Refinable tri-variate C1 splines for box-complexes including irregular points and irregular edges

Abstract

C1 splines over box-complexes generalize C1 degree 3 (cubic) tensor-product splines. A box-complex is a collection of 3-dimensional boxes forming an unstructured hexahedral mesh that can include irregular points and irregular edges where the layout deviates from the tensor-product grid layout. For example, an edge shared and enclosed by five boxes is irregular. Where the mesh is locally regular, the restriction of the space to each box is a polynomial piece of the C1 tri-cubic tensor-product spline, by default initialized as a C2 tri-cubic. Boxes containing irregularities have their polynomials binarily split into 23 pieces to isolate the irregularity. The pieces join with matching derivatives. The derivatives are zero at irregularities, but these singularities are removable by a local change of variables. The space consists of 23 linearly independent functions per box and is refinable.