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

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Koh, Kim Jie 
Toshniwal, Deepesh 
Cirak, Fehmi 

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 C0 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 bihar- monic problems on semi-structured quadrilateral and hexahedral meshes, and numerically establishing their optimal convergence in one and two dimensions.

Description

Keywords

cs.NA, math.NA, math.NA

Journal Title

Computer Methods in Applied Mechanics and Engineering

Conference Name

Journal ISSN

0045-7825
1879-2138

Volume Title

399

Publisher

Elsevier