Algorithms for determining invertible two‐ and three‐dimensional quadratic isoparametric finite element transformations

This paper presents an algorithm which determines the invertibility of any planar, triangular quadratic isoparametric finite element transformation. Extensions of the algorithm to three‐dimensional isoparametric finite element transformations yield conditions which guarantee invertibility of 10‐node tetrahedra and 8‐node bricks. The mathematical basis for the algorithm focuses on the Jacobian as a continuous function defined over a compact set where the Jacobian attains a maximum and a minimum value. The algorithm then determines whether these values are of opposite sign.