Density of Silicon Nit_ride Ceramics ....._. Krzysztof J. Cios and Leszek M. Sztandera. = University of Toledo. Toledo, Ohio and. George Y. Baaklini and Alex Vary.
NASA
Technical
Memorandum
106049
7,a3
Fuzzy Sets Predict F!exural Strength and Density of Silicon Nit_ride Ceramics ..... _
Krzysztof University
J. Cios and Leszek of Toledo
M. Sztandera
=
Toledo, Ohio and George Y. Baaklini and Alex Vary Lewis Research Center .... Cleveland,
Ohio
May 1993
r T7
(NASA-TM-I06049) PREOICT FLEXURAL DENSITY OF SILICON (NASA)
23
N93-27270
FUZZY SETS STRENGTH AND NITRIDE CERAMICS
Uncl
p
G3/24
fiJ/ A
as
0169404
__
I
m
__
n
_
It
FUZZY
SETS
PREDICT FLEXURAL OF SILICON NFrRIDE Krzysztof
J. Cios
and
STRENGTH CERAMICS
Leszek
University
AND
DENSITY
M. Sztandera
of Toledo
Toledo,
Ohio
43606
and George National
Y. Baaklini
Aeronautics Lewis
and
and Research
Cleveland,
Alex
Space
Vary
Administration
Center
Ohio
44135
SUMMARY
In this strength time,
work
we utilize
and
density
sintering
time,
Flexural
strength
modulus
of
Celsius
are
utilized
to build
exceed
3.3%,
11.34%
obtained
These
the
randomness
bars
the
and
6Y
are
such
fuzzy
as ceramics, variables
of manufacturing
7.1%,
networks,
that
The
of flexural
are
as an input
to the fuzzy
used
of the and
system.
135
operator
maximum
Data tested
and
Hamming
error
with
system.
273 Si3N 4
at 1370
for
the
of milling
from
bars
test
as compared
degrees
distance
density
errors
of
are
does 1.72%
not and
respectively. sets
theory
especially
and
predictions variables
mean
model.
make
Processing
parameters
Generalized
neural
and
ceramic.
temperature
strength
demonstrate
nitride
output
predictive
by using
to evaluate
pressure
at room
flexural
processing
silicon
the
study.
fuzzy
theory
nitrogen
tested
this
for
sets
sintering
density
in
materials,
between
and
rupture
results
designing
of NASA
and
used
fuzzy
can be incorporated for
parameters,
assessing like
into the
more
strength,
complex
which
process
of
relationships
are
governed
by
processes.
INTRODUCTION
High savings,
engine reduced
improvements to
leave
temperatures
weight,
and
for automotive
50 percent
*On sabbatical
operating
over
current
from the University
made
environmental engines
engine
of Toledo
with
benefits. structural
technology.
and NASA
possible
Resident
Estimates ceramic
Structural
Research
by ceramics
in energy
potential
efficiency
of
components
ceramics
Associate
will result
range
such as silicon
at Lewis
Research
Center.
from carbide
30
and
silicon
their
nitride
relatively
light
high-temperature when
mass
fracture parts
strength.
toughness
discrete
Therefore
fabrication and
only
physical
and
ceramic regarding
the
through
improved
expended
fabrication
reliable
structural
Scatter
scatter
silicon
work
a
silicon
nitride
Lewis
Research
strongly
nitride
affected temperature, processing procedures.
to by
and
taken
shorten
may the
metal
to
Current bulk
performance
NDE
[1].
available
regarding
time
engine
attributed
methods
the
that
material's
technology may
minimize
help
low
containing
affect
to
inexpensive
processing.
information
good
relatively
many
ceramics
(N'DE)
N'DE
be
and
during
information
be
and
generally
By incorporating
of ceramics
powder
variables
Center
a
obtained occurence
reduce
needed
into
be their
to
the
effort
to develop
sintering
conditions
were
trials
varied
conditions
processing time,
grinding
time,
21
batches
and
based
on
feedback
improved
of
during
strong,
3].
work
evident
Based at
NASA
The
investigation
of one
from
strength
radiography and
2
reduced
From
research
and density
the
gradients
sintered
silicon
a program
was
strength
relationships
as
described. to obtain
variables
position.
The
of powder
silicon
scatter.
Space
LeRC,
furnace
or exclusion
were
during
sintering
inclusion
material
that
upon
gradient-flexural
and
occuring
Aeoronautics
it was
[2,
contact,
a design/reliability
fabrication.
National
variables.
setter
of an extensive
of
concomitant
the
from
inhomogeneities
and
LeRC)
density
drawback
and
characterization
powder
In [4], the results
at
(NASA
sintering
overpressure, were
defects
compositions
investigate
and
nitrogen
to
is a great
composition
upon
systematically sintering
with
provide
are
adversely
evaluation
Thus,
pro_am
X-radiographic
involving
structures
can
can
attributed
nitride
dependent
preliminary
undertaken
produce
steps
procedures.
is
of
were
frequently
so
properties
processing
Administration
introduced
that
can
replacing
toughness
program,
that
that
of
ceramics.'
