Development of C1 smooth isogeometric functions for planar multi-patch domains for NURBS based analysis

This paper proposes a novel framework for constructing $C^1$ smooth isogeometric functions on the planar multipatch domain. We extend the concept of $C^1$ coupling, wherein the null space approach was used to construct geometrically continuous basis functions as linear combinations of $C^0$ basis functions near patch junctions. However, due to the lack of continuity constraints, the resulting approximate basis functions violated the partition of unity and non-negativity properties. The proposed framework enforces the partition of unity and non-negativity conditions through additional equations, preserving higher-order continuity across the interface. The patch coupling algorithm provided generates $C^1$ smooth isogeometric functions for arbitrarily shaped planar multipatch geometries. The advantage of this proposed approach is a reduced degree of approximation and a smooth transition from 1D to 2D patch coupling methodology. The computational effort to determine a new set of basis functions is significantly reduced due to the partition of unity property. Numerical studies are performed for the Kirchhoff–Love plate and biharmonic equations on various curved multipatch geometries, including an additional patch test. Enhanced numerical accuracy is observed for geometries with curved interfaces and boundaries. The accuracy and numerical efficiency of the proposed framework are analysed through $L^2$ and $H_0^2$ errors, showing optimal convergence behaviour for different polynomial orders. Furthermore, well-conditioned global matrices are observed with increasing refinement levels, demonstrating the efficiency of the methodology.