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A genetic algorithm approach to the optimization

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    A genetic algorithm approach to the optimization A genetic algorithm approach to the optimization Document Transcript

    • INTERNATIONALMechanical Engineering and Technology (IJMET), ISSN 0976 – International Journal of JOURNAL OF MECHANICAL ENGINEERING 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME AND TECHNOLOGY (IJMET)ISSN 0976 – 6340 (Print)ISSN 0976 – 6359 (Online) IJMETVolume 3, Issue 3, September - December (2012), pp. 459-470© IAEME: www.iaeme.com/ijmet.asp ©IAEMEJournal Impact Factor (2012): 3.8071 (Calculated by GISI)www.jifactor.com A GENETIC ALGORITHM APPROACH TO THE OPTIMIZATION OF PROCESS PARAMETERS IN LASER BEAM WELDING Dr. G HARINATH GOWD1* Professor, Department of Mechanical Engineering Madanapalle Institute of Technology & Science, Madanaaplle Andhra Pradesh., INDIA. Email: gowdmits@gmail.com E VENUGOPAL GOUD Associate Professor, Department of Mechanical Engineering G. Pullareddy Engineering college, Kurnool 1* Corresponding author Email: gowdmits@gmail.com ABSTRACT Laser beam welding (LBW) is a field of growing importance in industry with respect to traditional welding methodologies due to lower dimension and shape distortion of components and greater processing velocity. Because of its high weld strength to weld size ratio, reliability and minimal heat affected zone, laser welding has become important for varied industrial applications. LBW process is so complex in nature that the selection of appropriate input parameters (Pulse duration, Pulse frequency, Welding speed and Pulse energy) is not possible by the trial-and-error method. So there is a need to develop a methodology to find the optimal process parameters in ND-YAG Laser beam welding process thereby producing sound welded joints at a low cost. In view of this, research is carried on INCONEL to find the optimal process parameters. Accurate prediction mathematical models to estimate Bead width, Depth of Penetration & Bead Volume were developed from experimental data using Response Surface Methodology (RSM). These predicted mathematical models are used for optimization of the process. Total volume of the weld bead, an important bead parameter, is optimized (minimized), keeping the dimensions of the other important bead parameters as constraints, to obtain sound and superior quality welds. As the amount of data generated in the iterative process for optimization is enormous and each design cycle requires substantial calculations, the popular evolutionary algorithm Genetic Algorithm is used for the optimization. In summary, the proposed methodology enables the manufacturing engineers to compute the optimal control factor settings depending upon the production requirements. Consequently, the process could be automated based on the optimal settings. 459
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEKeywords: ND-YAG Laser Beam welding, Modeling, Genetic algorithm, Optimization.1. INTRODUCTION Laser Beam Welding (LBW) processes is a welding technique used to join multiplepieces of metal through the heating effect of a concentrated beam of coherent monochromaticlight. Light amplification by stimulated emission of radiation (LASER) is a mechanismwhich emits electromagnetic radiation, through the process of simulated emission. Lasersgenerate light energy that can be absorbed into materials and converted into heatenergy.LBW is a high-energy-density welding process and well known for its deeppenetration, high speed, small heat-affected zone, fine welding seam quality, low heat inputper unit volume, and fiber optic beam delivery [1]. The energy input in laser welding iscontrolled by the combination of focused spot size, focused position, shielding