scholarly journals Hexahedron Projection for Curvilinear Grids

2007 ◽  
Vol 12 (2) ◽  
pp. 33-45 ◽  
Author(s):  
Nelson Max
Keyword(s):  
Curvilinear Grids

On the construction of orthogonal curvilinear grids for regional oceanic modeling: algorithm and user guide

2020 ◽  
Vol 48 (4) ◽  
pp. 45-111
Author(s):  
A. F. Shepetkin
Keyword(s):  
Inverse Problem ◽  
Numerical Solution ◽  
Elliptic Problem ◽  
Software Package ◽  
Conformal Transformation ◽  
Iterative Procedure ◽  
Geometric Shape ◽  
Curvilinear Grids ◽  
Grid Points ◽  
Grid Nodes

A new algorithm for constructing orthogonal curvilinear grids on a sphere for a fairly general geometric shape of the modeling region is implemented as a “compile-once - use forever” software package. It is based on the numerical solution of the inverse problem by an iterative procedure -- finding such distribution of grid points along its perimeter, so that the conformal transformation of the perimeter into a rectangle turns this distribution into uniform one. The iterative procedure itself turns out to be multilevel - i.e. an iterative loop built around another, internal iterative procedure. Thereafter, knowing this distribution, the grid nodes inside the region are obtained solving an elliptic problem. It is shown that it was possible to obtain the exact orthogonality of the perimeter at the corners of the grid, to achieve very small, previously unattainable level of orthogonality errors, as well as make it isotropic -- local distances between grid nodes about both directions are equal to each other.

IZOGRID - A new tool for setting up orthogonal curvilinear grids for oceanic modeling

2021 ◽  
Author(s):  
Alexander Shchepetkin
Keyword(s):  
Iterative Procedure ◽  
Cartesian Grid ◽  
Reference Points ◽  
Curvilinear Coordinates ◽  
Grid Spacing ◽  
Curvilinear Grids ◽  
Curvilinear Grid ◽  
Curvilinear Contour ◽  
Grid Points ◽  
Grid Nodes

<p>Virtually all modern structured-grid ocean modeling codes are written in orthogonal curvilinear coordinates in horizontal directions, yet the overwhelming majority of modeling studies are done using very simple grid setups - mostly rectangular patches of Mercator grids rotated to proper orientation.  Furthermore, in communities like ROMS, we even observe decline in both interest and skill of creating curvilinear grids over long term.  This is caused primarily by the dissatisfaction with the existing tools and procedures for grid generation due to inability to achieve acceptable level of orthogonality errors.  Clearly, this causes underutilization of full potential of the modeling codes.</p><p>To address these issues, a new algorithm for constructing orthogonal curvilinear grids on a sphere for a fairly general geometric shape of the modeling region is implemented as a compile-once - use forever software package.  Theoretically one can use Schwartz-Christoffel conformal transform to project a curvilinear contour onto rectangle, then draw a Cartesian grid on it, and, finally, apply the inverse transform (the one which maps the rectangle back to the original contour) to the Cartesian grid in order to obtain the orthogonal curvilinear grid which fits the contour.  However, in the general case, the forward transform is an iterative algorithm of Ives and Zacharias (1989), and it is not easily invertible, nor it is feasible to apply it to a two-dimensional object (grid) as opposite to just one-dimensional (contour) because of very large number of operations.  To circumvent this, the core of the new algorithm is essentially based on the numerical solution of the inverse problem by an iterative procedure - finding such distribution of grid points along the sides of curvilinear contour, that the direct conformal mapping of it onto rectangle turns this distribution into uniform one along each side of the rectangle.  Along its way, this procedure also finds the correct aspect ratio, which makes it possible to automatically chose the numbers of grid points in each direction to yield locally the same grid spacing in both horizontal directions.  The iterative procedure itself turns out to be multilevel - i.e. an iterative loop built around another, internal iterative procedure.  Thereafter, knowing this distribution, the grid nodes inside the region are obtained solving a Dirichlet elliptic problem.  The latter is fairly standard, except that we use "mehrstellenverfahren" discretization, which yields fourth-order accuracy in the case of equal grid spacing in both directions.  The curvilinear contour is generated using splines (cubic or quintic) passing through the user-specified reference points, and, unlike all previous tools designed for the same purpose, it guarantees by the construction to yield the exact 90-degree angles at the corners of the curvilinear perimeter of grid.</p><p>Overall, with the combination of all the new features, it is shown that it is possible to achieve very small, previously unattainable level of orthogonality errors, as well as make it isotropic -- local distances between grid nodes in both directions are equal to each other.</p>

