Differences and similarities of different approaches to locally refined splines over hierarchical meshes

Hierarchical meshes are attractive in Isogeometric Analysis as additional degrees of freedom can be inserted locally in a uniform way in regions where needed. Three are three major approaches to locally refined splines over hierarchical meshes: Truncated Hierarchical B-splines (THB), T-spline (TS) and Locally Refined B-splines (LRB). For all three approaches the structures of polynomial elements produced are similar. In most cases the spline space of THB or TS can be exactly reproduced by LRB, in some case the spline spaces produced LRB over a given THB or TS element mesh are slightly bigger than those of THB and TS, In this presentation we address the differences and similarities of the three different approaches, as well as look at how the extra flexibility offered by LRB can be used for reducing the number of B-splines covering and element. We will also show the positive effect this has on condition numbers of mass and stiffness matrices.