Flattening 3D Triangulations for Quality Surface Mesh Generation

A method of flattening 3D triangulations for use in surface meshing is presented. The flattening method supports multiple boundary loops and directly produces planar locations for the vertices of the triangulation. The general nonlinear least-square fit condition for the triangle vertices includes conformal (angle preserving) and authalic (area preserving) conditions as special cases. The method of Langrange multipliers is used to eliminate rotational and translation degrees of freedom and enforce periodic boundary conditions. Using matrix partitioning, several alternative sets of constraints can be efficiently tested to find which produces the best domain. A surface boundary term is introduced to improve domain quality and break the symmetry of indeterminate multi-loop problems. The nonlinear problems are solved using a scaled conformal result as the initial input. The resulting 2D domains are used to generate 3D surface meshes. Results indicate that best mesh quality is achieved with domains generated using an intermediate altitude preserving condition. Apart from an admirable robustness and overall efficiency, the 2D developed domains are particularly suited for structured transfinite/mapped meshes which often reveal wiggly irregularities with most conventional developed domains. Flattening and meshing (both free and transfinite/mapped) results are presented for several 3D triangulations.