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  • Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems

     Giorgiani, Giorgio; Fernandez Mendez, Sonia; Huerta Cerezuela, Antonio
    International journal for numerical methods in fluids
    Vol. 72, num. 12, p. 1244-1262
    DOI: 10.1002/fld.3784
    Date of publication: 2013
    Journal article

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    A p-adaptive hybridizable discontinuous Galerkin method for the solution of wave problems is presented in a challenging engineering problem. Moreover, its performance is compared with a high-order continuous Galerkin. The hybridization technique allows to reduce the coupled degrees of freedom to only those on the mesh element boundaries, whereas the particular choice of the numerical fluxes opens the path to a superconvergent postprocessed solution. This superconvergent postprocessed solution is used to construct a simple and inexpensive error estimator. The error estimator is employed to obtain solutions with the prescribed accuracy in the area (or areas) of interest and also drives a proposed iterative mesh adaptation procedure. The proposed method is applied to a nonhomogeneous scattering problem in an unbounded domain. This is a challenging problem because, on the one hand, for high frequencies, numerical difficulties are an important issue because of the loss of the ellipticity and the oscillatory behavior of the solution. And on the other hand, it is applied to real harbor agitation problems. That is, the mild slope equation in frequency domain (Helmholtz equation with nonconstant coefficients) is solved on real geometries with the corresponding perfectly matched layer to damp the diffracted waves. The performance of the method is studied on two practical examples. The adaptive hybridizable discontinuous Galerkin method exhibits better efficiency compared with a high-order continuous Galerkin method using static condensation of the interior nodes.

  • High-order continuous and discontinuous Galerkin methods for wave problems

     Giorgiani, Giorgio; Modesto Galende, David; Fernandez Mendez, Sonia; Huerta Cerezuela, Antonio
    International journal for numerical methods in fluids
    Vol. 73, num. 10, p. 883-903
    DOI: 10.1002/fld.3828
    Date of publication: 2013-12-10
    Journal article

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    Three Galerkin methods-continuous Galerkin, Compact Discontinuous Galerkin, and hybridizable discontinuous Galerkin-are compared in terms of performance and computational efficiency in 2-D scattering problems for low and high-order polynomial approximations. The total number of DOFs and the total runtime are used for this correlation as well as the corresponding precision. The comparison is carried out through various numerical examples. The superior performance of high-order elements is shown. At the same time, similar capabilities are shown for continuous Galerkin and hybridizable discontinuous Galerkin, when high-order elements are adopted, both of them clearly outperforming compact discontinuous Galerkin.

    hree Galerkin methods—continuous Galerkin, Compact Discontinuous Galerkin, and hybridizable discontinuous Galerkin—are compared in terms of performance and computational efficiency in 2-D scattering problems for low and high-order polynomial approximations. The total number of DOFs and the total runtime are used for this correlation as well as the corresponding precision. The comparison is carried out through various numerical examples. The superior performance of high-order elements is shown. At the same time, similar capabilities are shown for continuous Galerkin and hybridizable discontinuous Galerkin, when high-order elements are adopted, both of them clearly outperforming compact discontinuous Galerkin

  • Adaptive hybrid discontinuous methods for fluid and wave problems  Open access

     Giorgiani, Giorgio
    Department of Applied Mathematics III, Universitat Politècnica de Catalunya
    Theses

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    Esta tesis propone una técnica de adaptividad p para el método de Galerkin discontinuo híbrido (HDG)El método HDG es un nuevo método de Galerkin discontinuo (DG) con características interesantes. Tiene todas las ventajas de los métodos discontinuos, como la estabilización inherente y las propiedades conservación local, y además permite reducir los grados de libertad acoplados del problema a aquellos de una aproximación de la solución definida sólo en las caras de la malla . Además, las propiedades de convergencia de la solución de HDG permiten postprocesar la solución elemento a elemento, obteniendo una solución superconvergente.Debido al carácter discontinuo de la aproximación en HDG, es posible implementar con simplicidad cálculos con p-variable. En esta tesis el postproceso es superconvergente se utiliza para definir un estimador de error fiable y computacionalmente barato, que se utiliza para guiar un proceso de adaptativo automatico. Se modifica el grado del polinomio en cada elemento con el objetivo de obtener una distribución de error uniforme por debajo de una tolerancia definida por usuario. No se aplican modificas topológicas a la discretización, entonces el proceso adaptativo es muy rapido.Al principio, la tecnica de HDG p-adaptativo se aplica a la solución de problemas de onda. En particular, la ecuación de ¿pendiente suave¿ se utiliza para modelar el problema de propagación de ondas marina en áreas costeras y puertos. El método de HDG se compara con el método de elementos finitos Galerkin continuo (CG), que es hoy en día el método más común utilizado en la práctica de la ingenieril para este tipo de aplicaciones. Experimentos numéricos revelan que la eficiencia de HDG es paracida a la eficiencia de CG para cálculos de grado uniforme, y claramente superior a otros métodos DG como el método de Galerkin discontinuo compacto.Cuando se considere p-adaptatividad, se obtiene un importante ahorro en el coste computacional.Luego, la metodología se aplica a la solución de las ecuaciones de Navier-Stokes para flujo incompresible laminare. Se consideran aplicaciones tanto en estado estacionario como en transitorio. Se presentan varios experimentos numéricos en 2D y 3D, incluyendo ejemplos académicos y aplicaciones de interese ingenieril. A pesar de la simplicidad y bajo coste del estimador de error, se demuestra su alta eficiencia usando ejemplos analíticos. Además, aunque la técnica adaptativa sea basada en una estimación de error para sólo el campo de velocidad, se alcanza alta precisión para todas las variables, y una evaluación precisa de las fuerzas fluidodinamicas. En particular, los altos grados polinomiales son automáticamente situado a lo largo de capas límite, lo que reduce la necesidad de elementos altamente distorsionados. Pruebas numéricas muestran una importante reducción en el coste computacional, comparado con cálculos de grado uniforme, para los cálculos estacionarios y transitorios.

