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

A practical method for computing with piecewise Chebyshevian splines

A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial splines, the related parametric curves exhibit the desired properties of convex hull inclusion, variation diminution and intuitive relation between the curve shape and the location of the control points. For a good-for-design space, in this paper we construct a set of functions, called transition functions, which allow for efficient computation of the B-spline basis, even in the case of nonuniform and multiple knots. Moreover, we show how the spline coefficients of the representations associated with a refined knot partition and with a raised order can conveniently be expressed by means of transition functions. This result allows us to provide effective procedures that generalize the classical knot insertion and degree raising algorithms for polynomial splines. We further discuss how the approach can straightforwardly be generalized to deal with geometrically continuous piecewise Chebyshevian splines as well as with splines having section spaces of different dimensions. From a numerical point of view, we show that the proposed evaluation method is easier to implement and has higher accuracy than other existing algorithms

WENO Schemes and Their Application as Limiters for RKDG Methods Based on Trigonometric Approximation Spaces

NSFC [10931004, 11002071]; ISTCP of China [2010DFR00700]In this paper, we present a class of finite volume trigonometric weighted essentially non-oscillatory (TWENO) schemes and use them as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods based on trigonometric polynomial spaces to solve hyperbolic conservation laws and highly oscillatory problems. As usual, the goal is to obtain a robust and high order limiting procedure for such a RKDG method to simultaneously achieve uniformly high order accuracy in smooth regions and sharp, non-oscillatory shock transitions. The major advantage of schemes which are based on trigonometric polynomial spaces is that they can simulate the wave-like and highly oscillatory cases better than the ones based on algebraic polynomial spaces. We provide numerical results in one and two dimensions to illustrate the behavior of these procedures in such cases. Even though we do not utilize optimal parameters for the trigonometric polynomial spaces, we do observe that the numerical results obtained by the schemes based on such spaces are better than or similar to those based on algebraic polynomial spaces