NLR Reports Repository

Ven, H. van der

Nationaal Lucht- en Ruimtevaartlaboratorium
National Aerospace Laborator y NLR
NLR TP 96424
NICE: do-it-yourself flow solutions for engineers
and scientists
H. van der Ven
217-02
DOCUMENT CONTROL SHEET
                     
ORIGINATOR'S REF.                    SECURITY CLASS.
                     NLR TP 96424 U                 Unclassified
ORIGINATOR 
National Aerospace Laboratory NLR, Amsterdam, The Netherlands
TITLE  
NICE: Do-It-Yourself flow solutions for engineers and scientists
PRESENTED AT
the International EUROSIM Conference HPCN challenges in telecomp and telecom, Delft, June 10-12,
1996.
AUTHORS                 DATE                 pp       ref
H. van der Ven
                  
960703 6 -
DESCRIPTORS   
Computational fluid dynamics Software tools
Computer aided design Supercomputers
Netherlands Technology utilization
Numerical flow visualization Three dimensional models
Parallel processing (computers)
ABSTRACT
This report contains the presentation of the NICE project at the scientific exhibition at the International
EUROSIM Conference HPCN challenges in telecomp and telecom, Delft, June 10-12, 1996.
- 3 -
TP 96424
Summary
This report contains the presentation of the NICE project at the scientific exhibition at the Inter-
national EUROSIM Conference HPCN challenges in telecomp and telecom, Delft, June 10-12,
1996.
- 4 -
TP 96424
Contents
1 Introduction 5
2 The presentation 6
(6 pages in total)
- 5 -
TP 96424
1 Introduction
In 1996 the NICE project, under a contract of the Dutch HPCN Foundation, started its work to
disseminate HPCN in the Netherlands. The NICE project is a consortium of the following partners:
National Aerospace Laboratory NLR, Delft Hydraulics, Maritime Research Institute MARIN,
TNO-Institute for Applied Physics, J.M. Burgers Centrum (including the Dutch universities TUD,
UT and RUG), and the Centre for Mathematics and Computer Science CWI.
Development and use of flow simulation techniques through HPCN is the purpose of the NICE
project. The title of this report is motivated by this purpose: in the end state-of-the-art flow
simulation techniques developed by scientists and engineers shall be easily attainable for engineers
in the application field. Necessary resources shall be easily accessible nation-wide.
The project presented itself at the scientific exhibition at the International EUROSIM Conference
HPCN challenges in telecomp and telecom, Delft, June 10-12, 1996. This report contains the
presentation in the next chapter. The material of the presentation was delivered by the partners in
the NICE project, NLR combined the contributions to the present presentation.
- 6 -
TP 96424
2 The presentation
The panels of the presentation are presented in the following pages.
NLR
Air
Apart from the development of HFS
NLR is involved in the field of parallel
computing in the following subjects.
Parallelization of the flow simulation
system ENFLOW.
Optimization of use on the newest
generation of supercomputers for a
wide range of applications.
WL
Water
The need to protect the
Netherlands against the continuous
threat of natural water systems has
made 'water' a factor of conside-
rable economic interest. In order to
deal with all information required to
design, construct, operate and
optimize major infrastructural
projects, HPCN is becoming an
indispensable tool.
Flow field and changes in bottom topography
in the coastal area of the Waddenzee
(The Netherlands)
Development of domain
decomposition methods for (time ac-
curate) adaptive, unstructured flow
solvers.
– dynamic load balancing
– message passing paradigm
– balance between integration,
adaptation and repar titioning
The scope of the sector 'Water' is
to realize an interactive information
system that is capable of gene-
