Discretization of the Region of Interest

[EN]The meccano method was recently introduced to construct simultaneously tetrahedral meshes and volumetric parameterizations of solids. The method requires the information of the solid geometry that is defined by its surface, a meccano, i.e., an outline of the solid defined by connected polyhedral pieces, and a tolerance that fixes the desired approximation of the solid surface. The method builds an adaptive tetrahedral mesh of the solid (physical domain) as a deformation of an appropriate tetrahedral mesh of the meccano (parametric domain). The main stages of the procedure involve an admissible mapping between the meccano and the solid boundaries, the nested Kossaczký’s refinement, and our simultaneous untangling and smoothing algorithm. In this chapter, we focus on the application of the method to build tetrahedral meshes over complex terrain, that is interesting for simulation of environmental processes. A digital elevation map of the terrain, the height of the domain, and the required orography approximation are given as input data. In addition, the geometry of buildings or stacks can be considered. In these applications, we have considered a simple cuboid as meccano.Ministerio de Economía y Competitividad, Gobierno de España; Fondos FEDER; Departamento de Educación, Junta de Castilla y León; CONACYT-SENER, Fondo Sectorial CONACYT SENER HIDROCARBUROS

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

This is the peer reviewed version of the following article: Ruiz Gironès , E., Oliver , A., Socorro, G., Cascón, J., Escobar, J.M., Montenegro, R., Sarrate, J. Insertion of triangulated surfaces into a meccano tetrahedral discretization by means of mesh refinement and optimization procedures. "International journal for numerical methods in engineering", 2018, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5706/pdf. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.In this paper, we present a new method for inserting several triangulated surfaces into an existing tetrahedral mesh generated by the meccano method. The result is a conformal mesh where each inserted surface is approximated by a set of faces of the final tetrahedral mesh. First, the tetrahedral mesh is refined around the inserted surfaces to capture their geometric features. Second, each immersed surface is approximated by a set of faces from the tetrahedral mesh. Third, following a novel approach, the nodes of the approximated surfaces are mapped to the corresponding immersed surface. Fourth, we untangle and smooth the mesh by optimizing a regularized shape distortion measure for tetrahedral elements in which we move all the nodes of the mesh, restricting the movement of the edge and surface nodes along the corresponding entity they belong to. The refining process allows approximating the immersed surface for any initial meccano tetrahedral mesh. Moreover, the proposed projection method avoids computational expensive geometric projections. Finally, the applied simultaneous untangling and smoothing process delivers a high-quality mesh and ensures that the immersed surfaces are interpolated. Several examples are presented to assess the properties of the proposed method.Peer ReviewedPostprint (author's final draft

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

[EN]In this paper, we present a new method for inserting several triangulated surfaces into an existing tetrahedral mesh generated by the meccano method. The result is a conformal mesh where each inserted surface is approximated by a set of faces of the final tetrahedral mesh. First, the tetrahedral mesh is refined around the inserted surfaces to capture their geometric features. Second, each immersed surface is approximated by a set of faces from the tetrahedral mesh. Third, following a novel approach, the nodes of the approximated surfaces are mapped to the corresponding immersed surface. Fourth, we untangle and smooth the mesh by optimizing a regularized shape distortion measure for tetrahedral elements in which we move all the nodes of the mesh, restricting the movement of the edge and surface nodes along the corresponding entity they belong to. The refining process allows approximating the immersed surface for any initial meccano tetrahedral mesh. Moreover, the proposed projection method avoids computational expensive geometric projections. Finally, the applied simultaneous untangling and smoothing process delivers a high‐quality mesh and ensures that the immersed surfaces are interpolated. Several examples are presented to assess the properties of the proposed method.FEDER y Gobierno de España (Ministerio de Economía y Competitividad) y ondo Sectorial CONACYT SENER HIDROCARBURO

Wind Forecasting Based on the HARMONIE Model and Adaptive Finite Elements

In this paper, we introduce a new method for wind field forecasting over complex terrain. The main idea is to use the predictions of the HARMONIE meso-scale model as the input data for an adaptive finite element mass-consistent wind model. The HARMONIE results (obtained with a maximum resolution of about 1 km) are refined in a local scale (about a few metres). An interface between both models is implemented in such a way that the initial wind field is obtained by a suitable interpolation of the HARMONIE results. Genetic algorithms are used to calibrate some parameters of the local wind field model in accordance to the HARMONIE data. In addition, measured data are considered to improve the reliability of the simulations. An automatic tetrahedral mesh generator, based on the meccano method, is applied to adapt the discretization to complex terrains. The main characteristic of the framework is a minimal user intervention. The final goal is to validate our model in several realistic applications on Gran Canaria island, Spain, with some experimental data obtained by the AEMET in their meteorological stations.This work has been supported by the Spanish Government, ‘‘Ministerio de Ciencia e Innovación’’, Grant Contracts: CGL2011-29396-C03-01 and CGL2011-29396-C03-02, and by ‘‘Junta de Castilla León’’, ‘‘Consejería de Educación’’, Grant Contract SA266A12-2

