George Y. Baaklini and Alex Vary ......... Lewis Research. Center .... by Sanders and Baaklini [3], we selected three input varaiables, namely, the milling time of ...
5/ NASATechnical
Radial -"
_°_-_
Memorandum
10,6.048 ........
Basis Function
Network
) _,e,f/m_/
......
7J-b
Learns
Ceramic Processing and Predicts Related Strength and Density
,_Krzysztof J. Cios ; ;Ur_v_r'-_-(y of Tbledo ........................ Toledo, Ohio George Y. Baaklini and Alex Vary Lewis Research Center Cleveland,
Ohio
.........
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::_
and .......
Robert E. Tjia University of Toledo Toledo, Ohio :
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_
=
May 1993
(NASA-TM-I06048) RADIAL BASIS FUNCTION NETWORK LEARNS CERAMIC PROCESSING AND PREDICTS RELATED • STRENGTH AND DENSITY (NASA) 20
N93-27129
Unclas
p
G3124
0169424
_=
RADIAL
BASIS
FUNCTION
NETWORK
PREDICTS
RELATED
LEARNS
CERAMIC
STRENGTH
AND
PROCESSING
AND
DENSITY
Krzysztof J. Cios _* University of Toledo Toledo, Ohio 43606 George Y. Baaklini and Alex Vary National Aeronautics arid Space Administration Lewis Research Center Cleveland,
Ohio 44135 and
Robert E. Tjia University of Toledo Toledo, Ohio 43606
ABSTRACT
Radial
basis
Si3N4
modulus
MOR
bars
pressure
which
density
were
points"
method
(RBF)
of rupture
were
were
the the
descent
less than
12%
demonstrated
tested
outputs used
ceramic
°C.
the
to set the
hidden
RBF
network
an average and
were
centers
and
predicted
error
sintering
input
networks
of less than the
the
data
temperature time,
features.
output
and
from
273
and
135
Sintering
gas
Flexural
strength
and
The
"nodes-at-data-
assessed.
strength
accelerating
using
at room
time,
as the
layer
trained
tested
Milling
RBF
for optimizing
were
were
used
by which
with
a potential
which
at 1370
The
density
bars
networks
parameters
method. and
neural
(MOR)
processing
was
gradient
emerging
function
layer with
training
an
used
average
2%. Further,
the RBF
development
and
the
error
of
network
processing
of
materials.
INTRODUCTION
Ceramics material weight, currently fracture processing :t
for
such
heat
resistance
as
engine
with which
[1, 2, 3].
On sabbatical at Lewis Research
In
nitride
applications
to oxidization,
encountered toughness,
silicon
occur their
due and
this
(Si3N4)
type due
work,
are
to their
thermal
to discrete Sanders
leave from the University Center.
high
shock
of ceramic
under
investigation
operating
resistance
is its widely defects
and
of Toledo
introduced
Baaklini
[3],
and NASA
as
a candidate
temperatures,
reduced
[1]. The varying into were
Resident
major
drawback
strength the
and
material
concerned
Research
low
during with
Associate
the
problem that
of designing
possesses
high
manufacturing
they
sintering
investigated
the effect
was superior
manufacturing
nitride
strength
with
tried
temperature,
sieving
a silicon
fully dense
the
In
to optimize time,
lowest
several
nitrogen
of sintering
amount
pressure
Hence,
by using sound
of
varaiables and
and temperature
to dry sieving.
process
ceramic with the goal of achieving scatter.
such
as milling
setter
contact.
variations
process
time,
In
judgement
trying
coupled
they
wet powder
to optimize
with
of
sintering
addition,
and whether
in their work, they were
engineering
the
material
the
trial and error
methodology.
In our work networks
to help
approximation output
we are interested in the process
making
parameter,
modelling
design
say strength,
for new materials. but it becomes
data collected
by Sanders
during
variables mentioned
Designers
inputs)
data
However,
we expected
despite
was
not
powder,
strength
there
variables
available
density.
not enough
such as temperature
originally that
of rupture
and
were
obtained
an RBF
nitrogen
pressure
we attempt
on resultant
make use of the data obtained network.
The original
different
combinations
sieving,
would
determine and density
how effectively
of milling
times,
bars tested a neural
of a batch of MOR
toward
a desired
MOR
rationale
training
pairs
2
namely,
the
pressure the output the
above
asociated
with
be noted that the network
accurate
analysis. predictions
in the input space.
of milling
time,
sintering
the aid of a neural
test bar strength times,
can be trained
bars.
It should
for neural
pressure,
on 273 MOR
bars
the purpose to predict
time
and
network.
We
and testing
and density
nitrogen
at 1370 °C. Therefore,
network
the
From
(outputs
of a
From
for using
study [3] for training
sintering
effects
varaiables,
give reasonably
with
up process
of variables.
test bars.
neural
in function
and the nitrogen
The
distributed
and density
from the previous
time,
intended
to find the effects
data had exhibited
and 135MOR
excel
most
input
(MOR)
etc. Thus, the data set used in this study is based
temperature
to utilize
the combined
and sieving.
nor
network
strength
three
the sintering
the fact that the data points are unevenly
In this paper
comprehend
[3], we selected
of the modulus
is that
for processing
networks
that contribute
can usually
and Baaklini
flexural
variables
Neural
very difficult to do so for a large number
sintering
we selected
variables
it is possible
from a few trials. This Should help in speeding
time of the Si3N4-Si02-Y203
employed
of ceramics.
it easy to identify
few variables
milling
in finding whether
the neural
variations
for
powder
wet
tested
at room
of this study is to
the resultant
strength
i f
/
RADIAL BASIS FUNCTION
One complex,
of the common
non-linear
functions.
any given function (nodes).
it employs
layer
proposed
the weights
network
(RBF)
approach
uses a combination
network,
data points
close to its center.
network
canbe
made
number
has been
is computationally
shown
elements
to be successful gradient
of
to approximate
of processing
demanding
iterative
error
reduction
linear outputs, slower iterative
of network
descent
in this method
and slow and results
because
the hidden
less
layer nodes
scheme,
direct approches methods.
as the radial
training
time
learning.
are RBF nodes
of it) and each node only responds
some
form
in
similar
to that
involving
used
matrix
learning
basis
because
the
The network centered
functions method,
in backpropagation.
inversion
has
is
at the
to an input which is
or sigmoidal
of supervised
layer
known
and supervised
layer nodes are usually linear
using
units in the hidden
[4]. Also
requires
of self-organization
The output
may be obtained
processing
to backpropagation
(or some subset
weights
and their such
as an
In the case of
can be used
in place
of the
of the RBF Network
Figure network
network
is the approximation
is the fact that the iterative
this type
as self-organized
Description
networks
has a sufficient
with "locally-tuned"
as an alternative
function
training
a neural
that the network
disadvantage
neural
times.
A three
considered
Theoretically,
backpropagation
its major
to optimize
long training
been
provided
The traditional
area. However,
uses of feedforward
NETWORKS
1 shows
performs
a general
a mapping
RBF
network
f: R n -- R
with n inputs
and one linear
given by the following
equation
output.
This
I I.II denotes
the
[5] :
11 r
f(x) -- a0 + 2E ,li_o(J
Ix-oil I)
(1)
i=l
where Euclidean
x _
R n is the norm,
Jli (0
nr) are the RBF centers,
input
vector,
_,(.) is a function
< = i < = nr) are
the weights
and nr is the number
from of the
of centers.
3
R n -- R, output
node,
As a variation
ci (0
< = i < =
of the linear
output,
I Bias
I
l_
