Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids N. Nguyen-Thanh a,⇑ , H. Nguyen-Xuan b,c , S.P.A. Bordas d , T. Rabczuk a a Institute of Structural Mechanics, Bauhaus-University Weimar, Marienstr. 15, D-99423 Weimar, Germany b Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science, VNU HCM, Viet Nam c Division of Computational Mechanics, Ton Duc Thang University, HCM, Viet Nam d Institute of Modelling & Simulation in Mechanics & Materials, School of Engineering Cardiff University, Queen’s Buildings, The Parade, CARDIFF CF24 3AA, Wales, UK article info Article history: Received 18 June 2010 Received in revised form 24 October 2010 Accepted 23 January 2011 Available online 1 February 2011 Keywords: Isogeometric analysis T-spline T-meshes PHT-spline abstract Isogeometric analysis has become a powerful alternative to standard finite elements due to its flexibility in handling complex geometries. One of the major drawbacks of NURBS-based isogeometric finite ele- ments is the inefficiency of local refinement. In this study, we present an alternative to NURBS-based iso- geometric analysis that allows for local refinement. The idea is based on polynomial splines and exploits the flexibility of T-meshes for local refinement. The shape functions satisfy important properties such as non-negativity, local support and partition of unity. Several numerical examples are used to demonstrate the reliability of the present method. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Isogeometric analysis was introduced by Hughes et al. [1] in or- der to improve and accelerate numerical analysis and to closely link Computer Aided Design (CAD) and Computer Aided Engineer- ing (CAE). The basic idea is to use CAD basis functions in the con- text of numerical analysis. While the finite element method is most popular in CAE, the most common CAD basis functions are Non Uniform Rational B-splines (NURBS). Therefore, most studies in the context of isogeometric analysis focus on NUBRS-based isogeo- metric finite element formulations [2–6]. One major advantage of CAD basis functions (e.g. NURBS) over finite elements is their ability to describe a larger class of geomet- ric objects, e.g. conic geometries. However, the requirements on basis functions in CAE are higher than in CAD. Besides their potential to unify CAD and CAE and therefore to reduce computational cost, NURBS-based isogeometric finite ele- ment formulations have other advantages over (Lagrange) polyno- mial based finite elements (FEs):  For many examples (see e.g. the results in [7–12]), it was found that NURBS-based isogeometric FEs give more accurate results than their traditional FE-counterparts. This was attributed to the higher smoothness and continuity of the isogeometric basis functions. Higher continuous formulations do not lead to jump in derivatives, e.g. in the strain field and stress field in mechan- ical analysis, inherent to C 0 continuous FE formulations.  The higher continuity of the isogeometric formulation can also be exploited in a different context, e.g. for thin plates and thin shells [13,14] or for gradient based constitutive models. We note that any order of continuity-even C 1 – can be created in NURBS basis functions through a simple procedure, i.e. knot insertion.  Initial studies [13] conjecture that isogeometric FEs based on NURBS show less sensitivity with respect to excessive mesh dis- tortion as compared to Lagrange polynomial based FEs making them particularly attractive for problems with large deforma- tions such as shear band formation, sheet metal stamping or crashworthiness, etc. (though mesh distortion can also be a con- sequence of poor mesh generation); this was again attributed to higher-order and higher-continuity of the approximation.  It was found that the natural eigenfrequencies of higher order NURBS-based isogeometric FEs are much lower compared to higher order Lagrange polynomial based FEs [15]. This is partic- ularly advantageous for explicit time integration where the sta- ble time step is inversely proportional to the maximum eigenfrequency.  Besides the conventional h-refinement and p-refinement, NURBS-based isogeometric FEs offer a more flexible k-refine- ment. The k-refinement is ideally suited for higher-order approximations. It maintains the polynomial degree and the 0045-7825/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2011.01.018 ⇑ Corresponding author. E-mail address: nhon.nguyen.thanh@uni-weimar.de (N. Nguyen-Thanh). Comput. Methods Appl. Mech. Engrg. 200 (2011) 1892–1908 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma