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

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

On curving high-order hexahedral meshes

We present a new definition of distortion and quality measures for high-order hexahedral (quadrilateral) elements. This definition leads to two direct applications. First, it can be used to check the validity and quality of a high-order hexahedral (quadrilateral) mesh. Second, it allows the generation of high-order curved meshes composed of valid and high-quality hexahedral (quadrilateral) elements. We describe a method to simultaneously smooth and untangle high-order hexahedral (quadrilateral) meshes by minimizing the proposed mesh distortion. Finally, we analyze the behavior of the proposed distortion measure and we present several results to illustrate the benefits of the mesh generation framework.Peer ReviewedPostprint (author's final draft

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

Wind field simulation with isogeometric analysis

[EN]For wind field simulation with isogeometric analysis, firstly it is necessary to generate a spline parameterization of the computational domain, which is an air layer above the terrain surface. This parameterization is created with the meccano method from a digital terrain model. The main steps of the meccano method for tetrahedral mesh generation were introduced in [1, 2]. Based on the volume parameterization obtained by the method, we can generate a mapping from the parametric T-mesh to the physical space [3, 4]. Then, this volumetric parameterization is used to generate a cubic spline representation of the physical domain for the application of isogeometric analysis. We consider a mass-consistent model [5] to compute the wind field simulation in the three-dimensional domain from wind measurements or a wind forecasted by a meteorological model (for example, WRF or HARMONIE). From these data, an interpolated wind field is constructed. The mass-consistent model obtains a new wind field approaching the interpolated one, but verifying the continuity equation (mass conservation) for constant density and the impermeabilitycondition on the terrain. This adjusting problem is solved by introducing a Lagrange multiplier, that is the solution of a Poisson problem. The resulting field is obtained from the interpolated one and the gradient of the Lagrange multiplier. It is well known that if we use classical Lagrange finite elements, the gradient of the numerical solution is discontinuous over the element boundary. The advantage of using isogeometric analysis with cubic polynomial basis functions [6, 7] is that we obtain a C2 continuity for the Lagrange multiplier in the whole domain. In consequence, the resulting wind field is better approximated. Applications of the proposed technique are presented.Ministerio de Economía y Competitividad del Gobierno de España; Fondos FEDER; CONACYT-SENE

Anisotropic Quadrilateral Mesh Optimization

In order to determine the validity and the quality of meshes, mesh optimization methods have been formulated with quality measures. The basic idea of mesh optimization is to relocate the vertices to obtain a valid mesh (untangling) or improve the mesh quality (smoothing), or both. We will look at a new algebraic way of calculating quality measure on quadrilateral meshes, based on triangular meshes in 2D as well as new optimization methods for simultaneous untangling and smoothing for severely deformed meshes. An innovative anisotropic diffusion method will be introduced for consideration of inner boundary deformation movements for quadrangle meshes in 2D

Spline parameterization method for 2D and 3D geometries based on T-mesh optimization

[EN]We present a method to obtain high quality spline parameterization of 2D and 3D geometries for their use in isogeometric analysis. As input data, the proposed method demands a boundary representation of the domain, and it constructs automatically a spline transformation between the physical and parametric domains. Parameterization of the interior of the object is obtained by deforming isomorphically an adapted parametric T-mesh onto the physical domain by applying a T-mesh untangling and smoothing procedure, which is the key of the method. Mesh optimization is based on the mean ratio shape quality measure. The spline representation of the geometry is calculated by imposing interpolation conditions using the data provided by one-to-one correspondence between the meshes of the parametric and physical domains. We give a detailed description of the proposed technique and show some examples. Also, we present some examples of the application of isogeometric analysis in geometries parameterized with our method.Secretaría de Estado de Universidades e Investigación del Ministerio de Economía y Competitividad del Gobierno de España y fondos FEDER; Programa de FPU 12/00202 del Ministerio de Educación, Cultura y Deporte; Programa de FPI propio de la Universidad de Las Palmas de Gran Canari