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NASA NTRS1997

Exact Integrations of Polynomials and Symmetric Quadrature Formulas over Arbitrary Polyhedral Grids

This paper is concerned with two important elements in the high-order accurate spatial discretization of finite volume equations over arbitrary grids. One element is the integration of basis functions over arbitrary domains, which is used in expressing various spatial integrals in terms of discrete unknowns. The other consists of quadrature approximations to those integrals. Only polynomial basis functions applied to polyhedral and polygonal grids are treated here. Non-triangular polygonal faces are subdivided into a union of planar triangular facets, and the resulting triangulated polyhedron is subdivided into a union of tetrahedra. The straight line segment, triangle, and tetrahedron are thus the fundamental shapes that are the building blocks for all integrations and quadrature approximations. Integrals of products up to the fifth order are derived in a unified manner for the three fundamental shapes in terms of the position vectors of vertices. Results are given both in terms of tensor products and products of Cartesian coordinates. The exact polynomial integrals are used to obtain symmetric quadrature approximations of any degree of precision up to five for arbitrary integrals over the three fundamental domains. Using a coordinate-free formulation, simple and rational procedures are developed to derive virtually all quadrature formulas, including some previously unpublished. Four symmetry groups of quadrature points are introduced to derive Gauss formulas, while their limiting forms are used to derive Lobatto formulas. Representative Gauss and Lobatto formulas are tabulated. The relative efficiency of their application to polyhedral and polygonal grids is detailed. The extension to higher degrees of precision is discussed.

Liu, Yen↗

NASA NTRS1958

Experimental Wind-Tunnel Investigation of the Transonic Damping-in-Pitch Characteristics of Two Wing-Body Combinations

The results of an experimental wind-tunnel investigation of the damping in pitch of two wing-body combinations are presented. The tests were conducted in the Ames 14-foot transonic wind tunnel over a Mach number range from 0.60 to 1.18. Reynolds numbers varied from 2.3 million to 5.5 million. One model with a triangular wing of aspect ratio 2 having NACA 0003-63 sections was oscillated at an amplitude of 1.5 and a frequency of 17 cycles per second. The second model with a straight, tapered wing of aspect ratio 3 having 3-percent biconvex circular-arc sections was oscillated at an amplitude of 1.0 deg and a frequency of 21 cycles per second. The tests were made with the models at a mean angle of attack of 0 deg. The models were oscillated with a dynamic balance that was actuated by an electrohydraulic servo valve. The results of this investigation indicate the usefulness of this new apparatus. The experimental results of a previous damping-in-pitch investigation conducted in the Ames 6- by 6-foot supersonic wind tunnel at Mach numbers from 1.2 to 1.7 are included along with the theoretical results for this Mach number range. In the region of Mach numbers available for comparison, good agreement is shown to exist between the data obtained in the two facilities, except for some inconsistency in the slopes of the curves at M = 1.2 for the triangular wing. The results of this investigation clearly show that for the models tested the maximum values of the damping in pitch occur at Mach numbers very close to 1.0, and that abrupt changes in the pitch damping are encountered near sonic velocity.

Emerson, Horace F.↗

NASA NTRS1958

Inclined Bodies of Various Cross Sections at Supersonic Speeds

To aid in assessing effects of cross-sectional shape on body aerodynamics, the forces and moments have been measured for bodies with circular, elliptic, square, and triangular cross sections at Mach numbers 1.98 and 3.88. Results for bodies with noncircular cross sections have been compared with results for bodies of revolution having the same axial distribution of cross-sectional area (and, thus, the same equivalent fineness ratio). Comparisons have been made for bodies of fineness ratios 6 and 10 at angles of attack from 0 deg to about 20 deg and for Reynolds numbers, based on body length, of 4.0 x 10(exp 6) and 6.7 x 10(exp 6). The results of this investigation show that distinct aerodynamic advantages can be obtained by using bodies with noncircular cross sections. At certain angles of bank, bodies with elliptic, square, and triangular cross sections develop considerably greater lift and lift-drag ratios than equivalent bodies of revolution. For bodies with elliptic cross sections, lift and pitching-moment coefficients can be correlated with corresponding coefficients for equivalent circular bodies. It has been found that the ratios of lift and pitching-moment coefficients for an elliptic body to those for an equivalent circular body are practically constant with change in both angle of attack and Mach number. These lift and moment ratios are given very accurately by slender-body theory. As a result of this agreement, the method of NACA Rep. 1048 for computing forces and moments for bodies of revolution has been simply extended to bodies with elliptic cross sections. For the cases considered (elliptic bodies of fineness ratios 6 and 10 having cross-sectional axis ratios of 1.5 and 2), agreement of theory with experiment is very good. As a supplement to the force and moment results, visual studies of the flow over bodies have been made by use of the vapor-screen, sublimation, and white-lead techniques. Photographs from these studies are included in the report.

