Optimum design based on mathematical model and neural network to predict weld parameters for fillet joints

Journal of Manufacturing Systems Vol. 16/No. 1 1907

Optimum Design Based on Mathematical Model and Neural Network to Predict Weld Parameters for Fillet Joints H y e o n g - S o o n M o o n and S u c k - J o o Na, Korea Advanced Institute of Science and Technology (KAIST),

Taejon, South Korea

Abstract

ly coupled, thus making it difficult to derive a mathematical relationship between them. The approach to mathematical modeling is to deepen the understanding of the basic phenomena involved in the process. Therefore, many attempts have been implemented to model the weld bead geometry, but these have been generally restricted to computational approaches 1,2or controls3'4 for a single input and output parameter. For computational approaches, the selection of the appropriate welding process variables for the desired weld bead shape can be achieved by setting up adequate boundary conditions. The boundary conditions, however, are applied to the computational approach by simplifying the real material characteristics. The single-input single-output system (SISO) 4,s is generally applied to the quality control of welding processes, which is not based on mathematical modeling but just on experimental results. Thus there are many drawbacks to estimating the WPVs for various weld bead shapes at the same time. To solve this problem, it is necessary to know the tendencies of the weld bead shape according to each weld parameter. Then, experiments based on the results of these tendencies should be performed for verifying the accuracy of the mathematical modeling between the WPVs and weld bead shape. However, it is insufficient to estimate the whole weld parameters for the required weld bead geometry by using just mathematical modeling because the relationship between the WPVs and weld bead shape generally shows a nonlinear characteristic. Due to the nonlinear characteristics, other methods such as the neural network and optimum design are needed for optimizing the WPVs. The particular characteristics of the neural network are its ability to learn, capacity for abstraction, and sensitivity to recognize faults in signal patterns. For several outstanding characteristics, the neural

The welding process variables of welding current, arc voltage, welding speed, gas flow rate, and offset distance, which influence weld bead shape, are coupled with each other but not directly connected with weld bead shape individually. Therefore, it is very difficult and time consuming to determine the welding process variables necessary to obtain the desired weld bead shape. Mathematical modeling in conjunction with many experiments must be used to predict the magnitude of weld bead shape. Even though experimental results are reliable, prediction is difficult because of the coupling characteristics. In this study, the 2 "-1 fractional factorial design method was used to investigate the effect of welding process variables on fillet joint shape. Finally, a neural network based on the backpropagation algorithm and an optimum design based on mathematical modeling were implemented to estimate the weld parameters for the desired fillet joint shape. Mathematical modeling based on multiple nonlinear regression analysis was used for modeling the gas metal arc welding (GMAW) parameters of the fillet joint. It was shown that the neural network and optimum design for estimating the weld parameters could be effectively implemented, which resulted in little error percentage difference between the estimated and experimental results.

Keywords: Optimum Design, Mathematical Model, Neural Network, Horizontal Fillet Welding, Weld Parameter

Introduction Welding is an operation in which two or more parts are united by means of heat or pressure or both in such a way that there is a continuity in the nature of the metal between these parts. A filler metal, whose melting temperature is of the same order as that of the parent metal, may or may not be used. Especially for gas metal arc welding (GMAW), heat and mass inputs are coupled while the heat is transferred from the welding arc and the wire to the molten weld pool. Because of the melting and metal transfer phenomena, GMAW processes are nonlinear and complex to analyze. Welding process variables (WPVs) such as welding current, arc voltage, welding speed, gas flow rate, and offset distance are high-

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Journal of Manufacturing Systems Vol. 16/No. 1 1997

Figure 1. Figure 2 shows the schematic diagram of a fillet welded joint shape with the offset distance. To achieve a satisfactory fillet welded joint geometry, it is necessary to study the relationship between the weld bead shape and welding conditions, which can be represented by a mathematical model based on the experimental results. In horizontal fillet welding, the WPVs included welding current, arc voltage, welding speed, gas flow rate, and offset distance. All other parameters except these under consideration were kept constant. Before the mathematical modeling for the fillet welded joint shape was formulated, the effect of the five variables on leg length, penetration, throat thickness, and reinforcement height were determined by the 2 "-1 fractional factorial design, where n means the number of WPVs. In this case, n is 5 and 2 "-1 can be represented by 2 s-1. The 2 s-1 fractional factorial design has the unique efficient property that the 5 individual effects and the 10 two-factor interaction effects can be determined with only 16 trials (2 s-1 = 2 4 = 16) [Ref. 13] by neglecting the higher level interaction effects (three, four, and fivefactor effects in this study). The individual effect of a variable is the independent effect on the fillet welded joint shape. An interaction effect is the dependent effect that two or more variables together have on the fillet joint shape. To evaluate the individual and interaction effects, the welding conditions for the electrode, shielding gas, and base metal were set as shown in Table 1, and a higher and lower level of each variable was chosen as shown in Table 2. To simplify calculations, a statistical coding system was used, where the low level of a variable is

network has recently been studied in many fields of imaging processes, control systems, medicine, and interpretation of acoustic signal processing, s7 Work related to the selection of the welding conditions has been done in gas tungsten arc welding (GTAW) processes and GMAW, s'9 but this research was confined to the selection of welding conditions without investigating the effectiveness of welding conditions on weld bead shape. Therefore, the welding parameters and ranges of welding conditions used in those research papers were randomly selected. Optimum design, which is based on the mathematical model, was successfully applied to the welding processes for selecting the optimal welding parameters for the desired weld bead shape in GTAW; l°'~z the controllable parameter was only welding current or the welding current/speed combination because of the limitation of the mathematical model. The purpose of this paper is, on the basis of the 2"-1 fractional factorial design method, to develop a system to estimate the WPVs' effect on the fillet welded joint shape using the neural network and optimum design. A merit of this method is the ability to select the appropriate WPVs, which leads to a reduction in production cost and time and thus can induce the increase of productivity. The main object of the neural network and optimum design is to determine the optimal WPVs without any subsequent experiments to achieve the desired weld bead shape.

Analysis of Variable Effects Generally, fillet welded joint shapes are classified into four parts, such as leg length, penetration, throat thickness, and reinforcement height, as shown in

=

leg length penetration throat thickness

L2

reinforcement height

Figure 1

Figure 2

Definition of Weld Bead Shape

Definition of Offset Distance

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Journal of Manufacturing Systems

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Table 1 Electrode, Shielding Gas, and Base Metal

Table 2 Experimental Variable Levels and Result Variables

ExperimentalCondition

Description

Electrode Shieldinggas Base metal

Solid wireAWSER70S-6(1.2ram) AR (80%)+ CO2(20%) Mild steel

Thickness

X1 X2 X3 X4 X5

designated by -1 and the high level by 1. The standardized variables were denoted by X 1, X2, X3, X4, and X5. The variables affected by the standardized variables were denoted by HI, H2, H3, H4, and H5. The difference between the high level and low level of a variable was defined as the range of the variable. The ranges for the welding speed and welding current were limited due to the possibility that the values below or above the values listed in Table 2 would cause weld defects such as undercut and overlap. The offset distance was chosen as a variable because a high current in conjunction with the gravitational force and the surface tension in the weld pool might bring about the overlap. Figure 3 shows the definition of the undercut and overlap in horizontal fillet welding. Table 3 was used to determine the effect of welding conditions on the experimental results, 14 where the experimental results were leg length 1, leg length 2, penetration, throat thickness, and reinforcement height. All trials were run in a random order to eliminate any time or sequence effect that might have been present. The individual and interaction effects were calculated by multiplying the appropriate standardized variable by the experimental result for each trial and summing the resultant values. For example,

H1 H2 H3 H4 H5

Welding speed (mm/sec) Arc voltage (V) Welding current (A) Gas flow rate (L/min) Offset distance (mm)

4 26 250 14 0

8 30 280 18 2

Leg length 1 (mm) Leg length2 (mm) Penetration (mm) Throat thickness(ram) Reinforcementheight(mm)

Table 4 shows the values of the individual effects and the two-factor interactional effects for each experimental result obtained from the 2 s-~ fractional factorial design. According to the data in Table 4, El, E2, and E3 show larger values than the other effects in leg length 1, leg length 2, penetration, and throat thickness, where El, E2, and E3 mean the effectiveness of X1 (welding speed), X2 (arc voltage), and X3 (welding current) on the weld surface profile. According to the above notation, E 12 means the effectiveness of the coupled variable of X1 and X2 on the weld surface profile. The larger the absolute value of these effects, the more dominant the effect on the weld surface profile. Therefore, the dominant variables on the fillet welded joint shape, except the reinforcement height, were welding

(1)

{ (-1)(8.0) + (1)(5.5) + (-1)(9.5) + (1)(5.1) + (-1)(9.0) + (1)(5.3) + (-1)(10.2) + (1)(8.5) + (-1)(7.5) + (1)(4.3) + (-1)(8.5) + (1)(6.0) + (-1)(9.5) + (1)(5.9) + (-1)(11.0) + (1)(5.5)}

=

High level (+1)

Figure 3 Definition of Undercut and Overlap

i=l

=

Low level (-1)

~ut

i=16

ZXli°Hli

Experimental variables

Experimental results

E1 (for leg lengthl)

:

7 mm

-27.1

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Journal o f Manufacturing Systems Vol. 16/No. 1 1997

Table 3 Standardized Variables of Welding Conditions and Experimental Results for Trials 1-16

