2D Quadrilateral Meshing
Automatic generation of 2D quadrilateral meshes is based on the subdivision of an arbrarily complex domain into simpler meshable subregions using the Medial Axis Transform or skeleton of the region (Tam & Armstrong, 1991). These simple subregions, which have at most 6 sides, are meshed with quad elements using a technique called midpoint subdivision. Compatibility between meshes in adjacent subrergions is enforced by Integer Programming (Tam & Armstrong, 1993).
3D Hexahedral Mesh Generation
In 3D, hexahedral mesh generation is accomplished using the Medial Surface of a solid to subdivide a complex body into a collection of simpler domains. Each face, edge and vertex of the Medial Surface corresponds to a meshable domain (Price, Armstrong & Sabin, 1994). A 3D midpoint subdivision algorithm is used in conjunction with Integer Programming to create a compatible mesh (Li, Armstrong & McKeag, 1994) Adaptive analysis based on this technique has been demonstrated (Armstrong et al, 1993).