Comparison of high‐order curved finite elements

Abstract

Several finite element techniques used in domains with curved boundaries are discussed and compared, with particular emphasis in two issues: the exact boundary representation of the domain and the consistency of the approximation. The influence of the number of integration points in the accuracy of the computation is also studied. Two dimensional numerical examples, solved with continuous and discontinuous Galerkin formulations, are used to test and compare all these methodologies. In every example shown, the recently proposed NURBS-enhanced finite element method (NEFEM) provides the maximum accuracy for a given spatial discretization, at least one order of magnitude more accurate than classical isoparametric finite element methods (FEM). Moreover, NEFEM outperforms Cartesian FEM and p-FEM, stressing the importance of the geometrical model as well as the relevance of a consistent approximation in finite element simulations.

Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier–Stokes equations

“…In this example a geometrical representation with constant degree k = 2 is used in all computations. However, it is worth mentioning that in practical applications a proper description of the geometry is crucial to attain the best accuracy for a given computational mesh [47,48]. In each iteration of the adaptive process the degree is updated for all elements in the computational mesh.…”

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.…”

3D NURBS‐enhanced finite element method (NEFEM)

“…To account for changes of NURBS parametrization, only the quadrature in Λ e must be modified as described in Section 3.3.1, see Figure 15. This represents an important advantage compared to numerical integration for p-FEM with NURBS, see [5] for further details. Similarly, for an element with one edge on a NURBS boundary, volume integrals are performed using parametrization (5) as…”

“…The standard FE technique in domains with curved boundaries is the isoparametric FEM, in which curved boundaries are approximated using piecewise polynomial parametrizations, see [4]. Geometric approximation induced by isoparametric FEs may lead to an important loss of accuracy, specially when high-order approximations are considered, see [5]. In this situation, mesh refinement to accurately capture geometry may compromise the benefits of using high-order approximations.…”

“…See also [13] for the application of NEFEM to the numerical solution of inviscid flow problems. Several highorder FE methodologies for the treatment of curved boundaries are discussed and compared in [5], including isoparametric FEM, cartesian FEM, p-FEM and NEFEM. Numerical examples show that NEFEM is not only more accurate than FE methods with an approximate boundary representation, but also outperforms p-FEM with an exact boundary representation, showing the advantages of combining cartesian approximation with exact boundary representation.…”