Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes

TL;DR

This paper presents a comprehensive method for constructing smooth Bezier surfaces over arbitrary quadrilateral meshes, ensuring $G^{1}$ or $C^{1}$ continuity for polynomial patches of order at least 5, with practical applications in interpolation and PDE solutions.

Contribution

It provides a complete solution to the problem of $G^{1}$/$C^{1}$ continuity over unstructured meshes, including conditions for polynomial degree and mesh restrictions, enabling practical surface approximation and PDE solving.

Findings

Always a solution exists for degree n≥5.

Solution exists for degree n=4 under certain mesh restrictions.

Method applies to meshes with cubic boundary curves.

Abstract

We solve the following problem: given a polynomial of order $n$ and the corresponding $B\'ezier$ tensor product patches over an unstructured regular quadrilateral mesh of any valence, find a solution to the $G^{1}1$ or $C^{1}1$ approximation (resp. interpolation) problem ! Constraints defining regularity conditions across patches have to be satisfied. The resulting number of free degrees of freedom must be such that for instance the interpolation problem has a solution. This is similar to studying the minimal determining set (MDS) for a $C^{1}$ continuity construction. The givenunstructured quadrilateral mesh can include a cubic boundary curve. The final surface approximation or PDE solution is obtained by energy methods. We completely solve the problem and show that there is always a solution for $n\ge 5$ and under some mesh restrictions for $n=4$. From a practical point of view, the present paper provides a way to build first order smooth interpolation/approximation and solutions to partial differential equations for arbitrary structures of quadrilateral meshes.