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The gasdynamic equations are transformed to new coordinates based on particle paths and an appropriate set of characteristics. The numerical integration of the transformed equations is accomplished by a scheme which is accurate, rapidly convergent, and free from shock oscillations. Complete flow solutions are presented in graphical form for several cases.
Blocked algorithms have much better properties of data locality and therefore can be much more efficient than ordinary algorithms when a memory hierarchy is involved. On the other hand, they are very difficult to write and to tune for particular machines. The reorganization is considered of nested loops through the use of known program transformations in order to create blocked algorithms automatically. The program transformations used are strip mining, loop interchange, and a variant of loop skewing in which invertible linear transformations (with integer coordinates) of the loop indices are allowed. Some problems are solved concerning the optimal application of these transformations. It is shown, in a very general setting, how to choose a nearly optimal set of transformed indices. It is then shown, in one particular but rather frequently occurring situation, how to choose an optimal set of block sizes.
A simple method of transforming vector magnetograms to heliographic coordinates is demonstrated. The merits of this transformation are illustrated using a vector magnetogram obtained with the MSFC vector magnetograph 80 minutes prior to a white light flare in active region AR 4474 on April 25, 1984. The original magnetogram shows strong magnetic shear along the neutral line at both the flare site and a nonflaring site. The transformation of the magnetogram to heliographic coordinates shows that the elimination of projection effects results in a much shorter length of the sheared region at the nonflaring site than what is inferred from the image plane vector magnetogram. The length of the sheared region at the flare site is relatively less affected by the transformation.
A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given. Some general mathematical framework and error analysis common to such coordinate systems is also included. The general categories of generating systems are those based on conformal mapping, orthogonal systems, nearly orthogonal systems, systems produced as the solution of elliptic and hyperbolic partial differential equations, and systems generated algebraically by interpolation among the boundaries. Also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation. Dynamically adaptive coordinate systems, coupled with the physical solution, and time-dependent systems that follow moving boundaries are treated. References reporting experience using such coordinate systems are reviewed as well as those covering the system development.
Inherent geographical hazards of Martian terrain may impede a safe landing for science exploration spacecraft. Surface visualization software for hazard detection and avoidance may accordingly be applied in vehicles such as the Mars Exploration Rover (MER) to induce an autonomous and intelligent descent upon entering the planetary atmosphere. The focus of this project is to develop an image transformation algorithm for coordinate system matching between consecutive frames of terrain imagery taken throughout descent. The methodology involves integrating computer vision and graphics techniques, including affine transformation and projective geometry of an object, with the intrinsic parameters governing spacecraft dynamic motion and camera calibration.
In order to provide a rational background for the analysis of experimental observations of blast wave phenomena, the conservation equations governing their nonsteady flow field are formulated in a general manner, without the usual restrictions imposed by an equation of state, and with proper account taken, by means of source terms, of other effects which, besides the inertial terms that conventionally dominate these equations, can affect the flow. Taking advantage of the fact that a blast wave can be generally considered as a spatially one-dimensional flow field whose nonsteady behavior can be regarded, consequently, as a function of just two independent variables, two generalized blast wave coordinates are introduced, one associated with the front of the blast wave and the other with its flow field. The conservation equations are accordingly transformed into this coordinate system, acquiring thereby a comprehensive character, in that they refer then to any frame of reference, being applicable, in particular, to problems involving either space or time profiles of the gas-dynamic parameters in the Eulerian system, or time profiles in the Lagrangian system.
Models used in geodesy to transform two sets of coordinates are studied and distortions in geodetic networks are investigated. Commonly used transformation models are first reviewed and most of them are interpreted. Differences between various models are discussed. Pitfalls in partial solutions are then considered. It is shown that only as many chords and/or directional elements can be used in the computation as are needed to completely determine the size or shape of the polyhedron implied in the set of Cartesian coordinates. Each additional element causes the normal matrix to be singular provided that all correlations between the chords are used. A number of tables and maps indicating distortions in the NAD 27, Precise Traverse M-R '72, AUS, and SAD 69 geodetic datums are also included. The residuals of the coordinates are scanned for systematic patterns after transforming each geodetic system to the NWL9D Doppler system. Also, an attempt is made to show scale distortions in the NAD 27.
Although the albedo of specific belts and zones varies as a function of time, there is evidence that wind maxima may be fixed in latitude. Before considering a standard notation for wind jets, it is necessary to establish a coordinate system within which the nomenclature would be defined. Traditionally, the BAA has used planetographic latitudes; however, this system is based not only on an accurate determination of the polar diameter but also on the assumption that the equipotential surfaces can be represented by biaxial ellipsoids. The International Astronomical Union strives to adopt unambiguous nomenclature that will be universally acceptable. It is proposed that planetocentric coordinates be utilized and that a standardized value of the ratio of the polar diameter to the equatorial diameter be established for each planet to facilitate transformation into planetographic coordinates.