rating and processing information
for the optimization of large engi-
neering projects and operations by
following an integrated approach of
design, construction and environ-
mental impact.
Activities include
– system definition for system
integrity, flexibility and
maintainability
– modeling of interactions between
hydrodynamics, transport of
pollutants and morphodynamics
– development of optimal parallel
numerical algorithms
Specific applications are aimed are
situations in the Southern Chinese
Seas.
System components for
integrated modeling and design
Vortex structure over delta wing
computed by adaptive unstructured flow
solver
Partitioning of single air foil by
dynamic load balancing tool
SME benefit from NICE
Biddle BV is an SME specialized in
high quality air technical equipment. In
close cooperation with NLR and RuG,
Biddle develops a new air heater de-
sign.
In this way NLR applies proven HPCN
techniques to SME's.
Temperature distribution
in heated room
Large-scale wave problems
Goals:
– Reducing computational costs and
turn-around time of simulations
– Enabling the computation of large-
scale wave problems by using
massively parallel systems
Domain decomposition is a technique
which introduces a natural division of
the computation and enables the use
of parallel processors with more
memory.
Results:
– Large speed-up using only a few
processors on a heterogeneous
cluster
– Possibilities for the application to
large-scale wave systems
CWI
HPCN actions
– interactive visualization
– computational steering
– presentation of massive data
– software tools for HPCN
– dynamic (re)distribution
– visual parallel programming
– multi platform coupling
– massive parallelism
– multi-media and virtual reality
suppor t
NICE:
Three snap shots of water surface
simulated using four subdomains
Wave reflecting on offshore
structure
Centre for Flow Simulation HFS
HFS is a digital working environment
for flow simulation and information
exchange.
This common working environment
serves four purposes:
– working environment for flow
simulations
– public relations
– project management
– knowledge transfer
Each contractor will develop and
maintain its own par t of the infra-
structure. Each of these local infra-
structure will be integrated in one
large virtual infrastructure.
air
ch
em
is
tr
y
industrialflow o
the
r
sec
tor
s
water
shipsHFS
NLR
VVL
TPD
JMBC
MARIN
High performance computers
The partners in the NICE project have
access to the following supercomputer
facilities:
– NEC SX4 with sixteen processors.
This is a publicly available shared
memory computer with a peak of
32 Gflop/s and 12 GB memory.
– Cray T3E with 64 processors, with
2 to 16 GB/s bandwidth and
128 MB memory per processor.
Detail of NICE home page
Organogram of NICE
For more information please contact: W. Loeve,
National Aerospace Laboratory NLR, +31 20 511 3422
You can find us at http://www.nlr.nl/public/projects/nice/
The required powerful computers are
available in the Netherlands. However,
it is necessary to make these
systems available from the different
locations where the simulation
techniques are developed and
applied.
Development and use of flow
simulation techniques through HPCN
is the purpose of the NICE project.
The NICE project is sponsored by the
Dutch HPCN Foundation.
The par tners in NICE are active in the
following application sectors:
The NICE project works in close
cooperation with the industry and
impor tant goal is the dissemination of
knowledge to SME's.
The HFS infrastructure
The Project
Flow simulation on computer systems
considerably enhances the effectivity
of product design in many industries.
For this purpose, commercially
available software is often inade-
quate. The development of suitable
simulation systems require
thorough knowledge of the physics of
flow patterns, a deep understanding
of numerical mathematics and
powerful computer systems.
The methods for the simulation of
different flow types for the different
kinds of industries in the Netherlands
have a lot in common, but also differ
in some crucial aspects.
Development of the simulation
techniques takes place in large
technological institutes, universities
and industries.
air
ch
em
is
tr
y
industrialflow o
the
r
sec
tor
s
water
ships
SP1
SP2
KP
NEC SX4
An HFS prototype
do-it-yourself flow solutions for
J.M. Burgerscentrum
Chemistry
Supported by HPCN, Philips,
Hoogovens and Shell Amsterdam
Motivation: turbulent flows with
chemical reactions or fluid-particle
interaction are important for many
industries and their accurate
simulation requires vast computer
resources.
Code development concerns LES co-
des for turbulent flow with reactions,
3D finite volume for high temperature
combustion, and 2D finite volume for
reacting flow with detailed chemistr y
and joint probability density functions.
Numerical methods and aspects
include scalable, massively parallel
solvers including load balancing
algorithms, domain decomposition
techniques with local grid refinement,
and special algorithms for LES, large
sets of equations, spectral radiative
heat transfer, Monte Carlo
representations, and fluid-particle
interaction.
TNO-TPD
Industrial flow
WISH3D:
– general purpose CFD code
– flow, combustion, radiation and heat
transfer
– process-product interaction
– applications (a.o.) glass, metal and
food industr y, furnaces, burners and
dryers
Parallel WISH3D:
– portability and maintainability more
important than per formance
– scalability
– distributed memory multiprocessor
system
– industrial applicability
LES of reacting flow
RuG
DNS rushes at full scale turbulence
Stormy developments in algorithm
development will shortly allow direct
numerical simulation (DNS) of fully
developed turbulent flow at engineer-
ing Reynolds numbers. DNS resolves
all the length and time-scales of the
flow, without a need for turbulence
modeling. Current state-of-the-ar t
algorithms already permit simulation
of internal flows at a Reynolds number
of 100,000.
The figure shows a snapshot of the
vorticity distribution in a three
dimensional cavity, driven from left to
right by the upper lid. Use of fourth-
order space discretization allows full
resolution with a 'mere' 10 million
grid points. The whole computation
fits within 1 GB of memory and was
performed at SARA's Cray C90 (with
support from NCF).
Flow about a ship
Photograph of glass-melting
furnace
Simulation of glass-melting furnace
Parallelization strategies:
on loop level:
– using PVM/MPI approach
– large programming effort
on loop level:
– HPF approach with PREPARE tool
– less programming effort
– parallel compilers not mature (yet)
domain decomposition:
– PVM/MPI approach for SPMD
– system specific routines isolated
– lowest amount of data exchange
– can be combined with local grid
refinement
MARIN
Ships
At MARIN the viscous flows around
ship sterns sailing at constant speed
in still water is computed with the CFD
tool PARNASSOS, solving the
parabolized Navier-Stokes equations.
Within the NICE project a domain
decomposition method will be deve-
loped. This will allow for application to
a wider variety of ship geometries and
for local grid refinement. New applica-
tions will include appendages to ship
hulls, and appendages to locally
influence the flow.
Domain decomposition will be
achieved using a Schwarz iteration
method on the level of subdomain
blocks. A flexible coupling strategy will
optimize the convergence speed.
The figure indicates streamlines of the
flow along the bare hull of a ship. It
clearly shows the formation of a
vor tex along the hull, coming off the
stern.
engineers and scientists