Jorgensen, Leland H.↗

NASA NTRS1959

Low-Speed Measurements of Static and Oscillatory Lateral Stability Derivatives of a 1/5 Scale Model of a Jet-Powered Vertical-Attitude VTOL Research Airplane

Force tests of the static and dynamic lateral stability characteristics of a VTOL airplane having a triangular wing mounted high on the fuselage with a triangular vertical tail on top of the wing and no horizontal tail have been made in the Langley free-flight tunnel. The static lateral stability parameters and the rolling, yawing, and sideslipping dynamic stability derivatives are presented without analysis.

Shanks, Robert E.↗

NASA NTRS1994

Hyperbolic Prismatic Grid Generation and Solution of Euler Equations on Prismatic Grids

A hyperbolic grid generation method is used to generate prismatic grids and an approach using prismatic grids to solve the Euler equations is presented. The theory of the stability and feasibility of the hyperbolic grid generation method is presented. The hyperbolic grid generation method of Steger et al for structured grids is applied to a three dimensional triangularized surface definition to generate a grid that is unstructured on each successive layer. The grid, however, retains structure in the body-normal direction and has a computational cell shaped like a triangular prism. In order to take advantage of the structure in the normal direction, a finite-volume scheme that treats the unknowns along the normal direction implicitly is introduced and the flow over a sphere is simulated.

Pandya, S. A.↗

NASA NTRS2002

Predicting Print-thru for the Sub-scale Beryllium Mirror Demonstrator (SBMD)

This document presents a finite element method for predicting print-thru or quilting for a lightweight mirror in a low temperature environment. The mirror is represented with quadrilateral and triangular plate finite elements. The SBMD (Sub-scale Beryllium Mirror Demonstrator) is circular with a diameter of 50 cm and one flat side. The mirror structure is a thin-wall triangular cell core with a single facesheet. There is a 4 mm radius fillet between the facesheet and cell walls. It is made entirely of Beryllium. It is assumed that polishing the mirror surface creates a thin surface layer with different material properties. Finite element results are compared with measured values at cryogenic temperatures.

Craig, Larry↗

NASA NTRS2002

Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids II: Extension to Two Dimensional Scalar Equation

The framework for constructing a high-order, conservative Spectral (Finite) Volume (SV) method is presented for two-dimensional scalar hyperbolic conservation laws on unstructured triangular grids. Each triangular grid cell forms a spectral volume (SV), and the SV is further subdivided into polygonal control volumes (CVs) to supported high-order data reconstructions. Cell-averaged solutions from these CVs are used to reconstruct a high order polynomial approximation in the SV. Each CV is then updated independently with a Godunov-type finite volume method and a high-order Runge-Kutta time integration scheme. A universal reconstruction is obtained by partitioning all SVs in a geometrically similar manner. The convergence of the SV method is shown to depend on how a SV is partitioned. A criterion based on the Lebesgue constant has been developed and used successfully to determine the quality of various partitions. Symmetric, stable, and convergent linear, quadratic, and cubic SVs have been obtained, and many different types of partitions have been evaluated. The SV method is tested for both linear and non-linear model problems with and without discontinuities.

Wang, Z. J.↗

NASA NTRS2002

Eigenvalues of Rectangular Waveguide Using FEM With Hybrid Elements

A finite element analysis using hybrid triangular-rectangular elements is developed to estimate eigenvalues of a rectangular waveguide. Use of rectangular vector-edge finite elements in the vicinity of the PEC boundary and triangular elements in the interior region more accurately models the physical nature of the electromagnetic field, and consequently quicken the convergence.

Deshpande, Manohar D.↗

NASA NTRS1998

Numerical Differentiation for Adaptively Refined Finite Element Meshes

Postprocessing of point-wise data is a fundamental process in many fields of research. Numerical differentiation is a key operation in computational electromagnetics. In the case of data obtained from a finite element method with automatic mesh refinement much work needs still to be done. This paper addresses some issues in differentiating data obtained from a finite element electromagnetic code with adaptive mesh refinement, and it proposes a methodology for deriving the electric field given the magnetic field on a mesh of linear triangular elements. The procedure itself is nevertheless more general and might be extended for numerically differentiating any point-wise solution based on triangular meshes.

