206 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
A nested hybridizable discontinuous Galerkin method for computing second-harmonic generation in three-dimensional metallic nanostructures
In this paper, we develop a nested hybridizable discontinuous Galerkin (HDG)
method to numerically solve the Maxwell's equations coupled with the
hydrodynamic model for the conduction-band electrons in metals. By means of a
static condensation to eliminate the degrees of freedom of the approximate
solution defined in the elements, the HDG method yields a linear system in
terms of the degrees of freedom of the approximate trace defined on the element
boundaries. Furthermore, we propose to reorder these degrees of freedom so that
the linear system accommodates a second static condensation to eliminate a
large portion of the degrees of freedom of the approximate trace, thereby
yielding a much smaller linear system. For the particular metallic structures
considered in this paper, the resulting linear system obtained by means of
nested static condensations is a block tridiagonal system, which can be solved
efficiently. We apply the nested HDG method to compute the second harmonic
generation (SHG) on a triangular coaxial periodic nanogap structure. This
nonlinear optics phenomenon features rapid field variations and extreme
boundary-layer structures that span multiple length scales. Numerical results
show that the ability to identify structures which exhibit resonances at
and is paramount to excite the second harmonic response.Comment: 31 pages, 7 figure
An interface-tracking space-time hybridizable/embedded discontinuous Galerkin method for nonlinear free-surface flows
We present a compatible space-time hybridizable/embedded discontinuous
Galerkin discretization for nonlinear free-surface waves. We pose this problem
in a two-fluid (liquid and gas) domain and use a time-dependent level-set
function to identify the sharp interface between the two fluids. The
incompressible two-fluidd equations are discretized by an exactly mass
conserving space-time hybridizable discontinuous Galerkin method while the
level-set equation is discretized by a space-time embedded discontinuous
Galerkin method. Different from alternative discontinuous Galerkin methods is
that the embedded discontinuous Galerkin method results in a continuous
approximation of the interface. This, in combination with the space-time
framework, results in an interface-tracking method without resorting to
smoothing techniques or additional mesh stabilization terms
eXtended Hybridizable Discontinous Galerkin (X-HDG) Method for Linear Convection-Diffusion Equations on Unfitted Domains
In this work, we propose a novel strategy for the numerical solution of
linear convection diffusion equation (CDE) over unfitted domains. In the
proposed numerical scheme, strategies from high order Hybridized Discontinuous
Galerkin method and eXtended Finite Element method is combined with the level
set definition of the boundaries. The proposed scheme and hence, is named as
eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the
Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted
domains; i.e, the computational mesh does not need to fit to the domain
boundary; instead, the boundary is defined by a level set function and cuts
through the background mesh arbitrarily. The original unknown structure of HDG
and its hybrid nature ensuring the local conservation of fluxes is kept, while
developing a modified bilinear form for the elements cut by the boundary. At
every cut element, an auxiliary nodal trace variable on the boundary is
introduced, which is eliminated afterwards while imposing the boundary
conditions. Both stationary and time dependent CDEs are studied over a range of
flow regimes from diffusion to convection dominated; using high order XHDG through benchmark numerical examples over arbitrary unfitted domains.
Results proved that XHDG inherits optimal and super
convergence properties of HDG while removing the fitting mesh restriction
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
- …