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

Duygu Sap

TL;DR

The paper addresses accurate PDE solutions on curved geometries while preserving computational efficiency by integrating NURBS geometry into a hex-element FE framework (NEFEM-HEX). It introduces a two-region decomposition (boundary layer and interior), a NURBS-enhanced boundary discretization, and a linear interior FE; it also develops a hybrid interpolation and a blended quadrature and demonstrates convergence through a priori $H^{1}$ estimates for a generic second-order elliptic problem with Poisson numerical tests. The contributions include stability and approximation theory for the interpolation operators, a priori $H^{1}$ error estimates, and validation via Poisson problem experiments across varying curvature, confirming the method’s accuracy and robustness. The proposed approach preserves exact geometric representation within a low-order finite element framework, offering a scalable technique for large-scale engineering simulations with potential extensions to local refinement and higher-order methods.

Abstract

In this paper, we present a NURBS-enhanced finite element method that integrates NURBS-based boundary representations of geometric domains into standard finite element frameworks applied to hexahedral meshes. We decompose an open, bounded, convex three-dimensional domain with a NURBS boundary into two parts, define the NURBS-enhanced finite elements over the boundary layer, and use piecewise-linear Lagrange finite elements in the interior region. We introduce a novel quadrature rule and a novel interpolation operator for the NURBS-enhanced elements. We derive the stability and approximation properties of the interpolation operators that we use. We describe 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 elliptic PDE and derive a priori error estimates in the $H^{1}$ norm. 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.

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

TL;DR

The paper addresses accurate PDE solutions on curved geometries while preserving computational efficiency by integrating NURBS geometry into a hex-element FE framework (NEFEM-HEX). It introduces a two-region decomposition (boundary layer and interior), a NURBS-enhanced boundary discretization, and a linear interior FE; it also develops a hybrid interpolation and a blended quadrature and demonstrates convergence through a priori estimates for a generic second-order elliptic problem with Poisson numerical tests. The contributions include stability and approximation theory for the interpolation operators, a priori error estimates, and validation via Poisson problem experiments across varying curvature, confirming the method’s accuracy and robustness. The proposed approach preserves exact geometric representation within a low-order finite element framework, offering a scalable technique for large-scale engineering simulations with potential extensions to local refinement and higher-order methods.

Abstract

In this paper, we present a NURBS-enhanced finite element method that integrates NURBS-based boundary representations of geometric domains into standard finite element frameworks applied to hexahedral meshes. We decompose an open, bounded, convex three-dimensional domain with a NURBS boundary into two parts, define the NURBS-enhanced finite elements over the boundary layer, and use piecewise-linear Lagrange finite elements in the interior region. We introduce a novel quadrature rule and a novel interpolation operator for the NURBS-enhanced elements. We derive the stability and approximation properties of the interpolation operators that we use. We describe 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 elliptic PDE and derive a priori error estimates in the norm. 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.

Paper Structure

This paper contains 22 sections, 9 theorems, 76 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Preliminaries
  4. B-Splines
  5. Non-Uniform Rational B-Splines (NURBS)
  6. Meshes
  7. NURBS-Enhanced Finite Element Method
  8. The Hybrid Mesh
  9. Transformations
  10. Function Spaces
  11. Basis functions
  12. Local Finite Element Spaces
  13. Global Finite Element Space
  14. Global Finite Element Spaces with Boundary Conditions
  15. Numerical Integration
  16. ...and 7 more sections

Key Result

Lemma 1

Figures (10)

  • Figure 1: A cylinder (on the left) depicted with elements in $\mathcal{T}_h^b$ and $\mathcal{T}_h^i$ (on the right).
  • Figure 2: A NURBS surface patch with its control mesh and knot-net.
  • Figure 3: The reference cube (on the left), and four $\mathcal{Q}\in \mathcal{T}_h^b$ and $\mathcal{Q}\in \mathcal{T}_h^i$ (on the right).
  • Figure 4: Fully NURBS-Enhanced element with the reference cube
  • Figure 6: Refinement of a boundary element.
  • ...and 5 more figures

Theorems & Definitions (25)

  • Remark 1
  • Remark 2
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 15 more