Journal of Minerals and Materials Characterization and Engineering, 2012, 11, 869-875
Published Online September 2012 (http://www.SciRP.org/journal/jmmce)
Application of Hooke & Jeeves Algorithm in Optimizing
Fusion Zone Grain Size and Hardness of Pulsed Current
Micro Plasma Arc W elded AISI 304L Sheets
Kondapalli Siva Prasad1*, Chalamalasetti Srinivasa Rao2, Damera Nageswara Rao3
1Department of Mechanical Engineering, Anil Neerukonda Institute of Technology & Sciences, Visakhapatnam, India
2Department of Mechanical Engineering, Andhra University, Visakhapatnam, India
3Centurion University of Technology & Management, Odisha, India
Received April 2, 2012; revised May 6, 2012; accepted May 27, 2012
ABSTRACT
AISI 304L is an austenitic Chromium-Nickel stainless steel offering the optimum combination of corrosion resistance,
strength and ductility. These attributes make it a favorite for many mechanical components. The paper focuses on de-
veloping mathematical models to predict grain size and hardness of pulsed current micro plasma arc welded AISI 304L
joints. Four factors, five level, central composite rotatable design matrix is used to optimize the number of experiments.
The mathematical models have been developed by Response Surface Method (RSM) and its adequacy is checked by
Analysis of Variance (ANOVA) technique. By using the developed mathematical models, grain size and hardness of the
weld joints can be predicted with 99% confidence level. The developed mathematical models have been optimized us-
ing Hooke and Jeeves algorithm to minimize grain size and maximize the hardness.
Keywords: Pulsed Current Micro Plasma Arc Welding; AISI 304L; Grain Size; Hardness; Hooke & Jeeves Algorithm
1. Introduction
In welding processes, the input parameters have greater
influence on the mechanical properties of the weld joints.
By varying the input process parameters, the output
could be changed with significant variation in their me-
chanical properties. Accordingly, welding is usually se-
lected to get a welded joint with excellent mechanical
properties. To determine these welding combinations that
would lead to excellent mechanical properties, different
methods and approaches have been used. Various opti-
mization methods can be applied to define the desired
output variables through developing mathematical mod-
els to specify the relationship between the input parame-
ters and output variables. One of the most widely used
methods to solve this problem is Response Surface
Methodology (RSM), in which the unknown mechanism
with an appropriate empirical model is approximated,
being the function of representing a RSM.
Welding thin sheets is quite different from welding
thick sections, because during welding of thin sheets
many problems are experienced. These problems are
usually linked with heat input. Fusion welding generally
involves joining of metals by application of heat for
melting of metals to be joined. Almost all the conven-
tional arc welding processes offer high heat input, which
in turn leads to various problems such as burn through or
melt trough, distortion, porosity, buckling warping &
twisting of welded sheets, grain coarsening , evaporation
of useful elements present in coating of the sheets, joint
gap variation during welding, fume generation form
coated sheets etc. Use of proper welding process, proce-
dure and technique is one tool to address this issue [1].
Micro Plasma arc Welding (MPAW) is a good process
for joining thin sheet, but it suffers high equipment cost
compared to Gas Tungsten Arc Welding (GTAW).
However it is more economical when compare with Laser
Beam welding and Electron Beam Welding processes.
Pulsed current MPAW involves cycling the welding
current at selected regular frequency. The maximum
current is selected to give adequate penetration and bead
contour, while the minimum is set at a level sufficient to
maintain a stable arc [2,3]. This permits arc energy to be
used effectively to fuse a spot of controlled dimensions
in a short time producing the weld as a series of overlap-
ping nuggets. By contrast, in constant current welding,
the heat required to melt the base material is supplied
only during the peak current pulses allowing the heat to
dissipate into the base material leading to narrower Heat
*Corresponding author.