gas, laserbeam power and welding speed. Because of the above advantages, LBW is widely used. Forthe laser beam welding of butt joint, the parameters of joint fit-up and the laser beam to jointalignment [2] becomes important. An inert gas, such as helium or argon, is used to protect theweld bead from contamination, and to reduce the formation of absorbing plasma. Dependingupon the type of weld required a continuous or pulsed laser beam may be used. There arethree basic types of lasers viz., the solid state laser, the gas laser and the semi conductor laser.Among all these variants Nd:YAG lasers are being used most extensively for industrialapplications because they are capable of durable multikilowatt operation. The principle of operation is that the laser beam is pointed on to a joint and the beamis moved along the joint. The process will melt the metals in to a liquid, fuse them togetherand then make them solid again thereby joining the two pieces. The beam provides aconcentrated heat source, allowing for narrow, deep welds and high welding rates. Theprocess is frequently used in high volume applications, such as in the automotive industry. Inany welding process, bead geometrical parameters play an important role in determining themechanical properties of the weld and hence quality of the weld [3]. In Laser Beam welding,bead geometrical variables are greatly influenced by the process parameters such as Pulsefrequency, Welding speed, Input energy, Shielding gas [4] and [5]. Therefore to accomplishgood quality it is imperative to setup the right welding process parameters. Quality can beassured with embracing automated techniques for welding process. Welding automation notonly results in high quality but also results in reduced wastage, high production rates withreduce cost to make the product. Some of the significant works in literature regard to themodeling and optimization studies of welding are as follows: Yang performed regressionanalysis to model submerged arc welding process [6]. Gunaraj and Murugan minimized weldvolume for the submerged arc welding process using an optimization tool in Matlab [7]. Beadheight, bead width and bead penetration were taken as the constraints. The Taguchi method was utilized by Tarng and Yang to analyze the affect of weldingprocess parameter on the weld-bead geometry [8]. Casalino has studied the effect of weldingparameters on the weld bead geometry in laser welding using statistical and taguchiapproaches [9]. Nagesh and Datta developed a back-propagation neural network, to establishthe relationships between the process parameters and weld bead geometric parameters, in ashielded metal arc welding process [10]. Young whan park has applied Genetic algorithmsand Neural network for process modeling and parameter optimization of aluminium laserwelding automation [11]. Mishra and Debroy showed that multiple sets of welding variablescapable of producing the target weld geometry could be determined in a realistic time frameby coupling a real-coded GA with and neural network model for Gas Metal Arc Fillet 460
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEWelding [12]. Saurav data has applied RSM to modeling and optimization of the features ofbead geometry including percentage dilution in submerged arc welding using mixture of freshflux and fused slag [13]. The literature shows that the most dominant modeling tools used till date are Taguchibased regression analysis and artificial neural networks. However, the accuracy andpossibility of determining the global optimum solution depends on the type of modelingtechnique used to express the objective function and constraints as functions of the decisionvariables. Therefore effective, efficient and economic utilization of laser welding necessitatesan accurate modeling and optimization procedure. In the present