This describes a tabulation method based on computing, retaining and accessing a large, on the order of millions, number of individual kinetic time step calculations and approximations. It is essentially an extension of Pope’s In Situ Adaptive Tabulation (ISAT) method. The primary differences lie in that not all configurations need be stored in memory and that a polynomial approximation is only calculated when enough points have accumulated within a localized area to be able to calculate the polynomial approximation. The latter increases efficiency because no extra points are evaluated to form an approximation (as is done in ISAT). The speed up is expected to be that of ISAT.

1. RESEARCH POSTER PRESENTATION DESIGN © 2012
www.PosterPresentations.com
This describes a tabulation method based on computing, retaining and accessing a
large, on the order of millions, number of individual kinetic time step calculations and
approximations. It is essentially an extension of Pope’s In Situ Adaptive Tabulation
(ISAT) method. The primary differences lie in that not all configurations need be
stored in memory and that a polynomial approximation is only calculated when
enough points have accumulated within a localized area to be able to calculate the
polynomial approximation. The latter increases efficiency because no extra points are
evaluated to form an approximation (as is done in ISAT). The speed up is expected to
be that of ISAT.
Abstract
Design
Principles
Calcula@on
of
Approxima@on
For a mechanism with n species a configuration is made up of a vector with n+2
numbers. The extra two numbers are the temperature and the pressure. To create a
polynomial approximation in a region n+3, i.e. one more than the dimension of the
system, is needed. In the traditional ISAT algorithm, when a new configuration is
needed, then a polynomial approximation is made. This means that n additional
calculations have to be made. In terms of efficiency and reuse, this means that this
configurations around the original point has to be used at least n times before the
algorithm breaks even, i.e. before the computational gain is seen. Experiments with
the method produced here indicates, that this gain is not realized. The
configurations, without the approximations, can be used.
Storage
Management
The total timing per cycle is influenced by basically three factors:
In Memory If the indexing nodes and configuration is already in memory, the retrieval
is the most efficient.
From Disk If some or all of the indexing nodes have to be retrieved from disk, the
retrieval time is dominated by the disk access time.
No Point on Node If the final indexed node does not have a corresponding
configuration point, then the configurations and nodes from the next level must be
accessed. This increases the number of nodes to be retrieved.
Thus
the
database
design
has
to
balance
these
four
principles:
Very
Large
Database
Every
calcula/on
is
stored
and
kept
for
direct
use
or
for
post
processing.
As
the
range
of
condi/ons
of
the
calcula/ons
expand,
so
does
the
database.
This
includes
point
single
points
and
the
hypercube
approxima/ons.
Efficient
Access
The
speed-­‐up
of
the
method
is
propor/onal
to
the
calcula/on
/me
of
the
solver
for
a
/me-­‐step
over
the
access
/me
of
the
database.
A
significant
amount
of
the
total
database
structure
deals
with
the
efficient
access
of
the
nearest
point.
Efficient
access
also
involves
search
a
tree.
Limited
Memory
The
most
efficient
usage
is
of
points
within
memory.
Part
of
the
database
management
is
deciding
what
informa/on
should
stay
within
main
memory
and
which
should
be
wriSen
to
disk
storage.
Response
Approxima@on
The
use
ISAT
with
the
first
order
approxima/on
can
be
used
to
consolidate
points.
However,
no
extra
calcula/ons
are
done
to
create
the
approxima/on.
An
approxima/on
is
calculated
only
when
enough
points
near
enough
each
other
are
accumulated.
On-­‐
and
Off-­‐Line
The
ini/al
database
can
be
set
up
through
off-­‐line
calcula/on
and
then
supplemented
by
on-­‐line
points
as
they
are
needed.
Reac/on,
Sweden
Blurock
Consul/ng
AB
Edward
S.
Blurock
Adap@ve
On-­‐the-­‐fly
Regression
Tabula@on:
Beyond
ISAT
.
.
.
.
.
.
.
ISAT:
Calculate
Extra
Points
. .. ..
Accumulate
Points
as
they
are
needed
by
computa/on
For
this
reason,
in
the
method
described
here,
these
extra
calcula/ons
are
not
done.
Instead,
the
individual
points
themselves
are
accumulated.
Only
when
a
sufficient
number
of
dis/nct
points,
defined
by
that
the
zeroth
order
approxima/on
is
not
accurate
enough,
are
present
is
a
approxima/on
calculated.