    This PhD thesis proposes a p-adaptive technique for the Hybridizable Discontinuous Galerkin method (HDG). The HDG method is a novel discontinuous Galerkin method (DG) with interesting characteristics. While retaining all the advantages of the common DG methods, such as the inherent stabilization and the local conservation properties, HDG allows to reduce the coupled degrees of freedom of the problem to those of an approximation of the solution de¿ned only on the faces of the mesh. Moreover, the convergence properties of the HDG solution allow to perform an element-by-element postprocess resulting in a superconvergent solution. Due to the discontinuous character of the approximation in HDG, p-variable computations are easily implemented. In this work the superconvergent postprocess is used to de¿ne a reliable and computationally cheap error estimator, that is used to drive an automatic adaptive process. The polynomial degree in each element is automatically adjusted aiming at obtaining a uniform error distribution below a user de¿ned tolerance. Since no topological modi¿cation of the discretization is involved, fast adaptations of the mesh are obtained. First, the p-adaptive HDG is applied to the solution of wave problems. In particular, the Mild Slope equation is used to model the problem of sea wave propagation is coastal areas and harbors. The HDG method is compared with the continuous Galerkin (CG) ¿nite element method, which is nowadays the common method used in the engineering practice for this kind of applications. Numerical experiments reveal that the e¿ciency of HDG is close to CG for uniform degree computations, clearly outperforming other DG methods such as the Compact Discontinuous Galerkin method. When p-adaptivity is considered, an important saving in computational cost is shown. Then, the methodology is applied to the solution of the incompressible Navier-Stokes equations for the simulation of laminar ¿ows. Both steady state and transient applications are considered. Various numerical experiments are presented, in 2D and 3D, including academic examples and more challenging applications of engineering interest. Despite the simplicity and low cost of the error estimator, high e¿ciency is exhibited for analytical examples. Moreover, even though the adaptive technique is based on an error estimate for just the velocity ¿eld, high accuracy is attained for all variables, with sharp resolution of the key features of the ¿ow and accurate evaluation of the ¿uid-dynamic forces. In particular, high degrees are automatically located along boundary layers, reducing the need for highly distorted elements in the computational mesh. Numerical tests show an important reduction in computational cost, compared to uniform degree computations, for both steady and unsteady computations.

  • Hybridizable discontinuous Galerkin with degree adaptivity for wave and flow problems

     Giorgiani, Giorgio; Fernandez Mendez, Sonia; Huerta Cerezuela, Antonio
    Computational Methods for Coupled Problems in Science and Engineering
    p. a970
    Presentation's date: 2013-06-17
    Presentation of work at congresses

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  • 2D comparative study of vertical axis wind and tidal turbines with a p-adaptive hybridizable Discontinuous Galerkin method and ANSYS Fluent

     de Villardi de Montlaur, Adeline; Giorgiani, Giorgio
    Congreso de Métodos Numéricos en Ingeniería
    Presentation's date: 2013-06-25
    Presentation of work at congresses

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  • On the numerical efficiency of model reduction and high-order discontinuous Galerkin for waves and fluid problems

     Modesto Galende, David; Giorgiani, Giorgio; Fernandez Mendez, Sonia; Huerta Cerezuela, Antonio
    Congreso de Métodos Numéricos en Ingeniería
    p. 1
    Presentation's date: 2013-06-26
    Presentation of work at congresses

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  • SIMULACIONES NUMERICAS DE ALTA FIDELIDAD PARA UNA INGENIERIA ASISTIDA POR ORDENADOR FIABLE

     Fernandez Mendez, Sonia; Sarrate Ramos, Jose; Roca Navarro, Francisco Javier; Sala Lardies, Esther; Casoni Rero, Eva; Giorgiani, Giorgio
    Competitive project

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    Hybridizable discontinuous Galerkin p-adaptivity for wave problems  Open access

