Constructing Explicit B-Spline
This provides a direct construction method for smooth spline bases on tetrahedral meshes, which is important for applications in geometric modeling and finite element analysis.
The paper introduces an explicit algorithm to construct multivariate B-spline bases on tetrahedral meshes with guaranteed C^r continuity, enabling analytic enforcement of smoothness across adjacent elements.
We introduce here a direct method to construct multivariate explicit B-spline bases. B-splines are piecewise polynomials, which are defined on adjacent tetrahedra and which are $C^{r}$ continuous throughout. The $C^{r}$ continuity is enforced by making sure that all directional derivatives of order $r$, and lower, on the boundaries of adjacent tetrahedra give the same values for both tetrahedra. The method presented here is explicit, in that we will provide an algorithm with which one can analytically construct the B-spline base that enforces $C^{r}$ continuity for a given geometry.