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NASA NTRS1985

The Construction of Curves and Surfaces Using Numerical Optimization Techniques

Numerical optimization techniques are playing an increasing role in curve and surface construction. Often difficult problems in curve and surface construction, especially when some aspect of shape control is involved, can be phrased as a constrained optimization problem. Four such classes of problems are explored: parametric curve fitting with non-linear shape constraints; explicit surface fitting with linear shape constraints; surface fitting to scattered data giving rise to ill-posed problems; finally, variable knot problems. In each of these problems there is a nonlinear aspect: either the shape of the curve or surface is important for manufacturing or engineering reasons or the shape affects the convergence of numerical algorithms which use the curve or surface or the placement of knots affects the accuracy of the fits. In all cases the class of functions used is that of parametric spline curves and tensor or direct product spline surfaces. The reason for choosing this class is that splines provide flexible models that are easily evaluated and stored. Furthermore, the B-spline representation of splines leads to convenient expressions for shape control over regions.

Ferguson, D. R.↗

NASA NTRS1985

Contouring Trivariate Data

In many applications, the data consist of discrete 3-D points at which one or more parameters are given. To display contours, the data are represented by a continuous function which is evaluated at any point as needed for contouring. Contouring results are presented which are applicable both to artitrarily spaced data and to data which lie on a topologically rectangular three dimensional grid. the contours are assumed to be described by 3-D display lists for viewing on a dynamic color graphics device; that is, they will not simply be projected into 2-D and viewed as a static image on a frame buffer. Dynamic viewing of color contours may be essential to the proper interpretation of results. The gridded data lie on a topologically rectangular grid although two or more nodes may be the same point. Parametric tensor product methods may be used to fit the gridded data and to generate the contours. Rectangular elements are convenient but not necessary. For example, there are other methods which are effective for contouring over tetrahedral elements.

Stead, S. E.↗

NASA NTRS1985

Solar physics applications of computer graphics and image processing

Computer graphics devices coupled with computers and carefully developed software provide new opportunities to achieve insight into the geometry and time evolution of scalar, vector, and tensor fields and to extract more information quickly and cheaply from the same image data. Two or more different fields which overlay in space can be calculated from the data (and the physics), then displayed from any perspective, and compared visually. The maximum regions of one field can be compared with the gradients of another. Time changing fields can also be compared. Images can be added, subtracted, transformed, noise filtered, frequency filtered, contrast enhanced, color coded, enlarged, compressed, parameterized, and histogrammed, in whole or section by section. Today it is possible to process multiple digital images to reveal spatial and temporal correlations and cross correlations. Data from different observatories taken at different times can be processed, interpolated, and transformed to a common coordinate system.

Altschuler, M. D.↗

NASA NTRS1985

Scalar/Vector potential formulation for compressible viscous unsteady flows

A scalar/vector potential formulation for unsteady viscous compressible flows is presented. The scalar/vector potential formulation is based on the classical Helmholtz decomposition of any vector field into the sum of an irrotational and a solenoidal field. The formulation is derived from fundamental principles of mechanics and thermodynamics. The governing equations for the scalar potential and vector potential are obtained, without restrictive assumptions on either the equation of state or the constitutive relations or the stress tensor and the heat flux vector.

Morino, L.↗

NASA NTRS1984

Relativistic dispersion, the cyclotron maser instability, and auroral kilometric radiation

It is demonstrated that relativistic effects can significantly modify the wave dispersion in auroral kilometric radiation (AKR), even for only mildly relativistic electrons, when the ratio of the square of the electron plasma frequeny omega(pe) to the square of the electron cyclotron frequency Omega(e) is much less than one, which is frequently the case in the AKR source region. The k-parallel dispersion relation for waves in a relativistic Maxwellian plasma is considered for the case of omega(pe) much less than Omega(e). The results of Shkarovsky (1966) are used to evaluate the relativistic corrections to the R-X mode cutoff. The general relativistic dispersion tensor is applied to evaluate the dispersion relation for a delta function ring distribution in p-perpendicular, again assuming omega(pe) much less than Omega(e). The effect of finite velocity spread is studied by analyzing the Dory-Guest-Harris distribution in the semirelativistic approximation. The results of computer simulations for ring and shell distributions are presented.

Pritchett, P. L.↗

NASA NTRS1984

The distribution of hot spots

Statistical analyses are compared for two published hot spot data sets, one minimal set of 42 and another larger set of 117, using three different approaches. First, the earths surface is divided into 16 equal-area fractions and the observed distribution of hot spots among them is analyzed using chi-square tests. Second, cumulative distributions about the principal axes of the hot spot inertia tensor are used to describe hot spot distribution. Finally, a hot spot density function is constructed for each of the two hot spot data sets. The methods all indicate that hot spots have a nonuniform distribution, even when statistical fluctuations are considered. To the first order, hot spots are concentrated on one half of of the earth's surface area; within that portion, the distribution is consistent with a uniform distribution. The observed hot spot densities for neither data set are explained solely by plate speed.