StandardizedVariables

Z6 ~ 1

3

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

5 10 15 1 4 9 11 2 12 16 8 13 14 7 6

X1

X2

-1 1 -1 1

-1 -1 1 1

-1

-1

1 -1 1 -1 1

X3

X5

X12 X13 X14 X15 X23 X24 X25 X34 X35 X45

-1 -1 -1 -1

1 -1 -1 1

1 -1 -1 1

1

-1

-1

-1 1 1 -1 -1

1 1 1 -1 -1

-1 -1 -1 1 1

-1

1

-1

1

1

-1 1

-1 -1

-1 1

-1

1

-1 -1 -1 -1

X4

Result

1 -1 1

1 -1 1

-1 -1 1

1 1 -1

1

-1 -1

-1 1

1 1

-1 -1

1 1 -1 -1 1

-1 -1 1 1 -1

1 -1 1 1 -1

-1 1 -1 -1 1

1 -1 -1 1 1

-1 1 1 1 1

1

1

-1

1

-1

-1

-1

1

1 1

-1 1

1 1

1 -1

-1 -1

-1 -1

1 -1

1

-1

-1

-1

1

1

1

1

1

1

1

1

-1

-1

1

1

1

-1

-1

1

1

1

1

-1 -1

1 -1

1

-1

1

1

y =

speed, welding current, and arc voltage, where a minus sign means that the experimental result would decrease in proportion to the increase o f a variable, and vice versa for a plus sign. To verify this result, a correlation parameter was adopted. The data used to calculate the correlation parameter were the same as the data o f Table 3, which was also used in the 2 ~-1 fractional factorial design. The correlation parameter, R, which stood for the correlation between the variable x and y, was represented by the following equation, ls,16 R =

1 1 -1 -1

-1 1 -1 1

1 1 1 1

-1 1 1 -1

1

1

-1

-1

1

1 -1 -1 -1 -1

-1 1 -1 1 -1

-1 -1 -1 -1 -1

1 1 -1 1 -1

-1 -1 1 -1 1

1

-1

-1

1

-1 1

1 1

-1 1

1

-1

-1

1

-1

-1

-1 -1

1 -1

1

1

1

-1 1 1 -1

1

H1

H2 H3 H4 H5

8 8.3 1.1 4.9 0.9 5.5 5.4 1.5 3.6 1.0 9.5 9.5 0.7 5.5 0.9 5.1 6.4 1.5 3.6 0.9 9 10.5 2.0 5.5 1.5 5.3 6.0 2.3 3.5 1.7 10.2 10.5 2.1 6.0 1.0 8.5 8.5 1.8 5.5 0.9 7.5 7.5 1.7 5.0 0.8 4.3 5.9 1.6 3.5 1.6 8.5 9.5 0.9 5.4 1.4 6.0 5.3 0.8 4.1 1.2 9.5 9.5 1.8 5.7 2.4 5.9 6.3 2.2 4.5 1.0 11.0 10.0 2.0 5.7 1.3 5.5 7.0 1.9 4.3 1.0

magnitude o f weld bead shape such as leg length, penetration, throat thickness, and reinforcement height (mm) m

variance x = 0 2 = E (xi - ~)2/(m - 1), i m

variance y = try2 = E ( y i _fi)z ~ ( m - l ) , i m

m

simple mean ~ = E xi / m, simple mean y = E Yi / m i

(2)

covariance (x,y) =

i

(xl - x)(yi - 2)

covariance of x and y square root of product of variance x and y m = number o f experiments (= 16 in 25-1 fractional factorial design experiments)

covariance (x,y) (variance x, variance y)l/z m

m

i=l

meg-

xl

i=l

The parameter x used in Eq. (2) means the value o f welding variables that correspond to (-1) or (+1) in Table 2.17 The graphical representation o f the correlation parameter is shown in Figure 4. In this figure, the perfect correlation means the perfect linear relationship between the x and y variables, while the good and negative correlation means the linear relationship between x and y including some errors. The perfect and good correlation means that the y value would increase in proportion to the increase o f the x value. On the contrary, the negative correlation means that the y value would decrease in proportion

i=1

mEY~-

Yi

where X =

value o f welding variables such as welding speed (mm/sec), arc voltage (V), welding current (A), gas flow rate (L/min), and offset distance (mm)

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Journal of Manufacturing Systems Vol. 16/No. 1

1997

Table 4 Variable Effects for Trials 1-16

Parameters for Fillet Weld SurfaceProfile

E1 E2 E3 E4 E5 El2 El3 El4 El5 E23 E24 E25 E34 E35 E45

Leg Length 1 (rank)

Leg Length 2 (rank)

-27.1(1) 9.3 (3) 10.5 (2) -2.9 (7) -6.5 (4) -1.1 (14) -1.9 (9) -2.5 (8) -4.9 (5) 1.7 (10) -1.7 (10) -4.9 (5) 0.7 (15) -1.3 (12) 1.3 (12)

