Multi-patch spline parametrizations are used in geometric design and
isogeometric analysis to represent complex domains. We deal with a particular
class of C0 planar multi-patch spline parametrizations called
analysis-suitable G1 (AS-G1) multi-patch parametrizations (Collin,
Sangalli, Takacs; CAGD, 2016). This class of parametrizations has to satisfy
specific geometric continuity constraints, and is of importance since it allows
to construct, on the multi-patch domain, C1 isogeometric spaces with optimal
approximation properties. It was demonstrated in (Kapl, Sangalli, Takacs; CAD,
2018) that AS-G1 multi-patch parametrizations are suitable for modeling
complex planar multi-patch domains.
In this work, we construct a basis, and an associated dual basis, for a
specific C1 isogeometric spline space W over a given AS-G1
multi-patch parametrization. We call the space W the Argyris
isogeometric space, since it is C1 across interfaces and C2 at all
vertices and generalizes the idea of Argyris finite elements to tensor-product
splines. The considered space W is a subspace of the entire C1
isogeometric space V1, which maintains the reproduction
properties of traces and normal derivatives along the interfaces. Moreover, it
reproduces all derivatives up to second order at the vertices. In contrast to
V1, the dimension of W does not depend on the domain
parametrization, and W admits a basis and dual basis which possess
a simple explicit representation and local support.
We conclude the paper with some numerical experiments, which exhibit the
optimal approximation order of the Argyris isogeometric space W and
demonstrate the applicability of our approach for isogeometric analysis