work, RSM is used fordeveloping the relationships between the weld bead geometry and the input variables. Themodels derived by RSM are utilized for optimizing the process by using the GeneticAlgorithm.2. EXPERIMENTAL WORK The experiments are conducted on High peak power pulsed Nd:YAG Laser weldingsystem with six degrees of freedom robot delivered through 300 um Luminator fiber asshown in Figure 1. Fig.1. Nd:YAG Robotic Laser Beam welding equipment In this research Butt welding of Inconel 600 is carried out at by varying the inputparameters. The size of each plate welded is 30mm long x 30mm width with thickness of2.5mm. The laser beam is focused at the interface of the joints. An inert gas such as argon isused to protect the weld bead from contamination, and to reduce the formation of absorbingplasma. Based on the literature survey and the trial experiments, it was found that the processparameters such as pulse duration (x1), pulse frequency (x2), speed (x3), and energy (x4) havesignificant effect on weld bead geometrical features such as penetration (P), bead width (W),and bead volume (V).In the present work, they are considered as the decision variables andtrial samples of butt joints are performed by varying one of the process variables to determinethe working range of each process variable. Absent of visible welding defects and at least halfdepth penetrations were the criteria of choosing the working ranges. 461
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME After conducting the experiments as per the design matrix, for measuring the outputresponses i.e bead geometry features such as Bead penetration & Bead width, welded joint issectioned perpendicular to the weld direction. The specimens are then prepared by the usualmetallurgical polishing methods and then etched. Then the bead dimensions were measuredusing Toolmaker’s microscope. For each response the readings were measured at threedifferent sections of the weld joint and the average value is taken. The study is focused toinvestigate the effects of process variables on the structures of the welds. An average of threemeasurements taken at three different places and the output responses are recorded for eachset. The output responses recorded are shown in the Table 1. Table 1. Experimental Observations Bead Bead Experiment x2 x3 x4 Penetration x1 (µs) width Volume No. (Hz) (mm/min) (J) (mm) (mm) (mm3) 1 2 10 300 12 1.800 1.200 0.469 2 4 10 300 12 2.230 1.020 0.460 3 2 18 300 12 1.900 1.150 0.500 4 4 18 300 12 2.280 1.210 0.500 5 2 10 700 12 1.700 0.819 0.330 6 4 10 700 12 2.070 0.910 0.340 7 2 18 700 12 1.800 0.805 0.495 8 4 18 700 12 2.010 0.856 0.500 9 2 10 300 18 1.840 1.010 0.444 10 4 10 300 18 2.240 0.990 0.486 11 2 18 300 18 1.950 1.015 0.531 12 4 18 300 18 2.180 1.015 0.580 13 2 10 700 18 1.770 0.768 0.318 14 4 10 700 18 2.180 0.950 0.352 15 2 18 700 18 2.010 0.756 0.500 16 4 18 700 18 2.155 0.978 0.520 17 1 14 500 15 1.500 0.900 0.400 18 5 14 500 15 2.260 1.060 0.420 19 3 6 500 15 1.912 0.940 0.150 20 3 22 500 15 2.170 1.015 0.540 21 3 14 100 15 2.250 1.269 0.555 22 3 14 900 15 1.940 0.701 0.400 23 3 14 500 9 1.800 1.015 0.390 24 3 14 500 21 2.250 0.984 0.500 25 3 14 500 15 2.050 0.980 0.512 26 3 14 500 15 2.070 0.958 0.500 27 3 14 500 15 2.080 0.950 0.520 28 3 14 500 15 2.105 0.936 0.491 29 3 14 500 15 2.098 0.928 0.487 30 3 14 500 15 2.050 0.916 0.490 31 3 14 500 15 1.961 0.900 0.490 462
