Geometrical validity of high-order triangular finite elements

This paper presents a method to compute accurate bounds on Jacobian determinants of high-order (curvilinear) triangular finite elements. This method can be used to guarantee that a curvilinear triangle is geometrically valid, i.e., its Jacobian determinant is strictly positive everywhere in its reference domain. It also provides an efficient way to measure the quality of triangles. The key feature of the method is to expand the Jacobian determinant using a polynomial basis, built using B,zier functions, that has both properties of boundedness and positivity. Numerical results show the sharpness of our estimates.