Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

This paper addresses the problem of constructing a structured quadrilateral grid inside a given four-sided 2D region by a particular boundary-conforming mapping scheme-Hermite Interpolation Mapping (HIM). When the four given boundary curves are concave and convoluted, all boundary-conform mapping methods suffer from potential self-overlapping problem. Under HIM, the geometry of the grid depends on both the four boundary curves and the tangent vector functions associated with the curves. While the four boundary curves are fixed, the tangent functions in HIM can be varied to suit the need of controlling the characteristics of the mesh inside the given region to prevent self-overlapping. Besides tangent functions, the four twist vectors at the corners of the region can also be adjusted to influence the distribution of the inner grid elements. In our approach, a constrained functional optimization scheme is adopted to adjust the tangent functions and the twist vectors, adaptive to the geometry of the boundary curves, so that the resulting HIM will be free of self-overlapping. The optimization is carried out on the shape control energy that measures the overall mesh quality of the underlying HIM while the self-overlapping is strongly prevented in the form of constraints to the optimization. Experimental results show the promise of the proposed method as a practical and effective solution for structured grid generation.