# Generalized T-splines and T-meshes guaranteeing their linear independence

#### Séminaire Modèles et Algorithmes Déterministes: EDP-MOISE-MGMI

22/01/2015 - 11:00
Mr Durkbin Cho *(Dongguk University, Republic of Korea)*

The T-spline functions, first introduced in [6] and in [7], are nowadays a relevant tool in isogeometric analysis (see, e.g., [1, 2]). The use of Tsplines gives some considerable improvements on the classical tensor-product splines and NURBS, such as the possibility to apply local refinements, a heavy reduction of the number of control points needed, and the ability to easily avoid gaps when joining several surfaces. The T-spline approach is mainly applied to polynomial splines, but in some situation the use of non-polynomial splines gives noteworthy advantages: for example, exactly reproducing some relevant shapes (such as cycloids and helices) which can only be approximated by using polynomial splines requires the use of non-polynomial splines (see, e.g., [5]). For this reason, we propose the extension of the T-spline framework to the generalized B-splines (see [5] for details), which are locally spanned both by polynomial and non-polynomial functions. This leads to the new concept of Generalized T-splines (see [3] for our results about the trigonometric case). In this talk, we will analyze the main properties of the Generalized T-splines [4]; in particular we will study their linear independence, which is a key point to be able to use them in the isogeometric framework. This study will allow us to define a class of T-meshes which guarantee the linear independence both of Generalized T-splines and of classical T-splines of the same bi-order associated to the same T-mesh; moreover, we will show that this class of T-meshes properly includes the well-known analysis-suitable (equivalently, dual compatible [8, 9]) T-meshes. References [1] Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott and T.W. Sederberg. Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Engrg., 199: 229-263, 2010. [2] J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs, Isogeometric analysis: toward integration of CAD and FEA, John Wiley & Sons, 2009. [3] C. Bracco, D. Berdisnky, D. Cho, M. Oh and T. Kim. Trigonometric Generalized T-splines. Comput. Methods Appl. Mech. Engrg., 268: 540-556, 2014. [4] C. Bracco and D. Cho. Generalized T-splines and VMCR T-meshes. Comput. Methods Appl. Mech. Engrg., 280: 176-196, 2014. [5] C. Manni, F. Pelosi and M.L. Sampoli. Generalized B-splines as a tool in isogeometric analysis. Comput. Methods Appl. Mech. Engrg., 200: 867-881, 2011. [6] T.W. Sederberg, J. Zheng, A. Bakenov and A. Nasri. T-splines and T-NURCCs. ACM Trans. Graph., 22(3): 477-484, 2003. [7] T.W. Sederberg, D.L. Cardon, G.T. Finnigan, N.S. North, J. Zheng and T. Lyche. T-spline simplification and local refinement. ACM Trans. Graph., 23(3): 276-283, 2004. [8] L. Beirao da Veiga, A. Bua, D. Cho, and G. Sangalli. Analysis-suitable T-splines are dual-compatible. Comput. Methods Appl. Mech. Engrg., 249-252: 42-51, 2012. [9] L. Beirao da Veiga, A. Bua, G. Sangalli and R. Vazquez. Analysis-suitable T-splines of arbitrary degree: definition and properties. Math. Mod. Meth. Appl. Sci., 23: 1979-2003, 2013.