Derivatives of Isogeometric Test Functions

This paper is motivated by the representation of the test functions in Isogeometric Analysis (IgA). IgA is a numerical method that uses the NURBS-based representation of a CAD model to generate the finite-dimensional space of test functions which is used for the simulation. More precisely, the test functions are obtained by composing the inverse of the geometry mapping with the NURBS basis functions. We derive a representation of the derivatives of these test functions in NURBS form. More precisely, given a (possibly piecewise) rational geometry mapping and rational test functions defined on it, we present a method to compute the derivatives of the test functions with respect to the global coordinate system. The derivatives are again given as a rational function defined on the rational geometry mapping. All computations can be described compactly using homogeneous coordinates. We then use these results to derive conditions on the isogeometric test functions which guarantee C smoothness, in particular for the interesting case of singularly parametrized domains. The conditions depend heavily on the given geometry mapping. We present C, C and C smoothness results for a special class of singularly parametrized domains in R 2 and compare them with existing H and H regularity results. The framework can be applied to all types of singularities and to derivatives of higher order.