808 research outputs found
Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations on deforming meshes
An overview is given of a space-time discontinuous Galerkin finite element method for the compressible Navier-Stokes equations. This method is well suited for problems with moving (free) boundaries which require the use of deforming elements. In addition, due to the local discretization, the space-time discontinuous Galerkin method is well suited for mesh adaptation and parallel computing. The algorithm is demonstrated with computations of the unsteady \ud
ow field about a delta wing and a NACA0012 airfoil in rapid pitch up motion
HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis
The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge-Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space-time discontinuous Galerkin discretization of the advection-diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for low and high cell Reynolds numbers and on highly stretched meshes
Multigrid Preconditioning for a Space-Time Spectral-Element Discontinuous-Galerkin Solver
In this work we examine a multigrid preconditioning approach in the context of a high- order tensor-product discontinuous-Galerkin spectral-element solver. We couple multigrid ideas together with memory lean and efficient tensor-product preconditioned matrix-free smoothers. Block ILU(0)-preconditioned GMRES smoothers are employed on the coarsest spaces. The performance is evaluated on nonlinear problems arising from unsteady scale- resolving solutions of the Navier-Stokes equations: separated low-Mach unsteady ow over an airfoil from laminar to turbulent ow. A reduction in the number of ne space iterations is observed, which proves the efficiency of the approach in terms of preconditioning the linear systems, however this gain was not reflected in the CPU time. Finally, the preconditioner is successfully applied to problems characterized by stiff source terms such as the set of RANS equations, where the simple tensor product preconditioner fails. Theoretical justification about the findings is reported and future work is outlined
Multigrid optimization for space-time discontinuous Galerkin discretizations of advection dominated flows
The goal of this research is to optimize multigrid methods for higher order accurate space-time discontinuous Galerkin discretizations. The main analysis tool is discrete Fourier analysis of two- and three-level multigrid algorithms. This gives the spectral radius of the error transformation operator which predicts the asymptotic rate of convergence of the multigrid algorithm. In the optimization process we therefore choose to minimize the spectral radius of the error transformation operator. We specifically consider optimizing h-multigrid methods with explicit Runge-Kutta type smoothers for second and third order accurate space-time discontinuous Galerkin finite element discretizations of the 2D advection-diffusion equation. The optimized schemes are compared with current h-multigrid techniques employing Runge-Kutta type smoothers. Also, the efficiency of h-, p- and hp-multigrid methods for solving the Euler equations of gas dynamics with a higher order accurate space-time DG method is investigated
h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems
In this work we exploit agglomeration based -multigrid preconditioners to
speed-up the iterative solution of discontinuous Galerkin discretizations of
the Stokes and Navier-Stokes equations. As a distinctive feature -coarsened
mesh sequences are generated by recursive agglomeration of a fine grid,
admitting arbitrarily unstructured grids of complex domains, and agglomeration
based discontinuous Galerkin discretizations are employed to deal with
agglomerated elements of coarse levels. Both the expense of building coarse
grid operators and the performance of the resulting multigrid iteration are
investigated. For the sake of efficiency coarse grid operators are inherited
through element-by-element projections, avoiding the cost of numerical
integration over agglomerated elements. Specific care is devoted to the
projection of viscous terms discretized by means of the BR2 dG method. We
demonstrate that enforcing the correct amount of stabilization on coarse grids
levels is mandatory for achieving uniform convergence with respect to the
number of levels. The numerical solution of steady and unsteady, linear and
non-linear problems is considered tackling challenging 2D test cases and 3D
real life computations on parallel architectures. Significant execution time
gains are documented.Comment: 78 pages, 7 figure
A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows
Both compressible and incompressible Navier-Stokes solvers can be used and
are used to solve incompressible turbulent flow problems. In the compressible
case, the Mach number is then considered as a solver parameter that is set to a
small value, , in order to mimic incompressible flows.
This strategy is widely used for high-order discontinuous Galerkin
discretizations of the compressible Navier-Stokes equations. The present work
raises the question regarding the computational efficiency of compressible DG
solvers as compared to a genuinely incompressible formulation. Our
contributions to the state-of-the-art are twofold: Firstly, we present a
high-performance discontinuous Galerkin solver for the compressible
Navier-Stokes equations based on a highly efficient matrix-free implementation
that targets modern cache-based multicore architectures. The performance
results presented in this work focus on the node-level performance and our
results suggest that there is great potential for further performance
improvements for current state-of-the-art discontinuous Galerkin
implementations of the compressible Navier-Stokes equations. Secondly, this
compressible Navier-Stokes solver is put into perspective by comparing it to an
incompressible DG solver that uses the same matrix-free implementation. We
discuss algorithmic differences between both solution strategies and present an
in-depth numerical investigation of the performance. The considered benchmark
test cases are the three-dimensional Taylor-Green vortex problem as a
representative of transitional flows and the turbulent channel flow problem as
a representative of wall-bounded turbulent flows
Efficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows
The present paper addresses the numerical solution of turbulent flows with
high-order discontinuous Galerkin methods for discretizing the incompressible
Navier-Stokes equations. The efficiency of high-order methods when applied to
under-resolved problems is an open issue in literature. This topic is carefully
investigated in the present work by the example of the 3D Taylor-Green vortex
problem. Our implementation is based on a generic high-performance framework
for matrix-free evaluation of finite element operators with one of the best
realizations currently known. We present a methodology to systematically
analyze the efficiency of the incompressible Navier-Stokes solver for high
polynomial degrees. Due to the absence of optimal rates of convergence in the
under-resolved regime, our results reveal that demonstrating improved
efficiency of high-order methods is a challenging task and that optimal
computational complexity of solvers, preconditioners, and matrix-free
implementations are necessary ingredients to achieve the goal of better
solution quality at the same computational costs already for a geometrically
simple problem such as the Taylor-Green vortex. Although the analysis is
performed for a Cartesian geometry, our approach is generic and can be applied
to arbitrary geometries. We present excellent performance numbers on modern,
cache-based computer architectures achieving a throughput for operator
evaluation of 3e8 up to 1e9 DoFs/sec on one Intel Haswell node with 28 cores.
Compared to performance results published within the last 5 years for
high-order DG discretizations of the compressible Navier-Stokes equations, our
approach reduces computational costs by more than one order of magnitude for
the same setup
A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow
We present an efficient discontinuous Galerkin scheme for simulation of the
incompressible Navier-Stokes equations including laminar and turbulent flow. We
consider a semi-explicit high-order velocity-correction method for time
integration as well as nodal equal-order discretizations for velocity and
pressure. The non-linear convective term is treated explicitly while a linear
system is solved for the pressure Poisson equation and the viscous term. The
key feature of our solver is a consistent penalty term reducing the local
divergence error in order to overcome recently reported instabilities in
spatially under-resolved high-Reynolds-number flows as well as small time
steps. This penalty method is similar to the grad-div stabilization widely used
in continuous finite elements. We further review and compare our method to
several other techniques recently proposed in literature to stabilize the
method for such flow configurations. The solver is specifically designed for
large-scale computations through matrix-free linear solvers including efficient
preconditioning strategies and tensor-product elements, which have allowed us
to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores.
We validate our code and demonstrate optimal convergence rates with laminar
flows present in a vortex problem and flow past a cylinder and show
applicability of our solver to direct numerical simulation as well as implicit
large-eddy simulation of turbulent channel flow at as well as
.Comment: 28 pages, in preparation for submission to Journal of Computational
Physic
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
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