HDG para problemas de ondas : Análisis de la dispersión numérica y aplicaciones

En este trabajo se realiza un análisis de dispersión para los métodos Continuous Galerkin (CG) y Hybridizable Discontinuous Galerkin (HDG) al ser aplicados a la ecuación de Helmholtz en 1D. Se calcula la expresión del error de dispersión con aproximaciones numéricas de alto orden para CG y HDG. Luego, se realizan comparativas entre los métodos en función del parámetro de estabilidad utilizando aproximaciones de segundo, tercer y quinto orden. Finalmente, se presentan pruebas numéricas para diferentes grados de aproximación numérica, parámetro de estabilidad y nodos por longitud de onda que confirman los resultados

Conformal n-dimensional bisection for local refinement of unstructured simplicial meshes

[English] In n-dimensional adaptive applications, conformal simplicial meshes must be lo cally modified. One systematic local modification is to bisect the prescribed simplices while surrounding simplices are bisected to ensure conformity. Although there are many conformal bisection strategies, practitioners prefer the method known as the newest vertex bisection. This method guarantees key advantages for adaptivity when ever the mesh has a structure called reflectivity. Unfortunately, it is not known (i) how to extract a reflection structure from any unstructured conformal mesh for three or more dimensions. Fortunately, a conformal bisection method is suitable for adap tivity if it almost fulfills the newest vertex bisection advantages. These advantages are almost met by an existent multi-stage strategy in three dimensions. However, it is not known (ii) how to perform multi-stage bisection for more than three dimensions. This thesis aims to demonstrate that n-dimensional conformal bisection is possible for local refinement of unstructured conformal meshes. To this end, it proposes the following contributions. First, it proposes the first 4-dimensional two-stage method, showing that multi-stage bisection is possible beyond three dimensions. Second, fol lowing this possibility, the thesis proposes the first n-dimensional multi-stage method, and thus, it answers question (ii). Third, it guarantees the first 3-dimensional method that features the newest vertex bisection advantages, showing that these advantages are possible beyond two dimensions. Fourth, extending this possibility, the thesis guarantees the first n-dimensional marking method that extracts a reflection struc ture from any unstructured conformal mesh, and thus, it answers question (i). This answer proves that local refinement with the newest vertex bisection is possible in any dimension. Fifth, this thesis shows that the proposed multi-stage method al most fulfills the advantages of the newest vertex bisection. Finally, to visualize four dimensional meshes, it proposes a simple tool to slice pentatopic meshes. In conclusion, this thesis demonstrates that conformal bisection is possible for local refinement in two or more dimensions. To this end, it proposes two novel methods for unstructured conformal meshes, methods that will enable adaptive applications on n-dimensional complex geometry. [Español] En aplicaciones adaptativas n-dimensionales, las mallas simpliciales conformes deben modificarse localmente. Una modificación local sistemática es bisecar los símplices prescritos mientras que los símplices circundantes se bisecan para garantizar la conformidad. Aunque existen muchas estrategias conformes de bisección, en aplicaciones prácticas se prefiere el método conocido como newest vertex bisection (NVB). Este método garantiza las propiedades deseadas para la adaptatividad siempre y cuando la malla tenga una estructura llamada reflectividad. Desafortunadamente, no se sabe (i) cómo extraer una estructura de reflexión de cualquier malla conforme no estructurada para tres o más dimensiones. Afortunadamente, un método de bisección conforme es adecuado para la adaptatividad si casi cumple con las propiedades de NVB. Estas propiedades son casi satisfechas por una estrategia existente de múltiples etapas en tres dimensiones. Sin embargo, no se sabe (ii) cómo realizar una bisección en múltiples etapas para más de tres dimensiones. Esta tesis tiene como objetivo demostrar que la bisección conforme n-dimensional es posible para el refinamiento local de mallas conformes no estructuradas. Para ello propone las siguientes aportaciones. Primero, propone el primer método de dos etapas de 4 dimensiones, que muestra que la bisección de múltiples etapas es posible en más de tres dimensiones. En segundo lugar, siguiendo esta posibilidad, la tesis propone el primer método n-dimensional de múltiples etapas y, por tanto, responde a la pregunta (ii). En tercer lugar, garantiza el primer método tridimensional que presenta las propiedades NVB, lo que demuestra que estas propiedades son posibles más allá de dos dimensiones. En cuarto lugar, ampliando esta posibilidad, la tesis garantiza el primer método de marcado n-dimensional que extrae una estructura de reflexión de cualquier malla conforme no estructurada y, por lo tanto, responde a la pregunta (i). Esta respuesta demuestra que el refinamiento local con NVB es posible en cualquier dimensión. Quinto, esta tesis muestra que el método de múltiples etapas propuesto casi cumple con las propiedades de NVB. Finalmente, para visualizar mallas de cuatro dimensiones, propone una herramienta simple para cortar mallas pentatópicas. En conclusión, esta tesis demuestra que la bisección conforme es posible para el refinamiento local en dos o más dimensiones. Con este fin, propone dos métodos novedosos para mallas conformes no estructuradas, métodos que harán posible aplicaciones adaptativas en geometría compleja n-dimensionalPostprint (published version

Control of robotic systems using differential flatness

In this work, a coordinate change of state variables is performed for drift-less systems of dimension m+2 with 2 inputs using Goursat Normal Form. Then, we define a feedback law that will allow us to convert the original system into chained form. Later on, we find the flat outputs and define a new feedback law. Finally, numerical simulations are presented for a planar space robot, a mobile robot with a trailer and a N-trailer

