Increased parallel efficiency for space-time multiscale computations
of turbulent flows
Gertjan van Zwieten
Site of the project:
2629 HT Delft
start of the project: September 2005
In December 2005 the
Thesis has been appeared.
The Master project has been finished in August 2006
by the completion of the
Masters Thesis and a final
presentation has been given.
For working address etc. we refer to our
Summary of the master project:
The prediction of interactions between compressible turbulent flows
and flexible structures is of great importance in Aerospace Engineering.
Such problems have been historically intractable, however, due
to the enormous difference in scale between turbulent fluctuations
and fluid-structure interactions. A promising approach is to make use
of Large-eddy simulation (LES) techniques, in which only the largest
scales of turbulence are simulated, while the effects of the
unresolved scales are modelled.
Recently, a new approach to LES, known as the variational
multiscale (VMS) method, has been developed [Hughes00,01a,01b].
This approach avoids some of the inconsistencies of traditional LES
techniques, and can be directly incorporated into h-p finite-element
discretisations [Collis02a]. The latter are advantageous in that they
can be locally adapted in heterogenous flows to minimize the total
number of degrees of freedom required for accurate solutions.
For problems with moving domains, space-time h-p finite-element
discretisations are particularly advantageous, as they inherently
incorporate mesh movement.
Unfortunately, space-time VMS discretisations of the compressible
Navier Stokes equations produce non-linear systems which can be quite
difficult to resolve. Firstly, space-time discretisations are
implicit, and introduce twice as many unknowns when time-discontinuous
formulations are used. Secondly, VMS discretisations
involve multiple basis functions per element, each of which can behave
quite differently. Finally, the compressible Navier-Stokes equations
are themselves a coupled system of equations which introduce
five unknowns per shape function. The resulting non-linear system
is thus large and heterogenous.
A typical LES simulation normally requires tens of thousands of time steps
to establish reliable turbulence statistics. For each time
step, an outer Newton iteration loop is used to resolve the
non-linear system, while an inner iteration loop is used to
solve the associated sparse linear Jacobian system.
For space-time VMS discretisations, the Jacobian matrix
tends to be ill-conditioned, particularly at larger time steps.
Numerical experimentation has shown, however, that often
the Jacobian matrix can be re-used, not only for several Newton
iterations, but also for several time steps. One is therefore
to store the Jacobian matrix to avoid re-computation, and to
construct and store an effective preconditioner to minimize the
number of inner iterations. For heterogenous systems,
ILU preconditioners are convenient, as they can be generated
Unfortunately the effectiveness of ILU preconditioning drops rapidly
increasing numbers of parallel processors are used. This currently
application of the store-and-reuse strategy to relatively small
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