Coordinate system

About: Coordinate system is a research topic. Over the lifetime, 22675 publications have been published within this topic receiving 269822 citations. The topic is also known as: system of coordinates.

A method for computing horizontal pressure-gradient force in an oceanic model with a nonaligned vertical coordinate

Abstract: [1] Discretization of the pressure-gradient force is a long-standing problem in terrain-following (or σ) coordinate oceanic modeling. When the isosurfaces of the vertical coordinate are not aligned with either geopotential surfaces or isopycnals, the horizontal pressure gradient consists of two large terms that tend to cancel; the associated pressure-gradient error stems from interference of the discretization errors of these terms. The situation is further complicated by the nonorthogonality of the coordinate system and by the common practice of using highly nonuniform stretching for the vertical grids, which, unless special precautions are taken, causes both a loss of discretization accuracy overall and an increase in interference of the component errors. In the present study, we design a pressure-gradient algorithm that achieves more accurate hydrostatic balance between the two components and does not lose as much accuracy with nonuniform vertical grids at relatively coarse resolution. This algorithm is based on the reconstruction of the density field and the physical z coordinate as continuous functions of transformed coordinates with subsequent analytical integration to compute the pressure-gradient force. This approach allows not only a formally higher order of accuracy, but it also retains and expands several important symmetries of the original second-order scheme to high orders [Mellor et al., 1994; Song, 1998], which is used as a prototype. It also has built-in monotonicity constraining algorithm that prevents appearance of spurious oscillations of polynomial interpolant and, consequently, insures numerical stability and robustness of the model under the conditions of nonsmooth density field and coarse grid resolution. We further incorporate an alternative method of dealing with compressibility of seawater, which escapes pressure-gradient errors associated with interference of the nonlinear nature of equation of state and difficulties to achieve accurate polynomial fits of resultant in situ density profiles. In doing so, we generalized the monotonicity constraint to guarantee nonnegative physical stratification of the reconstructed density profile in the case of compressible equation of state. To verify the new method, we perform traditional idealized (Seamount) and realistic test problems.

Calculation of material properties and ray tracing in transformation media.

Abstract: Complex and interesting electromagnetic behavior can be found in spaces with non-flat topology When considering the properties of an electromagnetic medium under an arbitrary coordinate transformation an alternative interpretation presents itself The transformed material property tensors may be interpreted as a different set of material properties in a flat, Cartesian space We describe the calculation of these material properties for coordinate transformations that describe spaces with spherical or cylindrical holes in them The resulting material properties can then implement invisibility cloaks in flat space We also describe a method for performing geometric ray tracing in these materials which are both inhomogeneous and anisotropic in their electric permittivity and magnetic permeability

Mathematical model of geometry and fibrous structure of the heart.

Abstract: We developed a mathematical representation of ventricular geometry and muscle fiber organization using three-dimensional finite elements referred to a prolate spheroid coordinate system. Within elements, fields are approximated using basis functions with associated parameters defined at the element nodes. Four parameters per node are used to describe ventricular geometry. The radial coordinate is interpolated using cubic Hermite basis functions that preserve slope continuity, while the angular coordinates are interpolated linearly. Two further nodal parameters describe the orientation of myocardial fibers. The orientation of fibers within coordinate planes bounded by epicardial and endocardial surfaces is interpolated linearly, with transmural variation given by cubic Hermite basis functions. Left and right ventricular geometry and myocardial fiber orientations were characterized for a canine heart arrested in diastole and fixed at zero transmural pressure. The geometry was represented by a 24-element ensemble with 41 nodes. Nodal parameters fitted using least squares provided a realistic description of ventricular epicardial [root mean square (RMS) error less than 0.9 mm] and endocardial (RMS error less than 2.6 mm) surfaces. Measured fiber fields were also fitted (RMS error less than 17 degrees) with a 60-element, 99-node mesh obtained by subdividing the 24-element mesh. These methods provide a compact and accurate anatomic description of the ventricles suitable for use in finite element stress analysis, simulation of cardiac electrical activation, and other cardiac field modeling problems.

Analyzing oriented patterns

Abstract: Oriented patterns, such as those produced by propagation, accretion, or deformation, are common in nature and therefore an important class for visual analysis. Our approach to understanding such patterns is to decompose them into two parts: the flow field, describing the direction of anisotropy; and the residual pattern obtained by describing the image in a coordinate system built from the flow field. We develop a method for the local estimation of anisotropy and a method for combining the estimates to construct a flow coordinate system. Several examples of the use of these methods are presented. These include the use of the flow coordinates to provide preferred directions for edge detection, detection of anomalies, fitting simple models to the straightened pattern, and detecting singularities in the flow field.