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NASA NTRS1981

Determination of the optimal level for combining area and yield estimates

Several levels of obtaining both area and yield estimates of corn and soybeans in Iowa were considered: county, refined strata, refined/split strata, crop reporting district, and state. Using the CCEA model form and smoothed weather data, regression coefficients at each level were derived to compute yield and its variance. Variances were also computed with stratum level. The variance of the yield estimates was largest at the state and smallest at the county level for both crops. The refined strata had somewhat larger variances than those associated with the refined/split strata and CRD. For production estimates, the difference in standard deviations among levels was not large for corn, but for soybeans the standard deviation at the state level was more than 50% greater than for the other levels. The refined strata had the smallest standard deviations. The county level was not considered in evaluation of production estimates due to lack of county area variances.

Bauer, M. E.↗

NASA NTRS1985

Flow structure capturing on overset patched meshes

The present paper describes the application of a simple, robust stable implicit approach to solutions of the conservative equations of gasdynamics on either composite or overset meshes to capture flow structures such as shocks with high resolution in complex geometric domains. Without requiring special flux conservative operators, but rather, interpolating conservative variable data at mesh boundaries, the well posed upwind method provides accurate and oscillation free solutions, even when shocks cross the interior patch boundaries. In three problems with flow complexities that require grid refinement, the paper demonstrates the capability to conveniently carry out for gasdynamics the adaptive refined meshing strategy in overset patches proposed by Berger and Oliger (1984), and it extends this technique to rapidly convergent implicit methods for the Euler and Navier-Stokes equations. The numerical experiments show concretely, in a realistic aerodynamic problem, the savings in mesh points (about an order of magnitude here in two dimensions) for similar accuracy that flow structure aligned adaptive patched meshing affords compared to uniform grid refinement.

Venkatapathy, E.↗

NASA NTRS1986

PROFILE user's guide

User information for program PROFILE, an aerodynamics design utility for refining, plotting, and tabulating airfoil profiles is provided. The theory and implementation details for two of the more complex options are also presented. These are the REFINE option, for smoothing curvature in selected regions while retaining or seeking some specified thickness ratio, and the OPTIMIZE option, which seeks a specified curvature distribution. REFINE uses linear techniques to manipulate ordinates via the central difference approximation to second derivatives, while OPTIMIZE works directly with curvature using nonlinear least squares techniques. Use of programs QPLOT and BPLOT is also described, since all of the plots provided by PROFILE (airfoil coordinates, curvature distributions) are achieved via the general purpose QPLOT utility. BPLOT illustrates (again, via QPLOT) the shape functions used by two of PROFILE's options. The programs were designed and implemented for the Applied Aerodynamics Branch at NASA Ames Research Center, Moffett Field, California, and written in FORTRAN and run on a VAX-11/780 under VMS.

Collins, L.↗

NASA NTRS1989

An embedded mesh procedure for leading-edge vortex flows

A cell-vertex scheme is outlined for solving the flow about a delta wing with M (sub infinity) is greater than 1. Embedded regions of mesh refinement allow solutions to be obtained which have much higher resolution than those achieved to date. Effects of mesh refinement and artificial viscosity on the solutions are studied, to determine at what point leading-edge vortex solutions are grid-converged. A macroscale and a microscale for the size of the vortex are defined, and it is shown that the macroscale (which includes the wing surface properties) is converged on a moderately refined grid, while the microscale is very sensitive to grid spacing. The level of numerical diffusion in the core of the vortex is found to be substantial. Comparisons with the experiment are made for two cases which have transonic cross-flow velocities.