K. S. PRASAD ET AL.
870
Affected Zone (HAZ). Advantages include improved
bead contours, greater tolerance to heat sink variations,
lower heat input requirements, reduced residual stresses
and distortion, refinement of fusion zone microstructure
and reduced width of HAZ. There are four independent
parameters that influence the process are peak current,
back current, pulse rate and pulse width.
From the literature review [4-18] it is understood that
in most of the works reported the effect of welding cur-
rent, arc voltage, welding speed, wire feed rate, magni-
tude of ion gas flow, torch stand-off, plasma gas flow
rate on weld quality characteristics like front melting
width, back melting width, weld reinforcement, welding
groove root penetration, welding groove width, front-side
undercut are considered. However, much effort was not
made to develop mathematical models to predict the
same especially when welding thin sheets in a flat posi-
tion. Hence an attempt is made to correlate important
pulsed current MPAW process parameters to grain size
and hardness of the weld joints by developing mathe-
matical models by using statistical tools such as design of
experiments, analysis of variance and regression analysis.
The grain size and hardness of the weld joints was opti-
mized using Hooke & Jeeves algorithm.
2. Experimental Procedure
Austenitic stainless steel (AISI 304L) sheets of 100 ×
150 × 0.25 mm are welded autogenously with square butt
joint without edge preparation. The chemical composi-
tion of AISI 304L stainless steel sheet is given in Table 1.
High purity argon gas (99.99%) is used as a shielding gas
and a trailing gas right after welding to prevent absorp-
tion of oxygen and nitrogen from the atmosphere. From
the literature four important factors of pulsed current
MPAW as presented in Table 2 are chosen. The welding
has been carried out under the welding conditions pre-
sented in Table 3. A large number of trail experiments
were carried out using 0.25 mm thick AISI 304L sheets
to find out the feasible working limits of pulsed current
MPAW process parameters. Due to wide range of factors,
it has been decided to use four factors, five levels, rotatable
central composite design matrix to perform the number
of experiments for investigation. Table 4 indicates the 31
set of coded conditions used to form the design matrix.
The first sixteen experimental conditions (rows) have
been formed for main effects. The next eight experimental
conditions are called as corner points and the last seven
experimental conditions are known as center points.
Table 1. Chemical composition of AISI 304L (wt%).
C Si Mn P S Cr Ni Mo Ti N
0.021 0.35 1.27 0.030 0.001 18.10 8.02 - - 0.053
Table 2. Important factors and their levels.
Levels
Serial noInput factorUnits –2 –1 0 +1+2
1 Peak currentAmperes 6 6.5 7 7.5 8
2 Back currentAmperes 3 3.5 4 4.5 5
3 Pulse ratePulses/second 20 30 40 50 60
4 Pulse width% 30 40 50 60 70
Table 3. Welding conditions.
Power source Secheron micro plasma arc machine
(Model: PLASMAFIX 50E)
Polarity DCEN
Mode of operation Pulse mode
Electrode 2% thoriated tungsten electrode
Electrode diameter 1 mm
Plasma gas Argon & hydrogen
Plasma gas flow rate 6 Lpm
Shielding gas Argon
Shielding gas flow rate 0.4 Lpm
Purging gas Argon
Purging gas flow rate 0.4 Lpm
Copper nozzle diameter 1 mm
Nozzle to plate distance 1 mm
Welding speed 260 mm/min
Torch position Vertical
Operation type Automatic
The method of designing such matrix is dealt elsewhere
[19,20]. For the convenience of recording and processing
the experimental data, the upper and lower levels of the
factors are coded as +2 and –2, respectively and the
coded values of any intermediate levels can be calculated
by using the expression [21].

 
max minmax min
22
i
XXXXXX



(1)
where Xi is the required coded value of a parameter X.
The X is any value of the parameter from Xmin to Xmax,
where Xmin is the lower limit of the parameter and Xmax is
the upper limit of the parameter.

K. S. PRASAD ET AL.
871
Table 4. Design matrix and experimental results.
Serial no Peak current
(amperes)
Back current
(amperes)
Pulse rate
(pulses/second)
Pulse width
(%)
Grain size
(micons)
Hardness
(VHN)
1 –1 –1 –1 –1 20.812 198
2 1 –1 –1 –1 30.226 190
3 –1 1 –1 –1 21.508 199
4 1 1 –1 –1 27.536 193
5 –1 –1 1 –1 27.323 193
6 1 –1 1 –1 25.206 195
7 –1 1 1 –1 25.994 195
8 1 1 1 –1 23.491 197
9 –1 –1 –1 1 26.290 194
10 1 –1 –1 1 29.835 190
11 –1 1 –1 1 20.605 200
12 1 1 –1 1 27.764 193
13 –1 –1 1 1 30.095 190
14 1 –1 1 1 26.109 194
15 –1 1 1 1 27.385 193
16 1 1 1 1 25.013 195
17 –2 0 0 0 20.788 196
18 2 0 0 0 25.830 195
19 0 –2 0 0 31.663 188
20 0 2 0 0 27.263 193
21
0 0 –2 0 25.270 195
22 0 0 2 0 26.030 194
23 0 0 0 –2 24.626 195
24 0 0 0 2 26.626 194
25 0 0 0 0 24.845 196
26 0 0 0 0 24.845 196
27 0 0 0 0 20.145 200
28 0 0 0 0 24.845 195
29 0 0 0 0 20.045 201
30 0 0 0 0 24.845 195
31 0 0 0 0 20.445 198
3. Recording the Responses
3.1. Measurement of Grain Size
Three metallurgical samples are cut from each joint, with
the first sample being located at 25 mm behind the trail-
ing edge of the crater at the end of the weld and mounted
using Bakelite. Sample preparation and mounting is done
as per ASTM E3-1 standard. The samples are surface
grounded using 120 grit size belt with the help of belt
grinder, polished using grade 1/0 (245 mesh size), grade
2/0 (425 mesh size) and grade 3/0 (515 mesh size) sand
paper. The specimens are further polished by using alu-
minum oxide initially and the by utilizing diamond paste
and velvet cloth in a polishing machine. The polished
specimens are etched by using 10% Oxalic acid solution
to reveal the microstructure as per ASTM E407. Micro-
graphs are taken using metallurgical microscope (Make:
Carl Zeiss, Model: Axiovert 40MAT) at 100× magnifica-
tion. The micrographs of parent metal zone and weld
fusion zone are shown in Figures 1 and 2.
Figure 1. Microstructure of parent metal zone.

