A hybrid isogeometric and finite element method: NURBS-enhanced finite element method for hexahedral meshes (NEFEM-HEX)

In this paper, we present a NURBS-enhanced finite element method that integrates the NURBS-based boundary representation of a geometric domain into a standard finite element framework for hexahedral meshes. We decompose an open, bounded, convex three-dimensional domain with a NURBS boundary into two parts, define NURBS-enhanced finite elements over the boundary layer, and use piecewise-linear Lagrange finite elements in the interior region. We introduce a special quadrature rule and a stable interpolation operator for the NURBS-enhanced elements. We discuss how the h-refinement in finite element analysis and the knot insertion in isogeometric analysis can be utilized in the refinement of the NURBS-enhanced elements. To illustrate an application of our methodology, we utilize a generic weak formulation of a second-order linear elliptic boundary value problem and derive a priori error estimates in the $H^{1}$ norm. In addition, we use the Poisson problem as a model problem and provide numerical results that support the theoretical results. The proposed methodology combines the efficiency of finite element analysis with the geometric precision of NURBS, and may enable more accurate and efficient simulations over complex geometries.