Powell, Kenneth G.↗

NASA NTRS1989

Zonal multigrid solution of compressible flow problems on unstructured and adaptive meshes

The simultaneous use of adaptive meshing techniques with a multigrid strategy for solving the 2-D Euler equations in the context of unstructured meshes is studied. To obtain optimal efficiency, methods capable of computing locally improved solutions without recourse to global recalculations are pursued. A method for locally refining an existing unstructured mesh, without regenerating a new global mesh is employed, and the domain is automatically partitioned into refined and unrefined regions. Two multigrid strategies are developed. In the first, time-stepping is performed on a global fine mesh covering the entire domain, and convergence acceleration is achieved through the use of zonal coarse grid accelerator meshes, which lie under the adaptively refined regions of the global fine mesh. Both schemes are shown to produce similar convergence rates to each other, and also with respect to a previously developed global multigrid algorithm, which performs time-stepping throughout the entire domain, on each mesh level. However, the present schemes exhibit higher computational efficiency due to the smaller number of operations on each level.

Mavriplis, Dimitri J.↗

NASA NTRS1989

Application of unstructured grid methods to steady and unsteady aerodynamic problems

The purpose is to describe the development of unstructured grid methods which have several advantages when compared to methods which make use of structured grids. Unstructured grids, for example, easily allow the treatment of complex geometries, allow for general mesh movement for realistic motions and structural deformations of complete aircraft configurations which is important for aeroelastic analysis, and enable adaptive mesh refinement to more accurately resolve the physics of the flow. Steady Euler calculations for a supersonic fighter configuration to demonstrate the complex geometry capability; unsteady Euler calculations for the supersonic fighter undergoing harmonic oscillations in a complete-vehicle bending mode to demonstrate the general mesh movement capability; and vortex-dominated conical-flow calculations for highly-swept delta wings to demonstrate the adaptive mesh refinement capability are discussed. The basic solution algorithm is a multi-stage Runge-Kutta time-stepping scheme with a finite-volume spatial discretization based on an unstructured grid of triangles in 2D or tetrahedra in 3D. The moving mesh capability is a general procedure which models each edge of each triangle (2D) or tetrahedra (3D) with a spring. The resulting static equilibrium equations which result from a summation of forces are then used to move the mesh to allow it to continuously conform to the instantaneous position or shape of the aircraft. The adaptive mesh refinement procedure enriches the unstructured mesh locally to more accurately resolve the vortical flow features. These capabilities are described in detail along with representative results which demonstrate several advantages of unstructured grid methods. The applicability of the unstructured grid methodology to steady and unsteady aerodynamic problems and directions for future work are discussed.

Batina, John T.↗

NASA NTRS1991

Combining global and local approximations

A method based on a linear approximation to a scaling factor, designated the 'global-local approximation' (GLA) method, is presented and shown capable of extending the range of usefulness of derivative-based approximations to a more refined model. The GLA approach refines the conventional scaling factor by means of a linearly varying, rather than constant, scaling factor. The capabilities of the method are demonstrated for a simple beam example with a crude and more refined FEM model.

Haftka, Raphael T.↗

NASA NTRS1992

Conceptual design of a fleet of autonomous regolith throwing devices for radiation shielding of lunar habitats

This report presents refinements in two areas of the initial design presented in the report entitled 'Conceptual Design of a Fleet of Autonomous Regolith Throwing Devices for Radiation Shielding of Lunar Habitats'. The first section presents an evaluation of the critical areas of the design and presents alternative solutions for these areas. The areas for design refinement are the traction required by the device and the stability of the device when throwing regolith. Several alternative methods are presented to solve these problems. First, the issue of required traction is covered. Next, the design is refined to provide a more stable device. The issue of stability is addressed both by presenting solutions for the configuration chosen for the computer simulation and by presenting two more device configurations. The next section presents the selected solutions. To prevent inadequate traction, the depth of dig-per-pass is reduced. A method combining a dynamic counterweight and an outrigger is chosen to provide a stable device.

Armstrong, Karem↗

NASA NTRS1993

A multigrid method for steady Euler equations on unstructured adaptive grids

A flux-difference splitting type algorithm is formulated for the steady Euler equations on unstructured grids. The polynomial flux-difference splitting technique is used. A vertex-centered finite volume method is employed on a triangular mesh. The multigrid method is in defect-correction form. A relaxation procedure with a first order accurate inner iteration and a second-order correction performed only on the finest grid, is used. A multi-stage Jacobi relaxation method is employed as a smoother. Since the grid is unstructured a Jacobi type is chosen. The multi-staging is necessary to provide sufficient smoothing properties. The domain is discretized using a Delaunay triangular mesh generator. Three grids with more or less uniform distribution of nodes but with different resolution are generated by successive refinement of the coarsest grid. Nodes of coarser grids appear in the finer grids. The multigrid method is started on these grids. As soon as the residual drops below a threshold value, an adaptive refinement is started. The solution on the adaptively refined grid is accelerated by a multigrid procedure. The coarser multigrid grids are generated by successive coarsening through point removement. The adaption cycle is repeated a few times. Results are given for the transonic flow over a NACA-0012 airfoil.

Riemslagh, Kris↗

NASA NTRS1994

Deflections of anisotropic sandwich beams with variable face sheets and core thicknesses

A sandwich construction consists of a low-density core material with high strength face sheets bounded to the top and bottom surfaces. The construction has been widely used in the aerospace and marine industries due to its outstanding characteristics such as noise absorption, weight minimization, heat insulation, and better bending stiffness. In sandwich structures used in high-performance aircraft, the face sheets are often made of fiber-reinforced composite materials and the core is made of honeycomb. The structures may also have variable thickness so as to satisfy aerodynamic requirements. In the stress analysis, the constant-thickness face sheets are usually considered as membrane and the core is assumed to be inextensible but deformable in the thickness direction. The static behavior of variable-thickness, isotropic and homogeneous sandwich beams was successfully studied by employing a constant-thickness theory but allowing stiffnesses to vary in accordance with local thickness variations. It has been recently found in a refined theory that the analyses based on the constant thickness theory locally can lead to significant errors in structural responses if the sandwich beam is thickness-tapered and the cores are deformable in transverse shear. The errors arise mainly from two factors: (1) the transverse shear components of the membrane forces in the face sheets alter the transverse shears carried by the core; and (2) the face-sheet membrane strains arise from transverse shear deformation of the core. In practice the variable thickness may not only exist in core but also in face sheets. The thickness-variations may even be a type of step function. In this case the transverse shear stress in the face sheets and bending stress in the core should be taken into account in the refined theory mentioned. In the present study, energy principles are employed in deriving governing equations for general bending of anisotropic sandwich beams with variable thickness in both face sheets and cores. Solutions to these equations are based on a finite difference scheme. As an example in application, a simply supported thickness-tapered sandwich beam subject to a concentrated load at its center is considered. Let W' be the maximum deflection of the beam in which face sheets are considered as membrane, while W'' is that based on using the modified refined theory. It is found that W' is always larger than W'', however, the magnitude of (W'- W'') appears to be insensitive to the change of the taper of the beam.

Lu, Chu-Ho↗

NASA NTRS1994

Structural Health Monitoring of Large Structures

This paper describes a damage detection and health monitoring method that was developed for large space structures using on-orbit modal identification. After evaluating several existing model refinement and model reduction/expansion techniques, a new approach was developed to identify the location and extent of structural damage with a limited number of measurements. A general area of structural damage is first identified and, subsequently, a specific damaged structural component is located. This approach takes advantage of two different model refinement methods (optimal-update and design sensitivity) and two different model size matching methods (model reduction and eigenvector expansion). Performance of the proposed damage detection approach was demonstrated with test data from two different laboratory truss structures. This space technology can also be applied to structural inspection of aircraft, offshore platforms, oil tankers, ridges, and buildings. In addition, its applications to model refinement will improve the design of structural systems such as automobiles and electronic packaging.