Stefanick, M.↗

NASA NTRS1984

Development and testing of stable, invariant, isoparametric curvilinear 2- and 3-D hybrid-stress elements

Linear and quadratic Serendipity hybrid-stress elements are examined in respect of stability, coordinate invariance, and optimality. A formulation based upon symmetry group theory successfully addresses these issues in undistorted geometries and is fully detailed for plane elements. The resulting least-order stable invariant stress polynomials can be applied as astute approximations in distorted cases through a variety of tensor components and variational principles. A distortion sensitivity study for two- and three-dimensional elements provides favorable numerical comparisons with the assumed displacement method.

Punch, E. F.↗

NASA NTRS1985

Chiral solidification of a phospholipid monolayer

The formation of chiral solidlike domains observed by Weiss and McConnell (1984) in monolayers of depalmitoylphosphatidylcholine (DPPC) floating on an air-water interface is investigated theoretically. It is proposed that the diffusion tensor for the two-dimensional fluidlike phase of the DPPC molecules has a chiral component acting perpendicular to the concentration gradient and coupled to the rotational motion of a pinwheellike molecule by the viscous forces. Diagrams are provided, and numerical estimates of the forces involved are shown to be in agreement with the observed behavior of the structures.

Langer, J. S.↗

NASA NTRS1985

Geometric effects of global lateral heterogeneity on long-period surface wave propagation

The present investigation has the objective to document examples of anomalous long-period surface wave amplitude behavior and to provide a preliminary appraisal of the effects of global lateral heterogeneity on surface wave propagation from a ray theory perspective. Attention is given to remarkable long-period surface wave anomalies described in literature, an equidistance azimuthal plot centered on the Iranian source region, Rayleigh wave and Love wave spectra for the 256-s period arrivals for the Tabas earthquake, constrained moment tensor and fault model inversion solutions ofr Iranian earthquakes, aspects of surface wave ray tracing, and a table of Rayleigh wave amplitude anomalies for Iranian earthquakes. Surface wave ray-tracing calculations for models of global phase velocity variations proposed by Nakanishi and Anderson (1984) are found to show that large-amplitude anomalies will be observed for Love and Rayleigh waves with periods of 100-250 s.

Lay, T.↗

NASA NTRS1985

Error analyses for a gravity gradiometer mission

This paper addresses the usefulness of an orbiting gravity gradiometer as a sensor for mapping the fine structure of the earth gravity field. The exact knowledge of this field is essential for studies of the solid earth and the dynamics of the oceans. Although the earth gravity tensor, measured by a gradiometer assembly, has nine components, only five components are independent. This latter fact is as a consequence of the symmetry and conservative nature of the earth's gravity field. The most dominant component is the radial one. The error analyses considered here are therefore based only upon a single axis gradiometer sensing this radial component. The expected global gravity and geoid errors for a 50 x 50-km (1/2 x 1/2 deg) area utilizing a spaceborne gradiometer with a precision of 0.001 EU in a 160-km circular polar orbit are about 3 mGAL and 5 cm, respectively.

Kahn, W. D.↗

NASA NTRS1985

Particle and field stress balance within a planetary magnetosphere

A technique is developed for experimentally estimating the local tensor stresses within a planetary magnetic field configuration characterized by local spacecraft measurements. Key to the technique is the determination of the shapes of field lines using the symmetry properties of the system coupled with local and instantaneous measurements of the field line inclination angles. The technique is applied here to the inner and middle Saturnian magnetosphere using data returned by the Magnetic Field Experiment on the Voyager 1 spacecraft. It is concluded that the ring current has substantial radial structure, heretofore not shown. Outside about 13 R(s) the newly derived field stresses match remarkably well the funtional variation of the centrifugal corotation stresses of the cool particle population measured previously by the Plasma Science Experiment. Inside about 13 R(s) the key structure in the derived field stresses, a prominent local maximum, matches the approximate position of an apparent strong pressure gradient in the energetic particles characterized by the Low-Energy Charged Particle detectors.

Mauk, B. H.↗

NASA NTRS1985

Alternatives for Jet Engine Control. Volume 1: Modelling and Control Design with Jet Engine Data

This document compiles a comprehensive list of publications supported by, or related to, National Aeronautics and Space Administration Grant NSG-3048, entitled "Alternatives for Jet Engine Control". Dr. Kurt Seldner was the original Technical Officer for the grant, at Lewis Research Center. Dr. Bruce Lehtinen was the final Technical Officer. At the University of Notre Dame, Drs. Michael K. Sain and R. Jeffrey Leake were the original Project Directors, with Dr. Sain becoming the final Project Director. Publications cover work over a ten-year period. The Final Report is divided into two parts. Volume i, "Modelling and Control Design with Jet Engine Data", follows in this report. Volume 2, "Modelling and Control Design with Tensors", has been bound separately.