The
previously
used
points
are
used
to
calculate
the
approxima/on.
No
extra
calcula/ons
are
made.
Tabula/on
methods
depend
on
finding
the
closest
configura/on
as
efficiently
as
possible.
The
purpose
and
philosophy
of
the
tabula/on
is
to
provide
full
kine/c
data
with
computa/onal
efficiency.
However,
it
is
imprac/cal,
both
for
the
tabula/on
and
for
the
transport
complexity
within
a
CFD
calcula/on,
to
use
the
full
set
of
species
and
condi/ons
of
the
full
mechanism.
For
this
reason,
a
reduced
set
of
progress
variables
is
chosen
to
represent
the
posi/on
in
configura/on
space,
i.e.
the
hypercube,
of
the
calcula/on.
The
progress
variables
are
used
to
find
the
closest
configura/on.
In
the
design
of
this
tabula/on
method,
this
is
done
in
two
stages:
Hypercube:
Using
a
tree
structure,
a
set
of
configura/ons
bounded
by
a
hypercube
of
progress
variable
values
is
isolated.
Nearest:
All
configura/ons
collected
are
compared
and
the
closest
(currently
using
the
normed
euclidean
distance)
is
found.
Two
types
of
comparisons
are
possible:
Progress
Variables:
If
the
calcula/on
only
involves
the
progress
variables,
then
only
these
are
used.
This
situa/on
occurs
when
the
tabulated
database
is
used
by
an
external
calcula/on.
Full:
If
the
full
set
of
variables
are
available,
then
all
are
used.
This
situa/on
occurs
when
se^ng
up
the
database.
Design
Principles
Choice
of
Progress
Variables
Two
physical
parameters,
namely
temperature
and
pressure,
are
used.
Based
on
chemical
intui/on,
species
from
the
four
major
regions
of
an
igni/on
process
were
chosen:
Ini/al
reactant
phase,
the
intermediate
or
pre-­‐igni/on
phase,
the
fast
igni/on
phase
and
the
final
products
phase.
The
choice
also
reflects
the
use
of
a
variety
of
different
equivalence
ra/os.
For
the
ini/al
reactants
phase,
methane
and
oxygen
were
used,
reflec/ng
also
the
equivalence
ra/o.
For
end
product
phase
the
primary
products
of
CO2
and
H2O,
both
taken,
once
again
to
reflect
the
different
equivalence
ra/os.
For
the
intermediate
and
igni/on
phases,
several
significant
intermediate
species
and
radicals
were
chosen.
Secondary
considera/ons
are
measurability
and
that
there
are
representa/ves
of
both
oxygenated
compounds
and
hydrocarbons.
The
species
chosen
are
HO2,
H2O2,
CH4
and
C2H4.
T
s1
P
sn
Hypercube
n+2
Progress
Variables
(one
for
each
level)
Progress
Variable
to
Hypercube
At
the
ith,
non-­‐leaf
level,
the
search
proceeds
as
follows:
Value:
Isolate
the
ith
progress
variable
from
the
target
configura/on.
Interval:
From
the
minimum,
maximum
and
number
of
branches,
compute
which
branch
posi/on,
bi,
the
value
corresponds
to.
Branch:
The
index
at
the
bith
value
within
the
node,
corresponds
to
the
node,
represen/ng
the
(n+1)th
level,
to
branch
to.
If
a
node
does
not
exist,
the
create
an
empty
node
of
the
(n+1)th
level.
Computa/on
Point
Response
In
memory
Move
to
memory
Move
to
Disc
2
scenarios:
point
in
memory
or
point
on
disc
Stored
on
disc
Factors
affec@ng
speed-­‐up
Although
it
is
computa/onally
more
efficient
to
have
the
en/re
tabula/on
in
memory,
if
the
computa/onal
resources
do
not
allow
this,
then
a
reasonable
possibility
is
to
provide
an
interac/on
of
memory
and
disc.
One
alterna/ve
is
to
rely
on
the
general
swapping
mechanisms
of
the
opera/ng
system.
However,
in
the
simplicity
and
generality
of
this
approach,
efficiency
suffers.
Another
alterna/ve
is
to
provide
a
more
direct
tuned
swapping.
The
method
used
here
is
to
have
the
en/re
database
on
disc
and
to
read
into
memory
only
that
data
which
is
called
for.
Make
approxima/on
from
accumulated
points
The
range
of
validity
is
determined
by
the
point
ranges