Dg And Hdg Methods For Curved Structures

We introduce and analyze discontinuous Galerkin methods for a Naghdi type arch model. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the arch. These methods are thus free from membrane and shear locking. We also prove that, when polynomials of degree $k$ are used, {\em all} the numerical traces superconverge with a rate of order h 2k+1. Based on the superconvergent phenomenon and we show how to post-process them in an element-by-element fashion to obtain a far better approximation. Indeed, we prove that, if polynomials of degree k are used, the post-processed approximation converges with order 2k+1 in the L2-norm throughout the domain. This has to be contrasted with the fact that before post-processing, the approximation converges with order k+1 only. Moreover, we show that this superconvergence property does not deteriorate as the thickness of the arch becomes extremely small. Since the DG methods suffer from too many degree of freedoms we introduce and analyze a class of hybridizable discontinuous Galerkin (HDG) methods for Naghdi arches. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to approximations to the transverse and tangential displacement and bending moment at the element boundaries. The error analysis of the methods is based on the use of a projection especially designed to fit the structure of the numerical traces of the method. This property allows to prove in a very concise manner that the projection of the errors is bounded in terms of the distance between the exact solution and its projection. The study of the influence of the stabilization function on the approximation is then reduced to the study of how they affect the approximation properties of the projection in a single element. Consequently, we prove that HDG methods have the same result as DG methods. At the end of the thesis, we talk a little bit of shell problems

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems

A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs

We investigated an hybridizable discontinuous Galerkin (HDG) method for a convection diffusion Dirichlet boundary control problem in our earlier work [SIAM J. Numer. Anal. 56 (2018) 2262-2287] and obtained an optimal convergence rate for the control under some assumptions on the desired state and the domain. In this work, we obtain the same convergence rate for the control using a class of embedded DG methods proposed by Nguyen, Peraire and Cockburn [J. Comput. Phys. vol. 302 (2015), pp. 674-692] for simulating fluid flows. Since the global system for embedded DG methods uses continuous elements, the number of degrees of freedom for the embedded DG methods are smaller than the HDG method, which uses discontinuous elements for the global system. Moreover, we introduce a new simpler numerical analysis technique to handle low regularity solutions of the boundary control problem. We present some numerical experiments to confirm our theoretical results

A hybridizable discontinuous Galerkin method for both thin and 3D nonlinear elastic structures

We present a 3D hybridizable discontinuous Galerkin (HDG) method for nonlinear elasticity which can be efficiently used for thin structures with large deformation. The HDG method is developed for a three-field formulation of nonlinear elasticity and is endowed with a number of attractive features that make it ideally suited for thin structures. Regarding robustness, the method avoids a variety of locking phenomena such as membrane locking, shear locking, and volumetric locking. Regarding accuracy, the method yields optimal convergence for the displacements, which can be further improved by an inexpensive postprocessing. And finally, regarding efficiency, the only globally coupled unknowns are the degrees of freedom of the numerical trace on the interior faces, resulting in substantial savings in computational time and memory storage. This last feature is particularly advantageous for thin structures because the number of interior faces is typically small. In addition, we discuss the implementation of the HDG method with arc-length algorithms for phenomena such as snapthrough, where the standard load incrementation algorithm becomes unstable. Numerical results are presented to verify the convergence and demonstrate the performance of the HDG method through simple analytical and popular benchmark problems in the literature

High-order hybridizable discontinuous Galerkin method for viscous compressible flows

