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

摘要

We introduce B\'{e}zier projection as an element-based local projectionmethodology for B-splines, NURBS, and T-splines. This new approach relies onthe concept of B\'{e}zier extraction and an associated operation introducedhere, spline reconstruction, enabling the use of B\'{e}zier projection instandard finite element codes. B\'{e}zier projection exhibits provably optimalconvergence and yields projections that are virtually indistinguishable fromglobal $L^2$ projection. B\'{e}zier projection is used to develop a unifiedframework for spline operations including cell subdivision and merging, degreeelevation and reduction, basis roughening and smoothing, and splinereparameterization. In fact, B\'{e}zier projection provides a\emph{quadrature-free} approach to refinement and coarsening of splines. Inthis sense, B\'{e}zier projection provides the fundamental building block for$hpkr$-adaptivity in isogeometric analysis.