-24.5 (I) 7.3 (3) 10.5 (2) -4.1 (6) 0.1 (14) -0.1 (14) -0.9 (9) 0.5 (10) -0.5 (10) 0.1 (12) -2.1 (7) 0.1 (12) -1.3 (8) -4.7 (5) 5.5 (4)

R = +1 (perfect correlation)

Penetration ThroatThickness (rank) (rank) 1.3 (4) -2.5 (2) 6.3 (1) -0.1 (15) 0.5 (12) -0.7 (10) -0.7 (10) -1.1 (8) 1.5 (6) 1.5 (6) -0.9 (9) 1.7 (5) -0.5 (13) -0.3 (14) -1.5 (3)

R ~ +1 (good correlation)

iiiiiii

III

X

X

I

IIIIII

Ililll

Illlil

R = 0 (no correlation)

-1 (negative correlation) Y

lil

Y

•

-0.9 (10) -2.3 (2) 2.1 (6) 1.9 (7) 2.3 (2) -0.3 (14) -2.3 (2) -1.3 (9) -0.1 (15) -2.5 (1) 0.5 (12) -2.3 (2) -0.7 (11) 0.5 (12) 1.9 (7)

ues of the correlation parameters for the weld bead geometry were -0.83, -0.85, 0.17, -0.80, and -0.14 for welding speed (El in 2 "-~ fractional factorial design), 0.29, 0.25, -0.32, 0.28, and --0.35 for arc voltage (E2 in 2 "-~ fractional factorial design), and 0.32, 0.36, 0.81, 0.37, and 0.33 for welding current (E3 in 2 "-I fractional factorial design). The values of the other correlation parameters for gas flow rate (E4 in 2 "-1 fractional factorial design) and offset distance (E5 in 2 "-1 fractional factorial design), except the value of the correlation parameter for reinforcement height, were smaller than those for welding speed, arc voltage, and welding current. Therefore, welding speed, arc voltage, and welding current are dominant factors in determining weld bead shape except for the reinforcement height.

Y

I

-11.1 (1) 3.9 (3) 5.1 (2) 0.1 (15) -2.5 (5) 0.9 (8) 0.9 (8) 0.3 (14) -3.1 (4) 0.7 (11) --2.3 (6) -0.5 (13) -0.7 (12) -0.9 (8) 1.7 (7)

Reinforcement Height (rank)

From Table 4, the tendency o f the welding condition

effect on the weld bead shape was found to be in a good accordance with the data in Table 5, which showed that the experiments to check the tendencies of welding

i i,

,i

,Hi

i ,, X

conditions were reliable. The subsequent experiments, based on these results, were carried out by changing the various ranges of the dominant variables, which were

............................. IL.....HL......IIIL ............................. Figure 4 Illustration o f Various Correlations Between x and y

the welding current, welding speed, and arc voltage.

Prediction of Welding Parameters

to the increase o f the x value. According to Eq. (2), the information could be obtained for the correlation between the welding condition and fillet w e l d e d joint shape. The correlation parameters from Eq. (2) are listed in Table 5. According to Table 5, the val-

Direct selection o f appropriate welding conditions for the desired weld bead geometry and dimension in GMAW is a very complex process. The most important reason is the coupling characteristic and

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Journal of Manufacturing Systems Vol. 16/No. 1 1997

Table 5 Correlation Parameters for Each Fillet Weld Surface Prof'fle

Correlation Parameter, R

Leg length 1 Leg length 2 Penetration Throat thickness Reinforcement height

Welding Speed

Arc Voltage

-0.83 -0.85 0.17 -0.80 -0.14

0.29 0.25 -0.32 0.28 -0.35

Welding Current

Gas Flow Rate

Offset Distance

4).09 -0.14 -0.01 0.00 0.29

-0.20 0.00 0.06 -0.18 0.35

0.32 0.36 0.81 0.37 0.33

the lack of comprehension of the physics of the welding process. Therefore, to solve this problem, some methods that have a special characteristic of decoupling the welding process variables are required. To meet this necessity, the neural network and optimum design method were adopted and the results were compared to each other. These methods have the same distinguishing characteristic with respect to the ability for selecting optimal welding conditions but a different characteristic with regard to theoretical background. The neural network does not necessitate the mathematical model, while the optimum design method can add the appropriate constraints for the variables so that an undesirable result will not occur.

output layer of the neural network, which can be expressed by a cost function as follows: is

J=(1/2)ZZ(dt, --ak) 2 p

(3)

k

where

dk = desired outputs ak = actual outputs from output layer The hidden layer, output layer, and the weights between the layers are expressed as follows:

(4)