     Giorgiani, Giorgio; Fernandez Mendez, Sonia; Huerta Cerezuela, Antonio
    European Community on Computational Methods in Applied Sciences Young Investigators Conference
    p. 1-10
    Presentation's date: 2012-04-24
    Presentation of work at congresses

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    A p-adaptive Hybridizable Discontinuous Galerkin (HDG) method is presented for the solution of wave problems. The HDG method allows to drastically reduce the coupled degrees of freedom of the computation seeking for an approximation of the solution that is defined only on the edges of the mesh. The particular choice of the numerical fluxes driven by the hybridization technique allows to obtain an optimally converging solution not only for the primal unknown but also for its derivative. This characteristic allows to perform a post-process of the solution that provides a super-convergent solution. The discontinuous character of the solution provides an optimal framework for a p-adaptive technique. The post-processed solution of the HDG method is used to construct a cheap and reliable error estimator that drives an element by element modification of the approximation degree. The proposed p-adaptive HDG method is compared with high-order CG computation with static condensation of the interior nodes. A challenging problem is considered for the comparison: a non-homogeneous scattering problem in an open domain.

  • Hybridizable discontinuous Galerkin p-adaptivity for fluid problems

     Giorgiani, Giorgio; Fernandez Mendez, Sonia; Huerta Cerezuela, Antonio
    European Congress on Computational Methods in Applied Sciences and Engineering
    p. 1-15
    Presentation's date: 2012-09-12
    Presentation of work at congresses

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  • Efficiency and accuracy of high-order computations and reduced order modelling in coastal engineering wave propagation problems

     Modesto Galende, David; Giorgiani, Giorgio; Zlotnik, Sergio; Huerta Cerezuela, Antonio
    European Community on Computational Methods in Applied Sciences Young Investigators Conference
    p. 1-10
    Presentation's date: 2012-04-24
    Presentation of work at congresses

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    Several numerical issues have to be considered when solving wave propagation problems, between which there are the artificial boundary conditions, the small geometrical features that can be influential or the variable coefficients. Apart from them, two issues are mainly addressed and discussed. Firstly, low order elements need very high wave resolution for capturing the solution in the area of interest, leading to extremely dense meshes. High-order finite elements are proposed to be an efficient and accurate solution for solving the problem. Secondly, the very large number of test cases. When designing harbour models, a huge number of incident waves, in term of wavelengths and directions, have to be studied. The excessive computational cost to carry out all the possible direct problems prevents the whole data evaluation, inducing the lost of important information. Reduced order models may be an alternative if they are computable, efficient and accurate. The applicability of Proper Generalized Decomposition (PGD) is exploited. Unlike previous PGD contributions, which deal with elliptic problems, the present work is focused on a more challenging scenario for the separable representation due to the loss of the elliptic behaviour. The proposed PGD involves a separable representation of the unknown reflected wave in space, wave number and angle of incidence. Such decomposition appears to be really interesting for practical purpose, where goal-oriented results are critical for a wide range of frequencies and incident waves. Moreover, when accuracy and efficiency are of concern, the number of terms in the reduced model are determined by means of an error estimation based on the dual formulation of the problem.

  • High-order discontinuous Galerkin methods: are they competitive?

     Huerta Cerezuela, Antonio; Casoni Rero, Eva; Giorgiani, Giorgio; Modesto Galende, David; Fernandez Mendez, Sonia; Peraire, Jaume; Angeloski, Aleksandar
    International Conference on Finite Elements in Flow Problems
    p. 48
    Presentation's date: 2011-03-25
    Presentation of work at congresses

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  • Cálculos p-adaptativos de Galerkin discontinuo de alto orden

     Huerta Cerezuela, Antonio; Giorgiani, Giorgio; Modesto Galende, David; Fernandez Mendez, Sonia
    Congreso en Métodos Numéricos em Engenharia
    p. 245-
    Presentation's date: 2011-06-24
    Presentation of work at congresses

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  • MÈTODES NUMÈRICS EN CIÈNCIES APLICADES I ENGINYERIA

     Roca Navarro, Francisco Javier; Giorgiani, Giorgio; Zlotnik, Sergio; Fernandez Mendez, Sonia; Rodriguez Ferran, Antonio; Muñoz Romero, Jose Javier; Arias Vicente, Irene; de Villardi de Montlaur, Adeline; Sarrate Ramos, Jose; Diez Mejia, Pedro; Arroyo Balaguer, Marino; Sevilla Cardenas, Ruben; Parés Mariné, Núria; Casoni Rero, Eva; Ruiz Girones, Eloi; Modesto Galende, David; Millan, Raul Daniel; Abdollahi Hosnijeh, Amir; Steffens, Lindaura Maria; Discacciati, Marco; Shen, Yongxing; Rahimi Lenji, Mohammad; Tamayo Mas, Elena; Diaz Cereceda, Cristina; Prat Robles, David; Verdugo Rojano, Francesc; Zhang, Kuan; Estela Carbonell, M. Rosa; Peco Regales, Christian; Huerta Cerezuela, Antonio
    Competitive project

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