### Gridding a Hemisphere in a Box with a Structured Butterfly Grid

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!

Regards,
Matthew Slaby
Applications Engineer
CFDRC Customer Support