# Analysis suitable T-splines of arbitrary degree

## Prof. L. Beirao da Veiga

**June 20, 2013, 9 a.m. MT 130**

T-splines are nowadays recognized as a promising tool for IGA and have been the object of recent interest in the literature. In particular, analysis-suitable (AS) T-splines have emerged: introduced for the bi-cubic case, they are a sub-class of T-splines for which we have fundamental mathematical properties needed in a PDE solver, such as linear independence of the associated blending functions.

Only very recently in the literature T-splines have been extended to general polynomial degree. In the present talk we study bi-variate T-spline functions of arbitrary polynomial degrees p and q in the two coordinate directions. Depending on the odd/even nature of (p,q) the T-spline blending functions are naturally associated to vertexes, edges or elements in the (index) mesh.

After a brief review of IGA and defining the T-spline spaces, we generalize the concept of AS T-splines to arbitrary degree and introduce the definition of Dual Compatible (DC) T-splines. A space of DC T-splines enjoys a mathematical structure that, as shown later in the talk, guarantees a long list of fundamental properties. As one of the main results here presented we show that AS T-splines of any degree are also DC, and that the opposite also holds.

Therefore, since the two concepts above are equivalent, by using the DC structure in the final part of the talk we are able to show that AS T-splines satisfy (possibly under additional minimal assumptions) important properties such as linear indipendence, partition of unity, optimal approximation capabilities, and others.