potential for damaged or debonded zones to be generated by an embedded crack is discussed, and stress concentration factors for fibers near the crack are given. Detailed comparisons are made between the present solution, the analogous two-dimensional problem, and corresponding shear-lag models.

Goree, J. G.↗

NASA NTRS1991

An improved error assessment for the GEM-T1 gravitational model

Several tests have been designed to estimate the correct error variances for the GEM-T1 gravitational solution that was derived exclusively from satellite tracking data. The basic method uses both independent and dependent subset data solutions and produces a coefficient-by-coefficient estimate of the model uncertainties. The GEM-T1 errors have been further analyzed using a method based on eigenvalue-eigenvector analysis, which calibrates the entire covariance matrix. Dependent satellite data sets and independent altimetric, resonant satellite, and surface gravity data sets all confirm essentially the same error assessment. The calibration test results yield very stable calibration factors, which vary only by approximately 10 percent over the range of tests performed. Based on these calibrated error estimates, GEM-T1 is a significantly improved solution, which to degree and order 8 is twice as accurate as earlier satellite derived models like GEM-L2. Also, by being complete to degree and order 36, GEM-T1 is more complete and has significantly reduced aliasing effects that were present in previous models.

Lerch, F. J.↗

NASA NTRS1989

Mesh refinement in finite element analysis by minimization of the stiffness matrix trace

Most finite element packages provide means to generate meshes automatically. However, the user is usually confronted with the problem of not knowing whether the mesh generated is appropriate for the problem at hand. Since the accuracy of the finite element results is mesh dependent, mesh selection forms a very important step in the analysis. Indeed, in accurate analyses, meshes need to be refined or rezoned until the solution converges to a value so that the error is below a predetermined tolerance. A-posteriori methods use error indicators, developed by using the theory of interpolation and approximation theory, for mesh refinements. Some use other criterions, such as strain energy density variation and stress contours for example, to obtain near optimal meshes. Although these methods are adaptive, they are expensive. Alternatively, a priori methods, until now available, use geometrical parameters, for example, element aspect ratio. Therefore, they are not adaptive by nature. An adaptive a-priori method is developed. The criterion is that the minimization of the trace of the stiffness matrix with respect to the nodal coordinates, leads to a minimization of the potential energy, and as a consequence provide a good starting mesh. In a few examples the method is shown to provide the optimal mesh. The method is also shown to be relatively simple and amenable to development of computer algorithms. When the procedure is used in conjunction with a-posteriori methods of grid refinement, it is shown that fewer refinement iterations and fewer degrees of freedom are required for convergence as opposed to when the procedure is not used. The mesh obtained is shown to have uniform distribution of stiffness among the nodes and elements which, as a consequence, leads to uniform error distribution. Thus the mesh obtained meets the optimality criterion of uniform error distribution.

Kittur, Madan G.↗

NASA NTRS1971

An analysis and comparison of several trajectory optimization methods

The sensitivities of the convergence characteristics of the methods to initially assumed parameters and trial solution, convergence times, computer logic, and storage requirements are discussed. Numerical comparison of the convergence characteristics is made by considering a minimum time, low thrust, Earth-Mars transfer trajectory. A modified quasi-linearization method reduces convergence time by approximately 70% when compared with the generalized Newton-Raphson method and allows the terminal boundary to be specified by a general function of the problem variables. A uniquely specified and easily determined, time dependent weighting matrix for the gradient techniques accelerates the shaping of the optimal control program and improves the convergence characteristics during the terminal iterations. Convergence envelopes, indicating how sensitive the convergence characteristics are to initially assumed parameters, are plotted for the perturbation and quasi-linearization methods. Several iteration schemes are proposed which increase the size of the convergence envelopes and decrease the sensitivity of the method to initially assumed parameters.

Lewallen, J. M.↗

NASA NTRS1972

Filament-wound graphite/epoxy rocket motor case

The fabrication procedures are described for a filament-wound rocket motor case, approximately 56 cm long x 71 cm diameter, utilizing high tensile strength graphite fibers. The process utilized Fiberite Hy-E-1330B prepreg tape which consists of Courtaulds HTS fibers in a temperature-sensitive epoxy matrix. This fabrication effort, with resultant design, material and process recommendations, substantiates the manufacturing feasibility of graphite/epoxy rocket motor cases in the 56 cm x 71 cm size range.

Humphrey, W. D.↗

NASA NTRS1973

The nature of dark-etching rims in meteoritic taenite.

