  We introduce manifold-based basis functions for isogeometric analysis of
surfaces with arbitrary smoothness, prescribed $C^0$ continuous creases and
boundaries. The utility of the manifold-based surface construction techniques
in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017).
The respective basis functions are derived by combining differential-geometric
manifold techniques with conformal parametrisations and the partition of unity
method. The connectivity of a given unstructured quadrilateral control mesh in
$\mathbb R^3$ is used to define a set of overlapping charts. Each vertex with
its attached elements is assigned a corresponding conformally parametrised
planar chart domain in $\mathbb R^2$, so that a quadrilateral element is
present on four different charts. On the collection of unconnected chart
domains, the partition of unity method is used for approximation. The
transition functions required for navigating between the chart domains are
composed out of conformal maps. The necessary smooth partition of unity, or
blending, functions for the charts are assembled from tensor-product B-spline
pieces and require in contrast to earlier constructions no normalisation.
Creases are introduced across user tagged edges of the control mesh. Planar
chart domains that include creased edges or are adjacent to the domain boundary
require special local polynomial approximants. Three different types of chart
domain geometries are necessary to consider boundaries and arbitrary number and
arrangement of creases. The new chart domain geometries are chosen so that it
becomes trivial to establish local polynomial approximants that are always
$C^0$ continuous across tagged edges. The derived non-rational manifold-based
basis functions are particularly well suited for isogeometric analysis of
Kirchhoff-Love thin shells with kinks.