This
on
low
because
resistance,
properties
toward
development
research
in mechanical
standpoint.
materials
in strength
uniformity.
components shock
cracks
also
and
defects
in a materials
and and
but
thermal
move
non-destructive
properties
of
variation
also
flaws
hot-section
nonstrategic
anomalies
to have
technology
source
of
inclusions,
discrete
materials
and
in strength
procedures
mechanical
oxidation
wide
microstructural
detect
engine
a large-scale
scatter
it is essential
not
their
as voids,
for
consist
precluded
The
such
variations
excellent
However,
ceramics.
cost-effective
candidates
They
have
defects
density
leading
weight,
produced.
with
can
are
nitride
were powder
wet-sieving composition
Sintering/processing high-density These
previous
uniform results,
in turn, were used in neural systems[5], and are usedin this paper to study the viability of using fuzzy sets for predicting strength and density, speeding the optimization of the manufacturingprocess,and for comparingfuzzy systemswith neural systems. In this work we are interested in finding whether it is possibleto utilize fuzzy systemsto help in the material developmentprocessof advancedceramics.Fuzzy systemsare good in function approximation, and if the trend could be easily noticed as to which variable contributesmost for the increaseof a desiredoutput parameter,say strength, then this may help in speedingup the processof manufacturingand optimizing a new material. Material developerscan easily notice suchchangesfor a few variablesbut it becomesvery difficult to do so for a large number of variables.From the data collected by Sandersand Baaklini [4], we selectedonly three input variables,namely, milling time of the composition powder, the sintering time of the modulus of rupture test bars, and the nitrogen pressure employed during sintering. From the output variables,flexural strengthand density were selected.The rationale for using only the above mentioned variables is that there were not enough training pairs (outputs associatedwith inputs) for all the variablesused in the experimental work [4]. In this paper, relationshipsbetweenthe milling time, sintering time and nitrogen pressureand the resultantstrengthand densityare establishedby using fuzzy systems.Fuzzy set results arecomparedwith thoseobtainedusing radial basisfunction neural network [5]. BASICS
Randomness
is not
mathematical became 11].
the
gradual class
was
to be known
Essentially,
which
transition than
membership
defined In the
for
as fuzzy set
abrupt, rather
dealing
sections
form
in which
to formulate set theory.
theory
between
to be susceptible next
only
developed
fuzzy
rather
methodology
this
tool
the
OF FUZZY
provides
than with
the
source
the presence phenomena
to analysis
we provide
basic
uncertainty and
deal
reveals
with
other
It was first introduced
membership and
SET THEORY
a natural and
of random that
by
are
conventional
definitions
paper.
3
of
is the
variables.
vague,
of the
the
published
the
fuzzy
classes
In other too
mathematical set theory
sixties
of uncertainty
absence
imprecise,
strict
In
[10,
problems
in
of objects
is
of sharply words,
defined
it renders
a
or too
ill
complex approaches which
a and
by Zadeh
to manipulating
nonmembership
of imprecision
forms
and
approach
itself.
are
[7, 9]. used
in
Definition
of Fuzzy
Let
R be
collection
Sets
the
of items
set
of reals
of interest.
defined
by a membership
interval
on the real
line.
Note
that
the
Then
U -> {0, 1}, where
the
gA : U ->
the fuzzy
subset
of discourse
element
A of U can
0 and
subset
[0, 1] denotes
be expressed
or grade
be considered
{0, 1} is the set of values
set)
the
which
is a
A of U is closed
unit
as:
g. [0,1]}.
to as the degree
set B can
(crisp
of U. A fuzzy
[0, 1], where
I.tA(u)
gA is referred
non-fuzzy
universe
u be a generic
{gA(u)/u;uEU,
value
a classical
Let
U be
function
A= In this case,
and
of membership
as a binary
1 rather
than
characteristic
of u in A. function
an interval.
Support Let
A be a fuzzy
memberships
subset
in A
of U. The
support
P.A ( U ), are positive. Supp(A)=
of A, Supp(A), That
is the set of elements
in U whose
is,
{ u /u e U,
gA ( u ) > 0}.
Normality A fuzzy That
subset
A is normal
is, the supremum
As an example
let us consider of U, and the
Then
subset
Much
less
Note the
a crisp
set
the items
above
itemizing
( _ A ( U ) ) = 1.
of U, called
which
U, where
sign " + " is used "much
have
zero
here
it is subnormal.
U =
value
+ 0.6/5
1 +
to express
less than 5", may
5 = 1/1 + 1/2 + 1/3 +1/4
that
Fuzzy
Let
than
if Sup
over U is unity; otherwise,
are elements a fuzzy
if and only
... +
10, where
membership
be expressed
+ 0.4/6
in the grade
2 +
+ 0.3/7
(union)
1, 2, ..., 10 in the
set.
as: + 0.1/8.
of membership
have
been
ignored
in
expression.
operations
A and
operations
B be fuzzy for fuzzy
subsets
of a crisp
set U, with membership
sets follow:
4
functions
gA and
I.tB. Major
Complement The complementof a fuzzy subsetA of a crisp setU, denotedA', is definedby A'=
_., (1-ktA(u))/u,ueU, U
where
_
is used
as a convenient
notational
form.
Union The
union
of A and
B, denoted
A k..)B = _ U
Ak.JB,
(Max
is defined
by
[_A(U),I.tB(U)])/U,
ueU.
Intersection The
intersection
of A and
B, denoted
A CkB,
is defined
by
A (-'IB = _.,(Min[_A(U),p.B(u)])/u,ueU. U The
union
connective
Examples
corresponds "AND",
of the
to the and
the
operations
connective
operation
defined
"OR",
of complementation
above
follow:
Let U={
1, 2, 3,4,
5, 6, 7 },
and
A = 0.8/3
+ 1/4 + 1/5 + 0.6/6,
and
B = 0.7/3
+ 1/4 + 0.5/6.
