An optimally convergent smooth blended B-spline construction for semi-structured quadrilateral and hexahedral meshes

TL;DR

This paper introduces SB-splines, a smooth blended B-spline construction for semi-structured quadrilateral and hexahedral meshes, enabling optimal convergence and smoothness in isogeometric analysis on complex domains.

Contribution

The paper presents a novel simple partition of unity method to construct globally smooth SB-splines on semi-structured meshes, extending B-spline applicability to complex topologies.

Findings

SB-splines are globally smooth and non-negative.

They achieve optimal convergence in Poisson and biharmonic problems.

The method generalizes to arbitrary degrees.

Abstract

Easy to construct and optimally convergent generalisations of B-splines to unstructured meshes are essential for the application of isogeometric analysis to domains with non-trivial topologies. Nonetheless, especially for hexahedral meshes, the construction of smooth and optimally convergent isogeometric analysis basis functions is still an open question. We introduce a simple partition of unity construction that yields smooth blended B-splines, referred to as SB-splines, on semi-structured quadrilateral and hexahedral meshes, namely on mostly structured meshes with a few sufficiently separated unstructured regions. To this end, we first define the mixed smoothness B-splines that are $C^0$ continuous in the unstructured regions of the mesh but have higher smoothness everywhere else. Subsequently, the SB-splines are obtained by smoothly blending in the physical space the mixed smoothness B-splines with Bernstein bases of equal degree. One of the key novelties of our approach is that the required smooth weight functions are assembled from the available smooth B-splines on the unstructured mesh. The SB-splines are globally smooth, non-negative, have no breakpoints within the elements and reduce to conventional B-splines away from the unstructured regions of the mesh. Although we consider only quadratic mixed smoothness B-splines in this paper, the construction generalises to arbitrary degrees. We demonstrate the excellent performance of SB-splines studying Poisson and biharmonic problems on semi-structured quadrilateral and hexahedral meshes, and numerically establishing their optimal convergence in one and two dimensions.