On Local Refinement in Isogeometric Analysis: A comparative study on Classical Hierarchical, Truncated Hierarchical and LR B-splines

Abstract

Smooth spline functions such as B-splines and NURBS are already an established technology in the field of computer-aided design (CAD) and have in recent years been given a lot of attention from the computer-aided engineering (CAE) community. The advantages of local refinement are obvious for anyone working in either field, and as such, several approaches have been proposed. Three strategies are the Classical Hierarchical B-splines, the Truncated Hierarchical B-splines and the Locally Refined B-splines. We will in this paper present these three frameworks and highlight similarities and differences between them. In particular, we will look at the function space they span and the support of the basis functions. We will then analyse the corresponding stiffness and mass matrices in terms of sparsity patterns and conditioning numbers. We show that the basis in general do not span the same space, and that conditioning numbers are comparable. Moreover we show that the weighting needed by the Classical Hierarchical basis to maintain partition of unity has extreme implications on the conditioning numbers.