Then A k..)B A _ A'
= 0.8/3
+
1/4+
1/5 + 0.6/6,
B -- 0.7/3
+
1/4 +
0.5/6,
= 1/1 + 1/2 + 0.2/3
+ 0.4/6
while
+ 1/7.
5
the
intersection corresponds
corresponds to negation.
to the
AGGREGATION
Aggregation
operations
combined
into
a
on
single
set.
fuzzy
sets
In
general,
OF
are
FUZZY
SETS
operations any
by
which
aggregegation
several
operation
fuzzy
is
sets
are
by
the
defined
function
h'[0,1]"410,1]
for some
n >
2. When
set A by operating
In order
Axiom
on the
to qualify
axiomatic
applied
as
1. Boundary
membership
an
requirements,
which
2. For
any pairai
additional
the fact Axiom
This
that
a noticeable Axiom
4.
equally
element
h must
the essence
of the
an aggregate
fuzzy
of U in the aggregated
satisfy
at
least
the
notion
of aggregation:
sets.
following
two
axioms they
.... ,0)=
0
are
usually
and
h(1,1
.....
e [0,1land
1.
bi e [0,1],ifai
nondecreasing
employed
1)=
>
bi for alli,
then
in all its arguments.
to characterize
aggregation
operations
despite
are not essential:
guarantees change
function.
that in the
h is a symmetric
an infinitesimal
variation
in any
argument
of h does
not produce
aggregate. function
in all its arguments,
that
is, the
aggregated
sets
are
important.
fuzzy
associativity arguments.
express
that is, h is monotonic
We can easily on
function,
, bi ,whereai
3. h is a continuous
axiom
of each
on U, h produces
conditions
h ( ai ) >- h ( bi ),
Two
sets def'med
grades
aggregation
h(O,O
Axiom
to n fuzzy
sets.
see that Although
provides Hence,
fuzzy they
a fuzzy
unions are
mechanism unions
and
and
intersections
defined
for
for
extending
intersections
6
qualify
only
two their
can
be
as aggregation
arguments, definition viewed
their to
as
any
special
operations property
of
number
of
aggregation
operationsthat are symmetric, usually continuous,and required to satisfy some additional boundary conditions. As a result of these additional requirements,the standard max and min operationsrepresentboundariesbetweenthe averagingoperationsand the fuzzy unions and intersections,respectively. There are severalclassesof averagingoperations[8, 9]. One of them that coversthe entire interval between the rain and max operationsconsistsof generalizedmeans.This class of operations will be usedin this paper. It is defined asfolows: 1 ha ( al , a2 , ....
an)=
(
aT+
a_ +n ""+
a_
)
-6
f
where
a
is a parameter
In our
application
consequently,
by which
we used
different
_x =
it represents
means
2. Function
are distin_ished,
ha clearly
a pararneterized
class
and
satisfies
o_ E R
Axioms
of continuous
and
\
_ (x _ 0 ).
1 through
symmetric
4, and,
aggregation
operations.
A METHOD
The were
fuzzy
tested
system
was built
at room
temperature
temperature, pressure listed
18 yielded
in Table
In order the
system
number
removed
from
into
the
of training
in the
ability
from
the 70
vectors
data. %
for
test
data
and
test
batch the
data
training set. This data
sets
was which
evaluate network
data
set
then
30
sintering
which
are
repeated
7
[4].
of the the
Thus,
testing. for
labeled
5 times
and
in Table
predictions.
number
values
6Y25
divided
into
number
6Y25
in order
as combinations
room
nitrogen
combination
batch
Batch
the
I. Also
we needed
output
optimium
which
°C.
predictions,
the error
the
listed
bars
C. For
time,
at 1370
pseudo-randomly
% for
were
system
to predict
represents
were and
fuzzy
of rupture
at 1370
for 9 combinations
then
available
tested
time,
densities
in the
of the
were
milling
densities
and
273 Si3N4 modulus
which
and
number
The
from
of
confidence
test
this
variables
approximately
our
and
PREDICTION
bars
strengths
strengths
known
as
135
combinations
to determine
6Y25,
the data
and
composition
I are the
interested
processing
pairs
the
using
particularly
inserted
different
using
FOR
to have A through
a
to test We
were
for
batch
for
the
was
first
ratio was
of then
5 different E (Table
Room temp Batch #
Number of
Millingtime,
specimen
[hr]
'6Y1B
30
24
6Y2B
30 15
24 100
15 15
300 100
14 19
300 ..... 24
10 10 10 10 15
100 100 100 100 100
15
100 100
6Y11 6Y12 6Y13 6Y14 6Y1516 6Y17 6Y18 6Y19 6Y20 6Y23 6Y24A 6Y24B 6Y25 6Y26A 6Y26B 6Y28 1370C Batch # 6Y9B 6Yll 6Y12 6Y13 6Y14 6Y1516 6Y17 6Y18 6Y25
Table
15 10
2.5 2.5 2.5
r
300
Actual strength, [MPa] 556
Nitrogen pressure, [MPa] 2.5 2.5
532 490
,
!
579 684 746 664 646
Actual density_ [g /cm "q 3.12 3.18 3.23 3.25 3.24
1 2
2.5 5
2 1.5
5 5
608
3.24 3.22 3.23 3.21
1.5 2 1.25
5 5 5
570 650 631
3.22 3.22 3.24
1.25 2 2
3.5 3.5
3.26 3.26
5
586 619 714
3.28
15 15 10
100 ..... 100 100
1 1 2
3.5 5 5
479 503 671
3.20 3.18 3.21
29
24 100
1 1
2.5 2.5
383 444
300 100
1
2.5 2.5
416 406 425 402
3.12 3.23 3.25
13 14 15 14 20 10 10 10
I: Strength/Density sintering
Sintedng time, [hr]
conditions.
