7,018 research outputs found
Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations
In this contribution, classical mixed methods for the incompressible Navier-Stokes equations that relax the divergence constraint and are discretely inf-sup stable, are reviewed. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970ies, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure-robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence-free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly
On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
An improved understanding of the divergence-free constraint for the
incompressible Navier--Stokes equations leads to the observation that a
semi-norm and corresponding equivalence classes of forces are fundamental for
their nonlinear dynamics. The recent concept of {\em pressure-robustness}
allows to distinguish between space discretisations that discretise these
equivalence classes appropriately or not. This contribution compares the
accuracy of pressure-robust and non-pressure-robust space discretisations for
transient high Reynolds number flows, starting from the observation that in
generalised Beltrami flows the nonlinear convection term is balanced by a
strong pressure gradient. Then, pressure-robust methods are shown to outperform
comparable non-pressure-robust space discretisations. Indeed, pressure-robust
methods of formal order are comparably accurate than non-pressure-robust
methods of formal order on coarse meshes. Investigating the material
derivative of incompressible Euler flows, it is conjectured that strong
pressure gradients are typical for non-trivial high Reynolds number flows.
Connections to vortex-dominated flows are established. Thus,
pressure-robustness appears to be a prerequisite for accurate incompressible
flow solvers at high Reynolds numbers. The arguments are supported by numerical
analysis and numerical experiments.Comment: 43 pages, 18 figures, 2 table
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
On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations
The present paper deals with the numerical solution of the incompressible
Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods
for discretization in space. For DG methods applied to the dual splitting
projection method, instabilities have recently been reported that occur for
coarse spatial resolutions and small time step sizes. By means of numerical
investigation we give evidence that these instabilities are related to the
discontinuous Galerkin formulation of the velocity divergence term and the
pressure gradient term that couple velocity and pressure. Integration by parts
of these terms with a suitable definition of boundary conditions is required in
order to obtain a stable and robust method. Since the intermediate velocity
field does not fulfill the boundary conditions prescribed for the velocity, a
consistent boundary condition is derived from the convective step of the dual
splitting scheme to ensure high-order accuracy with respect to the temporal
discretization. This new formulation is stable in the limit of small time steps
for both equal-order and mixed-order polynomial approximations. Although the
dual splitting scheme itself includes inf-sup stabilizing contributions, we
demonstrate that spurious pressure oscillations appear for equal-order
polynomials and small time steps highlighting the necessity to consider inf-sup
stability explicitly.Comment: 31 page
High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of and (div) methods
The accurate numerical simulation of turbulent incompressible flows is a
challenging topic in computational fluid dynamics. For discretisation methods
to be robust in the under-resolved regime, mass conservation as well as energy
stability are key ingredients to obtain robust and accurate discretisations.
Recently, two approaches have been proposed in the context of high-order
discontinuous Galerkin (DG) discretisations that address these aspects
differently. On the one hand, standard -based DG discretisations enforce
mass conservation and energy stability weakly by the use of additional
stabilisation terms. On the other hand, pointwise divergence-free
-conforming approaches ensure exact mass conservation
and energy stability by the use of tailored finite element function spaces. The
present work raises the question whether and to which extent these two
approaches are equivalent when applied to under-resolved turbulent flows. This
comparative study highlights similarities and differences of these two
approaches. The numerical results emphasise that both discretisation strategies
are promising for under-resolved simulations of turbulent flows due to their
inherent dissipation mechanisms.Comment: 24 pages, 13 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
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