Adaptive hybrid discontinuous methods for fluid and wave problems

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

Shock capturing computations with stabilized Powell-Sabin elements

International audienceIn the last recent years, thanks to the increasing power of the computational machines , the interest in more and more accurate numerical schemes is growing. Methods based on high-order approximations are nowadays the common trend in the computational research community, in particular for CFD applications. This work is focused on Powell-Sabin (PS) finite elements, a finite element method (FEM) based on PS splines. PS splines are piecewise quadratic polynomials with a global C1 continuity , defined on conforming triangulations. Despite its attractive characteristics, so far this scheme hasn't had the attention it deserves. PS splines are adapted to unstructured meshes and, contrary to classical tensor product B-splines, they are particularly suited for local refinement , a crucial aspect in the analysis of highly convective and anisotropic equations. The additional global smoothness of the C1 space has a beneficial stabilization effect in the treatment of advection-dominated equations and leads to a better capturing of thin layers. Finally, unlike most of other typology of high-order finite elements, the numerical unknowns in PS elements are located in the vertices of the triangulation, leading to an easy treatment of the parallel aspects. Some geometrical issues related to PS elements are discussed here, in particular, the generation of the control triangles and the imposition of the boundary conditions. The PS FEM method is used to solve the compressible Euler equation in supersonic regime. A classical shock-capturing technique is used to reduce the oscillation around the discontinuity, while a variational multiscale formulation is used to introduce numerical diffusion in the streamwise direction. Some typical numerical examples are used to evaluate the performance of the PS discretization

Avances en la simulación numérica de la propagación de oleaje en zonas costeras

Se presentan los últimos avances en la simulación numérica de la propagación de oleaje en zonas costeras. Las nuevas herramientas numéricas permiten obtener de manera precisa y eficiente las características de la ola en las zonas de interés. A partir de un modelo reducido es posible simular en tiempo real la propagación para cualquier oleaje predominante

Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier-Stokes equations

A degree adaptive Hybridizable Discontinuous Galerkin (HDG) method for the solution of the incompressible Navier-Stokes equations is presented. The key ingredient is an accurate and computationally inexpensive a posteriori error estimator based on the super-convergence properties of HDG. The error estimator drives the local modification of the approximation degree in the elements and faces of the mesh, aimed at obtaining a uniform error distribution below a user-given tolerance in a given output of interest. Three 2D numerical examples are presented. High efficiency of the proposed error estimator is found, and an important reduction of the computational effort is shown with respect to non-adaptive computations, both for steady state and transient simulations

Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems

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

Shock capturing computations with stabilized Powell-Sabin elements

International audienceIn the last recent years, thanks to the increasing power of the computational machines , the interest in more and more accurate numerical schemes is growing. Methods based on high-order approximations are nowadays the common trend in the computational research community, in particular for CFD applications. This work is focused on Powell-Sabin (PS) finite elements, a finite element method (FEM) based on PS splines. PS splines are piecewise quadratic polynomials with a global C1 continuity , defined on conforming triangulations. Despite its attractive characteristics, so far this scheme hasn't had the attention it deserves. PS splines are adapted to unstructured meshes and, contrary to classical tensor product B-splines, they are particularly suited for local refinement , a crucial aspect in the analysis of highly convective and anisotropic equations. The additional global smoothness of the C1 space has a beneficial stabilization effect in the treatment of advection-dominated equations and leads to a better capturing of thin layers. Finally, unlike most of other typology of high-order finite elements, the numerical unknowns in PS elements are located in the vertices of the triangulation, leading to an easy treatment of the parallel aspects. Some geometrical issues related to PS elements are discussed here, in particular, the generation of the control triangles and the imposition of the boundary conditions. The PS FEM method is used to solve the compressible Euler equation in supersonic regime. A classical shock-capturing technique is used to reduce the oscillation around the discontinuity, while a variational multiscale formulation is used to introduce numerical diffusion in the streamwise direction. Some typical numerical examples are used to evaluate the performance of the PS discretization

Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems

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.Peer ReviewedPostprint (author’s final draft

mantenimento orbitale presso punti di librazione tramite propulsione elettrica: studio preliminare di missione

L'esistenza dei punti di librazione per un sistema dinamico costituito da due corpi in orbita l'uno intorno all'altro è nota fin dalla metà del diciottesimo secolo; l'importanza di queste posizioni privilegiate del campo gravitazionale nell'ambito delle missioni spaziali fu chiara fin dagli albori dell'era spaziale. Fu evidente che lo sfruttamento dei cosiddetti punti di Lagrange avrebbero permesso lo svolgimento di tipologie di missioni non altrimenti realizzabili. La meccanica del volo sulle orbite caratteristiche attorno ai punti di librazione, le cosiddette orbite ad alone, impone l'utilizzo di un sistema propulsivo che effettui il mantenimento orbitale. Nelle missioni effettuate fino ad oggi sono sempre stati utilizzati allo scopo propulsori di tipo convenzionale: l'idea di utilizzare propulsione elettrica è stato fino al giorno d'oggi solo marginalmente toccato, soprattutto a causa del fatto che i propulsori che si prestano in modo particolare a questo tipo di operazione non hanno ancora raggiunto una maturità tale da poterne prevedere un utilizzo in un immediato futuro. Il proposito del presente lavoro è stato analizzare se sia possibile effettuare mantenimento orbitale su orbite presso i punti di librazione collineari del sistema Terra-Sole tramite propulsione elettrica, non solo da un punto di vista fisico ma soprattutto da un punto di vista sistemistico, vale a dire se esistano tipologie di missione per cui la propulsione elettrica sia una valida alternativa per questo compito

A Powell-Sabin finite element scheme for partial differential equations

In this paper are analyzed finite element methods based on Powell-Sabin splines, for the solution of partial differential equations in two dimensions. PS splines are piecewise quadratic polynomials defined on a triangulation of the domain, and exhibit a global C1 continuity. Critical issues when dealing with PS splines, and described in this work, are the construction of the shape functions and the imposition of the boundary conditions. The PS finite element method is used at first to solve an elliptic problem describing plasma equilibrium in a tokamak. Finally, a transient convective problem is also considered, and a stabilized formulation is presented