|
Since spheres, hemispheroids,
ellipsoids, and related geometries are very common in a great variety
of problems, some techniques have been developed to mesh this class of
geometries with structured grid systems. Just as a butterfly grid can
be used to grid a circular face, here we will demonstrate how this approach
is used to create a 3D butterfly grid for meshing a hemisphere.
Please recall the difference
between an H-grid, an O-grid, and a Butterfly-grid. An H-grid can be used
to topologically deform a single square grid to a circular shape. An O-grid
represents the circular shape by radial and circumfrential grid lines.
A butterfly grid system requires multiple blocks but generally has the
best grid quality in terms of orthogonality and mesh density.
|
|
|
|
|
H-Grid
|
O-Grid
|
Butterfly-Grid
|
Steps
1) The first step is to create
hemispherical surface by revolving an arc by 180 degrees as shown below.
2. Next a point is placed at
the hemisphere origin (0,0,0) which is extruded four times to get lines
which protrude past the hemisphere. This is done in the vector mode using
the following vectors: (1,1,1), (-1,1,1), (1,1,-1), (-1,1,-1).
3. The lines are then intersected
with the hemispherical surface to produce four points.
4. These intersection points
are used to make a square.
5. The construction lines are
split at the corners of the square and then the lower portions are removed
since they will not be needed anymore.
6. The arcs that make up the
base of the hemisphere need to be split at the corners of the square.
Use the "split curve at a point" tool to perform this operation.
7. Draw lines between the corners
of the square and the ends of the arcs.
8. Make the hemispherical surface
visible and project the lines created in step 7 and the square's lines
onto the hemisphere.
9. Showing just points, lines,
and curves we now have the following geometry.
10. At this point we could
make an H-type grid for the hemisphere, but better grid quality can be
obtained by adding elements to make a butterfly-type grid system. This
starts by createing an inner box and connect its corners to the arc ends.
11. Make the inner "cube"
region's edges, faces, and block by your favorite method (four sided faces
and six sided block, or create faces and blocks using the extrusion methods
for fewer mouse clicks).
12. Make the outer edges, faces,
blocks to fill the region between the cube and the hemisphere.
13. As can be seen in the image
above, the structured faces do not conform to the hemisphere surface due
to the nature of the trans-finite-interpolation scheme used to construct
the face grids. We can remedy that problem by projecting the structured
face grids to the hemisphere surface as shown below.
14. The sphere grid is completed.
If needed this grid system can be extended to include the "box"
around the hemisphere. Start by adding some construction lines.
15. Finish by making remaining
edges, faces, and blocks.
As you can see, it is not too
difficult to make these types of grid systems and of course the hemisphere
can be mirrored about the symmetry plane to obtain a full sphere model!
Click on hemisphere_box.GGD.zip
to download the CFD-GEOM file for this model. You will need CFD-GEOM V2003.0.2.2
or greater to read this file. The latest version of CFD-GEOM can always
be downloaded from the customer support
website.
Regards,
Matthew Slaby
Applications Engineer
CFDRC Customer Support
|