Selection of Appropriate Welding Conditions by Neural Network

a k = S ( ~ wjkhj

There are many special characteristics in the neural network. First, the processing unit is very simple, and the complex problems can be solved by using these simple units together. The second attractive feature is the ability to memorize the data by using the weights connected with each other in the overall structure. Therefore, the accuracy of the overall network can be satisfactorily maintained in spite of the break in some processing units. The neural network can adjust the weights by itself through the learning process and easily represent the nonlinear relationship between the input and output, which is difficult to express in mathematical modeling. The neural network based on the backpropagation algorithm consists of four layers--one input, one output, and two hidden layers. The purpose of the learning process is to decrease the error between the desired output and the actual one obtained from the

S(x) -- 1.0 / (1.0 + exp (-x

(5) - 0))

where p

= number of input patterns

Si

=

hj

-- output at hidden layers

input according to each pattern

wij, wjk = weights

18

0

= internal offset value

S(x)

= sigmoid function

(6)

Journal of Manufacturing Systems Vol. 16/No. 1 1997

At the first stage, the weights o f the neural network were not adjusted to accurately match the relationship between the input and output patterns. To update the weights, the gradient descent algorithm was used and each weight change was expressed as follows:

awij(t + 1) = arwi~(t)- 77Ow--~.~ awjk(t + 1) = arwjk(t ) -

w~ Leg length 1

(7)

Leg length 2

Welding speed Arc voltage

Penetration

Welding current

Throat thickness

Gas flow rate

Reinforcement height

Offset Distance

Thickness of base metal

3Jr

q o~jk

(8) Figure 5 Schematic Diagram of Neural Network for Estimating Welding Conditions for Desired Weld Bead Shape

The m o m e n t u m coefficient ot must be between 0 and 1. Using the momentum, the network tends to follow the bottom o f narrow gullies in the error surface rather than crossing rapidly from side to side. The architecture of the neural network for the selection of welding process variables consisted of three parts. The first part was the specification of the fillet welded joint shape: leg length, penetration, throat thickness, and reinforcement height. The second part was the hidden layer regarded as a black box. The last part was an output layer for setting up the welding conditions. The schematic diagram of the neural network for predicting the welding condition is shown in Figure 5. The range of the experimental data used to train the proposed neural network structure is shown in Table 6, and the number of the training data was 20 for each thickness of base metal. Therefore, 60 data were used to train the neural network. The appropriate WPVs, for a given weld bead shape, were selected by the structure of the neural network in Figure 5. The performance of the neural network depends on the number of hidden layers and the number o f nodes in the hidden layers. Therefore, many attempts should be carried out in choosing the optimal structure for the neural network by changing the number of nodes for the various hidden layers. The appropriate structure of the neural network for

estimating the welding conditions was chosen by the trial-and-error method. In this study, the adequate network structure for the hidden layers was 24-24, and consequently, the total structure of the neural network was 6-24-24-5. The comparison between the experimental and estimated data from this neural network structure was performed, and the error percentage differences between the experimental and estimated data are shown in Figure 6. The experiments were performed 12 times for estimating the performance o f the neural network, and the experimental data used for examining the reliability o f the neural network were shown in Table 7. The error percentages covered the range from +20% to -20%, which indicates that the neural network could successfully express the nonlinear welding process.

Selection of Appropriate Welding Conditions by Optimum Design Mathematical Modeling There are many cases in engineering where the nonlinear models must be fit to data, for example, the welding processes consisting of many nonlinear Table 6

Range of Experimental Data Used to Train Neural Network Thickness of

Range o f Weld Parameter

Base Metal (mm)

(from - to) Welding Speed (mm/sec)

4.5 6 7

4- 8 4- 8 4- 8

Arc Voltage (V) 20 - 26 23 - 29 26 - 31

19

WeldingCurrent (A)

Gas FlowRate (L/min)

OffsetDistance (mm)

200 - 260 220 - 300 250 - 350

14 - 18 14 - 18 14 - 18

0- 2 0- 2 0- 2

Journal of Manufacturing Systems Vol. 16/No. 1 1997

Y = f ( X l , X2, X3, X4, X5, X6)

(9)

120.0

In Eq. (9), Y is a variable indicating the weld bead shape, X l is the thickness o f the base metal, X2 is the welding speed, X3 is the arc voltage, X4 is the welding current, X5 is the gas flow rate, and X6 is the offset distance. Assuming that the relationship between welding conditions and weld bead geometry is nonlinear, Eq. (9) can be expressed as follows:

90.0 .~

60.0 30,0

" : . ; ° 0.0

o

- - . s

. . . .

o

= :

. . . . - . , . . - , . . - , . . , . . , . . . . . , . . , . . .

0

o,.