Three methods for performing Hankel transforms with optical or digital processors are described. The first method is applicable when the input data is available in Cartesian (x-y) format and uses the close connection between generalized Hankel transform and the two dimensional Fourier transform in Cartesian coordinates. The second method is useful when the input data is in polar (r - theta) format and uses change of variables to perform the nth order Hankel transform as a correlation integral. The third method utilizes the von Neumann addition theorem for Bessel functions to extract the Hankel coefficients from a correlation between the radial part of the input and a Bessel function. Initial experimental results obtained for optical implementation of the first two methods are presented.
We demonstrate the effectiveness of a relatively straightforward analysis of the complex 3D Fourier transform of galaxy coordinates derived from redshift surveys. Numerical demonstrations of this approach are carried out on a volume-limited sample of the Sloan Digital Sky Survey redshift survey. The direct unbinned transform yields a complex 3D data cube quite similar to that from the Fast Fourier Transform of finely binned galaxy positions. In both cases, deconvolution of the sampling window function yields estimates of the true transform. Simple power spectrum estimates from these transforms are roughly consistent with those using more elaborate methods. The complex Fourier transform characterizes spatial distributional properties beyond the power spectrum in a manner different from (and we argue is more easily interpreted than) the conventional multipoint hierarchy. We identify some threads of modern large-scale inference methodology that will presumably yield detections in new wider and deeper surveys.
We demonstrate the effectiveness of a relatively straightforward analysis of the complex 3D Fourier transform of galaxy coordinates derived from redshift surveys. Numerical demonstrations of this approach are carried out on a volume-limited sample of the Sloan Digital Sky Survey redshift survey. The direct unbinned transform yields a complex 3D data cube quite similar to that from the Fast Fourier Transform (FFT) of finely binned galaxy positions. In both cases deconvolution of the sampling window function yields estimates of the true transform. Simple power spectrum estimates from these transforms are roughly consistent with those using more elaborate methods. The complex Fourier transform characterizes spatial distributional properties beyond the power spectrum in a manner different from (and we argue is more easily interpreted than) the conventional multi-point hierarchy. We identify some threads of modern large scale inference methodology that will presumably yield detections in new wider and deeper surveys.
The feasibility of acoustic verification of microwave near field to far field transformation algorithms using the Phased Array Sonic Simulation System was studied. Existing electromagnetic near field measurement techniques and transformation algorithms (equations) were investigated. It was analytically determined that acoustic verification is valid. Acoustic simulation of electromagnetic near field to far field transformations is emphasized. The acoustic simulation of electromagnetic near field to far field transformation is verified for the modal expansion method. In the modal expansion method, data from antenna near field measurements are converted to a summation or spectrum of modes corresponding to wave numbers in the measurement coordinate system. Fourier transformation of those measurements preserves the far field information in a spectral form that is then readily extractable.
The flapping equation for a helicopter in forward flight are reported which have coefficients that are periodic in time, and this effect complicates the calculation of stability. A constant coefficient approximation which will allow the use of all the well known methods for analyzing constant coefficient equations are presented. The flapping equation is first transformed into the nonrotating coordinate frame, where some of the periodic coefficients are transformed into constant terms. The constant coefficient approximation is then made by using time averaged coefficients in the nonrotating frame. Stability calculations based on the approximation are compared to results from a theory which correctly includes all of the periodicity. The comparison indicates that the approximation is reasonably accurate at advance ratios up to 0.5.
The flapping equation for a helicopter in forward flight has coefficients which are periodic in time, and this effect complicates the calculation of stability. This paper presents a constant coefficient approximation which will allow the use of all the well known methods for analyzing constant coefficient equations. The flapping equation is first transformed into the nonrotating coordinate frame, where some of the periodic coefficients are transformed into constant terms. The constant coefficient approximation is then made by using time averaged coefficients in the nonrotating frame. Stability calculations based on the approximation are compared to results from a theory which correctly includes all of the periodicity. The comparison indicates that the approximation is reasonably accurate at advance ratios up to 0.5.
The use of tensors to provide a compact way of writing partial differential equations in a form valid in all coordinate systems is discussed. In order to find solutions to the equations with their boundary conditions they must be expressed in terms of the coordinate system under consideration. The process of arriving at these expressions from the tensor formulation was automated by a software system, TENSR. An allied system that analyzes the resulting expressions term by term and drops those that are negligible is also described.
