Abstract

An improvement of the classical finite element method is proposed. It is able to exactly represent the geometry by means of the usual CAD description of the boundary with Non-Uniform Rational B-Splines (NURBS). Here, the two-dimensional case is presented. For elements not intersecting the boundary, a standard finite element interpolation and numerical integration is used. But elements intersecting the NURBS boundary need a specifically designed piecewise polynomial interpolation and numerical integration. A priori error estimates are also presented. Finally, some examples demonstrate the applicability and benefits of the proposed methodology. NEFEM is at least one order of magnitude more precise than the corresponding isoparametric finite element in every numerical example shown.

This is the case for both continuous and discontinuous Galerkin formulations. Moreover, for a desired precision NEFEM is also more computational efficient, as shown in the numerical examples. The use of NEFEM is strongly recommended in the presence of curved boundaries and/or when the boundary of the domain has complex geometric details. The possibility of computing accurate solution with coarse meshes and high order interpolations, makes NEFEM a more efficient strategy than classical isoparametric FE.

Abstract

In this work, the NURBS-enhanced finite element method (NEFEM) is combined with a discontinuous Galerkin (DG) formulation for the numerical solution of Euler equations of gas dynamics. With NEFEM, numerical fluxes along curved boundaries are computed much more accurately due to the exact geometric representation of the computational domain. The proper implementation of the wall boundary condition and the exact geometry provide accurate results even with a linear approximation of the solution. A detailed comparison of NEFEM in front of isoparametric finite elements is presented, demonstrating the superiority of NEFEM approach for both linear and higher-order computations

Abstract

An improvement to the classical finite element (FE) method is proposed. It is able to exactly represent the geometry by means of the usual CAD description of the boundary with non-uniform rational B-splines (NURBS). Here, the 2D case is presented. For elements not intersecting the boundary, a standard FE interpolation and numerical integration are used. But elements intersecting the NURBS boundary need a specifically designed piecewise polynomial interpolation and numerical integration. A priori error estimates are also presented. Finally, some examples demonstrate the applicability and benefits of the proposed methodology. NURBS-enhanced finite element method (NEFEM) is at least one order of magnitude more precise than the corresponding isoparametric FE in every numerical example shown. This is the case for both continuous and discontinuous Galerkin formulations. Moreover, for a desired precision, NEFEM is also more computationally efficient, as shown in the numerical examples. The use of NEFEM is strongly recommended in the presence of curved boundaries and/or when the boundary of the domain has complex geometric details. The possibility of computing an accurate solution with coarse meshes and high-order interpolations makes NEFEM a more efficient strategy than classical isoparametric FE.