Bézier projection: A unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis

• DOI: 10.1016/j.cma.2014.07.014
• Publication Date: 2014-07-17T17:01:35Z
• Authors: D.C. Thomas; M.A. Scott; J.A. Evans; K. Tew; E.J. Evans
• Keywords: FOS: Mathematics; 0211 other engineering and technologies; Mathematics - Numerical Analysis; Numerical Analysis (math.NA); 02 engineering and technology

We introduce B��zier projection as an element-based local projection methodology for B-splines, NURBS, and T-splines. This new approach relies on the concept of B��zier extraction and an associated operation introduced here, spline reconstruction, enabling the use of B��zier projection in standard finite element codes. B��zier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global $L^2$ projection. B��zier projection is used to develop a unified framework for spline operations including cell subdivision and merging, degree elevation and reduction, basis roughening and smoothing, and spline reparameterization. In fact, B��zier projection provides a \emph{quadrature-free} approach to refinement and coarsening of splines. In this sense, B��zier projection provides the fundamental building block for $hpkr$-adaptivity in isogeometric analysis.<br/>56 pages, 28 figures<br/>