2.5
300 24
2
100 100 300
2 1.5 2
,r
at room
temperature
and
439 458 467
5 5
1370
C for
different
processing
3.24 3.24 3.22 3.23 3.21 3.28
and
Table
!I: Selected
batch
Room
numbers
Training
Temp.
B
A
for 70% and 60% training
sets..A through
Sets 70%
Traininq
C
D
E
w
*
}
*
*
I
*
I
*
I I I
"
A
8
E
Sets 60%
C
D
E
Batch 13o. r.
L
6YIB
I
6Y2B
I
6Yll
I
6Y12 6Y13 ! 6Y14 [
* * *
*
*
*
*
* *
*
*
*
*
*
*
*
*
*
nu
l
*
I *
* *
*
I ,! I
"
}
*
I
*
I
•
I
•
t
*
I
* *
l
" I
*
*
I I I
*
i
-
I
*
"
6Y15 6Y16 6Y17
6Y_8
I
I 6Y1£ 6Y20 I 6Y23 I 6Y24AI 6Y24B
I
6Y25
I
6Y26A
6Y2e8 6Y28
•
* *
*
•
*
*
•
I I I I
* * *
* •
'1 I I
W
•
* • *
" •
•
•
1 I
I *
*
I I
* *
* *
I
.
*
*
1
I I
. .,,.
* •
•
*
* *
it
•
I I I
*
I
•
t
"
I
t
•
I
I I I
" •
I
t
I
I I
-
1370°C Batch no. 6Y9B 6Y11
I
6Y12
I
I
.
•
•
*
6Y13 6Y14
I
*
*
6Y15 6Y16 6Y17
I
*
6Y18
I
"
6Y25
I
I
*
"
*
*
!,,,"
I
••
I
•
l
•
•
i I
*
-I•
I
I I
• -
'1 I I *
I--,
•
I I t I
i I
* -
*
I I I I
• •
I
•
iI
•
I
"
II). This
entire
training
process
a training
created.
Batch
batch
placed
data
composition
input
and
(nitrogen support
using
a ratio
of approximately
pressure,
sintering
elements
60 % data
used
for
elementwise,
and
membership
grades fuzzy
shown
shown
input
that
UI
we
fuzzy
sets
grinding
C. was
%) except
sole
made
vector.
6Y25
Finally,
predictions
density).
The
grades
repeated
for 70%
variables
sets
for each
for
was all the
the
for
while
every
of generalized and
60%
respectively.
The
Thus,
appropriate
models
parameters,
and
for the
Different of
were
for nitrogen
of the
input
ones
have
were
formed
sets
of
for
The
resulting
the
the
temperature
for
sintering
for
normalized
for the room
formed
their
to produce
membership
pressure,
of support
were
operation
data
(21
output
of prediction.
mean
material
values
fuzzy
step
of silicon
three
memberships
training grades
of 21 batches batch
are defined
time)
and
by means
sets
processing
IV,
VI.
(100
set as the
and
42 fuzzy
input and
1370
and
set
and
strength
combined
output
1370
flexural time
C are strength
and
grinding
parameters.
a measure
the
The time,
fuzzy
V and
as the
as the
between
The
in Tables
density
sets).
normalization
were
sets.
in Tables
After
the
data
to establish
and
numbers
the outputs.
(flexural
temperature
batch
in the test data
sintering
formed
fuzzy
room
placed
representing
was
of all the
in a training
21 output
of two
resulting
was
we do not know
collected
nitride
time
repeated
set consisting
6Y25
were
for which
The
and
data
number
numbers
vectors
are
then
and 40 % for testing.
Next,
the
was
actual
similar and
to Hamming
generalized
distance
fuzzy
sets
is proposed
of input
to calculate
parameters.
The
the
difference
measure
is defined
by:
ucS
where
S is an interval
in R, and
thenD(A,B/R)= Nextl from
the
values the
combination
B are
fuzzy
subsets
on S. In the case
when
S = R,
D(A,B).
the k-fraction
of the
generalized
for flexural
method
A and
grades strength
is depicted A with
70%
measure,
of memberships and densit3r,
on
where
Fig.
training
1. As at the
k e (0, of the
k was chosen an example room
1), is either
output
us consider
temperature.
10
parameters.
to be 0.1. The let
added
The
to or subtracted This
graphical 6Y12
generalized
results
in the
explanation test
batch
input
fuzzy
of from set
Generalized input fuzzy set
Actual input fuzzyset l.t
_t
Supp mt Generalized
output
st
p
fuzzy
set
Supp mt
Dissimilarity measure
Predicted
st
output
p
fuzzy
set
_t i
I
/
> Supp
Supp s
s
d
I'he generalized input fuzzy set consists of grades ol _embership obtained by generalized mean operation 3erformed on normalized values of input parameters: milling :ime, sintering time, and pressure (note that these fuzzy sets :lifter for different combinations of training data). The actual nput fuzzy set represents normalized values of a parficulat nput. The dissimilarity measure based on Hamminc :listance concept is then employed, that is, the elementwise :lifferences are summed up. k- fraction of the measure is :hen added to the grades of membership of the generalizec _utput fuzzy set ( a fuzzy set obtained by generalized meat _peration performed on normalized values of outpul 3arameters: strength and density), resulting in the predictec _utput fuzzy set. The grades of memberships of the latte_ are compared then with the actual ones obtained b normalization of the values of a particular output.
Fig.
1.
A graphical
explanation
of the method
used
11
Actual
d
output
fuzzy
set
I-t
Supp s
for fuzzy
set prediction.
d
consists of grades of membershipobtained by generalizedmean operation performed on normalized values of input parameters:milling time (mt), sintering time (st), and pressure (p). In this particular case the fuzzy set with the support of (mt, st, p) has grades of membership0.44, 0.73, and 0.82, respectively (Table III). Note that generalizedinput fuzzy setsdiffer for different combinationsof training data. The actual-input fuzzy set represents normalizedvaluesof the 6Y12 batch input, that is, a fuzzy set with supportof (mt, st, p) with gradesof membership1, 0.5, 0.5, respectively.The dissimilarity measurebasedon Hamming distance is then employed, that is, the elementwise differences between grades of membershipof actual and generalizedinput fuzzy sets are summed up. In this particular case the dissimilarity measureis 0.01. The k-fraction of the measure,which is 0.001 when k=0.1, is then added to the gradesof membershipof the generalizedoutput fuzzy set. The generalized output fuzzy set, obtained by generalized mean operation performed on normalized values of output parameters,strength(s) and density (d), has in this casegrades of membership0.82 and0.99, respectively,as shown in Table III. Addition of the k-fraction of dissimilarity measureresults in the predicted fuzzy set with gradesof membership0.821 and 0.991, respectively.The latter are then comparedwith the actual gradesof membership obtained by normalizationof the values of the 6Y12 batch output 0.781 and 0.99i, resulting in error of 4% for the strength and perfect prediction of density for the batch under consideration.
RESULTS
The
method
strength
and
prediction detailed
density
of the results
100%
ones
6Y25
Table
of
data.
the
for
and the
batch. 30%
optimum
E are
Table
data
variables. 6Y25.
room
sets
are
chosen and
listed
in Table
VIII
obtained
to predict
in Table
Resultant
strengths
reason
12
for 70%
made
being
C, II.
(A).
6Y25
and
densities fuzzy
as
well
Table
and
strength
selected
of the flexural
The
training,
for
is that
values
1370
the first combination
predictions
The
randomly
temperature
set, for
shown
shows
DISCUSSION
to predict
training
the results
XI
batch
in
The test
X shows
sintering
was used
samples
A through
training.
processing
above
batch
for the
for combinations 60%
described
AND
new are systems
as
VII
overall
shows results
in Table
and
for
IX for
density
combinations lower
than
are
bounded
using of the as
wasproven in our other work [6]. With 40% test data at room temperature,the strength and density values were predicted with an averagepercentageerror of less than or equalto 7.1% and 2.8%, respectively.When the slightly smaller test set of 30% was used,the averagepercentageerrors for strengthand density droppedslightly to less than or equal to 5.7% and 2.4%, respectively.Similar results were obtained for the 1370 °C data (Tables XII-XV). With 40% test data the average percentageerrors for strength and density were less than or equal to 5.4% and 3.3%, respectivelylWith 30% test data thesevalues were 4.6% and 2%, respectively.For 1370°C, prediction of the 25th batch was perfect (0% error) for all combinationsof 30% test data and even for all combinations of 40% test data, for both strength and density. So, even using only
60%
of the
data
The
reason
parameters. crisp
set
with
all
for
training,
for that
the
model
is that
membership
was
the fuzzy
functions
able
to exactly
set representing
equal'1
(it
predict
25th
the 25th
batch
reaches
batch
output
is actually
maximum
in
all
a
input
parameters). The
information
sintering that
and
obtained
in Tables
processing
variables
for
where
6Y25
sintering
time
can
sintering
time
of 2 hours,
strength 467
of 462.86
MPa,
materials
be used.
MPa
viewpoint.
sintered
grains,
which
can
temperature loss
pressure
required
strength
or density.
turned for
out
parameters.
to prevent
was
also
some
They
were
the
makes
with
large,
times
may
and
to find
thus
lead were
was
the
triples
of
not not
tried.
and
to have
room
and
1370
input
parameters (300,
more
of input
sense
1.75,
that
a
from
growth
to high
and
actual
beyond
that
on
either
parameters
which
In addition,
maximal
5 ), (300,
the
columnar
influence
the
of
powder,
resistant
C temperatures. gave
a
for 6Y25
pressure great
lower
predict
value
as
hours,
in a coarser
grain
Nitrogen
combination
p):
and
randomly-oriented
toughness
found
( mr, st,
13
results
to exaggerated
optimal
for both
combinations following
This
of
dense
of 250
sets
the
microstructure
fracture
time
fuzzy
also
and
pressure
less than
time
as for 6Y25
other
slightly
of 50 hours.
of higher
nitrogen
the
combinations
as strong
a milling
milling
decomposition
used
same
C
a
sintering
to evaporation,
to be the
1370
yield
using
shorter
the ceramics
Long
due
system
the
make
creep.
material
will
Namely,
almost
of 5 MPa,
This is only time
may be other
or lower
XVI,
pressure
in milling
there
material
time
in Table
a nitrogen
that
produce
grinding
can be obtained.
processing
suggests
will
shorter
and
a reduction
when
that
For example,
but with
which
The
XI and XVI
output
2, 4.5),
and
Table
III:
The
resulting
temperature
fuzzy
sets,
for 70%
Table
B C
.0.82 0.82 0.82
D E
0.83 0.79
IV: The
resulting
temperature
A B C D E
Table¥:
The
A B C I
D E
Pressure
0.99 0.99
0.50 0.49
0.69 0.75
0.80 0.79
0.99 0.99
0.49 0.39
0.67 0.76
generalized
mean
operation,
0.82
. 1
0.77 0.79
for the room
training.
