Lagrange extraction and projection for NURBS basis functions: A direct link between isogeometric and standard nodal finite element formulations

We introduce Lagrange extraction and projection that link a C⁰ nodal basis with a smooth B-spline basis. Our technology is equivalent to Bézier extraction and projection but offers an alternative implementation based on the interpolatory property of nodal basis functions. The Lagrange extraction operator can be constructed by simply evaluating B-spline basis functions at nodal points and eliminates the need for introducing Bernstein polynomials as new shape functions. The Lagrange projection operator is defined as the inverse of the Lagrange extraction operator and directly relates function values at nodal points to element-level B-spline coefficients of a local interpolant. For geometries based on polynomial B-splines, our technology allows the implementation of isogeometric analysis in standard nodal finite element codes with simple algorithms and minimal intrusion.