Computational Fluid Dynamics (CFD) is an essential tool for engineering design and analysis, especially in applications like aerospace, automotive and energy industries. Nowadays most commercial codes are based on Finite Volume (FV) methods, which are second order accurate, and simulation of viscous compressible flow around complex geometries is still very expensive due to large number of low-order elements required. One the other hand, some sophisticated physical phenomena, like aeroacoustics, vortex dominated flows and turbulence, need very high resolution methods to obtain accurate results. High-order methods with their low spatial discretization errors, are a possible remedy for shortcomings of the current CFD solvers. Discontinuous Galerkin (DG) methods have emerged as a successful approach for non-linear hyperbolic problems and are widely regarded very promising for next generation CFD solvers. Their efficiency for high-order discretization makes them suitable for advanced physical models like DES and LES, while their stability in convection dominated regimes is also a merit of them. The compactness of DG methods, facilitate the parallelization and their element-by-element discontinuous nature is also helpful for adaptivity. This PhD thesis focuses on the development of an efficient and robust high-order Hybridizable Discontinuous Galerkin (HDG) Finite Element Method (FEM) for compressible viscous flow computations. HDG method is a new class of DG family which enjoys from merits of DG but has significantly less globally coupled unknowns compared to other DG methods. Its features makes HDG a possible candidate to be investigated as next generation high-order tools for CFD applications. The first part of this thesis recalls the basics of high-order HDG method. It is presented for the two-dimensional linear convection-diffusion equation, and its accuracy and features are investigated. Then, the method is used to solve compressible viscous flow problems modelled by non-linear compressible Navier-Stokes equations; and finally a new linearized HDG formulation is proposed and implemented for that problem, all using high-order approximations. The accuracy and efficiency of high-order HDG method to tackle viscous compressible flow problems is investigated, and both steady and unsteady solvers are developed for this purpose. The second part is the core of this thesis, proposing a novel shock-capturing method for HDG solution of viscous compressible flow problems, in the presence of shock waves. The main idea is to utilize the stabilization of numerical fluxes, via a discontinuous space of approximation inside the elements, to diminish or remove the oscillations in the vicinity of discontinuity. This discontinuous nodal basis functions, leads to a modified weak form of the HDG local problem in the stabilized elements. First, the method is applied to convection-diffusion problems with Bassi-Rebay and LDG fluxes inside the elements, and then, the strategy is extended to the compressible Navier-Stokes equations using LDG and Lax-Friedrichs fluxes. Various numerical examples, for both convection-diffusion and compressible Navier-Stokes equations, demonstrate the ability of the proposed method, to capture shocks in the solution, and its excellent performance in eliminating oscillations is the vicinity of shocks to obtain a spurious-free high-order solution.Dinámica de Fluidos Computacional (CFD) es una herramienta esencial para el diseño y análisis en ingeniería, especialmente en aplicaciones de ingeniería aeroespacial, automoción o energía, entre otros. Hoy en día, la mayoría de los códigos comerciales se basan en el método de Volúmenes Finitos (FV), con precisión de segundo orden. Sin embargo, la simulación del flujo compresible y viscoso alrededor de geometrías complejas mediante estos métodos es todavía muy cara, debido al gran número de elementos de orden bajo requeridos. Algunos fenómenos físicos sofisticados, por ejemplo en aeroacústica, presentan vórtices y turbulencias, y necesitan métodos de muy alta resolución para obtener resultados precisos. Los métodos de alto orden, con bajos errores de discretización espacial, pueden superar las deficiencias de los actuales códigos de CFD. Los métodos Galerkin discontinuos (DG) han surgido como un enfoque exitoso para problemas hiperbólicos no lineales, y son ampliamente considerados muy prometedores para la próxima generación de códigos de CFD. Su eficiencia de alto orden los hace adecuados para modelos físicos avanzados como DES (Direct Numerial Simulation) y LES (Large Eddy Simulation), mientras que su estabilidad en problemas de convención dominante es también un mérito de ellos. La compacidad de los métodos DG facilita la paralelización, y su naturaleza discontinua es también útil para la adaptabilidad. Esta tesis doctoral se centra en el desarrollo de un método de alto orden, eficiente y robusto, basado en el método de elementos finitos Hybridizable Discontinuous Galerkin (HDG), para cálculos de flujo viscoso y compresible. HDG es un método novedoso, con los méritos de los métodos DG, pero con significativamente menos grados de libertad a nivel global en comparación con otros métodos discontinuos. Sus características hacen de HDG un candidato prometedor a ser investigado como una herramienta de alto orden de próxima generación para aplicaciones de CFD. La primera parte de esta tesis, recuerda los fundamentos del método HDG. Se presenta la aplicación del método para la ecuación de convección-difusión lineal en dos dimensiones, y se investiga su precisión y sus características. Posteriormente, el método se utiliza para resolver problemas de flujo viscoso compresible modelados por las ecuaciones de Navier-Stokes compresibles no lineales. Por último, se propone una nueva formulación HDG linealizada de alto orden y se implementa para este tipo de problemas. También se estudia su precisión y su eficiencia para problemas estacionarios y transitorios. La segunda parte es el núcleo de esta tesis. Se propone un nuevo método de captura de choque para la solución HDG de problemas de compresibles y viscosos, en presencia de choques o frentes verticales pronunciados. La idea principal es utilizar la estabilización que proporcionan los flujos numéricos, considerando un espacio discontinuo de aproximación en interior de los elementos, para disminuir o eliminar las oscilaciones en la proximidad de la discontinuidad o el frente. Las funciones de base nodales discontinuas, requieren una forma débil modificada del problema local de HDG en los elementos estabilizados. En primer lugar, el método se aplica a problemas de convección-difusión, con flujos numéricos de Bassi-Rebay y de LDG (Local Discontinuous Galerkin) dentro de los elementos. A continuación, la estrategia se extiende a las ecuaciones de Navier-Stokes compresibles utilizando flujos numéricos de LDG y de Lax-Friedrichs. Finalmente, varios ejemplos numéricos, tanto para convección-difusió, como para las ecuaciones de Navier-Stokes compresibles, demuestran la capacidad del método propuesto para capturar los choques o frentes verticales en la solución. Su excelente rendimiento, elimina o atenúa significativamente las oscilaciones alrededor de los choques, obteniendo una solución estable.Postprint (published version