A Bernstein-Bezier Sufficient Condition for Invertibility of Polynomial Mapping Functions
Stephen Vavasis
TL;DR
This work addresses the challenge of globally invertible polynomial mappings $F$ on reference domains $I^d$ or $\Delta^d$, which is crucial for curved isoparametric finite element meshes. It introduces a Bernstein-Bézier (BB) based sufficient condition that uses the convex hulls of derivative control points $G_\xi$, $G_\eta$, and $G_\zeta$ to guarantee the Jacobian $J$ is invertible on the entire domain, thereby ensuring global invertibility. A computable characterization is provided: in 2D, testing reduces to linear programs or angular separation of derivative directions, and in 3D to four halfspace tests with linear-time or $O(n\log n)$ convex-hull methods. The approach supports an affinity property and opens paths toward Ciarlet-Raviart-type bounds, while remaining general across $d=2,3$ and any polynomial degree $p$; it also offers practical guidance for enhanced meshing with curved geometries.
Abstract
We propose a sufficient condition for invertibility of a polynomial mapping function defined on a cube or simplex. This condition is applicable to finite element analysis using curved meshes. The sufficient condition is based on an analysis of the Bernstein-Bézier form of the columns of the derivative.