-30.0

0 = A • a

-60.0 -90.0

-120.0

,

,

1

2

speed voltage current gas flow rate offset distance i

3

(lO)

Y= (X1 a X2 b X3 c X4 d X5 e X6 r)

,

m

,

,

m

4

5

6

7

8

,

,

=

,

9 10 11 12

where the empirical coefficients a, b, c, d, e, and f are constants. The nonlinear regression method is based on determining the value o f parameters that minimize the sum o f squares o f the residuals. The GaussNewton method was used to minimize the sum o f squares o f the residuals between the data and nonlinear equations. 19 In determining the appropriate empirical coefficients, the standard statistical package program SAS was used. The following equations for the fillet welded joint shape were obtained by the SAS package using the nonlinear multiple regression, while the data used to determine the coefficient were the same as the data used to train the neural network.

Test data number Figure 6 Result of Error Percentage Between Experimental and Estimated Data Using Neural Network

parameters coupled with each other and having a complex relationship between the welding conditions and the w e l d quality. The interrelation between the six welding process parameters and the five variables indicating the welding surface profile can be represented by the nonlinear regression method. The basic function o f the interrelation is described as follows:

Table 7 Experimental Data Used for Examining Reliability of Neural Network and Optimum Design Desired Weld Bead Shape

Desired Welding Condition

(Input data used to examine reliability of neural network)

(Desired output data)

Test No.

T.P.

L1

L2

P

T.T.

R.H.

W.S.

A.V.

WC.

G.ER.

O.D.

1 2 3 4 5 6 7 8 9 10 11 12

4.5 4.5 4.5 4.5 6.0 6.0 6.0 6.0 7.0 7.0 7.0 7.0

7.7 7.7 6.7 5.5 7.5 5.8 7.8 4.7 8.0 8.5 8.5 5.9

7.6 8.3 6.5 5.5 7.5 6.5 7.4 5.2 8.3 8.5 9.5 6.3

1.8 2.0 1.5 1.7 1.9 1.1 1.9 2.0 1.1 1.8 0.9 2.2

5.0 5.0 4.5 3.5 4.8 3.9 6.0 4.0 4.9 5.5 5.4 4.5

1.4 1.0 1.2 0.9 1.2 1.0 1.0 1.0 0.9 0.9 1.4 1.0

4 4 6 8 4 6 6 8 4 8 4 8

24 26 26 26 28 25 29 26 26 30 30 26

230 250 250 260 240 230 270 260 250 280 250 280

18 18 14 18 14 14 14 14 14 14 18 18

0 2 0 0 0 2 0 0 2 0 2 0

T.E = thickness o f plate, L1 = leg length 1, L2 = leg length 2, P = penetration, T.T. = throat thickness, R.H. = reinforcement height, W.S. = welding speed, A.V. = arc voltage, WC. = welding current, G.ER. = gas flow rate, O.D. = offset distance

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Journal o f Manufacturing Systems

Vol. 16/No. I 1997

Leg Length 1 = Xl °'°gs X2 ~-°'Sz°)X3 °aS~ X4O.SS7X5(-0.367) X6O.OO36

where (11)

Leg Length 2 = X l 0"266 X 2 ~-°'ss3) X 3 0"473 X4O.338 X5(-o.331) X6O.OlO

(12)

Penetration = X l °'484 X2 ~-°'°°a)X3 t-°'s~7) X4O.S4o X5(~.595) X6O.oo~

(13)

Throat Thickness

= X l 0"192 X2 t-°'s17)

X3(-o.s6s) )(40.089 X5(-o.o99) X6O.OO5

Reinforcement Height - Xl °a59 X2 ~-°'227) X3(-2.652) X41.546X50ao4 X60.o0s

(14)

f(x)

= E(Xid i

"~"

desired value

Xi

-=

value determined by mathematical modeling

X

:

n dimensional vector of unknown

A

=

m × n matrix of given constants

E

=

p × n matrix of given constants

b

=

m dimensional vector of given constants

e

:

p dimensional vector of given constants

In this system, the cost function f and the constraint conditions were represented in the following equation:

(15)

minimize f(L1, L2, P, TT, RH) : ( L l d - L1) z + (L2d - L2) 2 + (Pd -- p)z + (TTd - TT) 2 + (RH~ - RH) 2

Feasible Direction M e t h o d The design of most engineering systems is a fairly complex process. Many assumptions must be made to develop models that can be subjected to analysis by the available methods and verified. One of the important problems to be solved in the welding process is to develop the mathematical model for the determination of optimum process parameters. The mathematical model is classified into two categories--analytical and experimental. The analytical model is simple, but it has limitations such as its need for many assumptions to formulate the equations. Especially, the welding processes are very complex and have many nonlinear parameters, such as arc phenomena, which are difficult to be analyzed in an analytical model. Therefore, experimental models such as nonlinear regression are required to exactly predict the welding phenomena. To some extent, the experimental modeling also has limitations, although it can describe the weld bead geometry according to the welding conditions. In this study, the quadratic program was used to determine the appropriate welding conditions. A quadratic programming problem has a quadratic cost function and linear constraints and is found in many real-world applications. The general form is: 2°

minimize

X id

subject to currentmin ~ current -< currentmax, voltagemin -< voltage --< voltagemax speed,.in -< speed -< speedmax, gaSmin --< gas --< gas=ax, offset=i. --< offset ~ offsetm~ where L1 and L2 are the leg length, P is the penetration, TT is the throat thickness, and RH is the reinforcement height. These parameters were determined by the mathematical modeling, Eqs. (5)-(9). To solve the nonlinear programming problems, the feasible direction method was utilized. The optimization of the welding conditions by the feasible direction method was to move from a feasible point to an improved feasible point. From among the methods to generate an improved feasible direction, Zoutendijk's method was implemented, whose algorithm is described as followsfl1

Initialization step Find a starting feasible solution xl with A r x -< b and E r x = e. Let k = 1, and go to the main step.

-- X i ) 2

subject to the linear inequality and equality constraints,

Main step 1. Given xk, suppose that At and bt are decomposed into (.41t, A{) and (bi t, b2t) so that A~xk = b and A2xk < b2. Let dk be an optimal solution to the following problem:

A r x <-- b Erx=e

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Journal of Manufacturing Systems Wol. 16/No. 1 1997

minimize

Vflxk)'d 120.0

subject to

90.0

Aid <- 0

60.0

Ed = 0

30.0

-l<-d<-lforj=l,...,n

0.0

If Vflx,)tdk = 0, stop; xk is a Kuhn-Tucker point. Otherwise, go to step 2

#

o

l

o

•

ua

O

o = =. • n

-60.0

-90.0 -120.0

&

aS

-30.0

2. Let h, be an optimal solution to the following line search (using the golden section method) problem: minimize

e

- - - = - - ~ - - ~ - - ~ - - o . . . . . . . E--~ . . . . . . ~ . . ~ . . | - . .

,

i

1

2

] 8

O

I

°

speed voltage current gas fl0w rate offset distance i i

3

i

iI

i

i

i

i

4

5

6

7

8

9

i

,

10 11

i

12

flxk + Xdk) Test data number

subject to

Figure 7 Result of Error Percentage Between Experimental and Estimated Data Using Optimal Design

0 -< h --< h,~,x Let xk÷l = xk + ht,dk, identify the new set of binding constraints at xk÷~,and update A] and Az accordingly. Replace k by k + 1 and repeat step 1. The results from this algorithm are shown in Figure 7. According to these results, the errors between the estimated and experimental results were somewhat larger than that of the neural network. Especially, the offset distance showed a large error percentage difference compared to the other welding parameters. However, as the result of the 2 s-~ fractional factorial design already revealed, the offset distance was not a dominant factor influencing weld bead geometry. A main merit of the optimum design method is a tendency not to induce undesirable results such as undercut and overlap by adopting the constraint conditions. Therefore, the optimum design method could also be effectively used to choose the appropriate welding parameters, although the errors between the estimated and experimental results were somewhat larger than those of the neural network.

based on the 25"1fractional factorial design was developed and used to determine the optimal welding conditions for the fillet welded joint shape. Based on the neural network and the optimum design, the appropriate welding process parameters could be effectively determined. Welding parameters selected from the neural network produced fillet joints very similar to the desired shapes but might cause weld defects such as undercut and overlap. In the case of the optimum design, the errors between the estimated and experimental values were somewhat larger than those for the neural network. However, the welding parameters selected from the optimum design would prevent weld defects because the appropriate constraints could be added to the cost function. The two proposed methods could be effectively used in determining the welding parameters for the desired weld bead shape, except for the weld bead shape, including weld defects, in horizontal fillet welding. Therefore, further studies considering weld defects in horizontal fillet welding should be continuously carried out.

Concluding Remarks Welding process variables are highly nonlinear characteristics and complex to analyze. Furthermore, the welding process variables, such as welding current, arc voltage, and welding speed, influencing weld bead shape are coupled with each other. Therefore, there is a limitation to derive the accurate relationship between welding conditions and weld bead shape. To overcome these difficulties, a mathematical model

References 1. K.C. Tsao and C.S. Wu, "Fluid Flow and Heat Transfer in GMA Weld Pools" WeldingJournal (v64, n3, 1988), pp70s-75s. 2. T. Zacharia, A.H. Eraslan, and D.K. Aidun, "Modeling of Autogenous Welding" WeldingJournal (v67, n3, 1988), pp53s-62s. 3. H. Nomura, Y. Sugitani, and Y. Suziki, "Automation Real-Time Bead Height Control with Arc Sensor in TIG Welding," Trans. of the Japan Welding Society (v18, n2, 1987), pp125-132.

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Journal of Manufacturing Systems Vol. 16/No. 1 1997

16. E.M. Bartee, Engineering Experimental Design Fundamentals (Englewood Cliffs, NJ: Prentice-Hall, Inc., 1968), pp292-294. 17. Y.H.Moon, K.H. Lee, and S.D. Hur, "Application of Plackett-Burman Model in Welding Experiments--Effects of Welding Parameters on Bead Shape in Cu-Ni PULSE MIG Process," Journal of the Korean Welding Society (v5, n4, 1987), pp47-53. 18. J. Hertz, A. Krogh, and R.G. Palmer, Introduction to the Theory of Neural Computation (vl) (Reading, MA: Addison-Wesley Publishing Co., 1991), pp115-123. 19. S.C. Chapra and R.P. Canale, Numerical Methodsfor Engineers, 2nd ed. (New York: McGraw-Hill Book Co., 1988), pp358-361. 20. J.S.Arora, Introduction to Optimum Design (NewYork: McGraw-Hill Book Co., 1989), pp377-378. 21. S. Bazaraa and C.M. Shetty, Nonlinear Programming--Theory and Algorithms (New York: John Wiley & Sons, 1979), pp360-366.

4. Y. Kozono, A. Onuma, and S. Kokura, "Control of Back Bead Width in Girth TIG Welding of Small Tubes Using Surface Temperature Monitoring," Weldinglnternational(n2, 1989), pp137-141. 5. W. Ameling, J. Borowka, U. Dilthey, L. Kreff, M. Raus, and W. Scheller, "Adaptive Arc Sensor for Varying Joint Geometry--The Use of Artificial Neural Network," Weldingand Cutting (v44, n2, 1992), ppE24-E25. 6. Y. Ichikawa and T. Sawa, "Neural Network Application for Direct Feedback Controllers," IEEE Trans. on Neural Networks (v3, n2, 1992), pp224-231. 7. R. Reilly, X. Xu, and J.E. Jones, "Neural Network Application to Acoustic Emission Signal Processing," Proc. of the International Conference on Computerization of Welding Information IV (Orlando, FL, Nov. 3-6, 1992), pp146-160. 8. Xiaoshu Xu, J.E. Jones, V. Sutter, and Xiaoling Xu, '% Neural Network Model of Weld Bead Geometry Control System," Proc. of the International Conference on Computerization of Welding Information IV (Orlando, FL, Nov. 3-6, 1992), pp267-294. 9. K. Andersen, G.E. Cook, G. Karssai, and K. Ramaswamy, "Artificial Neural Networks Applied to Arc Welding Process Modeling and Control," IEEE Trans. on IndustryApplication (v26, n5, 1990), pp824-830. 10. T-J. Lho and S-J. Na, "A Study on Parameter Optimization with Numerical Heat Conduction Model for Circumferential Gas Tungsten Are (GTA) Welding of Thin Pipes," Proc. of the Institution of Mechanical Engineers (v206, 1992), ppl01-111. 11. T. Ohji, K. Nishiguchi, and Y. Kometani, "Optimization of Welding Parameters by a Numerical Model--Thin Plate TIG Arc Welding," Technology Reports of Osaka University (v36, n1826, 1985), pp47-52. 12. T. Ohji, K. Nishiguchi, and Y. Yoshida, "Real Time Optimization of Thin Plate TIG Arc Welding," Technology Reports of Osaka University (v36, n1849, 1986), pp267-273. 13. D.A.J. Stegner, S.M. Wu, and N.R. Braton, "Prediction of Heat Input for Welding," WeldingJournal (v46, n3, 1967), pp122s-128s. 14, P.J. Ross, Taguchi Techniquesfor Quality Engineering (New York: McGraw-Hill Book Co., 1988), pp213-226. 15. I.N. Gibra, Probability and Statistical Inference for Scientists and Engineers (Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973), ppl07-115.

Authors' Biographies Hyeong-Soon Moon is currently a doctoral candidate at the Korea Advanced Institute of Science and Technology (KAIST). He received his BS in production engineering from Pusan National University (South Korea) in 1991 and his MS in mechanical engineering from KAIST in 1993. His research interests include automatic seam tracking, welding quality control, and process monitoring, neural networks, and fuzzy systems. Suck-Joo Na received his BS in mechanical engineering from Seoul National University (South Korea) in 1975, his MS in mechanical engineering from the Korea Advanced Institute of Science and Technology (KAIST) in 1977, and his Dr-Ing degree in welding engineering from the Technical University of Braunschweig (Germany) in 1983. In 1983, he joined KAIST, where he is a professor and researcher on welding and other thermal processes. He works mostly in the field of automation of welding and cutting processes, especially development of seam-tracking sensors, laser materials processing, development of CAD/CAM systems for laser and plasma cutting, and analysis of heat and mass flow in the welding process. Dr. Na is a member of the Korean Welding Society, the Korean Society of Mechanical Engineers, the American Welding Society, and the German Welding Society.

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