Algebraic mesh quality metrics

Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally-invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. the singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. Condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Combined metrics for shape and volume, shape-volume-orientation are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to non-simplical elements. A series of numerical tests verify the theoretical properties of the metrics defined

Validation and generation of high-order meshes on parameterized surfaces

We present a technique to extend Jacobian-based distortion (quality) measures for planar triangles to high-order isoparametric elements of any interpolation degree on CAD parameterized surfaces. The resulting distortion (quality) measures are expressed in terms of the parametric coordinates of the nodes. These extended distortion (quality) measures can be used to check the quality and validity of a high-order surface mesh. We also apply them to simultaneously smooth and untangle high-order surface meshes by minimizing the extended distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes. Thus, the nodes always lie on the surface. Finally, we include several examples to illustrate the application of the proposed techniquePeer ReviewedPostprint (published version

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

This is the accepted version of the following article: [Gargallo-Peiró, A., Roca, X., Peraire, J., and Sarrate, J. (2016) A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization. Int. J. Numer. Meth. Engng, 106: 1100–1130. doi: 10.1002/nme.5162], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5162/abstractA framework to validate and generate curved nodal high-order meshes on Computer-Aided Design (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, curve them to match the geometry, and 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.Peer ReviewedPostprint (author's final draft

A new procedure to smooth and untangle meshes on parameterized surfaces

We present a technique to extend any distortion (quality) measure for planar meshes to meshes on parameterized surfaces. The resulting distortion (quality) measure is expressed in terms of the parametric coordinates of the nodes. This extended distortion (quality) measure can be used to check the quality and validity of both triangle and quadrilateral surface meshes. We also apply it to simultaneously smooth and untangle surface meshes by minimizing the extended distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes and therefore, the nodes always lie on the surface. Finally, we include several examples to illustrate the applicability of the proposed technique. Specifically, we extend several Jacobian-based measures, and we us them to smooth and untangle triangle and quadrilateral meshes on CAD surfaces.Peer ReviewedPostprint (author’s final draft

Efficient Computation of the Extrema of Algebraic Quality Measures for Curvilinear Finite Elements

The development of high-order computational methods for solving partial differential equations on unstructured grids has been underway for many years. Such methods critically depend on the availability of high-quality curvilinear meshes, as one bad element can degrade the solution in the whole domain. The usual way of generating curved meshes is to first generate a (high-quality) straight-sided mesh. Then, mesh entities that are classified on the boundaries of the domain are curved. This operation introduces a "shape-distortion" that should be controlled. Quality measures allow to quantify to which point an element is well-shaped. They also provide tools to improve the quality of meshes through optimization. In this work we propose an efficient method to compute several quality measures for curved elements, based on the Jacobian of the mapping between the straight-sided elements and the curved ones. Contrary to the approach presented in "A. Gargallo-Peiró, X. Roca, J. Peraire, and J. Sarrate. Distortion and quality measures for validating and generating high-order tetrahedral meshes. Engineering with Computers, pages 1–15, 2014.", which relies on an L2-norm over the elements, we compute the actual minimum and maximum of the local quality measure for each element. The method is an extension of previous works on the validity of those elements (A. Johnen et al., 2013). The key feature is that we can adaptively expand functions based on the Jacobian matrix and its determinant in terms of Bézier functions. Bézier functions have both properties of boundedness and positivity, which allow sharp computation of minimum or maximum of the interpolated functions

Strategies for optimization of hexahedral meshes and their comparative study

In this work, we study several strategies based on different objective functions for optimization of hexahedral meshes. We consider two approaches to construct objective functions. The first one is based on the decomposition of a hexahedron into tetrahedra. The second one is derived from the Jacobian matrix of the trilinear mapping between the reference and physical hexahedral element. A detailed description of all proposed strategies is given in the present work. Some computational experiments have been developed to test and compare the untangling capabilities of the considered objective functions. In the experiments, a sample of highly distorted hexahedral elements is optimized with the proposed objective functions, and the rate of success of each function is obtained. The results of these experiments are presented and analyzed.Secretaría de Estado de Universidades e Investigación del Ministerio de Economía y Competitividad del Gobierno de España; Programa de FPU del Ministerio de Educación, Cultura y Deporte; Programa de FPI propio de la ULPGC; Fondos FEDE