An optimally convergent smooth blended B-spline construction for unstructured quadrilateral and hexahedral meshes
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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 on unstructured quadrilateral and hexahedral meshes. The new basis functions, referred to as SB-splines, are obtained by, first, defining a linearly independent set of mixed smoothness splines that are C^0 continuous in the unstructured regions of the mesh and have higher smoothness everywhere else. Next, globally smooth linearly independent splines are obtained by smoothly blending the set of mixed smoothness splines with Bernstein basis functions of equal degree. One of the key novelties of our approach is that the required smooth weight functions are obtained from the first set of mixed smoothness splines. The obtained SB-splines are smooth, non-negative, have no breakpoints within the elements and reduce to conventional B-splines away from the extraordinary features in the mesh. Although we consider only quadratic B-splines in this paper, the construction carries over to arbitrary degrees. We demonstrate the excellent performance and optimal convergence of the SB-splines studying Poisson and biharmonic problems in one, two and three dimensions.
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