  We present an isogeometric analysis technique that builds on manifold-based
smooth basis functions for geometric modelling and analysis. Manifold-based
surface construction techniques are well known in geometric modelling and a
number of variants exist. Common to all is the concept of constructing a smooth
surface by blending together overlapping patches (or, charts), as in
differential geometry description of manifolds. Each patch on the surface has a
corresponding planar patch with a smooth one-to-one mapping onto the surface.
In our implementation, manifold techniques are combined with conformal
parametrisations and the partition-of-unity method for deriving smooth basis
functions on unstructured quadrilateral meshes. Each vertex and its adjacent
elements on the surface control mesh have a corresponding planar patch of
elements. The star-shaped planar patch with congruent wedge-shaped elements is
smoothly parameterised with copies of a conformally mapped unit square. The
conformal maps can be easily inverted in order to compute the transition
functions between the different planar patches that have an overlap on the
surface. On the collection of star-shaped planar patches the partition of unity
method is used for approximation. The smooth partition of unity, or blending
functions, are assembled from tensor-product b-spline segments defined on a
unit square. On each patch a polynomial with a prescribed degree is used as a
local approximant. To obtain a mesh-based approximation scheme, the
coefficients of the local approximants are expressed in dependence of vertex
coefficients. This yields a basis function for each vertex of the mesh which is
smooth and non-zero over a vertex and its adjacent elements. Our numerical
simulations indicate the optimal convergence of the resulting approximation
scheme for Poisson problems and near optimal convergence for thin-plate and
thin-shell problems.
