Séminaire LJKModèles et Algorithmes Déterministes: EDPMOISEMGMI

Le Jeudi 22 Janvier 2015 à 11h00 en

Séminaire de Mr Durkbin CHO (Dongguk University, Republic of Korea)

Generalized Tsplines and Tmeshes guaranteeing their linear independence

Résumé:

The Tspline 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 tensorproduct 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 Tspline approach is mainly applied to polynomial splines, but in some situation the use of nonpolynomial 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 nonpolynomial splines (see, e.g., [5]). For this reason, we propose the extension of the Tspline framework to the generalized Bsplines (see [5] for details), which are locally spanned both by polynomial and nonpolynomial functions. This leads to the new concept of Generalized Tsplines (see [3] for our results about the trigonometric case).
In this talk, we will analyze the main properties of the Generalized Tsplines [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 Tmeshes which guarantee the linear independence both of Generalized Tsplines and of classical Tsplines of the same biorder associated to the same Tmesh; moreover, we will show that this class of Tmeshes properly includes the wellknown analysissuitable (equivalently, dual compatible [8, 9]) Tmeshes.
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 Tsplines. Comput. Methods Appl. Mech. Engrg., 199: 229263, 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 Tsplines. Comput. Methods Appl. Mech. Engrg., 268: 540556, 2014.
[4] C. Bracco and D. Cho. Generalized Tsplines and VMCR Tmeshes. Comput. Methods Appl. Mech. Engrg., 280: 176196, 2014.
[5] C. Manni, F. Pelosi and M.L. Sampoli. Generalized Bsplines as a tool in isogeometric analysis. Comput. Methods Appl. Mech. Engrg., 200: 867881, 2011.
[6] T.W. Sederberg, J. Zheng, A. Bakenov and A. Nasri. Tsplines and TNURCCs. ACM Trans. Graph., 22(3): 477484, 2003.
[7] T.W. Sederberg, D.L. Cardon, G.T. Finnigan, N.S. North, J. Zheng and T. Lyche. Tspline simplification and local refinement. ACM Trans. Graph., 23(3): 276283, 2004.
[8] L. Beirao da Veiga, A. Bua, D. Cho, and G. Sangalli. Analysissuitable Tsplines are dualcompatible. Comput. Methods Appl. Mech. Engrg., 249252: 4251, 2012.
[9] L. Beirao da Veiga, A. Bua, G. Sangalli and R. Vazquez. Analysissuitable Tsplines of arbitrary degree: definition and properties. Math. Mod. Meth. Appl. Sci., 23: 19792003, 2013.
