The generation of triangular meshes for NURBS‐enhanced FEM

Abstract

SUMMARYThis paper presents the first method that enables the fully automatic generation of triangular meshes suitable for the so-called non-uniform rational B-spline (NURBS)-enhanced finite element method (NEFEM). The meshes generated with the proposed approach account for the computer-aided design boundary representation of the domain given by NURBS curves. The characteristic element size is completely independent of the geometric complexity and of the presence of very small geometric features. The proposed strategy allows to circumvent the time-consuming process of de-featuring complex geometric models before a finite element mesh suitable for the analysis can be produced. A generalisation of the original definition of a NEFEM element is also proposed, enabling to treat more complicated elements with an edge defined by several NURBS curves or more than one edge defined by different NURBS. Three examples of increasing difficulty demonstrate the applicability of the proposed approach and illustrate the advantages compared with those of traditional finite element mesh generators. Finally, a simulation of an electromagnetic scattering problem is considered to show the applicability of the generated meshes for finite element analysis.

Solution of geometrically parametrised problems within a CAD environment via model order reduction

“…The inner radius R in is considered an extra parameter within the proposed PGD framework and the objective is to find, in the off-line stage, the the reference domain, with 251, 1,023 and 4,256 elements respectively, are represented in Figure 3. These meshes are generated using the technique proposed in [35] to guarantee that elements without an edge on a curved boundary can be mapped to a reference triangle using an affine mapping.…”

HDG-NEFEM for two dimensional linear elasticity

“…The next example, usually referred to as the Lamé problem [43,44,45], meshes are generated using the strategy described in [46,47] with an extra constraint to ensure that the internal triangular edges are not deformed. The NEFEM meshes are generated using the technique proposed in [48]. the polygonal description of the curved boundaries employed in the isoparametric formulation.…”

“…The process starts with a uniform linear approximation of the solution and the desired error is ε σ e = 0.5 × 10 −3 . After five iterations the degree of approximation required in each element to guarantee that the error is below ε σ e is represented in Figure 16 The extension to three dimensional problems does not introduce a conceptual difficulty but extra attention must be paid to the NEFEM mesh generation [48]. Finally, the extension to non-linear problems only needs a standard linearisation but it requires a careful selection of the stability parameter of the HDG formulation [49].…”

The application of a high-order discontinuous Galerkin time-domain method for the computation of electromagnetic resonant modes

“…High-order curvilinear meshes for the isoparametric FEM are obtained using the solid mechanics analogy proposed in [35,36], whereas the meshes for NEFEM are generated using the technique recently proposed in [37]. For curved elements, it is not possible to precompute the mass matrix in the reference element and scale it with the Jacobian for each physical element.…”