The Argyris isogeometric space on unstructured multi-patch planar domains

Multi-patch spline parametrizations are used in geometric design and isogeometric analysis to represent complex domains. We deal with a particular class of $C^0$ planar multi-patch spline parametrizations called analysis-suitable $G^1$ (AS-$G^{1}$) 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, $C^1$ isogeometric spaces with optimal approximation properties. It was demonstrated in (Kapl, Sangalli, Takacs; CAD, 2018) that AS-$G^1$ 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 $C^1$ isogeometric spline space $\mathcal{W}$ over a given AS-$G^1$ multi-patch parametrization. We call the space $\mathcal{W}$ the Argyris isogeometric space, since it is $C^1$ across interfaces and $C^2$ at all vertices and generalizes the idea of Argyris finite elements to tensor-product splines. The considered space $\mathcal{W}$ is a subspace of the entire $C^1$ isogeometric space $\mathcal{V}^{1}$, 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 $\mathcal{V}^{1}$, the dimension of $\mathcal{W}$ does not depend on the domain parametrization, and $\mathcal{W}$ 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 $\mathcal{W}$ and demonstrate the applicability of our approach for isogeometric analysis.