  Easy to construct and optimally convergent generalisations of B-splines to
unstructured meshes are essential for the application of isogeometric analysis
to domains with non-trivial topologies. Nonetheless, especially for hexahedral
meshes, the construction of smooth and optimally convergent isogeometric
analysis basis functions is still an open question. We introduce a simple
partition of unity construction that yields smooth blended B-splines, referred
to as SB-splines, on semi-structured quadrilateral and hexahedral meshes,
namely on mostly structured meshes with a few sufficiently separated
unstructured regions. To this end, we first define the mixed smoothness
B-splines that are $C^0$ continuous in the unstructured regions of the mesh but
have higher smoothness everywhere else. Subsequently, the SB-splines are
obtained by smoothly blending in the physical space the mixed smoothness
B-splines with Bernstein bases of equal degree. One of the key novelties of our
approach is that the required smooth weight functions are assembled from the
available smooth B-splines on the unstructured mesh. The SB-splines are
globally smooth, non-negative, have no breakpoints within the elements and
reduce to conventional B-splines away from the unstructured regions of the
mesh. Although we consider only quadratic mixed smoothness B-splines in this
paper, the construction generalises to arbitrary degrees. We demonstrate the
excellent performance of SB-splines studying Poisson and biharmonic problems on
semi-structured quadrilateral and hexahedral meshes, and numerically
establishing their optimal convergence in one and two dimensions.
