Comput Mech DOI 10.1007/s00466-013-0914-z ORIGINAL PAPER An adaptive three-dimensional RHT-splines formulation in linear elasto-statics and elasto-dynamics N. Nguyen-Thanh · J. Muthu · X. Zhuang · T. Rabczuk Received: 12 October 2012 / Accepted: 12 August 2013 © Springer-Verlag Berlin Heidelberg 2013 Abstract An adaptive three-dimensional isogeometric for- mulation based on rational splines over hierarchical T- meshes (RHT-splines) for problems in elasto-statics and elasto-dynamics is presented. RHT-splines avoid some short- comings of NURBS-based formulations; in particular they allow for adaptive h-refinement with ease. In order to drive the adaptive refinement, we present a recovery-based error estimator for RHT-splines. The method is applied to sev- eral problems in elasto-statics and elasto-dynamics including three-dimensional modeling of thin structures. The results are compared to analytical solutions and results of NURBS based isogeometric formulations. Keywords Isogeometric analysis · NURBS · PHT-splines · RHT-splines N. Nguyen-Thanh · X. Zhuang (B ) Department of Geotechnical Engineering, Tongji University, Shanghai, China e-mail: xiaoying.zhuang@gmail.com N. Nguyen-Thanh · T. Rabczuk (B ) Institute of Structural Mechanics, Bauhaus University Weimar, Marienstrasse 15, 99423 Weimar, Germany e-mail: timon.rabczuk@uni-weimar.de J. Muthu School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa T. Rabczuk School of Civil, Environmental and Architectural Engineering, Korea University, Seoul, Korea 1 Introduction Isogeometric analysis (IGA) was introduced by Hughes et al. [1] in order to unify Computer Aided Design (CAD) and Computer Aided Engineering (CAE). Non uniform ratio- nal B-splines (NURBS) are classically used in CAD though they have certain drawbacks in numerical analysis. One draw- back is related to adaptive h-refinement that is complex for NURBS-based isogeometric approaches. Recent approaches in IGA exploit different basis functions such as T-splines [2–4], volumetric solid T-spline modeling [5–9], polycube splines [10], Locally refined splines [11], polynomial splines over hierarchical T-meshes (PHT-splines) [12] and among others [13–21]. The PHT-splines inherits all important properties of NURBS such as linear independence of the basis functions, partition of unity, non-negativity and local support [22– 25]. In contrast to NURBS, PHT-splines have the capabil- ity of joining geometric objects without gaps, preserving higher order continuity everywhere and allow for simple and effective h-refinement strategies. From a linear algebra point of view, the NURBS space is a subspace of the PHT- spline space. Moreover, local refinement algorithms are rel- atively simple while the complexity of knot insertion with T-splines might be high, particularly in 3D. However, since PHT-splines are polynomial, they cannot exactly represent common engineering shapes of conic sections such as cir- cles, spheres, ellipsoids, etc. Therefore, we employ rational splines: RHT-splines. Previous, we presented such formula- tions for problems in linear elasto-statics in 2D [26] and for thin shell analysis [27]. In this paper, we extend our 2D formulation to 3D for prob- lems in elasto-statics and elasto-dynamics. We also present a stress recovery technique in isogeometric analysis to drive the adaptive h-refinement procedure. We employ the Super- 123