Kim, Hyoung M.↗

NASA NTRS1996

Parallel implementation of an adaptive scheme for 3D unstructured grids on the SP2

Dynamic mesh adaption on unstructured grids is a powerful tool for computing unsteady flows that require local grid modifications to efficiently resolve solution features. For this work, we consider an edge-based adaption scheme that has shown good single-processor performance on the C90. We report on our experience parallelizing this code for the SP2. Results show a 47.0X speedup on 64 processors when 10 percent of the mesh is randomly refined. Performance deteriorates to 7.7X when the same number of edges are refined in a highly-localized region. This is because almost all the mesh adaption is confined to a single processor. However, this problem can be remedied by repartitioning the mesh immediately after targeting edges for refinement but before the actual adaption takes place. With this change, the speedup improves dramatically to 43.6X.

Strawn, Roger C.↗

NASA NTRS1995

Adaptive Finite Element Methods for Continuum Damage Modeling

The paper presents an application of adaptive finite element methods to the modeling of low-cycle continuum damage and life prediction of high-temperature components. The major objective is to provide automated and accurate modeling of damaged zones through adaptive mesh refinement and adaptive time-stepping methods. The damage modeling methodology is implemented in an usual way by embedding damage evolution in the transient nonlinear solution of elasto-viscoplastic deformation problems. This nonlinear boundary-value problem is discretized by adaptive finite element methods. The automated h-adaptive mesh refinements are driven by error indicators, based on selected principal variables in the problem (stresses, non-elastic strains, damage, etc.). In the time domain, adaptive time-stepping is used, combined with a predictor-corrector time marching algorithm. The time selection is controlled by required time accuracy. In order to take into account strong temperature dependency of material parameters, the nonlinear structural solution a coupled with thermal analyses (one-way coupling). Several test examples illustrate the importance and benefits of adaptive mesh refinements in accurate prediction of damage levels and failure time.

Min, J. B.↗

NASA NTRS1996

Parallel Implementation of an Adaptive Scheme for 3D Unstructured Grids on the SP2

Dynamic mesh adaption on unstructured grids is a powerful tool for computing unsteady flows that require local grid modifications to efficiently resolve solution features. For this work, we consider an edge-based adaption scheme that has shown good single-processor performance on the C90. We report on our experience parallelizing this code for the SP2. Results show a 47.OX speedup on 64 processors when 10% of the mesh is randomly refined. Performance deteriorates to 7.7X when the same number of edges are refined in a highly-localized region. This is because almost all mesh adaption is confined to a single processor. However, this problem can be remedied by repartitioning the mesh immediately after targeting edges for refinement but before the actual adaption takes place. With this change, the speedup improves dramatically to 43.6X.

Oliker, Leonid↗

NASA NTRS1996

Solution-Adaptive Cartesian Cell Approach for Viscous and Inviscid Flows

A Cartesian cell-based approach for adaptively refined solutions of the Euler and Navier-Stokes equations in two dimensions is presented. Grids about geometrically complicated bodies are generated automatically, by the recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal cut cells are created using modified polygon-clipping algorithms. The grid is stored in a binary tree data structure that provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite volume formulation. The convective terms are upwinded: A linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The results of a study comparing the accuracy and positivity of two classes of cell-centered, viscous gradient reconstruction procedures is briefly summarized. Adaptively refined solutions of the Navier-Stokes equations are shown using the more robust of these gradient reconstruction procedures, where the results computed by the Cartesian approach are compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

