Embedding rectangular grids into square grids

2006 ◽  
pp. 181-190 ◽  
Author(s):  
John A. Ellis
Keyword(s):  
Rectangular Grids

scholarly journals Vector Arithmetic in the Triangular Grid

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 373
Author(s):  
Khaled Abuhmaidan ◽  
Monther Aldwairi ◽  
Benedek Nagy
Keyword(s):  
Coordinate System ◽  
Triangular Grid ◽  
Vector Addition ◽  
Plane Space ◽  
Physical Simulations ◽  
Rectangular Grids ◽  
Cartesian Coordinate ◽  
Zero Sum ◽  
Coordinate Geometry ◽  
Point Lattice

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

scholarly journals Topologically Rectangular Grids in the Parallel Simulation of Semiconductor Devices

VLSI Design ◽  
1998 ◽  
Vol 6 (1-4) ◽  
pp. 91-95 ◽  
Author(s):  
A. Asenov ◽  
A. R. Brown ◽  
S. Roy ◽  
J. R. Barker
Keyword(s):  
Simulated Annealing Algorithm ◽  
Semiconductor Device ◽  
Electronic Device ◽  
Device Simulation ◽  
Power Electronic ◽  
Two Dimensional ◽  
Solution Strategy ◽  
Rectangular Grids ◽  
Annealing Algorithm ◽  
2D And 3D

Topologically rectangular grids offer simplicity and efficiency in the design of parallel semiconductor device simulators tailored for mesh connected MIMD platforms. This paper presents several approaches to the generation of topologically rectangular 2D and 3D grids. The effects of the partitioning of such grids on different processor configurations are studied. A simulated annealing algorithm is used to optimise the partitioning of 2D and 3D grids on two dimensional arrays of processors. Problems related to the discretization, parallel matrix generation and solution strategy are discussed. The use of topologically rectangular grids is illustrated through the example of power electronic device simulation.

FRACTAL INTERPOLATION SURFACES ON RECTANGULAR GRIDS

2015 ◽  
Vol 91 (3) ◽  
pp. 435-446 ◽  
Author(s):  
HUO-JUN RUAN ◽  
QIANG XU
Keyword(s):  
General Framework ◽  
Vertical Scaling ◽  
Fractal Interpolation ◽  
Scaling Factors ◽  
Rectangular Grids

In this paper, we present a general framework to construct fractal interpolation surfaces (FISs) on rectangular grids. Then we introduce bilinear FISs, which can be defined without any restriction on interpolation points and vertical scaling factors.

scholarly journals Converting Molecular Meshes Into Smooth Interpolatory Spline Solid Models

2005 ◽  
Author(s):  
Joel Daniels ◽  
Elaine Cohen ◽  
David Johnson
Keyword(s):  
Continuous Model ◽  
Multiple Components ◽  
Spline Models ◽  
Solid Models ◽  
Spline Surfaces ◽  
Physical Form ◽  
Cad Cam ◽  
The Common ◽  
Rectangular Grids ◽  
Multiple Levels

The study and understanding of molecules, once the domain of blackboards and stick-and-ball models, has become more and more exclusively linked to the use of computer-aided visualizations. Our project seeks to return the physical facsimile to the biologists, allowing the use of tactile senses while interacting with and manipulating a physical model, thus aiding educational and research endeavors. To increase the effectiveness of such a tool, the model is constructed such that multiple levels of information are viewable within the single physical form, stressing the interaction between the assorted components within the molecule. We use the term 3-D physical visualizations to refer to the fabricated model, to avoid confusion with the common usage of model as a virtual representation on the computer. To effectively combine multiple components into a smooth manufacturable physical visualization, all components of the model must be in a homogeneous format. Our research sets forth a method for converting triangulated mesh data, as provided by the molecular modeling packages, into spline models. Spline models have the attractive qualities that they are smooth without triangular facets, can be combined using traditional boolean operations (and, or, not), and can be directly fabricated using modern CAD/CAM techniques. Our method divides the polyhedral representation into multiple rectangular grids, then fits interpolatory spline surfaces to the data in each region, while focusing on smoothly stitching the boundaries and corners of the spline surfaces in order to create a near G1 continuous model.

A Design of Coverage Area for Textured Surface of Sliding Journal Bearing Based on Genetic Algorithm

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
Hui Zhang ◽  
Mahshid Hafezi ◽  
Guangneng Dong ◽  
Yang Liu
Keyword(s):  
Genetic Algorithm ◽  
Journal Bearing ◽  
Journal Bearings ◽  
Textured Surface ◽  
The Novel ◽  
Eccentricity Ratio ◽  
Coverage Area ◽  
Elliptical Shape ◽  
Bearing Load ◽  
Rectangular Grids

This paper aims to improve the tribological performance of journal bearings by optimizing the coverage area of circular microtextures in divergent region of the sleeve. A numerical model is proposed to calculate the friction coefficient and bearing load of textured journal bearings. The surface of the sleeve is divided into rectangular squares. Textures that located at the center of rectangular grids are assumed to be present or absent, marked as 1 and 0, respectively. Afterward, different texture coverage area arrangements are evolved and selected based on the genetic algorithm (GA). The area of semi-elliptical shape is obtained as the novel and preferable textured coverage area design for journal bearings. Influences of width and eccentricity ratio are discussed, which confirm the semimajor and semiminor axes of the semi-elliptical shape of texture coverage area equal to one-third of the circumferential length and half of the width of the journal bearing, respectively.

scholarly journals Discrete Normal Vector Field Approximation via Time Scale Calculus

2020 ◽  
Vol 5 (1) ◽  
pp. 349-360
Author(s):  
Ömer Akgandüller ◽  
Sibel Paşalı Atmaca
Keyword(s):  
Time Scales ◽  
Vector Fields ◽  
Geometric Analysis ◽  
Field Approximation ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Time Scale Calculus ◽  
Vertex Set ◽  
Rectangular Grids ◽  
Mimetic Discretization

AbstractThe theory of time scales calculus have long been a subject to many researchers from different disciplines. Beside the unification and the extension aspects of the theory, it emerge as a powerful tool for mimetic discretization process. In this study, we present a framework to find normal vector fields of discrete point sets in ℝ3 by using symmetric differential on time scales. A surface parameterized by the tensor product of two time scales can be analogously expressed as the vertex set of non-regular rectangular grids. If the time scales are dense, then the discrete grid structure vanishes. If the time scales are isolated, then the further geometric analysis can be executed by using symmetric dynamic differential. Moreover, we present an algorithmic procedure to determine the symmetric dynamic differential structure on the neighborhood of points in surfaces. Our results indicate that the method we present has good approximation to unit normal vector fields of parameterized surfaces rather than the Delaunay triangulation for some points.

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