Grid Generation

Preamble

This page is meant for the discretization of geometrically simple domains, specifically rectangles and radial annulii. For more complicated domains you will need to use a more sophisticated grid generator like CUBIT or other such programs. CUBIT can be licensed free (at least to US institutions) for educational purposes. In case you do use CUBIT, please refer to Kate Hedstrom's manual

Grid Generation and manipulation

There are a number of program to generate simple spectral element grids. These programs are written in fortran, the executable is created by saying "make programname". The programs are the following:

rectelem: produces a spectral element grid for rectangular domains. It can also be used to produce a cubed-sphere mesh.
radelem: produces a spectral element grid for annular type domains
ablockmesh2: uses algebraic mapping to generate curved spectral element grids. The element partitioning is still structured.
periodize: takes a rectangular spectral element produced by "rectelem" and eliminate duplicated nodes.
stretch: takes a rectangular spectral element and stretches by a constant factor in the horizontal and a linearly changing factor in the vertical. This is useful to create linearly sloping bottoms.

Creating the executable

Here is the gzipped and tarred file for simple-domain grid generation. Follow the directions below to generate a grid:

Uncompress and untar the file. This will create a directory called Rect.
Edit the Makefile to set the right compiler and its flags. The code is written in F90.
Build the executables
```
 make rectelem
 make radelem
```
Run the code, for example "rectelem < rectelem.in". The grid is written to a file called fort.8; its format is explained in the grid section. By default, the boundary conditions listed are of type Dirichlet all around the domain boundaries. The format of the rectelem.in file is described below.

rectelem usage

"rectelem" accepts a few options; example of usage are

rectelem -h produces a short description of options
rectelem < rectelem.in to produce a SE in rectangular domains
rectelem -cs < rectelem.in to produce the 6 faces of a cubed sphere SE grid using the default equidistant mapping
rectelem -cs -ea < rectelem.in to produce the 6 faces of a cubed sphere SE grid using the equiangle mapping
rectelem -tr < rectelem.in to produce the 6 faces of a cubed sphere SE grid using the equiangle mapping

Format of rectelem.in

The file should have the following information:

nx ny npts	        ! line1
xc(1) xc(2) ... xc(nx)	! nx x-coordinates of element corners
yc(1) yc(2) ... yc(ny)	! ny y-coordinates of element corners

The example above will create a grid with (nx-1) elements in the x-direction, (ny-1) elements in the y-direction, and each element will have npts*npts collocation points at the Gauss-Lobatto roots of the Legendre polynomial of degree (npts-1). If a staggered pressure grid file needs to be generated the code can be run again with npts replaced by npts-2.

The format of the radelem.in file is the same except that the x-direction refers now to the radial direction, and the y-direction to the azimuthal direction. The coordinates in the azimuthal direction are in degrees. Do not duplicate the first and last node in the azimuthal direction since the domain is assumed periodic azimuthally. Notice that the elements are curved in this case, and care must taken to make sure the shape of the element can be well represented by the isoparametric mapping, i.e. the spectral expansion is high enough to represent the distortion (in general the mapping is tolerant of some pretty distorted elements).

The picture shows the elemental grid created with radelem.in. The element number is shown inside each element and the interior collocation points are omitted. This figure was created with a drawcoast script .

The cubed sphere

The cubed sphere can be tiled using a combination of programs available here. First one needs to generate a single face of the cubed sphere. This single face is then projected on the 6 faces of the cubed sphere to produce their lon-lat coordinates (the output is stored in fort.8_n for face n). The program merge_grid is then used to merge these 6 lon-lat spectral element grids into a single one (the output is in fort.8_all). Since the process is fairly simple but requires the user to pipe the output from one program to the input of another.

Luckily the process can be automated through the bash script cubedsh. This script takes two input arguments, the number of elements in the x-direction and the spectral truncation and produces at the end a lon-lat spectral element grid in file fort.8_all. It requires the programs "rectelem" and "merge_grid". The grid can be visualized through the NCAR graphics program "drawcoast" or through the matlab script segdrawm.m.