NICE: do-it-yourself flow solutions for engineers and scientists

Jonge, J.B. de

Hol, P.A.

Variation in load factor experience  - A re-analysis of Fokker F27 and F28 operational acceleration data-

Wiggenraad, J.F.M.

Damage propagation in composite structural elements - Analysis and experiments on structures -

Niemann-Tuitman, B.E.

Veldman, A.E.P.

Nationaal Lucht- en Ruimtevaartlaboratorium
National Aerospace Laborator y NLR
NLR TP 96431
An explicit multi time stepping algorithm for
aerodynamic flows
H. van der Ven, B.E. Niemann-Tuitman, A.E.P. Veldman
DOCUMENT CONTROL SHEET
ORIGINATOR’S REF. SECURITY CLASS.
NLR TP 96431 U Unclassified
ORIGINATOR
National Aerospace Laboratory NLR, Amsterdam, The Netherlands
TITLE
An explicit multi time stepping algorithm for aerodynamic flows
PRESENTED AT
The International Congress on Computational and Applied Mathematics,
July 21-26, 1996, Leuven, Belgium
AUTHORS DATE pp ref
H. van der Ven, B.E. Niemann-Tuitman,
A.E.P. Veldman
960708 15 5
DESCRIPTORS
Aerodynamics coefficients Finite volume method Step functions
Airfoils Multiblock grids Time dependence
Algorithms Numerical stability Turbulent flow
Boundary conditions Run time (computers)
Computational grids Runge-Kutta method
ABSTRACT
An explicit multi time stepping algorithm with applications to
aerodynamic flows is presented. In the algorithm in different parts of
the computational domain different time steps are taken, and the flow is
synchronized at so-called synchronization levels. The algorithm is
validated for aerodynamic turbulent flows. For two dimensional flows
speedups in the order of five with respect to single time stepping are
obtained.
217-02
- 3 -
TP 96431
Summary
An explicit multi time stepping algorithm with applications to aerodynamic flows is presented.
In the algorithm in different parts of the computational domain different time steps are taken,
and the flow is synchronized at so-called synchronization levels. The algorithm is validated for
aerodynamic turbulent flows. For two dimensional flows speedups in the order of five with respect
to single time stepping are obtained.
- 4 -
TP 96431
Contents
1 Introduction 5
2 The multi time stepping algorithm 6
2.1 The basic algorithm 6
2.2 The simplified algorithm 8
3 Experiments and validation 9
3.1 Stability 9
3.1.1 The basic algorithm 9
3.1.2 The simplified algorithm 10
3.2 Accuracy 10
3.3 Efficiency 12
4 Conclusions 14
1 Table
5 Figures
(15 pages in total)
- 5 -
TP 96431
1 Introduction
For time accurate simulations stability restrictions on the time step in finely resolved parts of the
computational domain decrease the performance of explicit methods applying one uniform time
step in the entire domain. Multi time stepping methods, where in different parts of the domain
different time steps are taken, are more efficient.
Crucial to multi time stepping methods is the information exchange between the different parts
of the domain. For an overview of previous work in the field of linear structural dynamics we
refer to Belytschko and Lu, Ref. 1. For convection dominated flow problems Maurits et al. Ref. 4
considered the one-dimensional convection-diffusion equation as a model problem. Maurits et
al. exchanged information only at synchronization levels, that is, at the largest time step. Time
accurate simulations were performed with a ratio of 400 between the different time steps. Kleb
et al. Ref. 3 considered point-wise multi time stepping: each grid point is advanced using its
own time step which fits a power of two times in the largest time step. The larger time steps are
advanced the earlier in order to generate necessary results at intermediate levels. The results at
intermediate levels are obtained using linear interpolation. Kleb et al. report speedups between 4
and 10 for oscillating airfoils.
The present investigation extends the multi time stepping method of Maurits et al. to three
dimensional aerodynamic simulations.
Apart from their efficiency, explicit multi time stepping algorithms are easy to implement in
existing explicit steady state solvers. Moreover, they have an inherent parallelism and their
simplicity leads one to expect excellent parallel efficiency.
The contents of the report is as follows. In Chapter 2 the multi time stepping algorithm used in this
paper is described. Two algorithms are described, the basic algorithm and a simplified algorithm
with less information exchange. In Chapter 3 the algorithms are validated. In the final chapter
conclusions are drawn.
- 6 -
TP 96431
2 The multi time stepping algorithm
Basically, any algorithm in which different time steps are taken in different parts of the grid will
be called a multi time stepping algorithm. Crucial are time levels at which (part of) the flow is
synchronized to obtain time-accurate behaviour. Only at these synchronization levels information
between the different parts is exchanged.
Two algorithms are tested in the present report. In the first algorithm, the basic algorithm, the