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME3. DEVELOPMENT OF EMPIRICAL MODELS The need in developing the mathematical relationships from the experimental data isto relate the measure output responses Penetration, Bead width and Bead volume to the inputprocess parameters such as pulse duration (x1), pulse frequency(x2), speed (x3), and energy(x4) thereby facilitating the optimization of the welding process. RSM is used to predict theaccurate models.P e n e tr a tio n = 2 .0 6 + 0 .3 4 x1 + 0 .0 8 1 x 2 - 0 .1 1 x 3 + 0 .1 2 x 4 - 0 .1 6 x1 x 2- 0 . 0 7 6 x 1 x 3 - 0 . 0 5 1 x 1 x 4 + 0 . 0 1 4 x 2 x 3 + 0 . 0 1 9 x 2 x 4 + 0 . 1 3 x 3 x 4 - 0 . 1 8 x 12- 0 . 0 2 0 x 22 + 0 . 0 3 4 x 32 - 0 . 0 3 6 x 42 Eq. (1)B e a d w id th = 0 .9 6 + 0 .0 6 0 x1 + 0 .0 2 2 x 2 - 0 .2 4 x 3 − 0 .0 4 6 x 4 + 0 .0 6 5 x1 x 2+ 0 .1 7 x1 x 3 + 0 .0 9 1 x1 x 4 − 0 .0 5 5 x 2 x 3 − 0 .0 0 6 5 x 2 x 4 + 0 .1 5 x 3 x 4 Eq. (2)B e a d v o lu m e = 0 .5 0 + 0 .0 1 6 x1 + 0 .1 4 x 2 - 0 .0 7 7 x 3 + 0 .0 3 0 x 4-0 .0 0 0 7 5 x1 x 2 - 0 .0 0 3 2 5 x1 x 3 + 0 .0 3 5 x1 x 4 + 0 .1 1 x 2 x 3 + 0 .0 3 4 x 2 x 4− 0 .0 2 2 x 3 x 4 - 0 . 0 6 3 x 12 - 0 . 1 3 x 2 2 + 0 .0 0 4 5 5 x 2 3 - 0 .0 2 8 x 2 4 Eq. (3) The developed mathematical models are checked for their adequacy using ANNOVAand normal probability plot of residuals. Then these models are used for Optimization ofprocess parameters using Genetic Algorithms.4. FORMULATION OF OPTIMIZATION PROBLEM In the present work, the bead geometrical parameters were chosen to be theconstraints and the minimization of volume of the weld bead was considered to be theobjective function. Minimizing the volume of the weld bead reduces the welding cost throughreduced heat input and energy consumption and increased welding production through a highwelding speed [14]. The present problem is formulated an optimization model as shownbelow:MinimizeB e a d v o lu m e = 0 .5 0 + 0 .0 1 6 x1 + 0 .1 4 x 2 - 0 .0 7 7 x 3 + 0 .0 3 0 x 4-0 .0 0 0 7 5 x1 x 2 - 0 .0 0 3 2 5 x1 x 3 + 0 .0 3 5 x1 x 4 + 0 .1 1 x 2 x 3 + 0 .0 3 4 x 2 x 4 2 2 2 2− 0 .0 2 2 x 3 x 4 - 0 .0 6 3 x 1 - 0 .1 3 x 2 + 0 .0 0 4 5 5 x 3 - 0 .0 2 8 x 4Subject to:P e n e tr a tio n = 2 .0 6 + 0 .3 4 x1 + 0 .0 8 1 x 2 - 0 .1 1 x 3 + 0 .1 2 x 4 - 0 .1 6 x1 x 2- 0 . 0 7 6 x 1 x 3 - 0 . 0 5 1 x 1 x 4 + 0 . 0 1 4 x 2 x 3 + 0 . 0 1 9 x 2 x 4 + 0 . 1 3 x 3 x 4 - 0 . 1 8 x 12- 0 . 0 2 0 x 22 + 0 . 0 3 4 x 32 - 0 . 0 3 6 x 4 ≥ 2 . 2 5 2&B e a d w id th = 0 .9 6 + 0 .0 6 0 x1 + 0 .0 2 2 x 2 - 0 .2 4 x 3 − 0 .0 4 6 x 4 + 0 .0 6 5 x1 x 2+ 0 .1 7 x1 x 3 + 0 .0 9 1 x1 x 4 − 0 .0 5 5 x 2 x 3 − 0 .0 0 6 5 x 2 x 4 + 0 .1 5 x 3 x 4 ≤ 0 .7 463
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEWith the parameter feasible ranges: 1 µs ≤ x1 ≤ 5 µs, 6 Hz ≤ x2 ≤ 22 Hz, 100 mm/min ≤ x3 ≤ 900 mm/min, 9 J ≤ x4 ≤ 21 J The bead parameters and the feasible ranges of the input variables were established with aview to have defect-free welded joint. Once the optimization problem is formulated, then it is solved using Geneticalgorithms (GA). The GA optimization module available in MATLAB is used to find out theoptimal parameters. Tables 2, 3 and 4 exhibit the implementation of GA for minimizing theBead volume as objective. Sample calculations are shown for one iteration of the algorithm.The bit lengths chosen for x1, x2, x3 and x4 are chosen 4, 4, 5 and 4 respectively. As a firststep, an initial population of 40 chromosomes is generated randomly as shown in Table 2.Chromosome strings of individual input variables are decoded and substituted to determinethe objective function value of Bead volume. From Table 2, the first string (0000 1101 111101101) is decoded to values equal to x1=1, x2=20, x3=874 and x4=19 using linear mappingrule. Then the objective function value is computed which is obtained as 0.4874. The fitnessfinal value at this point using the transformation rule F(x(1)) = 1.0/(1.0+0.4874) is obtained as0.6723. This fitness function value is used in the reproduction operation of GA. Similarly,other strings in the population are evaluated and fitness values are calculated. Table 2 showsthe objective function value and the fitness value for all the 40 strings in the initialpopulation. In the next step, good strings in the population are to be selected to form the