Generation of Curved High-order Meshes with Optimal Quality and Geometric Accuracy

We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric Full-size image (<1 K)-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the Full-size image (<1 K)-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.This research was partially supported by the Spanish Ministerio de Economía y Competitividad under grand contract CTM2014-55014-C3-3-R, and by the Government of Catalonia under grand contract 2014-SGR-1471. The work of the last author was supported by the European Commission through the Marie Sklodowska-Curie Actions (HiPerMeGaFlows project).Peer ReviewedPostprint (published version

Generation of curved high-order meshes with optimal quality and geometric accuracy

We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric L2-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the L2-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.Peer ReviewedPostprint (published version

Automatic generation of quadrilateral structured meshes using linear programming and transfinite interpolation

This paper presents an implementation of a structured quadrilateral algorithm called submapping. The meshing procedure splits the geometry into patches and then meshes each patch separately preserving the mesh compatibility between patches via an linear integer problem. The submapping algorithm can only be applied to geometries such that each angle between consecutive edges is, approximately, an integer multiple of ¼=2

Estudio numérico del control exacto desde el contorno de la ecuación Korteweg–de Vries lineal

En este artículo primero se desarrolla un esquema en diferencias finitas de segundo orden, tanto en la discretización espacial como temporal, para la resolución de la versión lineal de la ecuación de Korteweg–de Vries (KdV) en dominios acotados. Seguidamente, y basándose en el esquema descrito anteriormente, se presenta un modelo numérico para implementar el método de unicidad de Hilbert (HUM) al control de la ecuación KdV lineal donde el control actúa sobre el contorno del domino. En particular, se encuentra una solución al problema de control en el subespacio de funciones lineales a trozos definidas sobre la discretización del dominio. Finalmente, se presentan varios ejemplos numéricos que ponen de manifiesto tanto el rango de aplicación del método desarrollado como la exactitud del mismo.Peer Reviewe

Generación de mallas de hexaedros en geometrías múltiplemente conexas mediante submapping

En este trabajo se presenta una mejora del método de submapping para ampliar su aplicabilidad. El algoritmo de submapping tiene dos limitaciones principales. La primera de ellas es que sólo se puede aplicar en geometrías tales que los ángulos entre las aristas y las caras de la geometría tienen que ser, aproximadamente, múltiplos enteros de π/2. La segunda limitación es que la geometría tiene que ser simplemente conexa. En este trabajo se presenta una extensión del algoritmo de submapping para mallar geometrías con agujeros totalmente interiores.Peer Reviewe

Discretization of multiply connected surfaces using submapping

Una de las técnicas más utilizadas para generar mallas estructuradas de cuadriláteros es el método de submapping. Este método descompone la geometría en piezas lógicamente equivalentes a un cuadrilátero y después malla cada una de ellas por separado manteniendo la compatibilidad de la malla mediante la resolución de un problema lineal entero. El algoritmo de submapping tiene dos limitaciones principales. La primera de ellas es que sólo se puede aplicar en geometrías tales que el ángulo entre dos aristas consecutivas es, aproximadamente, un múltiplo entero de π/2. La segunda limitación es que la geometría tiene que ser simplemente conexa. Con el objetivo de mitigar estas restricciones, en este artículo se presentan dos modificaciones originales que permiten reducir el efecto de dichas limitaciones. Finalmente, se presentan diversos ejemplos numéricos que ponen de manifiesto la robustez y la aplicabilidad de los algoritmos desarrollados.Peer Reviewe

An interactive mesh generation environment for geometry-based simulations. Progress updated

We presented an overview of an interactive mesh generation environment in our previous work that it is being developed in the Laboratori de Càlcul Numèric (LaCàN). We started the development of the software in order to unify available legacy code, new developments and research algorithms in only one mesh generation package. This paper presents an update to the last environment overview after significant implementation and conceptual development. We provide a brief summary of: New features, as submapping and data output suitable for geometry-based methods as; current developments, as GUI improvement and 3D modelling tools; and future features, such as the command interpreter. Finally, we conclude that chosen mesh generation paradigm and software engineering concepts have allowed us to improve and scale the environment since last revision

Numerical representation of the quality measures of triangles and triangular meshes

In this note a new procedure to represent the quality measure for triangles is proposed. The triangles are identified by their three angles and are represented in a bounded domain, called angle representation region, according to the area co-ordinates, which are common and well known by finite element users. The developed representation can also be used in order to visualize the characteristics of any quality measure. This new procedure is extended to graphically represent triangular meshes in the angle representation region.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

An HDG formulation for incompressible and immiscible two-phase porous media flow problems

This is an Accepted Manuscript of an article published by Taylor & Francis Group in International journal of computational fluid dynamics on 27/05/2019, available online at: https://tandfonline.com/doi/abs/10.1080/10618562.2019.1617855We develop a high-order hybridisable discontinuous Galerkin (HDG) formulation to solve the immiscible and incompressible two-phase flow problem in a heterogeneous porous media. The HDG method is locally conservative, has fewer degrees of freedom than other discontinuous Galerkin methods due to the hybridisation procedure, provides built-in stabilisation for arbitrary polynomial degrees and, if the error of the temporal discretisation is low enough, the pressure, the saturation and their fluxes converge with order P+1 in L2-norm, being P the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P+2 in L2-norm for the scalar variables. All of these advantages make the HDG method suitable for solving multiphase flow trough porous media. We show numerical evidence of the convergences rates. Finally, to assess the capabilities of the proposed formulation, we apply it to several cases of water-flooding technique for oil recovery.Peer ReviewedPostprint (author's final draft