A note on invertible two‐dimensional quadratic finite element transformations

Numerical Meth Engineering

et al. 2015

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

“…(42), (43), and (44), we conclude that in the implementation we only need to compute the gradients, the Hessian, and the value of the local functionf introduced in Eq. (41).…”

Section: B Implementation: Submesh Distortionmentioning

confidence: 94%

Section: Related Workmentioning

confidence: 99%

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

Numerical Meth Engineering

et al. 2015

“…A first requirement to repair invalid elements is an algorithm to detect them and to quantify the level of distortion. For high-order elements, several approaches have been proposed to detect the validity of the mapping [36,17,21,22,23,24,37,38,39], and to define suitable quality measures [40,41,42,43,44,45,31,32,33,35].…”

Section: Related Workmentioning

confidence: 99%

Optimization of a regularized distortion measure to generate curved high‐order unstructured tetrahedral meshes

Numerical Meth Engineering

et al. 2015

SUMMARYWe present a robust method for generating high-order nodal tetrahedral curved meshes. The approach consists of modifying an initial linear mesh by first, introducing high-order nodes, second, displacing the boundary nodes to ensure that they are on the CAD surface, and third, smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high-order mesh. The smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh. This means that whenever possible, the resulting mesh preserves the geometrical features of the initial linear mesh such as shape, stretching and size. We present several examples to illustrate the performance of the proposed algorithm. Furthermore, the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. The meshes obtained are suitable for high-order finite element analyses.

“…Specifically, it has been studied how to detect non-positive Jacobian determinants for B-spline based mappings [7,6,30,28,43] and quadratic iso-parametric elements [31,9,1]. Moreover, for higher interpolation degrees, in References [20,21] it is proposed to compute accurate bounds on Jacobian determinants of 2D and 3D curvilinear polynomial finite elements.…”

Section: Related Workmentioning

confidence: 99%

Distortion and quality measures for validating and generating high-order tetrahedral meshes

Engineering with Computers

et al. 2014

A procedure to quantify the distortion (quality) of a high-order mesh composed of curved tetrahedral elements is presented. The proposed technique has two main applications. First, it can be used to check the validity and quality of a high-order tetrahedral mesh. Second, it allows the generation of curved meshes composed of valid and high-quality high-order tetrahedral elements. To this end, we describe a method to smooth and untangle high-order tetrahedral meshes simultaneously by minimizing the proposed mesh distortion. Moreover, we present a p-continuation procedure to improve the initial configuration of a high-order mesh for the optimization process. Finally, we present several results to illustrate the two main applications of the proposed technique.