Application of Hooke & Jeeves Algorithm in Optimizing Fusion Zone Grain Size and Hardness of Pulsed Current Micro Plasma Arc Welded AISI 304L Sheets

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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

Email: *[email protected]

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

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

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)





,,,

fABCD (3)

The second order polynomial equation used to repre-

sent the response surface “Y” is given by [19]:

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)

42 13

22.859 1.0521.0580.315

0.625 1.6402.320

GXX

XX XX

 



(5)

Hardness (H)

42 13

197.286 0.7081.2920.292

0.542 1.6032.188

XX XX





(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

xxxx



 

where,





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

xxxx



 .

Step 6: Perform exploratory research using





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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