Some Properties for Analysis-Suitable $T$-Splines (Journal of Computational Mathematics, 2018)
Analysis-suitable $T$-splines (AS $T$-splines) are a mildly topological restricted subsetof T-splines which are linear independent regardless of knot values [1–3]. The presentpaper provides some more iso-geometric analysis (IGA) oriented properties for AS $T$-splines and generalizes them to arbitrary topology AS $T$-splines. First, we prove thatthe blending functions for analysis-suitable T-splines are locally linear independent, whichis the key property for localized multi-resolution and linear independence for non-tensor-product domain. And then, we prove that the number of $T$-spline control points which contributeeach Bézier element is optimal, which is very important to obtain a bound for the numberof non zero entries in the mass and stiffness matrices for IGA with $T$-splines. Moreover,it is found that the elegant labeling tool for B-splines, blossom, can also be applied foranalysis-suitable $T$-splines.