Regular Triangular Grid Research Articles

Parametrizations for triangular G k spline surfaces of low degree

In this article, we present regularly parametrized G k free-form spline surfaces that extend box and half-box splines over regular triangular grids. The polynomial degree of these splines is max{4 k + 1, ⌈3k/2 + 1⌉r}, where r ∈ ℕ can be chosen arbitrarily and determines the flexibility at extraordinary points. The G k splines presented in this article depend crucially on low-degree (re-)parametrizations of piecewise polynomial hole fillings. The explicit construction of such parametrizations forms the core of this work and we present two classes of singular and regular parametrizations. Also, we show how to build box and half-box spline surfaces of arbitrarily high smoothness with holes bounded by only n patches, in principle.

Addressing, distances and routing in triangular systems with applications in cellular networks

Triangular systems are the subgraphs of the regular triangular grid which are formed by a simple circuit of the grid and the region bounded by this circuit. They are used to model cellular networks where nodes are base stations. In this paper, we propose an addressing scheme for triangular systems by employing their isometric embeddings into the Cartesian product of three trees. This embedding provides a simple representation of any triangular system with only three small integers per vertex, and allows to employ the compact labeling schemes for trees for distance queries and routing. We show that each such system with n vertices admits a labeling that assigns O(log2 n) bit labels to vertices of the system such that the distance between any two vertices u and v can be determined in constant time by merely inspecting the labels of u and v, without using any other information about the system. Furthermore, there is a labeling, assigning labels of size O(log n) bits to vertices, which allows, given the label of a source vertex and the label of a destination, to compute in constant time the port number of the edge from the source that heads in the direction of the destination. These results are used in solving some problems in cellular networks. Our addressing and distance labeling schemes allow efficient implementation of distance and movement based tracking protocols in cellular networks, by providing information, generally not available to the user, and means for accurate cell distance determination. Our routing and distance labeling schemes provide elegant and efficient routing and connection rerouting protocols for cellular networks.

On the optimal base-station density for CDMA cellular networks

In this paper, the minimal base-station density for a code-division multiple-access (CDMA) cellular radio network is determined such that the outage probability does not exceed a certain threshold. Base stations are assumed to be located on a regular triangular grid of minimum distance d, while mobiles are randomly distributed according to a two-dimensional Poisson point pattern. Each mobile may be connected to, at most, one of four surrounding base stations, effectively connecting and applying power control to the one with least attenuation. Thus, we model the use of macroscopic selection diversity. We obtain a normal approximation to the total interference power at a reference base station for a correlated log-normal shadowing law. The base station distance we obtain is proportional to the inverse of the square root of the traffic intensity, and we obtain the constant of proportionality, which is itself a function of the minimum acceptable carrier-to-interference (C/I) ratio and the maximum tolerable outage probability. Our formula for this distance can be used in network planning and design.

A numerical scheme based on mean value solutions for the Helmholtz equation on triangular grids

A numerical treatment for the Dirichlet boundary value problem on regular triangular grids for homogeneous Helmholtz equations is presented, which also applies to the convection-diffusion problems. The main characteristic of the method is that an accuracy estimate is provided in analytical form with a better evaluation than that obtained with the usual finite difference method. Besides, this classical method can be seen as a truncated series approximation to the proposed method. The method is developed from the analytical solutions for the Dirichlet problem on a ball together with an error evaluation of an integral on the corresponding circle, yielding O ( h 4 ) O(h^{4}) accuracy. Some numerical examples are discussed and the results are compared with other methods, with a consistent advantage to the solution obtained here.

Finite-Difference Schemes on Regular Triangular Grids

The phase error and isotropy properties of various finite-difference schemes on grids consisting of regular triangles are compared with similar schemes on square grids. The comparisons are based on a Fourier analysis of semidiscrete solutions to the two-dimensional linear convection equation. The finite-difference schemes presented on the triangular grid include a second-order method, a compact fourth-order method, and a modified compact method designed to extend the accurate wave number range of the numerical approximation. All of the schemes considered are centered and hence nondissipative. In each case, the finite-difference scheme on the triangular grid reduces the anisotropy of the phase error in comparison with a similar scheme on the square grid.

