B\'{e}zier 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

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