Isogeometric analysis with $C^1$ cubic Powell-Sabin splines

TL;DR

This paper explores the use of $C^1$ cubic Powell-Sabin splines in isogeometric analysis, demonstrating their effectiveness for solving boundary value problems on complex surface domains, offering a flexible alternative to traditional methods.

Contribution

It introduces the construction and application of $C^1$ cubic Powell-Sabin splines for isogeometric analysis on surface domains, highlighting their advantages over existing spline-based methods.

Findings

Effective for solving Poisson and biharmonic problems

Provides a flexible surface domain representation

Outperforms $C^0$ cubic Lagrange and bicubic NURBS

Abstract

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.