Chapter 3Geometric Elements and Geometric Validity
In this chapter, we consider isoparametric finite elements. The geometric validity of these elements is guaranteed if the determinant of the Jacobian matrix (for the matrix mapping the reference element on the physical element), the Jacobian, is strictly positive. As this Jacobian is a polynomial of a certain degree (related to, but in general different from the degree of the element), we may naively assume that it is enough that this polynomial is positive at the nodes, at the quadrature points used later or in an adequately large sample of points – but this is mathematically incorrect1.
By considering the Bézier form of these elements, we can show that their Jacobian polynomial is also in Bézier form and that the convex hull-like property may be used to exhibit conditions that are sufficient for positivity and, thus, geometric validity [Dey et al. 1999], [Johnen et al. 2014], [George, Borouchaki-2011], [George, Borouchaki-2014], [George, Borouchaki-2015a].
We will first look at complete two-dimensional as well as three-dimensional elements, and we give the expression for the control coefficients of the Jacobian polynomial for an arbitrary degree by making the case for degrees 1 and 2 thoroughly explicit. We also indicate how to validate a surface element (seen solely from a geometric point of view and thus, eventually only seen as a geometric patch). The conditions for validity are related to the respective positions of the ...