Isogeometric analysis based on T-splines
This chapter provides an introduction to the use of T-splines in isogeometric analysis. A simple definition of two-dimensional T-splines is given and Bézier extraction is introduced. The basic details for implementation of T-splines as finite element shap
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Department of Physics and Astronomy Brigham Young University Department of Civil and Environmental Engineering Brigham Young University ‡ common affil
Abstract This chapter provides an introduction to the use of Tsplines in isogeometric analysis. A simple definition of two-dimensional T-splines is given and B´ezier extraction is introduced. The basic details for implementation of T-splines as finite element shape functions are given. Two examples of integrated analysis and design based on commercial tools are given to illustrate the utility of T-spline-based IGA in a design-through-analysis workflow.
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Introduction
Isogeometric analysis (Hughes et al. (2005); Cottrell et al. (2009)) is a generalization of finite element analysis which improves the link between geometric design and analysis. The isogeometric paradigm is simple: the smooth spline basis used to define the geometry is used as the basis for analysis. As a result, exact geometry is introduced into the analysis. The smooth basis can be leveraged by the analysis (Evans et al. (2009); Hughes et al. (2014); Cottrell et al. (2007)) leading to innovative approaches to model design (Cohen et al. (2010); Wang et al. (2011); Liu et al. (2014)), analysis (Schillinger et al. (2012); Scott et al. (2013); Schmidt et al. (2012); Benson et al. (2010a)), optimization (Wall et al. (2008)), and adaptivity (Bazilevs et al. (2010); D¨orfel et al. (2009); Scott et al. (2014); Evans et al. (2014)). Many of the early isogeometric developments were restricted to NURBS but the use of T-splines as an isogeometric basis has gained widespread attention across a number of application areas Bazilevs et al. (2010); Scott et al. (2011, 2012); Verhoosel et al. (2011b,a); Borden et al. (2012); Benson et al. (2010a); Schillinger et al. (2012); Scott et al. (2013); Simpson et al. (2014); Dimitri et al. (2014); Hosseini et al. (2014); Bazilevs et al. (2012); Buffa et al. (2014); Ginnis et al. (2014). Particular focus has been placed G. Beer, S. Bordas (Eds.), Isogeometric Methods for Numerical Simulation, CISM International Centre for Mechanical Sciences DOI 10.1007/ 978-3-7091-1843-6_5 © CISM Udine 2015
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D.C. Thomas and M.A. Scott
on the use of T-spline local refinement in an analysis context Scott (2011); Scott et al. (2012); Borden et al. (2012); Verhoosel et al. (2011a,b). T-splines, introduced in the CAD community (Sederberg et al. (2003)), are a generalization of non-uniform rational B-splines (NURBS) which address fundamental limitations in NURBS-based design. For example, a T-spline can model a complicated design as a single, watertight geometry and are also locally refineable (Sederberg et al. (2004); Scott et al. (2012)). Since their advent they have emerged as an important technology across multiple disciplines and can be found in several major commercial CAD products (Autodesk (2012); Autodesk, Inc. (2014)). Recent developments include analysis-suitable T-splines (Li et al. (2012); Scott et al. (2012); Beir˜ ao da Veiga et al. (2012); da Veiga et al. (2013); Li and Sco
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