0.81 0.82 0.79 0.79
0.99 0.99 0.99 0.99
sets,
after
Milling time 0.45 0.53
Sintering time 0.72
Pressure 0.82
0.72 0.73 0.70 0.75
0.85 0.79 0.82 0.77
0.53 0.42 0.41
generalized
mean
operation,
for
1370
C
training.
Milling time 0.61
0.91 0.93
Density 0.99 0.99 0.99
0.92 0.92
0.98 0.99
0.62
Strength 0.92
for the room
Sint.ering time 0.73
Density 0.99
for 70%
operation,
Milling time 0.44
Strength 0.82
fuzzy
mean
Density 0.99
sets, after
for 60%
resulting
temperature
fuzzy
generalized
training.
Strength A
after
0.60 0.47 0.24
14
I. Sintering time 0.65 0.71 0.74 0.74 0.55
Pressure 0.71 0.71 0.79 0.79 0.61
......
Table
VI: The
resulting
temperature,
sets, after
for 60%
E
VII:
Predicted
Density 0.98 0.99 0.99 0.98 0.98
room temperature
6Y2B 6Y12 6Y17 6Y18 6Y24A 6Y25
Overall
Combination
operation,
Sintering time 0.68
0.65 0.26 0.26 0.67
"0.74 0.78 0.68 0.56
with 70%
training,
0.74 0.74 0.84 0.74 0.63
Combination
Actual
Predicted Density, g/cm2.93 3.26 3.27 3.23 3.28
537 602
1 4
646
633
608 586 714
614 604 686
2 1
3.23 3.21
3 4 3
3.26 3.28
for room
Pressure
% Error
532 579
results
for 1370 C
Milling time 0.65
Density, cj/cm _ 3.12 3.25
AveraQe Error
VIII:
mean
strength
Predicted Actual Strength, MPa Strength, M Pa
Batch Number
Table
generalized
training.
Strength 0.93 0.93 0.94 0.93 0.91
A B C D
Table
fuzzy
temperature,
A % Error
6 0
3.25
1 2 0
i|
70% training
Strength - % error for 6Y25
A
Strength - average % error for all test vectors 2.2
Density - % error for 6Y25
4
Density - average % error for all test vectors 2.0
B C D E
3.0 7.0 6.2 10.0
4 4 3 7
2.8 3.6 1.6 2.0
0 0 0 0
Combined Average % error
5.7
4.4
2.4
0
15
0
....
Table
IX: Overall
results
Combination
Table
60% training
Strength- % error for 6Y25
Density oaverage % error for all test vectors
A B
2.7 6.0
4 6
.2.5 4.3
C D E
7.3 10.5
4 7
3.6 2.2
0 0
9.0 7.1
7 5
1.3 2.8
0
X: Prediction
Batch Number
Xl:
for 6Y25
density
and strength
716
714
Prediction
of selected
temperature Milling Time, hr
strength
''
processing
Densit,_,, g/cm_ .... 3.28
and
and density,
sintering
....
temperature,
Actual
_0.3
Density - % error for6Y25
training
Predicted
% Error
Density/, g/cm_ 3.28
0
variables
100 % plus 6Y25
100%
0 0
for
optimum
room
training
Nitrogen Pressure, MPa 3
Predicted Strength, MPa 596
Predicted Density, g/cm 3 3.15
1.5 1.5 1.75 1.5
3 3
604 611
3.18 3.21
4
634 619
3.28
1.5 1.75
4 4
634 649
300
1.5
4
649
300 300
.1.75 2
4
656 686
15C; 175 200 200 250 250 250
Sintering Time, hr
at room
% Error
Actual Predicted Strength, MPa Strength, MPa
6Y25
Table
temperature,
Strength- average % error for all test vecto rs
CombinedAverage % error
o
for room
1.5
t6
3.25 3.28 3.28 3.28 3.28 3.28
Table
Xil:
Predicted
6Yll 6Y156Y16 6Y25 Average Error I
Xlll-
Batch Number
395 426
11 6
467
467
0 5.7
for 6Y25
density
and strength
Actual Predicted Strength, MPa Strength, MPa 467
XlV:
Overall
% Error
444 402
Prediction
6Y25
Table
at 1370 °C with 70% training,
Predicted Actual Strength, MPa Strength, MPa
Batch Number
Table
strength
results
467
% Error
0
Combination
Actual
Predicted
Density,, _cm_ 3.23
Density, g/cm3.04
3.22
3.25
3.28
3.28
A % Error
6 1 0 2.3
at 1370 °C with 100% training Actual
Predicted
% Error
Density, .... g/cm _ 3.28
Density, g/cm V 3.28
0
for 1370 °C, 70% training
Combination
Strength- average % error for all test vectors
Strength - % error for 6Y25
Density- average % error for all test vectors
Density - % error for 6Y25
A B C D
8.5 6.5 1.0 2.5
0 0 0 0
3.5 1.5 3.0 0.5
0 0 0 0
E
4.5
0
1.0
0
4.6
0
2.0
0
Density - % error for 6Y25
Combined Average % error
Table
XV:
Overall
Combination
results
for 1370 °C, 60% training
Strength - % error for 6Y25
A B
Strength - average % error for all test vectors 6.3 4.3
0 0
Density- average % error for all test vectors 5.7 4.3
C D E
3.3 6.3 6.7
0 0 0
3.0 0.7 2.7
0 0 0
Combined Average % error
5.4
0
3.3
0
17
0 0
Table XVI: Prediction of selected processing and sintering variables for optimum density and strength at 1370 °C with 100 % plus 6Y25 training Sinterincj Time
MillingTime 150 175
Nitrogen Pressure 4 4
1.5 1.5 1.5
200
4 5
200 200 250
1.5 1.75
300 300
1.5
5 5 4
1.5 1.75 2
5 5 5
30,0 300
Predicted Strength 425 430 434
Predicted Density 3.18 3.21 3.25 3.28
444 448
3.28 3.28
462 453
3.28
462 467 467
3.28 3.28 3.28
Average error
12
%
- RBF
60%
70%
Room
temperature
60%
70%
8
- FS
0
Fig. 2
Average
errors
(RBF)
and
in predicting
fuzzy
sets
60%
%
using
radial
basis
function
neural
network
(FS).
