Implementation of Worsey-Farin splines with applications to solution transfer

TL;DR

In the study of conservation, it is demonstrated how adaptive numerical quadrature rules on the tetrahedron used in conjunction with global L2-projection can improve the conservation of the solution transfer process.

Abstract

This work primarily focuses on providing full implementation details for Worsey-Farin (WF) spline interpolation over tetrahedral elements. While this spline space is not new and the theory has been covered in other works, there is a lack of explicit and comprehensive implementation details, which we hope to provide. In this paper, we also demonstrate the effectiveness of the WF-spline space through a simple target application: solution transfer. Moreover, we derive an error estimate for the WF spline-based, solution transfer process. We conduct numerical experiments quantifying the conservative nature and order of accuracy of the transfer process, and we present a qualitative evaluation of the visualization properties of the smoothed solution. Additionally, in our study of conservation, we demonstrate how adaptive numerical quadrature rules on the tetrahedron used in conjunction with global L2-projection can improve the conservation of the solution transfer process.

Figures (18)

• Figure 1: The domain points for a cubic polynomial on a tetrahedron generated using equation \ref{['eqn:domain_points']}.
• Figure 2: A visual representation of our proposed 3-step transfer process.
• Figure 3: The Worsey-Farin split process shown for a single face neighbor. Top left shows the identification of the face neighbor; center shows an exploded view of the calculation of the incenter of each tetrahedron; bottom right shows the face split point (red ellipse) produced by intersection of the line between incenters and the shared face.
• Figure 4: The macro-tetrahedron shown in the center with internal edges illustrated by dotted lines. Subtetrahedra are produced by connecting the vertices of the macro-tetrahedron to the incenter (shown by green lines). Subsequent subtetrahedra are produced by connecting the vertices of the subtetrahedra to the facial split points (shown by red lines). The incenter and facial split points were found from the process shown in Figure \ref{['fig:wf_split']}. The numbering of nodes in this figure is simply the degree-of-freedom numbering for the WF spline. This numbering is explained in the subsequent text.
• Figure 5: Left, the $L_2$-error of the Gaussian function $u_1$, and right, the $L_2$-error of the gradient magnitude of $u_1$, with linear interpolation for $\mathcal{P}_1$ and $\mathcal{P}_2$ data shown in shades of red and WF transfer for $\mathcal{P}_1$ and $\mathcal{P}_2$ data shown in shades of green.

Theorems & Definitions (17)

• Definition 2.1: Bernstein Basis Polynomials
• Definition 2.2: Bernstein-Bézier Form
• Definition 2.3: The de Casteljau Algorithm
• Definition 3.3: $H^2$ Smoothing Operator
• Lemma 3.4
• proof
• Lemma 3.5: Bound on $H^2$ Smoothing Operator
• proof
• Remark 3.6
• Definition 3.7: Projection Operator