Coirier, William J.↗

NASA NTRS1998

Analysis of Gamma-Ray Data from Solar Flares in Cycles 21 and 22

One of our primary accomplishments under grant NAGW-35381 was the systematic derivation and compilation, for the first time, of physical parameters for all gamma-ray flares detected by the SMM GRS during its ten year lifetime. The flare parameters derived from the gamma-ray spectra include: bremsstrahlung fluence and best-fit power-law parameters, narrow nuclear line fluence, positron annihilation line fluence, neutron capture line fluence, and an indication of whether or not greater than 10 MeV emissions were present. We combined this compilation of flare parameters with our plots of counting rate time histories and flare spectra to construct an atlas of gamma-ray flare characteristics. The atlas time histories display four energy bands: 56-199 kev, 298526 keV, 4-8 MeV, and 10-25 MeV. These energy bands respectively measure nonrelativistic bremsstrahlung, trans-relativistic bremsstrahlung, nuclear de-excitation, and ultra-relativistic bremsstrahlung. The atlas spectra show the integrated high-energy spectra measured for all GRS flares and dissects them into electron bremsstrahlung, positron annihilation and nuclear emission components. The atlas has been accepted for publication in the Astrophysical Journal Supplements and is currently in press. The atlas materials were also supplied to the Solar Data Analysis Center at Goddard Space Flight Center and were made available through a web site at the University of New Hampshire. Since a uniform methodology was adopted for deriving the flare parameters, this atlas will be very useful for future statistical and correlative studies of solar flares-three independent groups are presently using it to correlate interplanetary energetic particle measurements with our gamma-ray measurements. A better model for the response of the GRS instrument to high energy radiation was also developed. A refined response model was needed because the old model was not adequate for predicting the first and second escape peaks associated with strong nuclear lines nor could it accurately describe the Compton continuum shape. The new response was developed using a GEANT based simulation code and tested against preflight calibration data. The refinement of the response model and the removal of systematic errors now allow more detailed spectral studies of the GRS gamma-ray measurements. This refined response function was supplied to the Solar DAC at Goddard and was also made available via a web site at the University of New Hampshire.

Vestrand, W. Thomas↗

NASA NTRS1999

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

Barth, Timothy↗

NASA NTRS2000

Effects of Thermal and Mechanical Processing on Microstructures and Desired Properties of Particle-Strengthened Cu-Cr-Nb Alloys

Ternary Cu-Cr-Nb alloys, particularly Cu-8 Cr-4 Nb (in at.%), have demonstrated good thermal stability as well as high strength and conductivity at elevated temperatures. The initial powder material has a bimodal size distribution of Cr2Nb precipitates. Primary Cr2Nb precipitates are approx. 1 micron, and secondary Cr2Nb particles are 30-200 nm. The particle coarsening was analyzed and found to follow LSW-type behavior, This study provides a detailed examination of the stability and strengthening effects of Cr2Nb particles. This investigation also revealed that the primary particles provide direct grain boundary pinning and indirect grain boundary strengthening but virtually no Orowan strengthening. The secondary particles found within grains do provide Orowan strengthening. For extruded material, grain bound-ary strengthening (Hall-Petch effect) accounts for two-thirds of the strength with Orowan effects contributing the remainder. The proven advantages of Cu-Cr-Nb were the motivation to improve these attributes via microstructural refinement. Mechanical milling (MM) of Cu- 4 Cr-2 Nb and Cu-8 Cr-2 Nb produced an increase in hot pressed Vickers hardness of 122% and 96%, respectively. The increase in hardness was more due to Cu grain-size refinement than to Cr,,Nb refinement. This study also demonstrated enhanced stability of MM Cu-4 Cr-2 Nb. Hot pressed 4 h milled Cu-4 Cr-2 Nb experienced only a 22% drop in hardness when annealed at 1273 K for 50 h versus a 30% drop for extruded Cu-8 Cr-4 Nb. The goal of improving the strength and stability of Cu-4 Cr-2 Nb to better than such properties for as- extruded Cu-8 Cr-4 Nb has been met. In addition, a figure-of-merit (FOM) coupling hardness and thermal conductivity was maximized for the case of 4 h milled Cu-4 Cr-2 Nb material. Overall, Cu-Cr-Nb alloys not only possess high strength, conductivity and thermal stability but also can be further developed to improve strength and stability.

Anderson, Kenneth Reed↗