Average error
strength
1370 C
60%
70%
70%
3
2
0 Room Fig. 3
Average (RBF)
errors and
in predicting
fuzzy
sets
temperature density
using
(FS).
18
1370 C radial
basis
function
neural
network
(270,
2, 5).
The and
results
show
property
processing learning based
control.
the
host
pressure,
fact
whereas
microstructure,
and
with
neural
less
precise
Fig.
3) which
are
on
due
Developers
data
existing
between
the
and
to be applicable and
for
density
nitride. than
predicting time, on
sets with fuzzy
processing
variables
and
strength.
sintering
pore
This
time
and
morphology,
on
defects.
that
in the
development
those
sets
variables
manufacturing density
obtained
are
superior
and
strength
process.
was
modelled
previously in modeling (Fig.
The
2 and
more
better
(in
precise terms
of
networks. and composite
the
processing
will
help
to capture
in a manufacturing
process,
This
variations
processing
of ceramics
relationships.
the
modeling
strength
to milling
fuzzy
[5] indicates
found
dependent
causing
process
the
predicting
related
by using
the same
was
successful
additionally
of failure
for both speed
for silicon
more
is directly
obtained
system
features
is
and
consequently
was
strength
presence
shorten
former
density
to statistical
between neural
as input
density
results
relationships
using
and
on the
networks
relationship error)
bulk
fuzzy
and
tool
optimize
The
parameters
bulk
be a powerful
help
materials.
flexural
of the
can
should
variables
that
the
Comparison
logic
processing
predicting
to the
set theory
ceramic
processing
In general, due
fuzzy
Fuzzy
of emerging
on three
was
that
can
by
structures
utilizing
fuzzy
sets
imprecise
relationships
the
to capture
be seen
latter as the
can achieve
alternative
and
better
neural
which
strength
networks
are due
and density,
in
tandem.
to unavoidable
more
precise,
although
to the
Taguchi
method
still
The
variations
very
complex,
[12].
CONCLUSIONS
Fuzzy and
sets
density,
pressure.
theory and
The
combinations by The fuzzy
are
three
learned
the
maximum sets
the
of the
calculating
was applied
error
to learn
processing
relationships processing
errors for
superior
on
relationships
variables: were
for
variables.
The
reliability
data
encompassing
test
neural
was
7.1%,
networks
and
time,
predicting
for
exist
density
between
sintering strength
of these
either
in capturing
19
that
milling
used
strength to
the
30%
vague
time, and
or 40% 3.3%.
strength
and
nirogen
density
predictions
it was
the
was
for
validated
of available It was
relationships
new
data.
found
that
between
the
processingvariablesand strength.However, for density which is more directly related to the input variables,neural networksgave better results. Summarizing, developers of structural ceramics and ceramic composites, may utilize computational paradigms of fuzzy sets and neural networks to optimize the desired parametersand to shortenthe processingand manufacturingcycle:
ACKNOWLEDGEMENT
The LeRC,
authors
would
like to thank
for thoughtful
Mr. William
A. Sanders
from
Ceramics
Branch,
NASA
discussions.
REFERENCES [1] Klima SAMPE
S. J. and
[2] Klima [3]
Baaldini
Quarterly,
Vol.
S. J., NDE
Baaklini,
G.:
for Heat
N'DE
for Gas
Turbines
[4]
W.
and
with
the
Strength
Proceedings, [5] Cios Ceramic
A.
Vol.
and
Engine
and
Baaldini
and
Process
Power,
Vol.
G.
Correlation 1986.
G. Y., Vary and
Testing,
Predicts Gordon
A. and Tjia Related
and
[7] Cios
K. J., Shin
Stenosis,
IEEE
[8]
G.
I., and
Computer
J. and Cliffs,
Folger
T.
A.,
for
1987,
Ceramics,
Nitride,
Structural
Ceramic
R. E., Radial and
1992,
1984. Ceramics.
Basis
Density,
and
Sintering
Engineering
Function International
L.,
Using
24:3,
Fuzzy
Sets,
1988.
2O
of
Variables and
Science
Network
Learns
Advances
in
submitted.
Boundness and Equivalence of Fuzzy IEEE Transactions on Fuzzy Systems,
Vol.
Journal
pp. 263-266.
of Processing
Strength
Breach,
Goodenday
Magazine,
of Structural
TM-86949,
Control 109,
Silicon
K. J. and Sziandera L. M., Artificial Neural Networks,
Englewood
Y.,
Radiography
[6] Cios Forward
Klir
NASA
of
Processing
Characterization
Ceramics,
7, No. 7-8, pp. 839-860,
K. I., Baaklini
Nondestructive
Nondestructive 1986, pp. 13-19.
