Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations

We present a framework for geometric design and isogeometric analysis on unstructured quadrilateral meshes. Acknowledging the differing requirements posed by design (e.g., the convenience of an intuitive control net) and analysis (e.g., good approximation behavior), we propose the construction of a separate, smooth spline space for each while ensuring isogeometric compatibility - requiring the geometric models to be members of the analysis-suitable spaces. The methodology is simple and is presented for bi-cubic splines; extensions to higher degrees are possible, and are briefly discussed. The presentation has been structured to show compatibility with T-splines - a state-of-the-art CAD technology - but the approach should extend to other locally refinable spline technologies (based on local tensor-product structures). An instantiation of the framework is presented, and several numerical tests focused on geometric design and isogeometric analysis demonstrate the versatility of the developed framework, and show significantly higher convergence rates than attained previously in the considered setting.