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Two computer programs generate two-dimensional body-fitted coordinate systems and coordinate transformation. Programs are useful in fields requiring accurate numerical representation of boundary conditions and accurate numerical solutions of partial differential equations.
An approximate analysis, applicable to nonorthogonal coordinate systems having a curved centerline and planar transverse coordinate surfaces normal to the centerline, is presented for computation of three-dimensional subsonic flow in straight and curved diffusers. The formulation is intended to facilitate the use of constructed coordinates in circumstances where it is difficult to maintain smooth behavior in higher derivatives; the use of local Cartesian variables and fluxes leads to governing equations which require only first derivatives of the coordinate transformation. The analysis is applied to a particular family of duct and diffuser geometries having curved centerlines and superelliptic cross sections. Qualitative agreement with experimental measurements is observed with regard to streamwise vortices and distortion of the primary flow.
A method for automatic generation of boundary-fitted curvilinear coordinate systems, where the transformed coordinates are solutions of an elliptic differential system in the physical plane, and where the coordinate lines are coincident with all boundaries of a general multiply-connected, two-dimensional region containing any number of arbitrarily shaped bodies, and is described along with a suitable computer code for implementing the method. Any partial differential system can be solved on the boundary-fitted coordinate system by appropriate transformations. The transformed equations are approximated by finite differences and solved numerically in the transformed plane. All computations, whether for generating coordinate system or then solving the transformed equations, can be done on a rectangular field with square mesh with no interpolation required on the boundaries. The physical boundaries may even be time-dependent.
Coordinate measuring system for flight control, and trajectory optimization
Over the past 40 years, an enormous quantity of orbital remote sensing data has been collected for Mars from many missions and instruments. Unfortunately these datasets currently exist in a wide range of disparate coordinate systems, making it extremely difficult for the scientific community to easily correlate, combine, and compare data from different Mars missions and instruments. As part of our work for the PDS Imaging Node and on behalf of the USGS Astrogeology Team, we are working to solve this problem and to provide the NASA scientific research community with easy access to Mars orbital data in a unified, consistent coordinate system along with a wide variety of other key geometric variables. The Unified Planetary Coordinates (UPC) system is comprised of two main elements: (1) a database containing Mars orbital remote sensing data computed using a uniform coordinate system, and (2) a process by which continual maintainance and updates to the contents of the database are performed.
The proceedings of a colloquium on lunar dynamics and observational coordinate systems are presented. Discussions were held on the establishment of a fundamental reference system and on the lunar ephemerides. Abstracts of the subjects discussed at the meeting are submitted. Some of the topics discussed are: (1) coordinates of the Apollo retroreflectors, (2) determination of lunar baselines, (3) numerical series for the variations of lunar coordinates, (4) fundamental craters for establishing a lunar coordinate system, and (5) composite lunar gravity fields.
Hypervirial theorems and Hellmann-Feynman theorem in different coordinate systems
Various finite difference techniques used to solve Laplace's equation are compared. Curvilinear coordinate systems are used on two dimensional regions with irregular boundaries, specifically, regions around circles and airfoils. Truncation errors are analyzed for three different finite difference methods. The false boundary method and two point and three point extrapolation schemes, used when having the Neumann boundary condition are considered and the effects of spacing and nonorthogonality in the coordinate systems are studied.
The technique of body-fitted coordinate systems is applied in numerical solutions of the complete time-dependent compressible and incompressible Navier-Stokes equations for laminar flow and to the time-dependent mean turbulent equations closed by modified Kolmogorov hypotheses for turbulent flow. Coordinate lines are automatically concentrated near to the bodies at higher Reynolds number so that accurate resolution of the large gradients near the solid boundaries is achieved. Two-dimensional bodies of arbitrary shapes are treated, the body contour(s) being simply input to the program. The complication of the body shape is thus removed from the problem.
Coordinate systems generation for optimal trajectories, considering Lagrange multiplier as motion constant
A numerically generated orthogonal coordinate system (with the body surface and shock wave as opposite boundaries) was applied with a time asymptotic method to obtain steady flow solutions for axisymmetric inviscid flow over several blunt bodies including spheres, paraboloids, ellipsoids, hyperboloids, hemisphere cylinders, spherically blunted cones, and a body with a concavity in the stagnation region. Comparisons with experimental data and with the results of other computational methods are discussed. The numerically generated orthogonal coordinate system is described and applications of the method to complex body shapes, particularly those with concave regions, are discussed.
Introduction: When describing human motion, biomechanists generally report joint angles in terms of Euler angle rotation sequences. However, there are known limitations in using this method to describe complex motions such as the shoulder joint during a baseball pitch. Euler angle notation uses a series of three rotations about an axis where each rotation is dependent upon the preceding rotation. As such, the Euler angles need to be regarded as a set to get accurate angle information. Unfortunately, it is often difficult to visualize and understand these complex motion representations. It has been shown that using a spherical coordinate system allows Anthropometry and Biomechanics Facility (ABF) personnel to increase their ability to transmit important human mobility data to engineers, in a format that is readily understandable and directly translatable to their design efforts. Objectives: The goal of this project was to use innovative analysis and visualization techniques to aid in the examination and comprehension of complex motions. Methods: This project consisted of a series of small sub‐projects, meant to validate and verify a new method before it was implemented in the ABF's data analysis practices. A mechanical test rig was built and tracked in 3D using an optical motion capture system. Its position and orientation were reported in both Euler and spherical reference systems. In the second phase of the project, the ABF estimated the error inherent in a spherical coordinate system, and evaluated how this error would vary within the reference frame. This stage also involved expanding a kinematic model of the shoulder to include the rest of the joints of the body. The third stage of the project involved creating visualization methods to assist in interpreting motion in a spherical frame. These visualization methods will be incorporated in a tool to evaluate a database of suited mobility data, which is currently in development. Results: Initial results demonstrated that a spherical coordinate system is helpful in describing and visualizing the motion of a space suit. The system is particularly useful in describing the motion of the shoulder, where multiple degrees of freedom can lead to very complex motion paths.
A solution method of the time-accurate, incompressible Navier-Stokes equations in generalized curvilinear moving coordinate systems is presented in this paper. Accuracy is achieved by a conservative finite-volume discretization which satisfies the geometric conservation laws in generalized moving coordinate systems. The solution method is second-order accurate in space and first-order accurate in time. A fractional step solution method is used to efficiently solve the discrete equations. The unknowns, namely the pressure and the volume-fluxes, are chosen to facilitate the formulation of a consistent Poisson equation and to obtain a robust Poisson solver with favorable convergence properties. The method is validated by comparisons to other numerical and experimental solutions. The comparisons show good agreement.
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.
Satellite geodesy has yielded the locations of more than fifty stations in a single coordinate system referred to the earth's center of mass with accuracies in the five to ten meter range. The different methods used at Goddard to accomplish this are described, and estimates of the accuracies of the satellite determinations are discussed. Theoretical aspects of coordinate systems associated with the earth and the moon are also considered.
Coordinate system influence on regularized trajectory optimization problem
Lie group use in solution of equations resulting from separation of Helmholtz equations in special coordinate systems
Application of digital computers for deriving motion equations of point mass in curvilinear coordinate systems by use of tensor calculus