Reliability
Engineering Sanders
G. Y.,
17, No. 3,
Fuzzy
1991,
Sets
to Diagnose
Systems and Feed 1992, submitted. Artery
Coronary
pp. 57-63.
Uncertainty,
and
Information,
Prentice
Hall,
[9]
Sztandera
Department Columbia, [10]
L.M.,
of MO,
Zadeh
Michie
L.,
Zadeh
[12]
McEntire
J. E.,
and
Turbine Contractors' PA,
1990,
Electrical 1990. A Theory
D., and
[11]
Spatial
Mikulich
L., Fuzzy
sets,
Relations and
Engines,
Computer
C. L.,
Willey,
of
the
Component
27th
Society
of
pp. 341-349.
21
an of
1979, pp.
M.S.
Thesis,
Missouri-Columbia,
9, in Hayes
I. K.,
149-194.
1965. E., Yeckley Development
Automotive Automotive
Image,
Intelligence,
York,
D. N., Bright
Nitride
of
University
8, pp. 338-353,
A. P., Heichel
Meeting,
Subsets
Machine
New
Control,
Silicon
Proceedings
Coordination
Reasoning,
John
Information
Fuzzy
Engineering,
of Approximate L. I., Eds.,
B. J., Taglialavore
Quackenbush
Among
R. L., for
Technology Engineers,
Holowczak
Advanced
Gas
Development Inc.,
Warrendale,
i [-_::-4/_"_/ '"
REPORT
DOCUMENTATION
Form Approved OMB NO.0704.0188
PAGE
Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching ex;st;,_ data sources, gathering and maintaining the data needed., and completing _ reviewing the .c_.lection of information. Send comments regarding th_ bu_en e_nate or.any other asl_ect, of this collection of information, including suggestions for reducing th=s burden, to Wash=ngton Headquarters Servmes, D=rectorate for Information uperauor_ arm HepottS, 1Zlf_ ,Jenerson Davis Highway, Suite 1204, Arlington. VA 2220"2--4302, and to the Offee of Management and Budget, Paperwork Reduction Project (0704-.0188), Washington, DC 20503. 1.
AGENCY
USE
ONLY
(Leave
blank)
2.
REPORT
DATE
3.
REPORT
TYPE
May 1993 4.
TITLE
AND
AND
DATES
Technical
SUBTITLE
5.
Fuzzy Sets Predict
Flexural
Strength
and Density
of Silicon
Nitride
COVERED
Memorandum
FUNDING
NUMBERS
Ceramics WU-506--01-50
AUTHOR(S)
Krzysztof
J. Cios, Leszek
M. Sztandera,
George Y. Baaklini,
and Alex Vary
8.
7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)
PERFORMING REPORT
National Aeronautics and Space Administration Lewis Research Center Cleveland, I 9.
Ohio
ORGANBATION
NUMBER
E-7794
44135-3191
SPONSORING/MONITORING
AGENCY
NAIVE(S)
AND
10.
ADDRESS(ES)
SPONSORING/MONITORING AGENCY
National Aeronautics and Space Administration Washington, D.C. 20546-0001
REPORT
NUMBER
NASATM-106049
11. SUPPLEMENTARYNOTES Krzysztof J. Cios (on sabbatical leave from the University of Toledo and NASA Resident Research Associate at Lewis Research Center), Leszek M. Sztandera, University of Toledo, Toledo, Ohio 43606; George Y. Baaldini and Alex Vary, NASA Lewis Research Center. Responsbile person_ .G.eorge Y. Baaklini_ (216) 433-6016. 12¢ DISTRIBUTION/AVAILABILITY STATEMENT 12b. DIS_-RiBUTION CODE Unclassified
- Unlimited
Subject Categories
13.
ABSTRACT
(Maximum
24 and 38
200
words)
In this work we utilize fuzzy sets theory to evaluate and make predictions of flexural strength and density of NASA 6Y silicon nitride ceramic. Processing variables of milling time, sintering time, and sintering nitrogen pressure are used as an input to the fuzzy system. Flexural strength and density are the output parameters of the system. Data from 273 Si3N 4 modulus of rupture bars tested at room temperature and 135 bars tested at 1370 degrees Celsius are used in this study. Generalized mean operator and Hamming distance are utilized to build the fuzzy predictive model. The maximum test error for density does not exceed 3.3%, and for flexural strength 7.1%, as compared with the errors of 1.72% and 11.34% obtained by using neural networks, respectively. These results demonstrate that fuzzy sets theory can be incorporated into the process of designing materials, such as ceramics, especially for assessing more complex relationships between the processing variables and parameters, like strength, which are governed by randomness of manufacturing
processes.
15.
14. SUBJECTTERMS
SECURITY OF
NsN 7540-01-280-5500
materials;
PAGES
OF 18.
THIS
PAGE
Unclassified CLASSIFICATION
SECURITY
16.
PRICE
CODE
A03 19.
CLASSIFICATION
REPORT
Unclassified
OF
22
Ceramics; Processing; Fuzzy logic; High-temperature Neural networks; Silicon nitride; Monolithic ceramics 17.
NUMBER
SECURITY
CLASSIFICATION
20.
UMITATION
OF
ABSTRACT
OF ABSTRACT
Unclassified Standard
Fo_m
Prescribed 298-102
by ANSI
298 Std.
(Rev. Z39-18
2-89)
