Section: New Results

Validity of rational and nonrational Lagrange finite elements of degree 1 and 2

Participants : Paul-Louis George [correspondant] , Houman Borouchaki.

A finite element is valid if its jacobian is strictly positive everywhere. The jacobian is the determinant of the jacobian matrix related to the partials of the mapping function which maps the parameter space (reference element) to the current element. Apart when it is constant, the jacobian is a polynomial whose degree is related to the degree of the finite element (but not the same in general). The value of the jacobian varies after the point where it is evaluated. Validating an element relies in finding the sign of this polynomial when one traverses the element.

Various papesr and a synthesis of those reports, shows how to calculating the jacobian of the different usual Lagrange finite elements of degree 1 and 2. To this end, we take the form of this polynomial as obtained in the classical finite element framework (shape functions and nodes) or after reformulating the element by means of a Bezier form (Bernstein polynomials and control points) which makes easier the discussion. We exhibit sufficient (necessary and sufficient in some cases) conditions to ensure the validity of a given element.