Evolutionary fuzzy SVR modeling of weld residual stress

G Model

ARTICLE IN PRESS

ASOC 3454 1–8

Applied Soft Computing xxx (2016) xxx–xxx

Contents lists available at ScienceDirect

Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Evolutionary fuzzy SVR modeling of weld residual stress

1

Q1

2

J. Edwin Raja Dhas a,∗ , Somasundaram Kumanan b,1 a

3

b

4

Department of Automobile Engineering, Noorul Islam Centre for Higher Education, Nagercoil 629180, Tamil Nadu, India Department of Production Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamil Nadu, India

5

6 22

a r t i c l e

i n f o

a b s t r a c t

7 8 9 10 11 12 13

Article history: Received 27 May 2014 Received in revised form 21 December 2015 Accepted 30 January 2016 Available online xxx

14

21

Keywords: Manufacturing Welding Residual stress X-ray diffraction Support Vector Regression Genetic Algorithm

23

1. Introduction

15 16 17 18 19 20

Residual stresses are an integral part of the total stress acting on any component in service. It is important to determine and/or predict the magnitude, nature and direction of the residual stress to estimate the life of important engineering parts, particularly welded components. Researchers have developed many direct measuring techniques for welding residual stress. Intelligent techniques have been developed to predict residual stresses to meet the demands of advanced manufacturing planning. This research paper explores the development of Finite Element model and evolutionary fuzzy support vector regression model for the prediction of residual stress in welding. Residual stress model is developed using Finite Element Simulation. Results from Finite Element Method (FEM) model are used to train and test the developed Fuzzy Support Vector Regression model tuned with Genetic Algorithm (FSVRGA) using K-fold cross validation method. The performance of the developed model is compared with Support Vector Regression model and Fuzzy Support Vector Regression model. The proposed and developed model is superior in terms of computational speed and accuracy. Developed models are validated and reported. The developed model finds scope in setting the initial weld process parameters. © 2016 Published by Elsevier B.V.

Q2 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Due to intense concentration of heat in heat source of welding, the regions near the weld line undergo severe thermal cycles, thereby generating inhomogeneous plastic deformation and residual stresses in weldment. Welding-induced residual stresses play an important role in the function of welded structures. Different experimental methods for measuring welding residual stresses such as neutron diffraction [1], Barkhausen Noise analysis [2], X-ray diffraction [3], deep hole drilling [4], holographic interferometry [5], Ultrasonic technique [6] and incremental hole-drilling [7] are available. All these methods require special equipment and are expensive. These techniques are limited in obtaining the entire picture of the residual stress distribution in weldment. Later finite element methods were applied to solve welding residual stress problems [8–11]. Although the finite element method has emerged as one of the most attractive approach for computing residual stresses in welded joints, its practical application to design problems has been limited due to computational difficulties. Analytical

∗ Corresponding author. Tel.: +91 04651 250566/366297; fax: +91 04651 257266/250266. E-mail address: [email protected] (J. Edwin Raja Dhas). 1 Tel.: +91 0431 2503033/2503507; fax: +91 0431 2500133/2503502.

modeling [12] and numerical modeling [13] were applied to determine weld residual stress. Due to the inability of the mathematical models to explain the nonlinear properties existing between the input and output parameters, intelligent techniques such as fuzzy logic, support regression techniques, Artificial Neural Network (ANN) have emerged. Genetic Algorithm [14,15], ANN [16,17], and hybrid algorithms [18–20] are applied to optimize various parameters in manufacturing process. However, the constructed architecture of ANN model using backpropagation algorithm is based on the empirical risk minimization (ERM) principle. ERM pursues minimizing the error on the training data, so it may fall into a local optimal solution due to the overtraining problem. Support vector machine (SVM), by Vapnik [21] is based on statistic learning theory and the structural risk minimization principle that pursues minimizing the upper bound on the expected risk. In conventional Support Vector Regression (SVR), the training process is very sensitive to noise or outliers in the training samples [22]. Fuzzy logic, is a powerful tool to deal with uncertain, vague and ill-posed problems, has been introduced to overcome limitations of SVR [23]. Inspired by Lin and Wang’s fuzzy SVM [24], a Fuzzy SVR (FSVR) is proposed and developed for weld residual stress prediction. For accurate residual stress estimation, it is necessary to find a reliable optimization approach to select the optimal FSVR parameters settings. An evolutionary approach Genetic Algorithm (GA) [25] is introduced to choose optimal parameters for the Fuzzy Support Vector Regression (FSVR). This paper addresses the

http://dx.doi.org/10.1016/j.asoc.2016.01.050 1568-4946/© 2016 Published by Elsevier B.V.

Please cite this article in press as: J. Edwin Raja Dhas, S. Kumanan, Evolutionary fuzzy SVR modeling of weld residual stress, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.01.050

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

G Model

ARTICLE IN PRESS

ASOC 3454 1–8

J. Edwin Raja Dhas, S. Kumanan / Applied Soft Computing xxx (2016) xxx–xxx

2

Start Define: Population Size, Population generation Mg & Parameter ranges

2.1.1. Development of FSVR model In FSVR model, residual stress is estimated based arc efficiency, voltage, current and welding speed. The accuracy of the developed model depends on the membership functions chosen. The FSVR is modeled by minimizing following constrained risk function in Eq. (1).

gen = 1

Initialize population Pg

R(w) =

1 T w w+C 2

 N 

91 92 93 94 95 96

 i (i + i )

(1)

97

i=1

Perform FSVR in each individual in Pg

Evaluate Dummy Fitness

Reproduction according to dummy fitness

Crossover

Mutation No gen
subject to the constraints of Eq. (2). Subject to the constraints

99

⎧ ⎫ y − wT ϕ(xi ) ≤  + i , i = 1, 2, . . ., N ⎪ ⎪ ⎨ i ⎬ ⎪ ⎩

wT ϕ(xi ) − yi ≤  + i ,

i = 1, 2, . . ., N

  , i ≥ 0

i = 1, 2, . . ., N

(2)

⎪ ⎭

where C is a regularization constant, controlling a compromise between maximizing the margin and minimizing the number of training set errors, and  i and   i represents upper and lower constrains on the outputs of model and  and ε are empirical parameters and |yi − f(x,w)|ε represents the ε-insensitive loss function, xi is the input vector to the SVR model, yi is the corresponding actual output value, and N is the number of input data, (x) denotes a nonlinear mapping function from the input space x to a highdimensional feature space, w and b are the support vector weight and the bias. A smaller i corresponds to attaining data point xi with less important. This constrained optimization problem is defined as the following Lagrangian function in Eq. (3).

Stop Fig. 1. Sequence of steps in EFSVR modeling.

98

J(w,   , ˛, ˛ , ,  ) =

1 T w w+C 2

 N 

100

101 102 103 104 105 106 107 108 109 110 111 112

 i (i + i )

113

i=1

68 69 70

−

development of evolutionary support vector regression (EFSVR) model for weld residual stress prediction. The results are compared with the developed SVR and FSVR models and validated.

N 

( i i + i i ) −

i=1

−

N 

N 

˛i[W T (X )−y +ε+ ] i

i

i

114

i=1

˛ i [yi − wT (xi ) + ε + i ]

(3)

115

i=1 71 72

73 74 75 76

77

78 79 80 81 82 83 84 85 86 87 88 89 90

2. Proposed evolutionary FSVR model to predict residual stress The proposed scheme to predict weld residual stress goes through three stages of development: (a) data acquisition, (b) development of Evolutionary Fuzzy Support Vector Regression model (c) validation of the developed model.

2.1. Development of proposed Evolutionary FSVR model The generalization performance of the FSVR models seriously depends on a proper setting of several parameters, such as the regularization parameter C, the insensitivity zone ε, the kernel function parameter , and the fuzzy membership parameters T, a, T1, and T2. A separate optimization of each parameter cannot guarantee the optimization of the FSVR model. Although several optimization approaches have been reported for SVR models, a set of parameters optimized for SVR would not work well in the FSVR model. In addition, there are more parameters that should be chosen for the FSVR model than for the SVR model. For accurate weld residual stress estimation, it is necessary to find a reliable optimization approach to select the optimal FSVR parameters settings. The proposed scheme is depicted in Fig. 1.

where ˛, ˛ , , and are the Lagrange multipliers. The optimization of J(w, ,  , ˛, ˛ , , ) must satisfy the following conditions:

116 117

 ∂J(W, ,   , ˛, ˛ , ,  ) =w− (˛i − ˛1 )(xi ) = 0 ∂w

(4)

118

∂J(W, ,   , ˛, ˛ , ,  ) = i − (Ci − ˛i ) = 0 ∂i

(5)

119

∂J(W, ,   , ˛, ˛ , ,  ) = i − (Ci − ˛i ) = 0 ∂i

(6)

120

N

i=1

To formulate the corresponding dual problem of Eq. (3), substitute conditions of Eqs. (4)–(6) in Eq. (3). A convex function is obtained Q (˛i ˛i ) =

N 

N 

(˛i − ˛i ) − ε

i=1

122 123

124

i=1

1  (˛i − ˛i )(˛j − ˛j )K(xi , xj ) 2 N

−

(˛i + ˛i )

121

N

(7)

125

i=1 i=1

Please cite this article in press as: J. Edwin Raja Dhas, S. Kumanan, Evolutionary fuzzy SVR modeling of weld residual stress, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.01.050

126

G Model

ARTICLE IN PRESS

ASOC 3454 1–8

J. Edwin Raja Dhas, S. Kumanan / Applied Soft Computing xxx (2016) xxx–xxx 127 128 129

130

The solution to the constrained optimization problem can be obtained by maximizing Eq. (7) subject to a new set of the constraints

⎧ N  ⎪ ⎪ ⎪ (˛i − ˛i ) = 0 ⎪ ⎨ i=1

⎪ 0 ≤ ˛i ≤ i C, ⎪ ⎪ ⎪ ⎩ 

i = 1, 2, . . .., N

0 ≤ ˛1 ≤ i C,

131 132

133

(8)

i = 1, 2, . . .., N

With the Lagrange multipliers ˛i and ˛*i, the estimated output can be represented by y=

N 

(˛i − ˛i )K(x, xi ) + b

(9)

i=1 134 135 136 137 138 139 140

141

142 143 144 145 146 147 148 149 150

151

where K(x, xi ) is called the kernel function. According to the Karush–Kuhn–Tucker’s (KKT) conditions of solving quadratic programming problem, only some of the coefficients ˛i − ˛*I are not zeros, and the corresponding training data vectors are called support vectors. In this study, FSVR equipped with the RBF as the kernel function. Generally, RBF yields better prediction performance and is defined as



K(xi , xj ) = exp

||Xi − Xj ||2

The chief step before establishing the FSVR model is to address the choice of membership values i that falls in the unit interval [0,1]. The output of SVR is used as the pre estimation of the residual stress, and then, fuzzy membership is generated. Suppose the pre estimation value for input data xi is denoted by yi , and its corresponding expected goal is di . Compute the Euclidean distance between yi and di (denoted by ei ), and then the membership values is calculated as follows: e −e Linear : i = − i (11) e¯ − e- + T Exponential : i =

ei − e-



pow a,

154 155 156 157 158 159 160 161

ei −e e¯ −e+T

0,



(12)

-

e − T1 Piecewise : i = 1− i , T2 − T1 ⎪ ⎪

⎩

153

(10)

2

⎧ 1, ⎪ ⎪ ⎨

152

ei < T1 T1 ≤ ei ≤ T2

(13)

ei > T2

where e = mini = 1, . . .,N ei , e = maxi = 1, . . .,N ei and T, a, T1 , and T2 are constants, which are used to control the effect of fuzziness to the model training. For instance, in the linear membership function, when T (0 < T < +∞) is large enough, the FSVR degenerates to a classical SVR. In the exponential membership function, larger a (a > 1) corresponds to faster attenuation of membership values. In the piecewise membership function, T1 and T2 (e < T1 , T2 < e) determine which parts of data are totally used in the model training, which parts are partially used, and which parts are abandoned [26].

error penalty factor C and kernel function parameter , hence the parameters ε, C and are optimized. The training parameters are initialized as 0.01, 10 and 64 respectively. Sets of input–output pairs from Table 1 are K-fold cross validated using five folds and are used to train the proposed model. To achieve optimal prediction performance of the proposed SVR regression model radial basis function kernel function is employed. 2.1.3. Data acquisition Mechanical and thermal response of a weldment is a three dimensional problem that requires a considerable amount of computational time. A two dimensional analysis is carried out with appropriate assumptions to reduce computational time preserving computational accuracy. In this simulation a 2D finite element analysis is used to simulate the 3D welding since quasi-steady state exists under uniform welding speed. Dimensions of 100 mm × 45 mm × 10 mm two ASTM A36 mild steel plates are modeled and simulated using ANSYS software to form gas metal arc butt weld. Chemical compositions of ASTM A36 steel are given in Table 1. Assumptions followed during the thermal and stress analysis are listed below: (i) Welding is simulated as a single pass. (ii) Bottom surface of the weld area is fixed to prevent body movement. (iii) No penetration and overfill of the weldments are considered. (iv) Initial temperature conditions for both weld and base metal are at uniform temperature of 27 ◦ C. (v) Heat loss due to radiation and forced convection due to shielding gas is neglected. (vi) Heat loss coefficient is assumed for all surfaces is 0.0001 Btu/in2 ◦ F. (vii) Heat losses or gains from phase transformation are neglected. (viii) Heat losses due to radiation and convection influences on microstructures and cooling rates of the weld metal are neglected. Weld model is simulated by bilinear strain hardening properties. The model is meshed with thermal solid element PLANE 55 for the weld bead region and PLANE 35 for the work piece. The convection and radiation surfaces are meshed with LINK34 and LINK31. The mesh density is made very fine in fusion zone, fine in HAZ and course in other zones for accuracy. The real constant for the convection and radiation elements are specified. For the structural analysis, the thermal elements are converted to its corresponding structural elements i.e. Solid Quad 4 node 42 and Triangle 6 node 2. The mesh used in the stress analysis is identical to that in the thermal analysis and the meshed workpiece as depicted in Fig. 2. Due to high stress and temperature gradients near the weld, the finite element model has a fine mesh in both sides of the weld center line. The total number of elements used in the model is 623. Heat flux is applied over the weld bead as the load input to the thermal analysis. Eq. (14) gives the arc heat input q=

162 163 164 165 166 167 168 169 170 171 172

2.1.2. Development of proposed SVR Model SVR model is constructed using four welding parameters such as arc efficiency, voltage, current and welding speed as input variables and residual stress as the output variable. The input variables are transformed into high dimensional space using nonlinear transformation which are defined by inner product functions and get optimal hyper plane in this space. Output layer is the linear combination of the intermediate nodes, each intermediate node corresponds to a support vector. Convex quadratic programming is used in getting the solution to avoid getting struck in local minima. Performance of SVR regression model depends on error ε,

3

a ∗ V ∗ I A

(14)

Based on the requirement of the weld bead area the amount of heat flux supplied to the weld plate changes and is found by Eq. (15) [19,27]. Weld bead area, A=

I

173 174 175 176 177 178 179

180 181 182 183 184 185 186 187 188 189 190 191 192

193 194 195 196 197 198 199 200 201 202 203 204 205 206

207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222

223

224 225 226 227

0.55

(103.95 ∗ S 0.003 )

(15)

where q is the arc heat flux in w/m2 , a is the arc efficiency, V is the arc voltage in volts, I is the arc current in amps, S = welding speed, mm/min and A is the weld bead area in m2 . Welding speed

Please cite this article in press as: J. Edwin Raja Dhas, S. Kumanan, Evolutionary fuzzy SVR modeling of weld residual stress, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.01.050

228

229 230 231

G Model

ARTICLE IN PRESS

ASOC 3454 1–8

J. Edwin Raja Dhas, S. Kumanan / Applied Soft Computing xxx (2016) xxx–xxx

4

Table 1 A training data set obtained from FEA for the proposed neural network. Arc efficiency (%)

Welding speed (mm/s)

Welding voltage (V)

Welding current (A)

Heat input (W)

Heat flux (W/m2 )

Residual stress Work piece/weld interface (MN/m2 )

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3 4 2 2.5 3.3

21 21 21 21 22 22 22 22 24 24 24 24 28 28 28 28 21 21 21 21 22 22 22 22 24 24 24 24 28 28 28 28 21 21 21 21 22 22 22 22 24 24 24

170 170 170 170 150 150 150 150 140 140 140 140 130 130 130 130 170 170 170 170 150 150 150 150 140 140 140 140 130 130 130 130 170 170 170 170 150 150 150 150 140 140 140

2856 2856 2856 2856 2640 2640 2640 2640 2688 2688 2688 2688 2912 2912 2912 2912 2677.5 2677.5 2677.5 2677.5 2475 2475 2475 2475 2520 2520 2520 2520 2730 2730 2730 2730 2499 2499 2499 2499 2310 2310 2310 2310 2352 2352 2352

14,280,000 14,280,000 14,280,000 14,280,000 13,200,000 13,200,000 13,200,000 13,200,000 13,440,000 13,440,000 13,440,000 13,440,000 14,560,000 14,560,000 14,560,000 14,560,000 13,387,500 13,387,500 13,387,500 13,387,500 12,375,000 12,375,000 12,375,000 12,375,000 12,600,000 12,600,000 12,600,000 12,600,000 13,650,000 13,650,000 13,650,000 13,650,000 12,495,000 12,495,000 12,495,000 12,495,000 11,550,000 11,550,000 11,550,000 11,550,000 11,760,000 11,760,000 11,760,000

67.90 62.79 74.50 236.30 63.00 61.40 115.00 297.00 64.44 61.00 108.50 286.00 69.00 63.00 73.60 213.00 60.90 64.20 110.00 289.00 60.04 65.14 178.00 325.00 60.70 64.45 143.20 319.20 65.30 61.30 100.10 275.99 60.36 64.86 162.80 322.30 58.93 66.05 254.00 300.80 58.90 65.92 238.70

is incorporated using load step options in the analysis. Governing differential equation for two dimensional transient heat transfer during welding is given in Eq. (16).

−

Fig. 2. Meshed butt-welded structure.

∂qy ∂qx + ∂x ∂y

+ Q = cp

∂T ∂t

(16)

where qx and qy are the components of heat flow rate vector per unit area in the plate (x; y), Q the heat generation, the density, cp the specific heat and (∂T/∂t) represents the temperature distribution with respect to time which is expected as the output from thermal analysis. Four welding parameters are given as the input for this analysis and they are arc efficiency, voltage, current and welding speed. The working ranges of the parameters were taken from the American Welding Society handbook. Heat input to the model is the product of arc efficiency, voltage, and current [28]. Material properties of mild steel [29] are shown in Fig. 3(a–c). Poisson’s ratio is taken as 0.3. Boundary conditions are modeled by nonlinear heat convection and heat emissivity coefficients. Non-linear transient thermal analysis is conducted first to obtain the global temperature history generated during the welding process. In thermal analysis heat flux is given as the load (arc heat) to the workpiece incorporating welding speed in terms of load step time. Stress analysis is carried out

Please cite this article in press as: J. Edwin Raja Dhas, S. Kumanan, Evolutionary fuzzy SVR modeling of weld residual stress, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.01.050

232 233 234

235

236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252

G Model ASOC 3454 1–8

ARTICLE IN PRESS J. Edwin Raja Dhas, S. Kumanan / Applied Soft Computing xxx (2016) xxx–xxx

5

Table 2 Chemical compositions of ASTM A36 steel. C

Mn

P

0.16

0.69

0.033 0.039 0.21

S

Si

Ni

Cr

Mo

Cu

Fe

< 0.10

< 0.08

< 0.10.

< 0.10

Bal

Table 3 Parameter settings for GA–FSVR model. Notations

Range of values

C  ε T1 T2

1 × 103 to 1 × 108 1 to 7 1 × 10−5 to 0.1 0–0.5* emax 0.55* emax to emax

Fig. 4. Stress distribution over butt-welded structure.

the workpiece/weld bead interface. Results in Table 2 are used to train and test the proposed SVM models. In the development of GA–FSVR modeling the chromosome X is represented by X = {p1, p2, p3, p4, p5}, where p1–p5 are binary coded and denote parameters C, ε, , T1 , and T2 , respectively. Parameter settings for the GA–FSVR modeling is shown in Table 3. The GA begins with an initial population of coded solutions to the problem. Then, each chromosome is evaluated by several objective functions. Given the FSVR estimated value yi (i = 1, . . ., N), the expected value di (i = 1, . . ., N), and the support vector weight w, three objective functions are defined to evaluate the fitness of current chromosome F1 =

N 1 

N 3

|yi − di |

253 254 255 256 257

with the temperatures obtained from the thermal analysis as the loading to the stress model. The output obtained out of this analysis is longitudinal residual stress distribution over the workpiece after cooling. Residual stress is obtained at the required nodes using the time history postprocessor and recorded in the weld bead and in

F3 =

259 260 261 262 263 264 265 266 267 268 269

(17)

270

(18)

271

(19)

272

i=1

F2 = wT w Fig. 3. (a–c) Material properties of mild steel (17).

258

Count(|˛i − ˛∗i |

> 0),

i = 1, 2, . . ., N

where Count() represents the operation of number counting. The objective function F1 lets the FSVR model accord with the training data, while F2 and F3 guarantee the smoothness of the regression model and avoid the over fitting problem. In the Genetic Algorithm sorting procedure, the individuals in the current population are first identified. In order to maintain diversity in the population, these classified individuals are shared

Please cite this article in press as: J. Edwin Raja Dhas, S. Kumanan, Evolutionary fuzzy SVR modeling of weld residual stress, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.01.050

273 274 275 276 277 278 279

G Model ASOC 3454 1–8 6

ARTICLE IN PRESS J. Edwin Raja Dhas, S. Kumanan / Applied Soft Computing xxx (2016) xxx–xxx

Fig. 5. Enlarged view of meshed model.

280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298

299 300

301 302 303 304 305 306 307 308 309 310

with their dummy fitness values. Then, the individuals of the first front are ignored temporarily, and the rest of the population is processed to identify individuals for the second front. After the selection, operation based on the dummy fitness value, normal crossover, and mutation operators are carried out to create an offspring population. In GA, the term chromosome is referred to as a candidate solution that minimizes a cost function. As the generation proceeds, populations of chromosomes are iteratively altered by biological mechanisms inspired by natural evolution such as selection, crossover, and mutation. GA’s require a fitness function that assigns a score to each chromosome (candidate solution) in the current population, and maximize the fitness function value. The fitness function evaluates the extent to which each candidate solution is suitable for specified objectives. GA starts with an initial population of chromosomes, which represent possible solutions of the optimization problem. The fitness function is computed for each chromosome. New generations are produced by the genetic operators, such as selection, crossover, and mutation. The algorithm stops after the maximum allowed time has elapsed.

2.1.4. Implementation of FSVR model The implementation of FSVR model is as follows: Step 1: Normalize training and testing data. Step 2: Obtain the membership for each training data. 1) Estimate the weld residual stress using SVR. 2) Compute the Euclidean distance between every output and its corresponding goal. 3) Compute the membership using Eqs. (11)–(13). Step 3: Optimize parameters of FSVR using GA. Step 4: FSVR. 1) Solving the quadratic programming problem. 2) Obtain the weld residual stress.

3. Results and discussion This paper addresses the development of finite element and SVR models for the prediction of residual stress in butt-welding. Finite element analysis is used to model weld residual stress. This study provides a good knowledge about residual stress distribution along with thermal history in the element. Data from finite element simulation is used to develop the proposed SVR models. Stress Distribution over butt-welded structure is displayed in Fig. 4. Residual stress result from finite element modeling at a particular node is explained below. As shown in Fig. 5 which is an enlarged view of Fig. 2 with nodes. Node 11 is taken from the meshed model for the study. For the weld condition with arc efficiency 0.8%, welding speed 2.5 mm/s, welding voltage 21 V, welding current 170 A as shown in Table 1 (second trial) the temperature distribution over time at a user defined node can plotted as graph and displayed in Fig. 6. For this condition the value of residual stress work piece at weld interface is 62.79 MN/m2 from Fig. 4. The performance analysis of the developed models in terms accuracy and speed are evaluated and compared. Models are developed using MATLAB platform. Confirmatory experiments are done using X-ray diffraction method to prove the effectiveness of this approach. Two ASTM A36 mild steel plates of dimensions 100 mm × 45 mm × 10 mm are but weld using metal inert gas welding. AWS ER70S – 6, 0.45 inch diameter of chemical composition (carbon, silicon and manganese) was used as an electrode. Shielding gas of Helium 90%, Argon 7.5% and CO2 2.5% were used to protect the weld area from atmosphere. After welding the residual stress on the weldment was determined by an advanced solid state X-ray stress AST × 2001 analyzer. Stresses were measured with Cr-Kct radiation yielding Cr 21 l-reflection at an angle of 2 = 145◦ 81 . Xray voltage is 30 kV and the X-ray current is 5.8 mA. Calibration distance ‘D’ is 49.49 mm. Results from confirmatory tests are shown in Table 4.

Please cite this article in press as: J. Edwin Raja Dhas, S. Kumanan, Evolutionary fuzzy SVR modeling of weld residual stress, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.01.050

311

312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343

G Model

ARTICLE IN PRESS

ASOC 3454 1–8

J. Edwin Raja Dhas, S. Kumanan / Applied Soft Computing xxx (2016) xxx–xxx

7

Table 4 Predicted values of residual stress from developed models. Residual stress in MN/m2

Process parameters Welding current (A)

Arc voltage (V)

Arc efficiency (%)

Weld speed (Mm/min)

Experiment

SVR model

FSVR model

EFSVR model

140 170 140

24 21 24

0.7 0.8 0.75

2 3.3 2.5

58 79 63

59.99 76.73 64.86

60.98 72.88 60.04

59.2 80.9 64.1

3.2. Computational speed Different developed SVR models are compared for their performance in terms of number of epochs required for training the network. To test the developed models for the prediction of weld residual stress actual data is taken from Table 1. Fig. 7 reveals the variation of the root mean square error of the trained models as a function of epochs. The computational time consumed by evolutionary FSVR model is comparatively less than other developed models as other model takes large time to converge. Tested model is forwarded to predict weld residual stress under different weld conditions. 4. Conclusions

Fig. 6. Time–temperature graph at node 11.

Fig. 7. Variation of the root mean square error of the trained models as a function of epochs.

344

345 346

347

3.1. Model accuracy Mean absolute percentage error (MAPE) is used to evaluate the performance of the resultant models. 1 MAPE = N i

348 349 350 351

 |ERS − PRS|  i PRSi

× 100%

(20)

where ERS is experimental residual stress, PRS is predicted residual stress and N is the number of test data. The test data is shown in Table 2. From the illustrated results, the MAPE values for all the predicted models are summarized below,

Evolutionary FSVR model is successfully employed to predict weld residual stress. Data sets from finite element analysis are validated by five fold cross validation method and then used to develop the SVR model. Meanwhile fuzzy mechanisms are introduced, to deal with the problem of vague data and FSVR model is developed. The generalization performance of the FSVR model is improved by setting several empirical parameters using Genetic Algorithm. Genetic Algorithm is introduced to choose optimal parameters for the FSVR. GAs are a class of structured random search procedures that mimics the process of the biological mechanisms of reproduction. GA starts with a population of candidate solutions to an optimization problem, with each solution encoded as a string of symbols (called a chromosome). An objective function is used to evaluate the fitness of each chromosome. In each round of optimization, a fitter chromosome receives a higher probability to be selected to bear the subsequent generation. After the selection, chromosomes are subjected to crossover and mutation, which represent the operations of exchanging portions between chromosomes and altering parts of a chromosome string, respectively. When a termination criterion is reached, the iteration of optimization is stopped, and the fittest chromosome is the solution of current optimization problem. The trained and tested SVR models will predict the requisite values of residual stress for given set of parameter combinations in real time without any extensive and expensive computations. The performance characteristics of the developed models are compared in terms of computational speed and accuracy. Results from confirmatory experiment prove that the developed evolutionary FSVR model achieves the best prediction performance prediction than the other models. The developed models are validated. As a result utilization of the developed model in manufacturing brings good quality of products reducing the residual stress developed due to efficient decision making in selection of process parameters. The scope of the paper is to aid the process planners to make intelligent decision making in choosing the appropriate process parameters to reduce the residual stress induced during welding. References

352 353 354

1. MAPE (%) of SVR is 3.71. 2. MAPE (%) of SVR using fuzzy logic is 6.489. 3. MAPE (%) of FSVR using genetic optimization is 2.812.

[1] A. Stacey, H. Macgillyivary, J. Webster, P.J. Webster, K.R.A. Ziebeck, Measurement of residual stress by Neutron diffraction, J. Strain Anal. 20 (2) (1985) 93–100.

Please cite this article in press as: J. Edwin Raja Dhas, S. Kumanan, Evolutionary fuzzy SVR modeling of weld residual stress, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.01.050

355

356 357 358 359 360 361 362 363 364 365

366

367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402

403

404 405 406

G Model ASOC 3454 1–8 8 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450

ARTICLE IN PRESS J. Edwin Raja Dhas, S. Kumanan / Applied Soft Computing xxx (2016) xxx–xxx

[2] J. Gauthier, T.W. Krause, D.L. Atherton, Measurement of residual stress in steel using the magnetic Barkhausen noise technique, Nondestruct. Test. Eval. Int. 31 (1998) 23–31. [3] K. Hariharan, V. Raghu Prakash, M. Sathya Prasad, Influence of yield criteria in the prediction of strain distribution and residual stress distribution in sheet metal formability analysis for a commercial steel, Mater. Manuf. Process. 25 (8) (2010) 828–836. [4] A.R. Kohandehghan, S. Serajzadeh, A.H. Kokabi, A study on residual stresses in gas tungsten arc welding of AA5251, Mater. Manuf. Process. 25 (11) (2010) 1242–1250. [5] D. Nelson, E. Fuchs, A. Makino, D. Williams, Residual-stress determination by single-axis holographic interferometry and hole drilling. Part II: experiments, Exp. Mech. 34 (1994) 79–88. [6] K.J. Kang, N. Yao, M.Y. He, A.G. Evans, A method for in situ measurement of residual stress in thin films by using the focused ion beam, Thin Solid Films 443 (1–2) (2003) 71–77. [7] M. Sedighi, M. Mahmoodi, Channel angular rolled Al 5083 by IHD technique: investigation of two calculation methods, Mater. Manuf. Process. 28 (1) (2012) 85–90. [8] T. Nomoto, Finite element analysis of thermal stress during welding, PhD Thesis. University of Tokyo, 1971. [9] S. Kumanan, J. Edwin Raja Dhas, R. Ashok Kumar, Development of welding residual stress predictor using a function-replacing hybrid system, Int. J. Adv. Manuf. Technol. 31 (2007) 1083–1091. [10] J. Edwin Raja Dhas, S. Kumanan, Weld residual stress prediction using artificial neural network and fuzzy logic modeling, Indian J. Eng. Mater. Sci. 18 (2011) 351–360. [11] N. Siva Shanmugam, G. Buvanashekaran, K. Sankaranarayanasamy, Some studies on weld bead geometries for laser spot welding process using finite element analysis, Mater. Des. 34 (2012) 412–426. [12] Tso-Liang Teng, Chin-Ping Fung, Peng-Hsiang Chang, Effect of weld geometry and residual stresses on fatigue in butt-welded joints, Int. J. Press. Vessel. Pip. 79 (2002) 467–482. [13] Peng-Hsiang Chang, Tso-Liang Teng, Numerical and experimental investigations on the residual stresses of the butt-welded joints, Comput. Mater. Sci. (29) (2004) 511–522. [14] D.S. Nagesh, G.L. Datta, Genetic algorithm for optimization of welding variables for height to width ratio and application of ANN for prediction of bead geometry for TIG welding process, Appl. Soft Comput. 10 (2010) 897–907. [15] D.K. Somak Datta, P.P. Pratihar, Bandyopadhyay, modeling of input–output relationships for a plasma spray coating process using soft computing tools, Appl. Soft Comput. (2012) 3356–3368. [16] Hsuan-Liang Lin, The use of the Taguchi method with grey relational analysis and a neural network to optimize a novel GMA welding process, J. Intell. Manuf. 23 (5) (2012) 1671–1680.

[17] Liang Tian, Yu Luo, Yang Wang, Xiao Wu, Prediction of transverse and angular distortions of gas tungsten arc bead-on-plate welding using artificial neural network, Mater. Des. 54 (2014) 458–472. [18] C. Ahilan, S. Kumanan, N. Sivakumaran, J. Edwin Raja Dhas, Modeling and prediction of machining quality in CNC turning process using intelligent hybrid decision making tools, Appl. Soft Comput. 13 (2013) 1543–1551. [19] J. Edwin Raja Dhas, S. Kumanan, Neuro evolutionary model for weld residual stress prediction, Appl. Soft Comput. 14 (2014) 461–468. [20] J. Edwin Raja Dhas, S. Kumanan, Modelling of residual stress in butt welding, Mater. Manuf. Process. 26 (2011) 1–6. [21] V. Vapnik, The Nature of Statistical Learning Theory, Springer, New York, 1995. [22] V. Kecman, Learning and Soft Computing, Springer-Verlag, New York, 1995. [23] R. Khemchadani, J.S. Jayadeva Chandra, Regularized least squares fuzzy support vector regression for financial time series forecasting, Exp. Syst. Appl. 36 (2009) 132–138. [24] C. Lin, S. Wang, Fuzzy support vector machines, IEEE Trans. Neural Netw. 13 (2) (2002) 464–471. [25] K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multi objective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput. 6 (2) (2002) 182– 197. [26] Jinhua Yu, Yuanyuan Wang, Ping Chen, Fetal weight estimation using the evolutionary fuzzy support vector regression for low-birth-weight fetuses, IEEE Trans. Inf. Technol. Biomed. 13 (1) (2009) 57–66. [27] N.R. Mandal, Welding Distortion and Control, Narosa Publishing House, New Delhi, India, 2004. [28] A. Wu, S. Syngellakis, B.G. Mellor, Finite element analysis of residual stresses in a butt weld, Materials Research Group, Computational Engineering & Design Group, School of Engineering Sciences, University of Southampton, Highfield, UK, 2001, pp. 37–38. [29] Y. Shim, Z. Feng, S. Lee, D. Kim, J. Jaeger, J.C. Papritan, C.L. Tsai, Determination of residual stresses in thick section weldments, Weld. J. (1992) 305– 312.

451

Dr. J. Edwin Raja Dhas is a Professor and Head in the Department of Automobile Engineering at Noorul Islam University, India. He obtained his Doctoral Degree from National Institute of Technology, Tiruchirappalli, India. His research interests are Non traditional simulation and optimization of Manufacturing Systems.

483

Dr. Somasundaram Kumanan is a Professor and Head in the Department of Production Engineering at National Institute of Technology, Tiruchirappalli, India. He obtained his Doctorate Degree in Manufacturing Management from Indian Institute of Technology, Madras, India. His research interests are Intelligent Manufacturing Systems, Modelling, Simulation and Optimization of Manufacturing Systems.

487

Please cite this article in press as: J. Edwin Raja Dhas, S. Kumanan, Evolutionary fuzzy SVR modeling of weld residual stress, Appl. Soft Comput. J. (2016), http://dx.doi.org/10.1016/j.asoc.2016.01.050

452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482

484 485 486

488 489 490 491 492