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Structured and Unstructured Grid Generation
Published in Theo C. Pilkington, Bruce Loftis, Joe F. Thompson, Savio L-Y. Woo, Thomas C. Palmer, Thomas F. Budinger, High-Performance Computing in Biomedical Research, 2020
Joe F. Thompson, Nigel P. Weatherill
It is possible to extend the use of an independent grid generation technique to include grid superposition and successive subdivision. The basic idea is to superimpose a regular grid over the domain. The regular grid can be generated using a quadtree or octree data structure which allows point density in the regular grid to be consistent with point spacing at the boundary. An example of this approach is shown in Figure 11. In general, this approach results in good spatial discretization in the interior of the domain, although in the vicinity of boundaries the grid quality can be poor.
Partitions
Published in Michael Hann, The Grammar of Pattern, 2019
This chapter discussed the nature of the unit cells associated with the three common regular grid types used on their own in regular all-over pattern construction: equal-size squares, equal-size equilateral triangles and equal-size regular hexagons. Various further partitions of the plane were identified and discussed briefly, including Penrose-type arrangements, created from only two tiles and arranged specifically to be aperiodic (or non-repeating).
Microstructural model of the behavior of a ferroalloy with shape memory in a magnetic field
Published in Mechanics of Advanced Materials and Structures, 2022
The numerical solution of Eqs. (3.3) and (3.6) was obtained using the FeniCS package (http://fenicsproject.org). The problem was solved by the finite element method on a regular grid having 5184 finite elements. The white square in Figure 1 was divided into 1296 equal squares, each of which was divided diagonally into four identical triangles. A quadratic approximation and type of finite elements P2 were set for the vector linear approximation and type of finite elements P1 were set for and λ (description for finite element types P1 and P2 can be found in https://www-users.cse.umn.edu/∼arnold/femtable). The periodicity conditions of the solution are imposed (see, for example, [11]). To implement these conditions, it is necessary to consider the domain already, in the middle of which the domain L × L (white square in Figure 1) is located.
Sparsifying spherical radial basis functions based regional gravity models
Published in Journal of Spatial Science, 2022
Haipeng Yu, Guobin Chang, Shubi Zhang, Nijia Qian
where zj represents the centre of the SRBF, x denotes the centre of the observation point. The positions and number of the point mass are on a regular grid which is empirically determined. The point mass can correspond to the same depth or different depths (compared to the Earth’s average radius R), which is the so-called multi-layer method when it corresponds to different depths. If the depths are D1 and D2, the radial distance of RBF are . When the number of RBF layers is m and the number of nodes in the grid is N, the number of parameters is N × m, and the corresponding observation equation is
Experimental and numerical analysis of flow hydraulics in triangular and rectangular piano key weirs
Published in Water Science, 2020
Rezvan Ghanbari, Mohammad Heidarnejad
FLOW-3D software is a suitable model with extensive application in the analysis of complicated fluid problems such as free-surface transient three-dimensional flow problems with complicated geometries. This software uses the finite volume method along with regular rectangular grids. Due to the use of the finite volume model in a regular grid, the form of the discrete equations used in the software is similar to those in the finite difference method. Usually, the two-equation models are used to simulate turbulence in hydraulic problems. In this study, the RNG model was used to obtain time-averaged Reynolds equations. Moreover, the FLOW-3D software was also used to solve the numerical problems and the transient governing equations were numerically solved using the finite volume model. In this software, the Fractional Area-Volume Obstacle Representation (FAVOR) algorithm was used to define the geometry in the finite volume method. This algorithm considers the in-field obstacles in the computational cells as a fractional value between 0 and 1, so that if the entire cell is filled by the obstacles, the fractional value of area-value would be equal to 1. Free surface of the flow is determined using the volume of fluid (VOF) algorithm. Velocity and pressure terms are implicitly coupled with the continuity and momentum equations using the pressure and velocity values at earlier times. In this software, the resulting quasi–implicit equations are solved iteratively using reduction techniques. In this article, the GMRES technique is used as the implicit solver of pressure (Maroosi, Roshan, & Sarkardeh, 2014).