A Bernstein-Bezier Sufficient Condition for Invertibility of Polynomial Mapping Functions

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. 1 Invertibility of polynomial mapping functions In finite element analysis, it is common to subdivide the domain into elements that are images of a reference domain under polynomial functions. This approach gives rise to the popular isoparametric elements [6]. The reference domain is the unit square I = {(ξ, η) : 0 ≤ ξ, η ≤ 1} or triangle ∆ = {(ξ, η) : 0 ≤ ξ, η; ξ + η ≤ 1} in R or the unit cube I = {(ξ, η, ζ) : 0 ≤ ξ, η, ζ ≤ 1} or tetrahedron ∆ = {(ξ, η, ζ) : 0 ≤ ξ, η, ζ ; ξ + η + ζ ≤ 1} in R. Polynomials defined on ∆, d = 2, 3, generally include monomial terms up to degree p for some p > 0. On the other hand, for the unit cube I, the monomial terms are generally up to degree p individually in each coordinate. Therefore, for the rest of the paper we use p to denote the maximum total ∗Supported in part by NSF ITR Award number 0085969.