An Automaton Model on Curvilinear Grids for the Simulation of Pedestrian Flow

2009 ◽  
Author(s):  
Hartmut Schwandt ◽  
Minjie Chen ◽  
Günter Bärwolff ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
Keyword(s):  
Pedestrian Flow ◽  
Curvilinear Grids

scholarly journals The Parcels v2.0 Lagrangian framework: new field interpolation schemes

2019 ◽  
Author(s):  
Philippe Delandmeter ◽  
Erik van Sebille
Keyword(s):  
Three Dimensional ◽  
Analysis Framework ◽  
Interpolation Scheme ◽  
Physical Processes ◽  
North West ◽  
The North ◽  
Lagrangian Framework ◽  
Virtual Particles ◽  
Curvilinear Grids ◽  
Computing Infrastructure

Abstract. With the increasing amount of data produced by numerical ocean models, so increases the need for efficient tools to analyse these data. One of these tools is Lagrangian ocean analysis, where a set of virtual particles are released and their dynamics is integrated in time based on fields defining the ocean state, including the hydrodynamics and biogeochemistry if available. This popular methodology needs to adapt to the large variety of models producing these fields at different formats. This is precisely the aim of Parcels, a Lagrangian ocean analysis framework designed to combine (1) a wide flexibility to model particles of different natures and (2) an efficient implementation in accordance with modern computing infrastructure. In the new Parcels v2.0, we implement a set of interpolation schemes to read various types of discretised fields, from rectilinear to curvilinear grids in the horizontal direction, from z- to s- levels in the vertical and different variable distributions such as the Arakawa's A-, B- and C- grids. In particular, we develop a new interpolation scheme for a three-dimensional curvilinear C-grid and analyse its properties. Parcels v2.0 capabilities, including a suite of meta-field objects, are then illustrated in a brief study of the distribution of floating microplastic in the North West European continental shelf and its sensitivity to different physical processes.

A FULLY IMPLICIT SOLVER OF 3-D EULER EQUATIONS ON MULTIBLOCK CURVILINEAR GRIDS

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1483-1486 ◽  
Author(s):  
HAI-QING SI ◽  
TONG-GUANG WANG ◽  
XIAO-YUN LUO
Keyword(s):  
Euler Equations ◽  
Three Dimensional ◽  
Implicit Time ◽  
Tvd Scheme ◽  
Time Step ◽  
Vector Operation ◽  
Fully Implicit ◽  
Curvilinear Grids ◽  
The Matrix ◽  
Implicit Solver

A fully implicit unfactored algorithm for three-dimensional Euler equations is developed and tested on multi-block curvilinear meshes. The convective terms are discretized using an upwind TVD scheme. The large sparse linear system generated at each implicit time step is solved by GMRES* method combined with the block incomplete lower-upper preconditioner. In order to reduce the memory requirements and the matrix-vector operation counts, an approximate method is used to derive the Jacobian matrix, which only costs half of the computational efforts of the exact Jacobian calculation. The comparison between the numerical results and the experimental data shows good agreement, which demonstrates that the implicit algorithm presented is effective and efficient.

A Numerical Shallow Water Model Based on the Non-Orthogonal Curvilinear Grids

2006 ◽  
Author(s):  
Chaofeng Tong ◽  
Yanqiu Meng
Keyword(s):  
Shallow Water ◽  
Curvilinear Coordinates ◽  
Water Model ◽  
Staggered Grid ◽  
Shallow Water Model ◽  
Governing Equations ◽  
Cartesian Coordinates ◽  
Independent Variables ◽  
Discrete Equations ◽  
Curvilinear Grids

According to the transformation relationships between the Cartesian coordinates and the general curvilinear coordinates, the governing equations of the model are derived as the forms in the general curvilinear coordinates from those in the Cartesian coordinates. In the model, the contravariant velocities are adopted as the independent variables in non-orthogonal grids. The momentum equations keep strongly conservative forms and the boundary conditions can be given easily. The model used a staggered grid arrangement. The discrete equations are solved using the SIMPLIC algorithms. The numerical model has been validated against the bifurcated flow of which the diversion angle is 30 degree. Compared with the measured values, the numerical shallow water model is shown to be capable of simulating the water domains with irregular boundaries.

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