matingpool. In this work, roulette-wheel selection procedure is used to select the good strings. As apart of this procedure, average fitness [15] of the population is calculated by adding thefitness values of all strings and dividing the sum by the population size and the average _fitness of the population ( F ) is obtained as 0.7772. The expected count is subsequentlycalculated by dividing each fitness value with the average fitness;  F( x)     _   F  For the first string, the expected count is (0.6723/0.7772) = 0.8649. Similarly, theexpected count values are calculated for all other strings in the population and shown inTable 3. Then, the probability of each string being copied in the mating pool can be computeddividing the expected count values with the population size. For instance, the probability offirst string is (0.8649/40) = 0.02. Similarly, the values of probability of selection for all thestrings are calculated and cumulative probability is henceforward computed. Theprobabilities of selection are listed in Table 3. Next random numbers between zero and oneare generated in order to form the mating pool. From Table 3, random number generated for the first string is 0.30 which means thetwelfth string from the population gets a copy in the mating pool, because that string occupiesthe probability interval (0.27, 0.30) as shown in the column of cumulative probability in theTable 3. In a similar manner, other strings are selected according to the random numbersgenerated in Table 3 and the complete mating pool is formed. The mating pool is displayed in 464
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMETable 7.14. By adopting the reproduction operator, the inferior points have beenautomatically eliminated from further consideration. As a next step in the generation, thestrings in the mating pool are used for the crossover operation. Table 2. Initial population with fitness values in GA S.No Chromosomes x1 x2 x3 x4 Objective Fitness values 1 0000 1101 11110 1101 1 20 874 19 0.4874 0.6723 2 0100 0111 10101 1001 2 13 642 16 0.2485 0.8010 3 1101 1101 10111 1011 4 20 694 18 0.3236 0.7555 4 1010 0111 11010 0111 4 13 771 15 0.3429 0.7447 5 1110 0010 11100 0100 5 8 823 12 0.3397 0.7464 6 0111 0111 10110 1101 3 13 668 19 0.2598 0.7937 7 1100 1011 11100 1000 4 18 823 15 0.4195 0.7045 8 0110 0111 11001 1001 3 13 745 16 0.3225 0.7562 9 1111 0100 01101 0100 5 10 435 12 0.1123 0.8991 10 1011 1111 01101 0110 4 22 435 14 0.1642 0.8589 11 0110 0111 10000 0111 3 13 513 15 0.1690 0.8554 12 0011 1110 11110 1010 2 21 874 15 0.4997 0.6668 13 0101 1011 01101 1101 2 18 435 19 0.1425 0.8753 14 1001 0010 10011 0101 3 8 590 13 0.1827 0.8455 15 1001 0110 10110 0010 3 12 668 11 0.2643 0.7909 16 1010 0111 01111 1000 4 13 487 15 0.1521 0.8680 17 1110 0100 11100 0010 5 10 823 11 0.3615 0.7345 18 1001 0111 10101 1110 3 13 642 20 0.2399 0.8065 19 1100 0000 10011 0000 4 6 590 9 0.1729 0.8526 20 0111 1111 10001 0010 3 22 539 11 0.2353 0.8095 21 0010 0110 11111 0000 2 12 900 9 0.4605 0.6847 22 1101 0000 10011 0010 4 6 590 11 0.1703 0.8545 23 1010 1111 11101 0111 4 22 848 15 0.4836 0.6740 24 0110 1001 11111 1110 3 16 900 20 0.4657 0.6823 25 0001 0110 10001 0001 1 12 539 10 0.1860 0.8432 26 0100 0111 01110 1110 2 13 461 20 0.1358 0.8804 27 1111 1011 11110 1011 5 18 874 18 0.4597 0.6851 28 0100 0111 11001 0111 2 13 745 15 0.3264 0.7539 29 0000 1011 11101 1101 1 18 848 19 0.4439 0.6926 30 0101 0110 11010 0110 2 12 771 14 0.3388 0.7469 31 1100 1011 01101 0111 4 18 435 15 0.1446 0.8737 32 1001 0000 11011 0100 3 6 797 12 0.3047 0.7664 33 0011 1111 11101 0101 2 22 848 13 0.4917 0.6704 34 0010 1000 11110 0100 2 15 874 12 0.4507 0.6893 35 1111 1111 11001 0001 5 22 745 10 0.3936 0.7175 36 0011 1111 01100 1010 2 22 410 17 0.1493 0.8701 37 1111 1000 11010 0011 5 15 771 11 0.3537 0.7387 38 0111 1011 10001 1011 3 18 539 18 0.2034 0.8309 39 0101 0101 10011 0110 2 12 590 14 0.2105 0.8261 40 1110 1011 10110 0011 5 18 668 11 0.2970 0.7710 465