Borgioli, Andrea↗

NASA NTRS2004

Web Growth Used to Confine Screw Dislocations to Predetermined Lateral Positions in 4H-SiC Epilayers

Silicon-carbide- (SiC-) based power devices could enable substantial aerospace electronics benefits over today's silicon-based electronics. However, present-day SiC wafers contain electrically harmful dislocations (including micropipes) that are unpredictably distributed in high densities across all commercial 4H- and 6H-SiC wafers. The NASA Glenn Research Center recently demonstrated a crystal growth process that moves SiC wafer dislocations to predetermined lateral positions in epitaxial layers so that they can be reproducibly avoided during subsequent SiC electronic device fabrication. The process starts by reactive ion etching mesa patterns with enclosed trench regions into commercial on-axis (0001) 4H- or 6H-SiC substrates. An example of a pregrowth mesa geometry with six enclosed triangular-shaped trench regions is shown. After the etch mask is stripped, homoepitaxial growth is carried out in pure stepflow conditions that enable thin cantilevers to grow laterally from the tops of mesas whose pregrowth top surfaces are not threaded by substrate screw dislocations. The image in the bottom figure shows the postgrowth structure that forms after the lateral cantilevers expand to coalesce and completely roof over each of the six triangular trench regions. Atomic force microscope (AFM) measurements of the roof revealed that three elementary screw dislocation growth spirals, each shown in the AFM insets of the bottom image on the previous page, formed in the film roof at three respective points of cantilever film coalescence. The image above shows the structure following an etch in molten potassium hydroxide (KOH) that produced surface etch pits at the dislocation defects. The larger KOH etch pits--S1, S2, and S3--shown in this image correspond to screw dislocations relocated to the final points of cantilever coalescence. The smaller KOH etch pits are consistent with epilayer threading edge dislocations from the pregrowth substrate mesa (P1, P3, and P4) and a final cantilever coalescence point (P2). No defects (i.e., no etch pits) are observed in other cantilevered portions of the film surface. On the basis of the principle of dislocation Burgers vector conservation, we hypothesize that all vertically propagating substrate dislocations in an enclosed trench region become combined into a single dislocation in the webbed film roof at the point of final roof coalescence. The point of final roof coalescence, and therefore the lateral location of a webbed roof dislocation, can be designed into the pregrowth mesa pattern. Screw dislocations with predetermined lateral positions can then be used to provide the new growth steps necessary for growing a 4H/6H-SiC epilayer with a lower dislocation density than the substrate. Devices fabricated on top of such films can be positioned to avoid the preplaced dislocations.

Powell, J. Anthony↗

NASA NTRS2005

Fabrication and Deployment Testing of Solar Sail Quadrants for a 20-Meter Solar Sail Ground Test System Demonstration

A 20-meter Scalable Square Solar Sail (S(sup 4)) System was produced and successfully completed functional vacuum testing in NASA Glenn's Space Power Facility at Plum Brook Station Ohio in May 2005. The S(sup 4) system was designed and developed by ATK Space Systems, and the design and production of the Solar Sails for this system was carried out by SRS Technologies. The S(sup 4) system consists of a central structure with four deployable carbon fiber masts that support four triangular sails. SRS has developed an effective and efficient design for triangular sail quadrants that are supported at three points and provide a flat reflective surface with a high fill factor. This sail design is robust enough for deployments in a one atmosphere, one gravity environment and incorporates several advanced features including adhesiveless seaming of membrane strips, compliant edge borders to allow for film membrane cord strain mismatch without causing wrinkling and low mass (3% of total sail mass) ripstop. This paper will outline some of the sail design and fabrication processes and the mature production, packaging and deployment processes that have been developed. This paper will also detail the successful ambient and vacuum testing of the sails and the ATK spacecraft structure. Based on recent experience and testing, SRS is confidant that high Technology Readiness Level (TRL) 5-6 solar sails in the 40-120-meter size range with areal density in the 4-5 grams per square meters (sail minus structure) range can be produced with existing technology. Additional film production research will lead to further reductions in film thickness to less than 1 micron enabling production of sails with areal densities as low as 2.0 grams per square meters using the current design, resulting in a system areal densities as low as 5.3 grams per square meters (sail and structure). These areal densities are low enough to allow nearly all of the Solar Sail missions that have been proposed by the scientific community. The fundamental technologies required to produce these systems has been demonstrated on the 20-meter S(sup 4) sails that have recently completed ground testing demonstrating a mature and technology suitable for incorporation into future flight validation and future mission. Solar Sails can support NASA's Vision for Space Exploration by allowing communication satellite orbits that can maintain continuous communication with the polar regions of the Moon and Mars and to support solar weather monitoring to provide early warning of solar flares and storms that could threaten the safety of astronauts and other spacecraft.