Sain, M. K.↗

NASA NTRS1985

Least squares collocation applied to local gravimetric solutions from satellite gravity gradiometry data

An autonomous spaceborne gravity gradiometer mission is being considered as a post Geopotential Research Mission project. The introduction of satellite diometry data to geodesy is expected to improve solid earth gravity models. The possibility of utilizing gradiometer data for the determination of pertinent gravimetric quantities on a local basis is explored. The analytical technique of least squares collocation is investigated for its usefulness in local solutions of this type. It is assumed, in the error analysis, that the vertical gravity gradient component of the gradient tensor is used as the raw data signal from which the corresponding reference gradients are removed to create the centered observations required in the collocation solution. The reference gradients are computed from a high degree and order geopotential model. The solution can be made in terms of mean or point gravity anomalies, height anomalies, or other useful gravimetric quantities depending on the choice of covariance types. Selected for this study were 30 x 30 foot mean gravity and height anomalies. Existing software and new software are utilized to implement the collocation technique. It was determined that satellite gradiometry data at an altitude of 200 km can be used successfully for the determination of 30 x 30 foot mean gravity anomalies to an accuracy of 9.2 mgal from this algorithm. It is shown that the resulting accuracy estimates are sensitive to gravity model coefficient uncertainties, data reduction assumptions and satellite mission parameters.

Robbins, J. W.↗

NASA NTRS1986

A study of tectonic activity in the Basin-Range Province and on the San Andreas Fault. No. 1: Kinematics of Basin-Range intraplate extension

Strain rates assessed from brittle fracture and total brittle-ductile deformation measured from geodetic data were compared to estimates of paleo-strain from Quaternary geology for the intraplate Great Basin part of the Basin-Range, western United States. These data provide an assessment of the kinematics and mode of lithospheric extension that the western U.S. Cordillera has experienced from the past few million years to the present. Strain and deformation rates were determined by the seismic moment tensor method using historic seismicity and fault plane solutions for sub-regions of homogeneous strain. Contemporary deformation in the Great Basin occurs principally along the active seismic zones. The integrated opening rate across the entire Great Basin is accommodated by E-E extension at 8 to 10 mm/a in the north that diminishes to NW-SE extension of 3.5 mm/a in the south. Zones of maximum lithospheric extension correspond to belts of thin crust, high heat flow, and Quaternary basaltic volcanism, suggesting that these parameters are related through mechanism of extension such as a stress relaxation, allowing bouyant uplift and ascension of magmas.

Eddington, P. K.↗

NASA NTRS1986

A study of tectonic activity in the Basin-Range Province and on the San Andreas Fault. No. 3: Kinematics of Great Basin intraplate extension from earthquake, geodetic and geologic information

Strain rates assessed from brittle fracture, associated with earthquakes, and total brittle-ductile deformation measured from geodetic data were compared to paleostrain from Quaternary geology for the intraplate Great Basin of the western United States. These data provide an assessment of the kinematics and mode of lithospheric extension that the western U.S. Cordillera has experienced in the last 5 to 10 million years. Strain and deformation rates were determined by the seismic moment tensor method using historic seismicity and fault plane solutions. Contemporary deformation of the Great Basin occurs principally along the active seismic zones. The earthquake related strain shows that the Great Basin is characterized by regional E-W extension at 8.4 mm/a in the north that diminishes to NW-SE extension of 3.5 mm/a in the south. Zones of maximum extension correspond to belts of shallow crust, high heat flow, and Quaternary basaltic volcanism, suggesting that these parameters are related through an effect such as a stress relaxation allowing bouyant uplift and ascension of magmas.

Eddington, P. K.↗

NASA NTRS1986

The algorithms for rational spline interpolation of surfaces

Two algorithms for interpolating surfaces with spline functions containing tension parameters are discussed. Both algorithms are based on the tensor products of univariate rational spline functions. The simpler algorithm uses a single tension parameter for the entire surface. This algorithm is generalized to use separate tension parameters for each rectangular subregion. The new algorithm allows for local control of tension on the interpolating surface. Both algorithms are illustrated and the results are compared with the results of bicubic spline and bilinear interpolation of terrain elevation data.

Schiess, J. R.↗

NASA NTRS1985

Conservation laws for relativistic guiding-center plasma

A covariant relativistic formulation is used for self-consistent dynamics of a guiding-center plasma in an electromagnetic field. The reciprocal interactions appear as a magnetization in Maxwell's equations and as field gradient forces in the guiding-center dynamical equation. The ten local conservation laws corresponding to translational and rotational invariance of the Minkowski space are expressed in terms of a symmetric energy-momentum tensor.

Similon, P. L.↗

NASA NTRS1985

A granular flow model for dense planetary rings

In the present study of the viscosity of a differentially rotating particle disk, in the limiting case where the particles are densely packed and their collective behavior resembles that of a liquid, the pressure tensor is derived from both the equations of hydrodynamics and a simple kinetic model of collisions due to Haff (1983). Density waves and narrow circular rings are unstable if the liquid approximation applies, and the consequent nonlinear perturbations may generate 'splashing' of the ring material in the vertical direction. These results are pertinent to the origin of the ellipticities of ringlets, the nonaxisymmetric features near the outer edge of the Saturn B ring, and unexplained residuals in kinematic models of the Saturn and Uranus rings.

Borderies, N.↗