We wanted to make a satellite altitude magnetic anomaly map of the large magnetic anomaly in the Central African Republic, the Bangui magnetic anomaly, with data from the Swarm satellites. In the first part of our study, we summarize the earlier investigations and their interpretation. In the second we discuss our data processing applied to produce a magnetic anomaly map. We used the IGRF 12th to remove the long-wavelength regional anomalies. We will use an inverse procedure, which always requires a solution of the direct problem, and a horizontal polygonal prism given in the Descartes coordinate system. For this, reason the total magnetic anomaly was transformed into the Descartes coordinate system. The magnetization and its direction were used from our previous paper. The inverse problem is solved by the Simplex procedure. Our selected polygon has 14 geometrical parameters however, the inverse problem that is the numerical determination of the minimum problem is solved in the 14 dimensions. The result of our inverse problem was the 12 horizontal coordinates and the two upper and lower data of the polygon. The origin of the Bangui anomaly has been discussed in several scientific reports, either as a deep crustal tectonic feature or the result of a large external impactor. However, according to our inversion computations we cannot make any unambiguous finding for the origin of this feature. The inaccuracy in our total anomaly map is given by the Gaussian error propagation.
The flow field is simulated on the surface of a given delta wing (Butler wing) at zero incident in a uniform stream. The simulation is done by integrating a set of flow field equations. This set of equations governs the unsteady, viscous, compressible, heat conducting flow of an ideal gas. The equations are written in curvilinear coordinates so that the wing surface is represented accurately. These equations are solved by the finite difference method, and results obtained for high-speed freestream conditions are compared with theoretical and experimental results. In this study, the Navier-Stokes equations are solved numerically. These equations are unsteady, compressible, viscous, and three-dimensional without neglecting any terms. The time dependency of the governing equations allows the solution to progress naturally for an arbitrary initial initial guess to an asymptotic steady state, if one exists. The equations are transformed from physical coordinates to the computational coordinates, allowing the solution of the governing equations in a rectangular parallel-piped domain. The equations are solved by the MacCormack time-split technique which is vectorized and programmed to run on the CDC VPS 32 computer.
We present a theoretical foundation for relativistic astronomical measurements in curved space-time. In particular, we discuss a new iterative approach for describing the dynamics of an isolated astronomical N-body system in metric theories of gravity. To do this, we generalize the Fock-Chandrasekhar method of the weak-field and slow-motion approximation (WFSMA) and develop a theory of relativistic reference frames (RF's) for a gravitationally bounded many-extended-body problem. In any proper RF constructed in the immediate vicinity of an arbitrary body, the N-body solutions of the gravitational field equations are formally presented as a sum of the Riemann-flat inertial space-time, the gravitational field generated by the body itself, the unperturbed solutions for each body in the system transformed to the coordinates of this proper RF, and the gravitational interaction term. We develop the basic concept of a general WFSMA theory of the celestial RF's applicable to a wide class of metric theories of gravity and an arbitrary model of matter distribution. We apply the proposed method to general relativity. Celestial bodies are described using a perfect fluid model; as such, they possess any number of internal mass and current multipole moments that explicitly characterize their internal structures. The obtained relativistic corrections to the geodetic equations of motion arise because of a coupling of the bodies' multiple moments to the surrounding gravitational field. The resulting relativistic transformations between the different RF's extend the Poincare group to the motion of deformable self-gravitating bodies. Within the present accuracy of astronomical measurements we discuss the properties of the Fermi-normal-like proper RF that is defined in the immediate vicinity of the extended compact bodies. We further generalize the proposed approximation method and include two Eddington parameters (gamma, Beta). This generalized approach was used to derive the relativistic equations of satellite motion in the vicinity of the extended bodies. Anticipating improvements in radio and laser tracking technologies over the next few decades, we apply this method to spacecraft orbit determination. We emphasize the number of feasible relativistic gravity tests that may be performed within the context of the parameterized WFSMA. Based on the planeto-centric equations of motion of a spacecraft around the planet, we suggested a new null test of the Strong Equivalence Principle (SEP). The experiment to measure the corresponding SEP violation effect could be performed with the future Mercury Orbiter mission. We discuss other relativistic effects, including the perihelion advance and the redshift and geodetic precession of the orbiter's orbital plane about Mercury, as well as the possible future implementation of the proposed formalism in software codes developed for solar-system orbit determination. All the important calculations are completely documented, and the references contain an extensive list of cited literature.