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

• Published in: Computer methods in applied mechanics and engineering Vol. 327; pp. 411 - 458
• Main Authors: Toshniwal, Deepesh, Speleers, Hendrik, Hughes, Thomas J.R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.12.2017; Elsevier BV
• Subjects: CAD; Compatibility; Computer aided design; Design analysis; Extraordinary points; Geometric modeling; Geometry; Isogeometric analysis; Mathematical models; Numerical analysis; Smooth splines; Splines; Studies; Unstructured quadrilateral meshes; Isogeometric analysis; Smooth splines; Geometric modeling; Unstructured quadrilateral meshes; Extraordinary points
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2017.06.008

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. •We present a framework for building smooth splines on meshes with extraordinary points.•The spline spaces possess several desirable properties for both CAD and IGA.•Vertex-based, smooth, linearly independent splines are used for modeling geometries.•Compatible design and analysis spaces imply exact satisfaction of all patch tests.•Optimal or almost-optimal convergence rates are achieved in typical analysis situations.