192 research outputs found
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
eXtended hybridizable discontinuous Galerkin for incompressible flow problems with unfitted meshes and interfaces
The eXtended hybridizable discontinuous Galerkin (X-HDG) method is developed for the solution of Stokes problems with void or material interfaces. X-HDG is a novel method that combines the hybridizable discontinuous Galerkin (HDG) method with an eXtended finite element strategy, resulting in a high-order, unfitted, superconvergent method, with an explicit definition of the interface geometry by means of a level-set function. For elements not cut by the interface, the standard HDG formulation is applied, whereas a modified weak form for the local problem is proposed for cut elements. Heaviside enrichment is considered on cut faces and in cut elements in the case of bimaterial problems. Two-dimensional numerical examples demonstrate that the applicability, accuracy, and superconvergence properties of HDG are inherited in X-HDG, with the freedom of computational meshes that do not fit the interfacesPeer ReviewedPostprint (author's final draft
Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations
We present optimal preconditioners for a recently introduced hybridized
discontinuous Galerkin finite element discretization of the Stokes equations.
Typical of hybridized discontinuous Galerkin methods, the method has
degrees-of-freedom that can be eliminated locally (cell-wise), thereby
significantly reducing the size of the global problem. Although the linear
system becomes more complex to analyze after static condensation of these
element degrees-of-freedom, the pressure Schur complement of the original and
reduced problem are the same. Using this fact, we prove spectral equivalence of
this Schur complement to two simple matrices, which is then used to formulate
optimal preconditioners for the statically condensed problem. Numerical
simulations in two and three spatial dimensions demonstrate the good
performance of the proposed preconditioners
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 Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow
We present a comparison between hybridized and non-hybridized discontinuous
Galerkin methods in the context of target-based hp-adaptation for compressible
flow problems. The aim is to provide a critical assessment of the computational
efficiency of hybridized DG methods. Hybridization of finite element
discretizations has the main advantage, that the resulting set of algebraic
equations has globally coupled degrees of freedom only on the skeleton of the
computational mesh. Consequently, solving for these degrees of freedom involves
the solution of a potentially much smaller system. This not only reduces
storage requirements, but also allows for a faster solution with iterative
solvers. Using a discrete-adjoint approach, sensitivities with respect to
output functionals are computed to drive the adaptation. From the error
distribution given by the adjoint-based error estimator, h- or p-refinement is
chosen based on the smoothness of the solution which can be quantified by
properly-chosen smoothness indicators. Numerical results are shown for
subsonic, transonic, and supersonic flow around the NACA0012 airfoil.
hp-adaptation proves to be superior to pure h-adaptation if discontinuous or
singular flow features are involved. In all cases, a higher polynomial degree
turns out to be beneficial. We show that for polynomial degree of approximation
p=2 and higher, and for a broad range of test cases, HDG performs better than
DG in terms of runtime and memory requirements
Implicit large-eddy simulation of compressible flows using the Interior Embedded Discontinuous Galerkin method
We present a high-order implicit large-eddy simulation (ILES) approach for
simulating transitional turbulent flows. The approach consists of an Interior
Embedded Discontinuous Galerkin (IEDG) method for the discretization of the
compressible Navier-Stokes equations and a parallel preconditioned Newton-GMRES
solver for the resulting nonlinear system of equations. The IEDG method arises
from the marriage of the Embedded Discontinuous Galerkin (EDG) method and the
Hybridizable Discontinuous Galerkin (HDG) method. As such, the IEDG method
inherits the advantages of both the EDG method and the HDG method to make
itself well-suited for turbulence simulations. We propose a minimal residual
Newton algorithm for solving the nonlinear system arising from the IEDG
discretization of the Navier-Stokes equations. The preconditioned GMRES
algorithm is based on a restricted additive Schwarz (RAS) preconditioner in
conjunction with a block incomplete LU factorization at the subdomain level.
The proposed approach is applied to the ILES of transitional turbulent flows
over a NACA 65-(18)10 compressor cascade at Reynolds number 250,000 in both
design and off-design conditions. The high-order ILES results show good
agreement with a subgrid-scale LES model discretized with a second-order finite
volume code while using significantly less degrees of freedom. This work shows
that high-order accuracy is key for predicting transitional turbulent flows
without a SGS model.Comment: 54th AIAA Aerospace Sciences Meeting, AIAA SciTech, 201
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