information exchange is such that at all block boundaries the local stability conditions are satisfied.
In the second algorithm, which is a simplification of the first, information exchange only takes
place after the largest time step has been performed. The motivation to also consider the second
algorithm is its simplicity: it requires less organization and is easier to implement.
The spatial discretization of the solver to which multi time stepping is added, is a cell-vertex
Jameson scheme and block boundaries are part of both blocks they bound. Around each block
two layers of dummy cells are added which contain the flow status in the bounding blocks (if
any). Since the block boundaries belong to both blocks they bound, the block boundaries are
multi-valued after an integration pass. After the integration pass the multi-valuedness is removed
by averaging the different values. Moreover, the dummy cells at the internal block boundaries are
refreshed.
The time integration scheme is the standard Runge-Kutta 4 algorithm. For linear problems
this scheme is third order accurate over a fixed time interval. The accuracy of the multi-block
implementation of this scheme has been measured for turbulent flows to be between second and
third order.
2.1 The basic algorithm
In the description of the algorithms it is assumed that the computational domain is subdivided into
blocks. The time step that is taken in one block will be called a block time step. The block time
steps ∆t
b
are calculated in the following way. First the locally stable time steps are determined.
Let ∆tb
c
be the locally stable time step in cell c of block b. Let ∆tbmin be the minimum of ∆tbc over
the cells in block b. The synchronization time step ∆tsyn is defined as the largest of these time
steps. Finally, the time steps ∆tbmin for all blocks are decreased as to fit in the synchronization time
step an integer number of times. The decreased time steps are the block time steps.
In each block the flow is advanced over as many block time steps that fit in the synchronization
time step. At each synchronization level the block time steps are determined again.
- 7 -
TP 96431
In order to satisfy the local stability conditions at the block boundaries the block time steps are
performed in a certain order. The dummy variables are refreshed after each block time step. The
order of the time step computations is roughly determined by the elapsed physical time, which is
measured by the so-called master clock. The master clock is advanced in time with steps equal to
the smallest block time step ∆tmin, that is, ∆tmin = minb ∆tb. Below is a description of the part of
the algorithm that advances the flow one synchronization time step.
repeat
do for all blocks
if time step should be set
integrate flow in block one block time step
endif
enddo
refresh dummies for all blocks
apply boundary conditions
advance time of master clock with ∆tmin
until one synchronization time step is elapsed at the master clock
The condition ‘time step should be set’ is true if the elapsed time T
b
in block b lags too far behind
with respect to the elapsed time on the master clock. Here we define T
b
as the time on the master
clock at which the latest time step in block b has been set. This implies that the next time step
in this block should be set no later than T
b
+ ∆t
b
. Let T denote the elapsed physical time on the
master clock, then we update block b at the time T for which
T < T
b
+ ∆t
b
 T + ∆tmin;
that is, the block update is made at the latest possible moment. Using this strategy it follows that
for any two blocks b and b0
jT
b
  T
b
0
j  max(∆t
b
;∆t
b
0
):
This means that at each loop index the elapsed time difference between two blocks is less than the
maximum of the block time steps in the two blocks. This implies that at the block boundaries the
local stability conditions are always satisfied.
Notice that in between synchronization levels the flow in the different blocks is not synchronized
as it is at the synchronization levels: the block time steps do not necessarily fit an integer time
in each other. Hence, the exchanged info in between synchronization levels is not necessarily at
- 8 -
TP 96431
the same time level. There is, however, information exchange in between synchronization levels:
the ‘refresh dummies’ command exchanges information between bounding blocks. An example
of the order of the time steps is given in Figure 1.
t
x
medium block coarse blockfine block
syn
1
∆ tb
2
3
4
5
6
7
8
9
2
3
5
6
8
9
9
∆ t
Fig. 1 Order of the block time step integration. The numbers refer to the order of the block time
integration.
Finally note that in the above algorithm no averaging takes place at the multi-valued block
boundaries. Since the boundaries should be accurately integrated in time, averaging is felt to be
unnecessary. Moreover, averaging is an accelaration technique for steady flow computations, and
