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.

💡 Research Summary

This paper introduces a novel hybrid discretization technique called the NURBS‑Enhanced Finite Element Method for Hexahedral meshes (NEFEM‑HEX), which integrates the exact NURBS representation of a domain’s curved boundary into a conventional finite element framework based on hexahedral elements. The authors consider an open, bounded, convex three‑dimensional domain Ω with a Lipschitz continuous boundary ∂Ω that can be described by a single NURBS patch P. The computational domain is partitioned into two sub‑regions: a boundary layer Ω_B consisting of hexahedral elements that touch the NURBS boundary, and an interior region Ω_int composed of standard interior hexahedra. Elements in Ω_int are constructed by the usual isoparametric mapping from the reference cube