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

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.

Figures (12)

• Figure 1: An excerpt from a triangulation $\triangle$ (depicted by solid lines) that is refined by a Powell--Sabin triangulation ${\triangle}_{\mathrm{PS}}$ (depicted by dotted lines). The gray colored triangles corresponding to the vertices of $\triangle$ represent a configuration for construction of cubic Powell--Sabin B-splines.
• Figure 2: Examples of cubic Powell--Sabin B-splines associated with vertices of the triangulation shown in Figure \ref{['fig:pstri']}.
• Figure 3: Examples of cubic Powell--Sabin B-splines associated with edges of the triangulation shown in Figure \ref{['fig:pstri']}.
• Figure 4: A cubic Powell--Sabin spline parametrization $F: \Theta \rightarrow \Omega$ described in Example \ref{['ex:puzzle']}.
• Figure 5: A cubic Powell--Sabin spline parametrization $F: \Theta \rightarrow \Omega$ described in Example \ref{['ex:annulus']}.

Theorems & Definitions (11)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5
• Example 6
• Example 7
• Example 8
• Example 9
• Example 10