K. S. PRASAD ET AL.
872
Grain size of parent metal and weld joint is measured
by using Scanning Electron Microscope (Make: INCA
Penta FETx3, Model: 7573). Figures 3 and 4 indicate the
measurement of grain size for parent metal zone and
weld fusion zone. Average values of grain size are pre-
sented in Table 4.
Figure 2. Microstructure of weld fusion zone.
Figure 3. Grain size of parent metal.
Figure 4. Grain size of weld fusion zone.
The grain size at the weld fusion zone is smaller than
parent metal zone, which indicates sound weld joint.
3.2. Measurement of Hardness
Vickers’s micro hardness testing machine (Make: MET-
SUZAWA CO. LTD., JAPAN, Model: MMT-X7) was
used to measure the hardness at the weld fusion zone by
applying a load of 0.5 Kg as per ASTM E384. Average
values of three samples of each test case are presented in
Table 4.
4. Developing Mathematical Models
The grain size and hardness of the weld joint is a func-
tion of peak current (A), back current (B), pulse (C) and
pulse width (D). It can be expressed as [22-24].
Grain size (G)


,,,GfABCD (2)
Hardness (H)


,,,
H
fABCD (3)
The second order polynomial equation used to repre-
sent the response surface “Y” is given by [19]:
2
0iiiiiiji j
Yb bxxbxx 

 

 (4)
Using MINITAB 14 statistical software package, the
significant coefficients were determined and final models
are developed using significant coefficients to estimate
grain size and hardness values of weld joint.
The final mathematical models are given by
Grain Size (G)
12
2
42 13
22.859 1.0521.0580.315
0.625 1.6402.320
GXX
XX XX
 

3
X
3
(5)
Hardness (H)
12
2
42 13
197.286 0.7081.2920.292
0.542 1.6032.188
H
XX
XX XX


X
(6)
where X1, X2, X3 and X4 are the coded values of peak
current, back current, pulse rate and pulse width respec-
tively.
5. Checking the Adequacy of the Developed
Models
The adequacy of the developed models was tested using
the ANOVA technique. As per this technique, if the cal-
culated value of the Fratio of the developed model is less
than the standard Fratio (from F-table) value at a desired
level of confidence (say 99%), then the model is said to
be adequate within the confidence limit. ANOVA test
results are presented in Table 5 for all the models. From
the table it is understood that the developed mathematical
models are found to be adequate at 99% confidence level.
The value of co-efficient of determination “R2” for the
above developed models is found to be about 0.85.

K. S. PRASAD ET AL.
873
Table 5. ANOVA test results for grain size and hardness.
Grain Size
Source DF Seq SS Adj SS Adj MS F P
Regression 14 249.023 249.023 17.7873 6.10 0.000
Linear 4 65.207 65.207 16.3018 5.59 0.005
Square 4 91.443 91.443 22.8608 7.84 0.001
Interaction 6 92.372 92.372 15.3954 5.28 0.004
Residual error 16 46.639 46.639 2.9149
Lack-of-fit 10 9.750 9.750 0.9750 0.16 0.994
Pure error 6 36.889 36.889 6.1481
Total 30 295.661
Hardness
Source DF Seq SS Adj SS Adj MS F P
Regression 14 228.18 228.18 16.299 5.67 0.001
Linear 4 61.17 61.17 15.292 5.32 0.006
Square 4 83.64 83.64 20.910 7.27 0.002
Interaction 6 83.38 83.38 13.896 4.83 0.005
Residual error 16 46.01 46.01 2.876
Lack-of-Fit 10 10.58 10.58 1.058 0.18 0.991
Pure Error 6 35.43 35.43 5.905
Total 30 274.19
Where DF = degrees of freedom; SS = sum of squares; MS = mean squares; F = fishers ratio.
6. Optimizing Using Hooke & Jeeves Method
Hooke and Jeeves algorithm [25] is used to search the
optimum values of the process variables. In this paper the
algorithm is developed to optimize the pulsed current
MPAW process variables. The objective is to minimize
grain size & maximize hardness. The coding for the Hoo-
ke Jeeves algorithm is written in MATLAB software.
The Hooke and Jeeves algorithm incorporates the past
history of a sequence of iterations into the generation of a
new search direction. It combines exploratory moves
with pattern moves. The exploratory moves examine the
local behavior of the function & seek to locate the direc-
tion of any stepping valleys that might be present. The
pattern moves utilize the information generated in the
exploration to step rapidly along the valleys.
Exploratory Move:
Given a specified step size which may be different for
each coordinate direction and change during search. The
exploration proceeds from an initial point by the speci-
fied step size in each coordinate direction. If the function
value does not increased the step is considered successful.
Otherwise the step is retracted and replaced by a step in
the opposite direction which in turn is retained in de-
pending upon whether it success or fails. When all N
coordinates have been investigated, the exploration move
is completed. The resulting point is termed a base point.
Pattern Move:
A pattern move consists of a single step from the pre-
sent base point along the line from the previous to the
current base point.
A new pattern point is calculated as:
 

