In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost-C¹, Analysis-Suitable G¹ and the Approximate C¹ constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C¹ and Analysis-Suitable G¹ converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost-C¹ and D-Patch provide relatively easy construction on complex geometries. The Almost-C¹ method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C¹ and Analysis-Suitable G¹ applicable to more complex geometries.

Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-C¹ splines are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are C¹ smooth at all regular and extraordinary vertices. Moreover, they are C¹ smooth across all edges between regular vertices and C⁰ smooth across all edges that are adjacent to an extraordinary vertex. The splines thus form H²-nonconforming analysis-suitable discretization spaces. This is the lowest-degree unstructured spline construction that can be used to solve fourth-order problems. The associated spline basis is non-singular and has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), the almost-C¹ splines are described in an explicit Bézier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behaviour.

With the aim of a seamless integration with Computer-Aided Design, Isogeometric Analysis has been proposed by Hughes et al. (2005) as a numerical technique for the solution of partial differential equations. Indeed, isogeometric analysis is based on splines, the same functions that are adopted for geometry parametrizations in CAD. Smooth splines yield two important benefits when compared to C⁰ piecewise polynomial approximations: superior accuracy and stability, and the possibility to directly discretize high-order differential equations, such as the ones arising in thin-shell theory, in fracture models, in phase-field based multiphase flows, and in geometric flows on surfaces. In this chapter we review three different methods to construct C¹ isogeometric spaces on multi-patch domains or unstructured quadrilateral meshes. The first, proposed in Collin et al. (2016), is based on the concept of geometric continuity, well-known in geometric design. The second, from Toshniwal et al. (2017c), uses a specific singular construction (the D-patch construction) at extraordinary points. The third is a polar construction from Toshniwal et al. (2017a). Such constructions possess properties that make them suitable for performing both computational analysis and geometric modeling.