NURBS-Enhanced Finite Element Method (NEFEM)

NURBS-Enhanced Finite Element Method (NEFEM) is a new and efficient technique to exactly treat curved boundaries in a finite element context. NEFEM is applied to the numerical solution of electromagnetic scattering and compressible flow applications. One of the challenges of these problems is that the size of the obstacle is sometimes subsidiary of the geometrical complexity, and not only on the solution itself. When the geometry of the obstacle has complex details classical finite element methods need extremely refined meshes to properly capture the geometry because small changes in the geometrical model cause global changes in the scattered field. Simplifying the geometry to avoid mesh refinement may cause global changes in the solution. One of the advantages of NEFEM is that allows to mesh the domain independently of the geometrical complexity.

Scattering by a complex aircraft profile: simplifying the geometry to avoid mesh refinement cause global changes in the scattered field distribution

Goal

The main goal of NEFEM is to work with the exact geometric model, with no dependence of the spatial discretization. NEFEM is able to exactly represent the geometry by means of the usual CAD description of the boundary with Non-Uniform Rational B-Splines (NURBS). The efficiency of classical finite element techniques is preserved considering standard procedures for the interpolation and the numerical integration for interior elements. For elements intersecting the NURBS boundary specific techniques for the interpolation and numerical integration are proposed.

A cut through an unstructured tetrahedral mesh with exact boundary representation: showing the independence of the spatial discretization with respect to the NURBS boundary representation

Strategy

The parametric space of the NURBS is used to define geometric entities affected by the NURBS boundary representation of the domain. The polynomial interpolation for curved NEFEM elements is defined in cartesian coordinates, directly in the physical space. For the numerical integration, efficient strategies are proposed. The key idea is to design a mapping decoupling the complexity of the NURBS surfaces with respect to the direction towards the interior of the element. For interior elements -not affected by the NURBS boundary representation- standard finite element techniques are used.

Curved faces on the NURBS boundary are defined as the image of a straight-sided triangle on the parametric space, and curved interior faces with an edge on the NURBS boundary are defined as a convex linear combination of the curved edge and the interior node

With NEFEM, it is neither necessary to locate nodes at boundary corners or edges nor to refine the mesh near the boundary to capture the geometry. It is exactly represented in NEFEM independently on the spacial discretization.Therefore, with NEFEM, small is influential does not imply elements!

Scattering by a complex aircraft profile: NEFEM is able to compute accurate solutions with coarse meshes and high-order approximations, avoiding the excessive mesh refinement

Applications and performance

Second-order elliptic problem

A simple second-order elliptic problem is solved in a sphere discretized with only eight curved elements. A comparison of FEM and NEFEM solutions with quadratic and cubic approximation reveals the deficiencies of the isoparametric mapping when ultra-coarse meshes are considered. Even if the degree of the isoparametric approximation is increased the solution error is dramatically controlled by the geometric error introduced by the isoparametric mapping. With NEFEM the exact geometric model is always considered, with no dependence on the spatial discretization. This translates in faster convergence; for very high-order approximations NEFEM is several orders of magnitude more precise than methods with an approximated boundary representation.

Second-order elliptic problem: comparison of FEM and NEFEM solutions, and error as the degree of the approximation is increased

Compressible flow

An accurate description of the geometrical model is crucial in the numerical solution of Euler equations of gas dynamics. In a Discontinuous Galerkin (DG) framework, using a linear approximation for the geometry it is not possible to converge to the steady state solution, even if the mesh is drastically refined near the curved boundary. NEFEM is proposed as an efficient alternative for a proper treatment of curved wall boundaries in compressible flow problems using a DG formulation.

Inviscid subsonic flow around a circle: comparison of FEM and NEFEM Mach number distributions for linear approximation

NEFEM also allows to obtain accurate solutions with ultra-coarse meshes and very high-order approximations. For instance, in the simulation of inviscid flows around airfoils standard methods need to perform excessive refinement towards the leading edge of the airfoil in order to capture the high variations of the curvature. With NEFEM, the spatial discretization is independent on the geometrical complexity and accurate solutions are obtained with only two elements to discretize the leading edge of the airfoil.