as such obsolete for unsteady computations.
2.2 The simplified algorithm
In the simplified algorithm the information exchange in the loop over the block time steps is moved
outside the repeat-until loop. Hence, in between synchronzation time steps the information at the
block boundaries is frozen.
The above algorithm applied to arbitrary block decompositions may cause violation of local
stability conditions at block boundaries bounding blocks with block time steps strictly smaller
than the synchronization time step.
- 9 -
TP 96431
3 Experiments and validation
The algorithm described in Chapter 2 is implemented in the structured, multi-block solver EDDS,
developed at NLR Ref. 5.
3.1 Stability
3.1.1 The basic algorithm
Turbulent flow is simulated around the RAE2822 airfoil under the flow conditions of Case 10 of
Ref. 2. The nonuniform grid that is used is divided in 18 blocks, in three layers normal to the
solid, each layer consisting of six blocks. The flow conditions are Re = 6:2  106,  = 2:8,
and the Mach number M = 0:75. The transition points are located at 3% chord. An initial (non
converged) solution was made with local time stepping. Then ten multi-time steps using the
simplified algorithm were taken.
The ratio r between the synchronization time step and the smallest block time step is 915. The
flow remains stable for ten multi time steps (elapsed non-dimensional time: 0.11); flow results are
displayed in Figure 2.
Fig. 2 Pressure distribution after 10 multi time steps with elapsed physical time 0.11.
- 10 -
TP 96431
3.1.2 The simplified algorithm
The same case as in the previous section is simulated using the simplified algorithm. Purpose of this
simulation was to examine to what extent information exchange can be postponed. Unfortunately,
judging from the wiggles in the shock region, the flow becomes instable.
To investigate the instability the synchronization level is decreased without affecting the smallest
block time steps. Consequently, the ratio r between the synchronization time step and the smallest
block time step decreases. Tested cases are shown in Table 1. All computations with smaller
synchronization time steps are stable.
Table 1 Synchronization time steps and ratio’s for RAE2822
∆tsyn ratio r stable
112:6  10 4 915 no
40:8  10 4 332 maybe
20:4  10 4 166 yes
2:1  10 4 18 yes
In the case of ∆tsyn = 2:1  10 4 the block time steps in the blocks in the two outer layers are
all equal to the synchronization time step. Hence in this case, the local stability conditions at the
block boundaries in normal direction are satisfied. In all other cases the synchronization level is
larger than the locally stable time steps at one or more block boundaries. Apparently, the flow in
the block can damp instabilities at block boundaries to a certain degree for a certain time. If the
information exchange is postponed too long, the computation becomes instable.
3.2 Accuracy
Because of the results of the stability tests, the accuracy is tested using the basic algorithm.
The flow obtained in Section 3.1.1 with multi time stepping is used for restarts using multi time
stepping with different time steps. Two experiments are performed.
In the first experiment the block time steps in the outer blocks are decreased in such a way that the
synchronization time step is halved, and successively quartered. The block time steps in the inner
blocks near the solid are unaltered. Hence this experiment tests the dependence of the accuracy
on the synchronization level.
In the second experiment the block time steps in all blocks are halved and successively quartered.
Hence this experiment tests the dependence of the accuracy on the block time steps.
- 11 -
TP 96431
In all experiments the flow is integrated from the non dimensional time 0.113 to 0.117, which
corresponds with one multi time step with unaltered block time step, or with 1000 single time
steps. The flow results of the experiments are compared to the flow results using single time
stepping.
In Figures 3 and 4 the time evolution of the aerodynamic coefficients for the different experiments
is monitored. The agreement for both coefficients is excellent.
Fig. 3 Time evolution of the aerodynamic coefficients for the experiment in which the synchro-
nization time step has been halved successively. Solid line: single time stepping; : ∆tsyn;
4: 12 ∆tsyn; 2:
1
4 ∆tsyn.
Fig. 4 Time evolution of the aerodynamic coefficients for the experiment in which all block time
steps have been halved successively. Solid line: single time stepping; : ∆tsyn;4: halved
block time steps; 2: quartered block time steps.
- 12 -
TP 96431
In order to determine the formal accuracy of the multi time stepping method the following
quotient was computed: (u1   u 1
2
)=(u 1
2
  u 1
4
). Here, u1 refers to the solution with the default
synchronization time step, u 1
2
with half the default synchronization time step, resp. half the block
time step, and u 1
4