11kkkk
p
xxxx

 
where,

1k
p
x

is temporary base point for a new ex-
ploratory move.
If the result of this exploration move is a better point

K. S. PRASAD ET AL.
874
then the previous base point x(k) then this is accepted as
the new base point x(k+1). If the exploratory move does not
produce improvement, the pattern move is discarded and
the search returns to x(k), where an exploratory search is
undertaken to find a new pattern.
Steps:
Step 1: Starting point x(0).
The increments ∆i for i = 1, 2, 3, ···, N.
Step reduction factor α > 1.
A termination parameter ε > 0.
Step 2: Perform exploratory search.
Step 3: Was exploratory search successful (i.e. was a
lower point found).
If yes go to Step 5.
Else continue.
Step 4: Check for the termination ||∆|| < ε current pint
approximation x0.
∆i = ∆i/α for i = 1, 2, 3, ···, N.
Go to Step 2.
Step 5: Perform pattern move
 

11kkkk
p
xxxx

 .
Step 6: Perform exploratory research using

1k
p
x

as
the base point; let the result be x(k+1).
Step 7: This step decides whether you are doing this
operation for minimization or maximization.
a) If you applied the condition “Is f(x(k+1)) < f(x(k))?”
then it is to find the maximum hardness.
b) If “Is f(x(k+1)) < f(x(k))?” then it is to find minimum
grain size.
Step a) & b) results either Yes or No basing on the re-
quirement of minimum grain size or maximum tensile
strength. After getting the result continue with the fol-
lowing process.
If Yes set x(k-1) = x(k).
x(k) = x(k+1) go to Step 5.
Else go to Step 4.
From Tables 6 and 7 it is understood that the values
predicted by Hooke and Jeeves algorithm and experi-
mental values are very close to each other.
Table 6. Optimized pulsed current MPAW parameters for
grain size.
Hooke & Jeeves Experimental
Peak current (amperes) 7.1299 7
Back current (amperes) 4.1299 4
Pulse rate (pulses/second) 42.5981 40
Pulse width (%) 52.5981 50
Grain size (microns) 21.1640 20.045
Table 7. Optimized pulsed current MPAW parameters for
hardness.
Hooke & Jeeves Experimental
Peak current (amperes) 7.1127 7
Back current (amperes) 4.1127 4
Pulse rate (pulses/second) 42.2539 40
Pulse width (%) 52.2539 50
Hardness (VHN) 220.5633 201
7. Conclusion
Empirical relations are developed to predict grain size
and hardness of pulsed current micro plasma arc welded
AISI 304 L using Response Surface Method. The devel-
oped model can be effectively used to predict grain size
and hardness values of pulsed current micro plasma arc
welded joints. From the experiments conducted the
minimum grain size of 20.045 microns and maximum
hardness of 201 VHN are obtained for the input parame-
ter combination of peak current of 7 Amperes, back cur-
rent of 4 Amperes, pulse rate of 40 pulses/second and
pulse width of 50%. From Hooke and Jeeves algorithm
the minimum value of grain size obtained is 21.1640
microns for the input parameter combination of peak
current of 7.1299 Amperes, back current of 4.1299 Am-
peres, pulse rate of 42.5981 pulses/second and pulse
width of 52.5981%. Whereas maximum hardness ob-
tained is 220.5633 VHN for the input parameter combi-
nation of peak current of 7.1127 Amperes, back current
of 4.1127 Amperes, pulse rate of 42.2539 pulses/second
and pulse width of 5.2539%. The values of grain size and
hardness obtained experimentally and predicted using
Hooke & Jeeves algorithm are within the limit.
8. Acknowledgements
The authors would like to thank Shri. R. Gopla Krishnan,
Director, M/s Metallic Bellows (I) Pvt Ltd., Chennai,
India for his support to carry out experimentation work.
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