Linear independent T-splines and their applications in two and three spatial dimensions
This presentation focuses on T-splines which arise as a method of decreasing the number of basis functions in computer-aided geometry design while maintaining the original design. Through the concept of isogeometric analysis it is possible to apply them as ansatz functions for the solution of finite element methods. In general, isogeometric analysis allows error reduction on curved boundaries through an exact representation and the use of high order basis functions with ease. T-splines then allow for local refinement. However, linear independence is not guaranteed and has to be checked. Upon their introduction this was done geometrically in 2D by extending hanging edges (T-junctions), and was later extended to arbitrary dimensions through an analytical criterion but only for odd polynomial degrees.
This work fills the gaps by extending the geometric definition to arbitrary dimensions and the analytical definition to arbitrary degrees. The definitions are unified by giving proper inclusions of the resulting mesh-classes. Further, we are interested in a user-friendly application of T-splines to partial differential equations to make the concept of isogeometric analysis with local refinement through T-splines available to the broad user-field of standard finite element methods. Based on the geometric definitions we have derived data structures and given in-depth explanations thereof. Lastly, we give some model problems with numerical solutions obtained through T-splines. A detailed comparison to standard methods is made as well.