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

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

Structural Shape Optimization Based On The Use Of Cartesian Grids

Tesis por compendioAs ever more challenging designs are required in present-day industries, the traditional trial-and-error procedure frequently used for designing mechanical parts slows down the design process and yields suboptimal designs, so that new approaches are needed to obtain a competitive advantage. With the ascent of the Finite Element Method (FEM) in the engineering community in the 1970s, structural shape optimization arose as a promising area of application. However, due to the iterative nature of shape optimization processes, the handling of large quantities of numerical models along with the approximated character of numerical methods may even dissuade the use of these techniques (or fail to exploit their full potential) because the development time of new products is becoming ever shorter. This Thesis is concerned with the formulation of a 3D methodology based on the Cartesian-grid Finite Element Method (cgFEM) as a tool for efficient and robust numerical analysis. This methodology belongs to the category of embedded (or fictitious) domain discretization techniques in which the key concept is to extend the structural analysis problem to an easy-to-mesh approximation domain that encloses the physical domain boundary. The use of Cartesian grids provides a natural platform for structural shape optimization because the numerical domain is separated from a physical model, which can easily be changed during the optimization procedure without altering the background discretization. Another advantage is the fact that mesh generation becomes a trivial task since the discretization of the numerical domain and its manipulation, in combination with an efficient hierarchical data structure, can be exploited to save computational effort. However, these advantages are challenged by several numerical issues. Basically, the computational effort has moved from the use of expensive meshing algorithms towards the use of, for example, elaborate numerical integration schemes designed to capture the mismatch between the geometrical domain boundary and the embedding finite element mesh. To do this we used a stabilized formulation to impose boundary conditions and developed novel techniques to be able to capture the exact boundary representation of the models. To complete the implementation of a structural shape optimization method an adjunct formulation is used for the differentiation of the design sensitivities required for gradient-based algorithms. The derivatives are not only the variables required for the process, but also compose a powerful tool for projecting information between different designs, or even projecting the information to create h-adapted meshes without going through a full h-adaptive refinement process. The proposed improvements are reflected in the numerical examples included in this Thesis. These analyses clearly show the improved behavior of the cgFEM technology as regards numerical accuracy and computational efficiency, and consequently the suitability of the cgFEM approach for shape optimization or contact problems.La competitividad en la industria actual impone la necesidad de generar nuevos y mejores diseños. El tradicional procedimiento de prueba y error, usado a menudo para el diseño de componentes mecánicos, ralentiza el proceso de diseño y produce diseños subóptimos, por lo que se necesitan nuevos enfoques para obtener una ventaja competitiva. Con el desarrollo del Método de los Elementos Finitos (MEF) en el campo de la ingeniería en la década de 1970, la optimización de forma estructural surgió como un área de aplicación prometedora. El entorno industrial cada vez más exigente implica ciclos cada vez más cortos de desarrollo de nuevos productos. Por tanto, la naturaleza iterativa de los procesos de optimización de forma, que supone el análisis de gran cantidad de geometrías (para las se han de usar modelos numéricos de gran tamaño a fin de limitar el efecto de los errores intrínsecamente asociados a las técnicas numéricas), puede incluso disuadir del uso de estas técnicas. Esta Tesis se centra en la formulación de una metodología 3D basada en el Cartesian-grid Finite Element Method (cgFEM) como herramienta para un análisis numérico eficiente y robusto. Esta metodología pertenece a la categoría de técnicas de discretización Immersed Boundary donde el concepto clave es extender el problema de análisis estructural a un dominio de aproximación, que contiene la frontera del dominio físico, cuya discretización (mallado) resulte sencilla. El uso de mallados cartesianos proporciona una plataforma natural para la optimización de forma estructural porque el dominio numérico está separado del modelo físico, que podrá cambiar libremente durante el procedimiento de optimización sin alterar la discretización subyacente. Otro argumento positivo reside en el hecho de que la generación de malla se convierte en una tarea trivial. La discretización del