Isogeometric analysis with $C^1$ cubic Powell-Sabin splines
This work addresses the need for more flexible and precise surface domain descriptions in computational engineering, though it appears incremental as it builds on existing spline and isogeometric analysis frameworks.
The paper tackles the numerical solution of boundary value problems on surfaces by using C^1 cubic Powell-Sabin splines in isogeometric analysis, demonstrating that they are a powerful alternative to existing methods like C^0 cubic Lagrange elements and bicubic NURBS.
In this paper, we consider $C^1$ cubic Powell-Sabin splines for the numerical solution of boundary value problems on planar and spatial surface domains. We first review the construction and basic properties of polynomial and rational $C^1$ cubic Powell-Sabin spline representations on unstructured triangulations. Then, we discuss how these flexible representations can be exploited to create geometry mappings suited for a precise description of (classes of) surface domains. This is illustrated with several examples. Finally, the obtained domain descriptions are utilized in the isogeometric analysis framework for solving various Poisson and biharmonic problems. It is demonstrated that $C^1$ cubic Powell-Sabin splines form a powerful alternative to $C^0$ cubic Lagrange elements and bicubic NURBS.