with quart default synchronization time step, resp. quart block time step. The
2 log of this quotient is the formal order of the accuracy of the multi time stepping method.
For the successive halvenings of the synchronization time step the quotient varied between 6.4
for the density and 15 for the energy. For the successive halvenings of the block time steps the
quotient varied between 1.7 for the energy and 2.1 for the x-velocity.
Hence, the present multi time stepping method is first order accurate in the block time steps, and
effect of the synchronization time step is negligible. In comparison with the single time stepping
integration scheme the multi time stepping method loses one order of accuracy.
In principle, the multi time stepping method allows for accurate simulation of physical quantities
within each block with an accuracy of the block time step. In order to assess this accuracy the time
evolution during a synchronization time step of the normal velocity was monitored at a point below
the airfoil and near the trailing edge. In Figure 5 the time evolution of this quantity is displayed
together with the time evolution as given by single time stepping. For multi time stepping the
velocity is printed every block time step, which is roughly a thousandth of the synchronization
time step. The agreement between single and multi time stepping is excellent.
3.3 Efficiency
The single time stepping run of the previous section takes 3276 seconds, while the multi time
stepping run with the default synchronization time step takes 731.2 seconds. Hence, a speedup of
4.5 is obtained.
When the block layer around the airfoil is halved in the normal direction to create a 24 blocks
topology, the speedup with respect to single time stepping is increased to 5.5.
- 13 -
TP 96431
Fig. 5 Time evolution of the normal velocity at a point below the airfoil near the trailing edge. —
single time stepping, - - - multi time stepping. The size of the block time step of the block
in which the point lies is roughly equal to the single time step. The extent of the x-axis is
equal to one synchronization time step.
- 14 -
TP 96431
4 Conclusions
The multi time stepping algorithm presented in this paper has been proven to be an accurate,
efficient and easy-to-implement algorithm.
The algorithm is stable whenever the block time steps are determined by the local stability
conditions of the time integration scheme in the block. A simplification of the algorithm where
the information exchange is postponed to the synchronization levels proved to be unstable.
The accuracy is determined by the block time steps and not by the synchronization time step. This
allows for the accurate simulation of phenomena with a smaller time scale than the synchronziation
time step. The accuracy of the multi time stepping method over a fixed time interval is first order
in the block time steps.
The efficiency is expressed in a speedup of five with respect to single time stepping.
The algorithm can be simply implemented in any block-structured flow solver.
- 15 -
TP 96431
References
1. T. Belytschko and Y.Y. Lu. Explicit multi-time step integration for first and second order finite
element semidiscretizations. Comp. Methods Appl. Mech. Eng., 108:353–383, 1993.
2. P.H. Cook, M.A. McDonald, and M.C.P. Firmin. Aerofoil RAE2822. Pressure distributions,
and boundary layer and wake measurements. Technical Report AR-138, AGARD, 1979.
3. W. L. Kleb, J.T. Batina, and M.H.Williams. Temporal adaptive Euler/Navier-Stokes algorithm
involving unstructured dynamic meshes. AIAA Journal, 30(8):1980–5, 1992.
4. N.M. Maurits, H. van der Ven, and A.E.P. Veldman. Explicit multi time stepping methods for
convection-dominated flow problems. submitted to Comp. Methods in Appl. Mech. and Eng.,
1995.
5. M.E.S. Vogels. Technical design of edds, a system for the solution of the compressible
flow equations on a three-dimensional block-structured grid. Technical Report TR 93235 L,
Netherlands National Aerospace Laboratory NLR, 1993.


An explicit multi time stepping algorithm for aerodynamic flows

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Kuijpers, E.A.

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Space automation and robotics activities past, present and future: next steps and robotics laboratory

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Not a member yet

NICE: do-it-yourself flow solutions for engineers and scientists

Variation in load factor experience - A re-analysis of Fokker F27 and F28 operational acceleration data-

Damage propagation in composite structural elements - Analysis and experiments on structures -

An explicit multi time stepping algorithm for aerodynamic flows

Dynamics and control of a spacecraft with a moving, pulsating ball in a spherical cavity

Space automation and robotics activities past, present and future: next steps and robotics laboratory

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