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. This talk will present new spline constructions that enable model reconstruction, as well as simulation of high-order PDEs on the reconstructed models.

The proposed almost-$C^1$ splines are finite piecewise-biquadratic splines on fully unstructured quadrilateral meshes (i.e., without restrictions on placements or number of extraordinary vertices). They can be used to build piecewise-biquadratic surfaces that are $C^1$ smooth at all vertices and across most edges, and approximately $C^1$ smooth across the remaining edges. Moreover, the proposed refinement scheme yields a $C^1$ smooth limit surface. The described basis for the spline space has no parametric singularities, has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), and can be implemented using Bézier-extraction. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behaviour.