Derivatives of isogeometric functions on n-dimensional rational patches in [formula omitted

Abstract

We consider isogeometric functions and their derivatives. Given a geometry mapping, which is defined by an n-dimensional NURBS patch in Rd, an isogeometric function is obtained by composing the inverse of the geometry mapping with a NURBS function in the parameter domain. Hence an isogeometric function can be represented by a NURBS parametrization of its graph. We take advantage of the projective representation of the NURBS patch as a B-spline patch in homogeneous coordinates.We derive a closed form representation of the graph of a partial derivative of an isogeometric function. The derivative can be interpreted as an isogeometric function of higher degree and lower smoothness on the same piecewise rational geometry mapping, hence the space of isogeometric functions is closed under differentiation. We distinguish the two cases n=d and n<d, with a focus on n=d−1 in the latter one.As a first application of the presented formula we derive conditions which guarantee C1 and C2 smoothness for isogeometric functions on several singularly parametrized planar and volumetric domains as well as on embedded surfaces. It is interesting to note that the presented conditions depend not only on the general structure of the patch, but on the exact representation of the interior of the given geometry mapping.