Discontinuous Finite Element Method for Solving the Two-Dimesional Neutron Transport Equation on Lagrangian Meshes

It is often desirable to solve the two-dimensional multigroup transport equation for (r-z) geometries directly given by hydrodynamic calculations. Usually, only Monte Carlo codes can compute kappa or k eigenvalues on such geometries. Most deterministic codes use an orthogonal mesh or restrict the mesh to a regular triangular grid. Other methods were developed at Commissariat a l'Energie Atomique and Los Alamos National Laboratory but do not solve the problem of sliding between two Lagrangian blocks. Thus, a production code has developed that solves these problems and is able to obtain kappa or k eigenvalues with a good degree of accuracy for such geometries.

Temperature field in heat-liberating piles of fast reactors

The accuracy of thermophysical calculation of heat-liberatlng piles (HLP) of fast reactors is determined to a considerable extent by the accuracy of the initial calculational model of the fleld-element bundle and also by,taking correct account of all the factors influencing the formation of a temperature field in the bundle. The calculation is usually conducted for nominal parameters of the fuel elements and the HLP, under the assumption of a regular arrangement of fuel elements at the points of a regular triangular grid. For this model, the distribution of the heat-carrier flow rate over the individual channels, the heating of the heat carrier in the most stressed channels, and the corresponding temperature of the fuel-element casing are calculated for this model.

The design of optimal sampling schemes for local estimation and mapping of of regionalized variables—I: Theory and method

Surveys of materials at the earth's surface, especially soil, can be planned to make the best use of the resources for survey or to achieve a certain minimum precision provided the nature of spatial dependence is known already. A method is described for designing optimal sampling schemes. It is based on the theory of regionalized variables, and assumes that spatial dependence is expressed quantitatively in the form of the semi-variogram. It assumes also that the maximum standard error of a kriged estimate is a reasonable measure of the goodness of a sampling scheme. By sampling on a regular triangular grid, the maximum standard error is kept to a minimum for any given sampling, but a square grid is approximately equivalent where variation is isotropic. Given the semi-variogram for a variable, the sampling density for any prescribed maximum standard error is determined. Where variation is geometrically anisotropic, the same method is employed to determine sample spacing in the direction of maximum change, and the grid mesh elongated in the perpendicular direction in proportion to the anisotropy ratio.

The design of optimal sampling schemes for local estimation and mapping of regionalized variables—II: Program and examples

A FORTRAN IV program, OSSFIM, is presented for calculating estimation variances when interpolating by kriging from regular rectangular and triangular grids of data and previously-determined semi-variogram. The variances are computed for a range of grid spacings and block sizes, and the results graphed. The user chooses a block size, and can read from the appropriate graph the sample spacing corresponding to any prescribed maximum tolerable error. This is the optimal sampling scheme. Use of the program is illustrated with two examples showing different types of variation in soil. In one, the pH of topsoil is isotropic with a spherical semi-variogram and negligible nugget variance. An equilateral triangular grid is the best sampling scheme; it is approximately 10 per cent more efficient than a square grid. In the other example, variation is linear but anisotropic with a large nugget variance. In these circumstances, a triangular grid has no advantage over a rectangular one, which should be elongated in the ratio 1.88 to I in the direction of minimum variation.

Computable finite element error bounds for Poisson's equation

AbstractTheoretical error bounds of the form equation image are often available for finite element solutions U of elliptic problems. In this form the estimates give the order of convergence of the method but are of little practical value for estimating the size of the error because the magnitudes of the constant K and the theoretical solution u are unknown. An exception occurs in the case of the equation ∂2u/∂x2 + ∂2u/∂y2 + f = 0 in a rectangle where the Ritz–Galerkin finite element solution involves piecewise linears over a regular triangular grid. In this case where α = 1 and equation image Barnhill and Gregory1 have obtained the theoretical value 0·93√2 for K. In this note calculations are carried out for a variety of problems and the quantity K* = ∥u – U∥E/h∥f∥L2 measured and compared with K. The values of K* obtained fit into a well defined pattern from which we conclude that the theoretical constant K is of the correct order of magnitude.

Operators for Solution of Discrete Dirichlet and Plateau Problems Over a Regular Triangular Grid

Operators for Solution of Discrete Dirichlet and Plateau Problems Over a Regular Triangular Grid