The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeoemtric analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. In this paper we address the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (MS B-splines) and for Locally Refined B-splines (LR B-splines) on LR-meshes. We show that the minimal number is six for MS Bsplines and eight for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent.
We present a new refinement strategy for locally refined B-splines which ensures the local linear independence of the basis functions, the spanning of the full spline space on the underlying locally refined mesh and nice grading properties which grant the preservation of shape regularity and local quasi uniformity of the elements in the refining process.
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