Local bisection for conformal refinement of unstructured 4D simplicial meshes

We present a conformal bisection procedure for local refinement of 4D unstructured simplicial meshes with bounded minimum shape quality. Specifically, we propose a recursive refine-to-conformity procedure in two stages, based on marking bisection edges on different priority levels and defining specific refinement templates. Two successive applications of the first stage ensure that any 4D unstructured mesh can be conformingly refined. In the second stage, the successive refinements lead to a cycle in the number of generated similarity classes and thus, we can ensure a bound over the minimum shape quality. In the examples, we check that after successive refinement the mesh quality does not degenerate. Moreover, we refine a 4D unstructured mesh and a space-time mesh (3D + 1D) representation of a moving object

Visualization of pentatopic meshes

We propose a simple tool to visualize 4D unstructured pentatopic meshes. The method slices unstructured 4D pentatopic meshes (fields) with an arbitrary 3D hyperplane and obtains a conformal 3D unstructured tetrahedral representation of the mesh (field) slice ready to explore with standard 3D visualization tools. The results show that the method is suitable to visually explore 4D unstructured meshes. This capability has facilitated devising our 4D bisection method, and thus, we think it might be useful when devising new 4D meshing methods. Furthermore, it allows visualizing 4D scalar fields, which is a crucial feature for our space-time application

Local bisection for conformal refinement of unstructured 4D simplicial meshes

We present a conformal bisection procedure for local refinement of 4D unstructured simplicial meshes with bounded minimum shape quality. Specifically, we propose a recursive refine to conformity procedure in two stages, based on marking bisection edges on different priority levels and defining specific refinement templates. Two successive applications of the first stage ensure that any 4D unstructured mesh can be conformingly refined. In the second stage, the successive refinements lead to a cycle in the number of generated similarity classes and thus, we can ensure a bound over the minimum shape quality. In the examples, we check that after successive refinement the mesh quality does not degenerate. Moreover, we refine a 4D unstructured mesh and a space-time mesh (3D + 1D) representation of a moving object.This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 715546. This work has also received funding from the Generalitat de Catalunya under grant number 2017 SGR 1731. The work of X. Roca has been partially supported by the Spanish Ministerio de Economía y Competitividad under the personal grant agreement RYC-2015-01633.Peer ReviewedPostprint (author's final draft

Bisecting with optimal similarity bound on 3D unstructured conformal meshes

We propose a new method to mark for bisection the edges of an arbitrary 3D unstructured conformal mesh. For these meshes, the approach conformingly marks all the tetrahedra with coplanar edge marks. To this end, the method needs three key ingredients. First, we propose a specific edge ordering. Second, marking with this ordering, we guarantee that the mesh becomes conformingly marked. Third, we also ensure that all the marks are coplanar in each tetrahedron. To demonstrate the marking method, we implement an existent marked bisection approach. Using this implementation, we mark and then locally refine 3D unstructured conformal meshes. We conclude that the resulting marked bisection features an optimal bound of similarity classes per tetrahedron.This project has received funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme under grant agreement No 715546. This work has also received funding from the Generalitat de Catalunya under grant number 2017 SGR 1731. The work of X. Roca has been partially supported by the Spanish Ministerio de Economía y Competitividad under the personal grant agreement RYC-2015-01633.Peer ReviewedPostprint (published version

HDG para problemas de ondas : Análisis de la dispersión numérica y aplicaciones

En este trabajo se realiza un análisis de dispersión para los métodos Continuous Galerkin (CG) y Hybridizable Discontinuous Galerkin (HDG) al ser aplicados a la ecuación de Helmholtz en 1D. Se calcula la expresión del error de dispersión con aproximaciones numéricas de alto orden para CG y HDG. Luego, se realizan comparativas entre los métodos en función del parámetro de estabilidad utilizando aproximaciones de segundo, tercer y quinto orden. Finalmente, se presentan pruebas numéricas para diferentes grados de aproximación numérica, parámetro de estabilidad y nodos por longitud de onda que confirman los resultados

HDG para problemas de ondas : Análisis de la dispersión numérica y aplicaciones

En este trabajo se realiza un análisis de dispersión para los métodos Continuous Galerkin (CG) y Hybridizable Discontinuous Galerkin (HDG) al ser aplicados a la ecuación de Helmholtz en 1D. Se calcula la expresión del error de dispersión con aproximaciones numéricas de alto orden para CG y HDG. Luego, se realizan comparativas entre los métodos en función del parámetro de estabilidad utilizando aproximaciones de segundo, tercer y quinto orden. Finalmente, se presentan pruebas numéricas para diferentes grados de aproximación numérica, parámetro de estabilidad y nodos por longitud de onda que confirman los resultados

Local bisection for conformal refinement of unstructured 4D simplicial meshes

We present a conformal bisection procedure for local refinement of 4D unstructured simplicial meshes with bounded minimum shape quality. Specifically, we propose a recursive refine-to-conformity procedure in two stages, based on marking bisection edges on different priority levels and defining specific refinement templates. Two successive applications of the first stage ensure that any 4D unstructured mesh can be conformingly refined. In the second stage, the successive refinements lead to a cycle in the number of generated similarity classes and thus, we can ensure a bound over the minimum shape quality. In the examples, we check that after successive refinement the mesh quality does not degenerate. Moreover, we refine a 4D unstructured mesh and a space-time mesh (3D + 1D) representation of a moving object