Taenite fields when etched develop a cloudy brown rim with approximate compositional limits of 25 and 40% Ni. In iron meteorites this cloudy zone is only a few microns wide, with a sharp, high-Ni edge about 1 micron from the kamacite interface and a diffuse edge several microns from the central plessite. It is always present in irons unless the meteorite has been cosmically or terrestrially reheated. X-ray and electron diffraction of grains scratched from exceptionally larger areas of cloudy taenite in the mesosiderite Estherville show that this etching zone contains a fine exsolution of kamacite. Detailed electron-probe investigations of taenite in Estherville show that there is a step in the M-shaped Ni profile at the sharp, high-Ni edge of the cloudy region, the Ni dropping suddenly from approximately 45 to 42%. It is proposed that exsolution in the cloudy region effectively froze in the Ni profile at that temperature. On subsequent cooling only the clear outer taenite continued to equilibrate with the kamacite matrix, producing the kink in the M profile. Cloudy taenite is therefore a variety of plessite differing from the usual varieties in that it forms at lower temperatures in areas much richer in Ni, and the morphology is not crystallographically oriented. Its absence can provide a sensitive indication of reheating.

Scott, E. R. D.↗

NASA NTRS1987

Nonlinear programming extensions to rational function approximations of unsteady aerodynamics

This paper deals with approximating unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these approximations are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is of lower order than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.

Tiffany, Sherwood H.↗

NASA NTRS1986

Composite tubes for the Space Station truss structure

The reference configuration of NASA's Space Station includes a large truss structure to support the various modules and solar arrays. This truss structure will be constructed from tubular members approximately 2 in. in diameter and up to 23 ft in length. The important design considerations for this structure are light weight, high stiffness, dimensional stability, and long-term durability. Continuous graphite fiber reinforced polymer matrix composite materials can meet the structural requirements, and are leading candidates for the tubular truss members. However, there are concerns regarding the durability of composites during the long-term exposure to atomic oxygen and thermal cycling that will be encountered during the Space Station service life. This paper discusses space environmental factors and their effect on composite materials, and provides estimates of the changes in mechanical and thermal properties of composites exposed to long-term Space Station conditions. The effect of low velocity impact and handling damage on composite tube properties is also discussed.

Bowles, David E.↗

NASA NTRS1988

Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces

The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.

Tiffany, Sherwood H.↗

NASA NTRS1989

Solution of steady-state one-dimensional conservation laws by mathematical programming

Solution techniques for a class of steady-state scalar conservation laws are developed analytically. Discretization by finite-volume formulas is employed to obtain an overdetermined system of algebraic equations, which are then perturbed nonsingularly (with perturbation coefficient = epsilon) and solved using the l(1) mathematical-programming algorithm of Seneta and Steiger (1984); this approach limits the matrix bandwidth to two, so that an explicit solution can be found efficiently. It is shown that, for small values of epsilon, the l(1) solutions exhibit sharp correctly located shocks and are nonoscillatory O(epsilon) approximations of the physically relevant solutions.

Lavery, J. E.↗

NASA NTRS1993

Fe/Mn in olivine of carbonaceous meteorites

Olivines in primitive meteorites show a range of Fe/Mn both within one grain and among grains suggesting that they have recorded changing conditions during or after growth. Because olivine should be an early forming phase, Fe/Mn is used here to infer these earliest conditions. Initial Fe/Mn in cores of isolated, euhedral forsterite in both C2 and C3 meteorites ranges from 25 to 35 but differs at grain edge. Murchison (C2) forsterites show Fe/Mn approaching 1.0 at the grain edge while Ornans Fe/Mn is near 60 at grain edge. These values are lower than the matrix Fe/Mn for both meteorites and the distinct difference in zoning profile indicates different processes operating during and after grain growth. The Fe/Mn of bulk samples from a particular source such as the Moon is nearly constant. Individual samples show variation suggesting that there is some fractionation of Mn from Fe. Minerals have their individual ranges of Fe/Mn which has been used to recognize different types of olivine within one meteorite. Extreme values of Fe/Mn below 1.0 occur in forsterite from some IDP's, UOC matrix, and C1 meteorites. There are apparently no detailed studies of Fe/Mn variation within single olivine grains. Forsterite grains in C2 and C3 carbonaceous chondrites show complex zoning, and the nearly pure forsterites (Fo greater than approximately 99.5) have high levels of some minor elements including Ti, Al, V, and Sc. There is disagreement on the original source of these grains and both chondrule and vapor growth have been proposed. In addition, there is clear evidence that diffusion has affected the outer margins but in some cases the whole grain. Within the cores, the FeO range is limited, and if growing under constant conditions, the Fe/Mn should be near constant as there is little fractionation of Mn from Fe by forsterite. Additionally, there are apparently no co-crystallizing phases as evidenced by a lack of common inclusions in the forsterites. These observations are now followed by analyses of isolated olivine grains in C2 and C3 meteorites.

Steele, Ian M.↗

NASA NTRS2007

Atomic Data and Spectral Line Intensities for CA XVII

Electron impact collision strengths, energy levels, oscillator strengths and spontaneous radiative decay rates are calculated for Ca XVII. The configurations used are 2s(sup 2), 2s2p, 2p(sup 2), 2l3l', 214l' and 2s5l', with l = s,p and l' = s,p, d giving rise to 92 fine-structure levels in intermediate coupling. Collision strengths are calculated at seven incident energies (15, 30, 75, 112.5, 150, 187.5 and 225 Ry) for the transitions within the three lowest configurations corresponding to the 10 lowest energy levels, and five incident energies (75, 112.5, 150, 187.5 and 225 Ry) for transitions between the lowest five levels and the n = 3,4,5 configurations. Calculations have been carried out using the distorted wave approximation. Excitation rate coefficients are calculated as a function of electron temperature by assuming a Maxwellian electron velocity distribution. Using the excitation rate coefficients and the radiative transition rates of the present work, and R-Matrix results for the 2s2, 2s2p, 2p2 configurations available in the literature, statistical equilibrium equations for level populations are solved at electron densities covering the range of 10(exp 8)-10(exp 14)/cu cm at an electron temperature of log Te(K)=6.7, corresponding to the maximum abundance of Ca XVII. Spectral line intensities are calculated, and their diagnostic relevance L; discussed. This dataset will be made available in the next version of the CHIANTI database.

Bhatia, A.K.↗

NASA NTRS1985

The accuracy of approximate solutions in the analysis of fracture of composites

This paper concerns the accuracy of three related mathematical models (developed by Hedgepeth, Eringen and Sendeckyj and Jones) used in the stress analysis and in fracture studies of continuous-fiber composites. These models have particular application in the investigation of fiber and matrix stresses in unidirectional composites in the region near a crack tip. The interest in such models is motivated by the desire to be able to simplify the equations of elasticity to the point that they can be solved in a relatively easy manner.

Goree, J. G.↗

NASA NTRS1985

The accuracy of approximate solutions in the analysis of fracture of composites

Goree, J. G.↗

NASA NTRS1977

Vibrational and rotational transitions in low-energy electron-diatomic-molecule collisions. I - Close-coupling theory in the moving body-fixed frame. II - Hybrid theory and close-coupling theory: An /l subscript z-prime/-conserving close-coupling approximation

A detailed vibrational-rotational (V-R) close-coupling formulation of electron-diatomic-molecule scattering is developed in which the target molecular axis is chosen to be the z-axis and the resulting coupled differential equation is solved in the moving body-fixed frame throughout the entire interaction region. The coupled differential equation and asymptotic boundary conditions in the body-fixed frame are given for each parity, and procedures are outlined for evaluating V-R transition cross sections on the basis of the body-fixed transition and reactance matrix elements. Conditions are discussed for obtaining identical results from the space-fixed and body-fixed formulations in the case where a finite truncated basis set is used. The hybrid theory of Chandra and Temkin (1976) is then reformulated, relevant expressions and formulas for the simultaneous V-R transitions of the hybrid theory are obtained in the same forms as those of the V-R close-coupling theory, and distorted-wave Born-approximation expressions for the cross sections of the hybrid theory are presented. A close-coupling approximation that conserves the internuclear axis component of the incident electronic angular momentum (l subscript z-prime) is derived from the V-R close-coupling formulation in the moving body-fixed frame.

Choi, B. H.↗

NASA NTRS1984

A finite element method for nonlinear forced vibrations of rectangular plates

The finite element method has been extended to determine the response of large amplitude forced vibrations of thin plates. A harmonic force matrix of a rectangular element under uniform harmonic excitation is developed for nonlinear forced vibration analysis. Inplane deformation and inertia are both considered in the formulation. Results obtained are compared with simple elliptic response, perturbation and other approximation solutions.

Mei, C.↗

NASA NTRS1986

Implicit solution methods in computational fluid dynamics

A class of implicit finite difference schemes of the Beam and Warming approximate factorization type will be addressed. The development and analysis of various aspects of this class of schemes will be given along with the motivations behind many of the choices. Various acceleration and efficiency modifications such as matrix reduction, diagonalization and flux split schemes will be presented. The methods are demonstrated in fully vectorized codes for a CRAY type architecture. The emphasis will be on the Euler equations in generalized coordinates.

Pulliam, Thomas H.↗