Inviscid subsonic flow around a RAE2822 airfoil: ultra-coarse mesh and Mach number distribution computed with NEFEM and a degree of approximation 7

Electromagnetic scattering

An accurate geometric description is also critical in electromagnetic scattering problems. Even with simple obstacles, high-order isoparametric and cartesian finite elements suffer from an important loss of accuracy due to the approximation boundary representation.

Scattering by a perfect conducting sphere: ultra-coarse mesh and RCS error convergence comparison between cartesian FEM and NEFEM

Scattering by a perfect conducting NACA0012 airfoil: ultra-coarse mesh and RCS pattern as the degree of the approximation is increased illustrating the deficiencies of the isoparametric mapping

Moreover, the cartesian approximation considered in NEFEM combined with the exact boundary representation provides more accurate results than methods working with an exact geometric model such as the p-version of the FEM.

Scattering by a perfect conducting circle: ultra-coarse mesh and RCS error convergence comparison between several high-order finite element techniques

Selected publications and talks

Journal papers

• R. Sevilla, S. Fernández-Méndez and A. Huerta. NURBS-Enhanced Finite Element Method (NEFEM), International Journal for Numerical Methods in Engineering, Vol. 76, Issue 1, pp. 56--83, 2008.
• R. Sevilla, S. Fernández-Méndez and A. Huerta. NURBS-Enhanced Finite Element Method (NEFEM) for Euler equations, International Journal for Numerical Methods in Fluids, Vol. 57, Issue 9, pp. 1051--1069, 2008.
• R. Sevilla, S. Fernández-Méndez and A. Huerta. 3D NURBS-Enhanced Finite Element Method (NEFEM), International Journal for Numerical Methods in Engineering, Vol. 88, Issue 2, pp. 103--125, 2011
• R. Sevilla and S. Fernández-Méndez. Numerical integration over 2D NURBS-shaped domains with applications to NURBS-enhanced FEM, Finite Elements in Analysis and Design, Vol. 47, Issue 10, pp. 1209--1220, 2011
• R. Sevilla, S. Fernández-Méndez and A. Huerta. Comparison of high-order curved finite elements, International Journal for Numerical Methods in Engineering, Vol. 87, Issue 8, pp. 719--734, 2011
• R. Sevilla, S. Fernández-Méndez and A. Huerta. NURBS-Enhanced Finite Element Method (NEFEM). A seamless bridge between CAD and FEM , Archives of Computational Methods in Engineering (invited), Vol. 18, Issue 4, pp. 441--484, 2011

Talks

• NURBS-Enhanced Finite Element Method (NEFEM), proceedings of the European Conference on Computational Mechanics (ECCM), Paris (France), 2010.
• 3D NURBS-Enhanced Finite Element Method (NEFEM), proceedings of the IX Congreso de Métodos Numéricos en la Ingeniería, Barcelona (Spain), 2009.
• 3D NURBS-Enhanced Finite Element Method (NEFEM), proceedings of the 8th World Congress on Computational Mechanics - 5th European Congress on Computacional Methods in Applied Sciences and Engineering, Venice (Italy), 2008.
• NURBS-Enhanced Finite Element Method (NEFEM), proceedings of the 14th International Conference on Finite Elements (FEF), Santa Fe, New Mexico (USA), 2007.
• NURBS-Enhanced Finite Element Method (NEFEM), proceedings of the European Conference on Computational Fluid Dynamics, Egmond and Zee (The Netherlands), 2006.

PhD Thesis

• R. Sevilla. NURBS-Enhanced Finite Element Method (NEFEM), Universitat Politècnica de Catalunya, July 2009. (download PDF, see presentation)
• SEMNI award for the best PhD thesis in 2009 on Numerical Methods in Engineering.
• ECCOMAS award for the best PhD theses in 2009 on Computational Methods in Applied Sciences and Engineering. See the ECCOMAS newsletter.
• EMERALD highly commended award from the Outstanding Doctoral Research Awards.

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