Laue, Greg↗

NASA NTRS2006

Effect of Reinforcement Architecture on Fracture of Selectively Reinforced Metallic Compact Tension Specimens

A computer-based parametric study of the effect of reinforcement architectures on fracture response of aluminum compact-tension (CT) specimens is performed. Eleven different reinforcement architectures consisting of rectangular and triangular cross-section reinforcements were evaluated. Reinforced specimens produced between 13 and 28 percent higher fracture load than achieved with the non-reinforced case. Reinforcements with blunt leading edges (rectangular reinforcements) exhibited superior performance relative to the triangular reinforcements with sharp leading edges. Relative to the rectangular reinforcements, the most important architectural feature was reinforcement thickness. At failure, the reinforcements carried between 58 and 85 percent of the load applied to the specimen, suggesting that there is considerable load transfer between the base material and the reinforcement.

Abada, Christopher H.↗

NASA NTRS2006

Fracture Response Enhancement Of Aluminum Using In-Situ Selective Reinforcement

A computer-based parametric study of the effect of reinforcement architectures on fracture response of aluminum compact-tension (CT) specimens is performed. Eleven different reinforcement architectures consisting of rectangular and triangular cross-section reinforcements were evaluated. Reinforced specimens produced between 13 and 28 percent higher fracture load than achieved with the unreinforced case. Reinforcements with blunt leading edges (rectangular reinforcements) exhibited superior performance relative to the triangular reinforcements with sharp leading edges. Relative to the rectangular reinforcements, the most important architectural feature was reinforcement thickness. At failure, the reinforcements carried between 58 and 85 percent of the load applied to the specimen, suggesting that there is considerable load transfer between the base material and the reinforcement.

Abada, Christopher H.↗

NASA NTRS2006

Minimum mass design of large-scale space trusses subjected to thermal gradients

Lightweight, deployable trusses are commonly used to support space-borne instruments including RF reflectors, radar panels, and telescope optics. While in orbit, these support structures are subjected to thermal gradients that vary with altitude, location in orbit, and self-shadowing. Since these instruments have tight dimensional-stability requirements, their truss members are often covered with multi-layer insulation (MLI) blankets to minimize thermal distortions. This paper develops a radiation heat transfer model to predict the thermal gradient experienced by a triangular truss supporting a long, linear radar panel in Medium Earth Orbit (MEO). The influence of self-shadowing effects of the radar panel are included in the analysis, and the influence of both MLI thickness and outer covers/coatings on the magnitude of the thermal gradient are formed into a simple, two-dimensional analysis. This thermal model is then used to size and estimate the structural mass of a triangular truss that meets a given set of structural requirements.

thermal truss analysis↗

NASA NTRS1992

Method of fabricating lightweight honeycomb structures

A process is disclosed for fabricating lightweight honeycomb type structures out of material such as silicon carbide (SiC) and silicon (S). The lightweight structure consists of a core to define the shape and size of the structure. The core is coated with an appropriate deposit such as SiC or Si to give the lightweight structure strength and stiffness and for bonding the lightweight structure to another surface. The core is fabricated from extremely thin ribs of appropriately stiff and strong material such as graphite. First, a graphite core consisting of an outer hexagonal cell with six inner triangular cells is constructed from the graphite ribs. The graphite core may be placed on the back-up side of a SiC faceplate and then coated with SiC to produce a monolithic structure without the use of any bonding agent. Cores and methods for the fabrication thereof in which the six inner triangular cells are further divided into a plurality of cells are also disclosed.

