3D NURBS‐enhanced finite element method (NEFEM)

Abstract

This paper presents the extension of the recently proposed NURBS-Enhanced Finite Element Method (NEFEM) to 3D domains. NEFEM is able to exactly represent the geometry of the computational domain by means of its CAD boundary representation with Non-Uniform Rational B-Splines (NURBS) surfaces. Specific strategies for interpolation and numerical integration are presented for those elements affected by the NURBS boundary representation. For elements not intersecting the boundary, a standard finite element rationale is used, preserving the efficiency of classical FEM. In 3D NEFEM special attention must be paid to geometric issues that are easily treated in the 2D implementation. Several numerical examples show the performance and benefits of NEFEM compared to standard isoparametric or cartesian finite elements. NEFEM is a powerful strategy to efficiently treat curved boundaries and it avoids excessive mesh refinement to capture small geometric features.

Comparison of high‐order curved finite elements

“…The proposed strategy to perform the numerical integration in 2D NE-FEM can be easily extended to 3D domains, see [22]. For instance, let Ω e be a tetrahedral element with a face on the curved boundary, and S its NURBS parametrization.…”

“…Moreover, if singular NURBS are present in the boundary description (that is, a NURBS containing a point where a directional derivative in the parametric space is zero, and thererfore knot lines converge towards a so-called singular point), Λ e may be a quadrilateral element in the parametric space of the NURBS. In such situations the mapping of Equation (8) is also valid, and it is only necessary to modify the quadrature of Λ e , see [22].…”

Blending isogeometric analysis and local maximum entropy meshfree approximants

“…The integration points for the interior cells are gener ated with standard Gaussian quadrature rules in a background Del aunay mesh supported on the nodes. On the other hand, for the boundary cells, we use quadrature rules recently developed for high order curved elements [36] and in NEFEM applications [26]. Subdivision of the boundaries cells into smaller cells is another op tion, which is nevertheless computationally more expensive.…”

“…In the same spirit of the method presented here, the NURBS en hanced finite element method (NEFEM) [26] adopts a NURBS boundary representation, coupled to standard finite elements in the interior of the domain. This approach exploits the high fidelity geometry representation of isogeometric analysis, but does not in sist in preserving the smoothness and positivity of the basis func tions, placing more emphasis in the high order reproducibility conditions.…”

Stabilized method of imposing Dirichlet boundary conditions using a recovered stress field

“…In order to maintain the optimal convergence rate of the FE solution, the degree of approximation to the boundary must be at least of the same order as the degree of the FE interpolation [7]. Transfinite mapping techniques commonly used in the p-version of the FEM, or the integration techniques described in [8] to consider the exact geometry given by a NURBS representation of the boundary, can be used in the elements cut by the boundary to obtain an exact representation of the domain. Boundary conditions: as the FE nodes do not generally lie on the boundary, the procedures used in the standard FEM to apply the boundary conditions cannot be used.…”