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NURBS-enhanced finite element method (NEFEM). (English) Zbl 1162.65389
Summary: An improvement to the classical finite element (FE) method is proposed. It is able to exactly represent the geometry by means of the usual computer-aided design 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.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A45 Diffraction, scattering
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
76M10 Finite element methods applied to problems in fluid mechanics
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