Goela, Jitendra S.↗

NASA NTRS2010

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes

Cell-centered and node-centered approaches have been compared for unstructured finite-volume discretization of inviscid fluxes. The grids range from regular grids to irregular grids, including mixed-element grids and grids with random perturbations of nodes. Accuracy, complexity, and convergence rates of defect-correction iterations are studied for eight nominally second-order accurate schemes: two node-centered schemes with weighted and unweighted least-squares (LSQ) methods for gradient reconstruction and six cell-centered schemes two node-averaging with and without clipping and four schemes that employ different stencils for LSQ gradient reconstruction. The cell-centered nearest-neighbor (CC-NN) scheme has the lowest complexity; a version of the scheme that involves smart augmentation of the LSQ stencil (CC-SA) has only marginal complexity increase. All other schemes have larger complexity; complexity of node-centered (NC) schemes are somewhat lower than complexity of cell-centered node-averaging (CC-NA) and full-augmentation (CC-FA) schemes. On highly anisotropic grids typical of those encountered in grid adaptation, discretization errors of five of the six cell-centered schemes converge with second order on all tested grids; the CC-NA scheme with clipping degrades solution accuracy to first order. The NC schemes converge with second order on regular and/or triangular grids and with first order on perturbed quadrilaterals and mixed-element grids. All schemes may produce large relative errors in gradient reconstruction on grids with perturbed nodes. Defect-correction iterations for schemes employing weighted least-square gradient reconstruction diverge on perturbed stretched grids. Overall, the CC-NN and CC-SA schemes offer the best options of the lowest complexity and secondorder discretization errors. On anisotropic grids over a curved body typical of turbulent flow simulations, the discretization errors converge with second order and are small for the CC-NN, CC-SA, and CC-FA schemes on all grids and for NC schemes on triangular grids; the discretization errors of the CC-NA scheme without clipping do not converge on irregular grids. Accurate gradient reconstruction can be achieved by introducing a local approximate mapping; without approximate mapping, only the NC scheme with weighted LSQ method provides accurate gradients. Defect correction iterations for the CC-NA scheme without clipping diverge; for the NC scheme with weighted LSQ method, the iterations either diverge or converge very slowly. The best option in curved geometries is the CC-SA scheme that offers low complexity, second-order discretization errors, and fast convergence.

Diskin, Boris↗

NASA NTRS2010

Measuring Cyclic Error in Laser Heterodyne Interferometers

An improved method and apparatus have been devised for measuring cyclic errors in the readouts of laser heterodyne interferometers that are configured and operated as displacement gauges. The cyclic errors arise as a consequence of mixing of spurious optical and electrical signals in beam launchers that are subsystems of such interferometers. The conventional approach to measurement of cyclic error involves phase measurements and yields values precise to within about 10 pm over air optical paths at laser wavelengths in the visible and near infrared. The present approach, which involves amplitude measurements instead of phase measurements, yields values precise to about .0.1 microns . about 100 times the precision of the conventional approach. In a displacement gauge of the type of interest here, the laser heterodyne interferometer is used to measure any change in distance along an optical axis between two corner-cube retroreflectors. One of the corner-cube retroreflectors is mounted on a piezoelectric transducer (see figure), which is used to introduce a low-frequency periodic displacement that can be measured by the gauges. The transducer is excited at a frequency of 9 Hz by a triangular waveform to generate a 9-Hz triangular-wave displacement having an amplitude of 25 microns. The displacement gives rise to both amplitude and phase modulation of the heterodyne signals in the gauges. The modulation includes cyclic error components, and the magnitude of the cyclic-error component of the phase modulation is what one needs to measure in order to determine the magnitude of the cyclic displacement error. The precision attainable in the conventional (phase measurement) approach to measuring cyclic error is limited because the phase measurements are af-

Ryan, Daniel↗

NASA NTRS2003

Noncircular Cross Sections Could Enhance Mixing in Sprays

A computational study has shown that by injecting drops in jets of gas having square, elliptical, triangular, or other noncircular injection cross sections, it should be possible to increase (relative to comparable situations having circular cross section) the entrainment and dispersion of liquid drops. This finding has practical significance for a variety of applications in which it is desirable to increase dispersion of drops. For example, in chemical-process sprays, increased dispersion leads to increases in chemical- reaction rates; in diesel engines, increasing the dispersion of drops of sprayed fuel reduces the production of soot; and in household and paint sprays, increasing the dispersion of drops makes it possible to cover larger surfaces. It has been known for some years that single-phase fluid jets that enter flow fields through noncircular inlets entrain more fluid than do comparable jets entering through circular inlets. The computational study reported here was directed in part toward determining whether and how this superior mixing characteristic of noncircular single phase jets translates to a similar benefit in cases of two-phase jets (that is, sprays). The study involved direct numerical simulations of single- and two-phase free jets with circular, elliptical, rectangular, square, and triangular inlet cross sections. The two-phase jets consisted of gas laden with liquid drops randomly injected at the inlets. To address the more interesting case of evaporating drops, the carrier gas in the jets was specified to be initially unvitiated by the vapor of the liquid chemical species and the initial temperature of the drops was chosen to be smaller than that of the gas. The mathematical model used in the study was constructed from the conservation equations for the two-phase flow and included complete couplings of mass, momentum, and energy based on thermodynamically self-consistent specification of the enthalpy, internal energy, and latent heat of vaporization of the vapor.

Bellan, Josette↗