dominio numérico y su manipulación, en coalición con la eficiencia de una estructura jerárquica de datos, pueden ser explotados para ahorrar coste computacional. Sin embargo, estas ventajas pueden ser cuestionadas por varios problemas numéricos. Básicamente, el esfuerzo computacional se ha desplazado. Del uso de costosos algoritmos de mallado nos movemos hacia el uso de, por ejemplo, esquemas de integración numérica elaborados para poder capturar la discrepancia entre la frontera del dominio geométrico y la malla de elementos finitos que lo embebe. Para ello, utilizamos, por un lado, una formulación de estabilización para imponer condiciones de contorno y, por otro lado, hemos desarrollado nuevas técnicas para poder captar la representación exacta de los modelos geométricos. Para completar la implementación de un método de optimización de forma estructural se usa una formulación adjunta para derivar las sensibilidades de diseño requeridas por los algoritmos basados en gradiente. Las derivadas no son sólo variables requeridas para el proceso, sino una poderosa herramienta para poder proyectar información entre diferentes diseños o, incluso, proyectar la información para crear mallas h-adaptadas sin pasar por un proceso completo de refinamiento h-adaptativo. Las mejoras propuestas se reflejan en los ejemplos numéricos presentados en esta Tesis. Estos análisis muestran claramente el comportamiento superior de la tecnología cgFEM en cuanto a precisión numérica y eficiencia computacional. En consecuencia, el enfoque cgFEM se postula como una herramienta adecuada para la optimización de forma.Actualment, amb la competència existent en la industria, s'imposa la necessitat de generar nous i millors dissenys . El tradicional procediment de prova i error, que amb freqüència es fa servir pel disseny de components mecànics, endarrereix el procés de disseny i produeix dissenys subòptims, pel que es necessiten nous enfocaments per obtindre avantatge competitiu. Amb el desenvolupament del Mètode dels Elements Finits (MEF) en el camp de l'enginyeria en la dècada de 1970, l'optimització de forma estructural va sorgir com un àrea d'aplicació prometedora. No obstant això, a causa de la natura iterativa dels processos d'optimització de forma, la manipulació dels models numèrics en grans quantitats, junt amb l'error de discretització dels mètodes numèrics, pot fins i tot dissuadir de l'ús d'aquestes tècniques (o d'explotar tot el seu potencial), perquè al mateix temps els cicles de desenvolupament de nous productes s'estan acurtant. Esta Tesi se centra en la formulació d'una metodologia 3D basada en el Cartesian-grid Finite Element Method (cgFEM) com a ferramenta per una anàlisi numèrica eficient i sòlida. Esta metodologia pertany a la categoria de tècniques de discretització Immersed Boundary on el concepte clau és expandir el problema d'anàlisi estructural a un domini d'aproximació fàcil de mallar que conté la frontera del domini físic. L'utilització de mallats cartesians proporciona una plataforma natural per l'optimització de forma estructural perquè el domini numèric està separat del model físic, que podria canviar lliurement durant el procediment d'optimització sense alterar la discretització subjacent. A més, un altre argument positiu el trobem en què la generació de malla es converteix en una tasca trivial, ja que la discretització del domini numèric i la seua manipulació, en coalició amb l'eficiència d'una estructura jeràrquica de dades, poden ser explotats per estalviar cost computacional. Tot i això, estos avantatges poden ser qüestionats per diversos problemes numèrics. Bàsicament, l'esforç computacional s'ha desplaçat. De l'ús de costosos algoritmes de mallat ens movem cap a l'ús de, per exemple, esquemes d'integració numèrica elaborats per poder capturar la discrepància entre la frontera del domini geomètric i la malla d'elements finits que ho embeu. Per això, fem ús, d'una banda, d'una formulació d'estabilització per imposar condicions de contorn i, d'un altra, desevolupem noves tècniques per poder captar la representació exacta dels models geomètrics Per completar la implementació d'un mètode d'optimització de forma estructural es fa ús d'una formulació adjunta per derivar les sensibilitats de disseny requerides pels algoritmes basats en gradient. Les derivades no són únicament variables requerides pel procés, sinó una poderosa ferramenta per poder projectar informació entre diferents dissenys o, fins i tot, projectar la informació per crear malles h-adaptades sense passar per un procés complet de refinament h-adaptatiu. Les millores proposades s'evidencien en els exemples numèrics presentats en esta Tesi. Estes anàlisis mostren clarament el comportament superior de la tecnologia cgFEM en tant a precisió numèrica i eficiència computacional. Així, l'enfocament cgFEM es postula com una ferramenta adient per l'optimització de forma.Marco Alacid, O. (2017). Structural Shape Optimization Based On The Use Of Cartesian Grids [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/86195TESISCompendi

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal