A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

Abstract

A framework to validate and generate curved nodal high-order meshes on CAD surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin-shell and 3D finite element analysis with unstructured high-order methods. First, we define a distortion (quality) measure for high-order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high-order), and handles with low-quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high-order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, we curve them to match the geometry, and we optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process.high-order methods; high-order mesh generation; quality measure; mesh optimization; curved elements; CAD ; parameterized surfaces 1

A CFD framework for offshore and onshore wind farm simulation

“…1(a)). For this reason, the 3D surface mesh is optimized maximizing the elemental quality presented in [12,13] and imposing the nodes position over the exact topography to avoid lost of real geometry representation, using the process presented in [14,15]. Fig.…”

“…However, this iterative scheme diverged in some of the tested complex cases, even using relaxation methods. Herein, introducing (15) in (14) we rewrite the iterative problem in terms of U ∞ and we translate it into solving the non-linear equation…”

Insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures

“…To overcome this drawback, we use the regularized distortion measure introduced in Escobar et al and Garanzha and Kaporin for linear elements, in which σ is replaced by where δ is a small parameter that depends on the problem and can be calculated according to Escobar et al and Gargallo‐Peiró et al This regularization is also used for optimizing curved high‐order meshes. ()…”

A unified approach for a posteriori high-order curved mesh generation using solid mechanics

“…In a standard high-order finite element formulation, measures involving the Jacobian of the isoparametric mapping have been extensively used [29,77], in particular the so-called scaled Jacobian. This measure only quantifies volumetric deformations and alternative measures that exploit different modes of deformation and account for shape, skewness and degeneracy of elements have only been recently considered [27,28]. However, it is worth noting that not all of these quality measures can be regarded as independent quantities.…”

“…Next, the quality of the generated meshes in terms of different measures is studied, namely the measures defined in Equation (27) that are defined in terms of the invariants in Equation (17). The two anisotropic mesh quality measures Q 4 and Q 5 are dropped from the comparison because they are only valid for the transversely isotropic material model, which has been shown to produce low quality meshes in the examples considered.…”