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

Abstract

We introduce Bézier projection as an element-based local projection methodology for B-splines, NURBS, and T-splines. This new approach relies on the concept of Bézier extraction and an associated operation introduced here, spline reconstruction, enabling the use of Bézier projection in standard finite element codes. Bézier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global $L^2$ projection. Bézier 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, Bézier projection provides a \emph{quadrature-free} approach to refinement and coarsening of splines. In this sense, Bézier projection provides the fundamental building block for $hpkr$-adaptivity in isogeometric analysis.