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME Table 3. Selection in GA Expected Cumulative Random Selected string S.No Probability Count Probability number number 1 0.8649 0.02 0.02 0.30 12 2 1.0305 0.03 0.05 0.68 27 3 0.9720 0.02 0.07 0.91 36 4 0.9580 0.02 0.09 0.61 24 5 0.9603 0.02 0.12 0.32 13 6 1.0212 0.03 0.14 0.70 28 7 0.9063 0.02 0.17 0.73 30 8 0.9728 0.02 0.19 0.56 22 9 1.1567 0.03 0.22 0.92 37 10 1.1050 0.03 0.25 0.36 14 11 1.1005 0.03 0.27 0.25 10 12 0.8579 0.02 0.30 0.44 17 13 1.1261 0.03 0.32 0.17 7 14 1.0878 0.03 0.35 0.40 16 15 1.0176 0.03 0.38 0.38 15 16 1.1167 0.03 0.40 0.94 38 17 0.9449 0.02 0.43 0.75 30 18 1.0376 0.03 0.45 0.52 20 19 1.0969 0.03 0.48 0.47 19 20 1.0415 0.03 0.51 0.54 21 21 0.8809 0.02 0.53 0.80 32 22 1.0993 0.03 0.56 0.89 36 23 0.8672 0.02 0.58 0.85 34 24 0.8778 0.02 0.60 0.99 40 25 1.0848 0.03 0.63 0.59 24 26 1.1327 0.03 0.66 0.65 26 27 0.8814 0.02 0.68 0.62 25 28 0.9699 0.02 0.70 0.10 4 29 0.8910 0.02 0.72 0.39 16 30 0.9610 0.02 0.75 0.33 13 31 1.1240 0.03 0.78 0.86 35 32 0.9861 0.02 0.80 0.96 39 33 0.8625 0.02 0.82 0.11 5 34 0.8868 0.02 0.85 0.51 20 35 0.9231 0.02 0.87 0.08 3 36 1.1194 0.03 0.90 0.05 2 37 0.9504 0.02 0.92 0.34 14 38 1.0690 0.03 0.95 0.15 6 39 1.0628 0.03 0.97 0.37 15 40 0.9919 0.02 1.00 0.64 25 466
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEIn the crossover operation, two strings are selected at random and crossed at a random site.Since the mating pool contains strings at random, pairs of strings are picked up form top ofthe list as shown in Table 4. Table 4. Crossover and Mutation in GAS.N Mating pool Crossover? Crossover Offspring Mutation Mutated chromosome Chromosomes site site1 0011 1110 11110 1010 No -- 0011 1110 11110 1010 8, 13 0011 1111 11111 10102 1111 1011 11110 1011 No -- 1111 1011 11110 1011 6 1111 1111 11110 10113 0011 1111 01100 1010 Yes 6, 12 0011 1001 11110 1010 8 0011 1000 11110 10104 0110 1001 11111 1110 Yes 6, 12 0110 1111 01101 1110 -- 0110 1111 01101 11105 0101 1011 01101 1101 No -- 0101 1011 01101 1101 -- 0101 1011 01101 11016 0100 0111 11001 0111 No -- 0100 0111 11001 0111 8 0100 0110 11001 01117 0101 0110 11010 0110 No -- 0101 0110 11010 0110 -- 0101 0110 11010 0110 8 1101 0000 10011 0010 No -- 1101 0000 10011 0010 7 1101 0010 10011 0010 9 1111 1000 11010 0011 No -- 1111 1000 11010 0011 -- 1111 1000 11010 001110 1001 0010 10011 0101 No -- 1001 0010 10011 0101 -- 1001 0010 10011 010111 1011 1111 01101 0110 No -- 1011 1111 01101 0110 -- 1011 1111 01101 011012 1110 0100 11100 0010 No -- 1110 0100 11100 0010 -- 1110 0100 11100 001013 1100 1011 11100 1000 Yes 9, 11 1100 1011 01100 1000 -- 1100 1011 01100 100014 1010 0111 01111 1000 Yes 9, 11 1010 0111 11111 1000 -- 1010 0111 11111 100015 1001 0110 10110 0010 No -- 1001 0110 10110 0010 3, 7 1011 0100 10110 001016 0111 1011 10001 1011 No -- 0111 1011 10001 1011 4 0110 1011 10001 101117 0101 0110 11010 0110 Yes 15, 17 0101 0110 11010 0010 -- 0101 0110 11010 001018 0111 1111 10001 0010 Yes 15, 17 0111 1111 10001 0110 -- 0111 1111 10001 011019 1100 0000 10011 0000 No -- 1100 0000 10011 0000 -- 1100 0000 10011 000020 0010 0110 11111 0000 No -- 0010 0110 11111 0000 -- 0010 0110 11111 000021 1001 0000 11011 0100 No -- 1001 0000 11011 0100 12 1001 0000 11001 010022 0011 1111 01100 1010 No -- 0011 1111 01100 1010 8 0011 1110 01100 101023 0010 1000 11110 0100 Yes 9, 12 0010 1000 10110 0100 3, 14 0000 1000 10110 110024 1110 1011 10110 0011 Yes 9, 12 1110 1011 11110 0011 -- 1110 1011 11110 001125 0110 1001 11111 1110 No -- 0110 1001 11111 1110 17 0110 1001 11111 111126 0100 0111 01110 1110 No -- 0100 0111 01110 1110 -- 0100 0111 01110 111027 0001 0110 10001 0001 No -- 0001 0110 10001 0001 -- 0001 0110 10001 000128 1010 0111 11010 0111 No -- 1010 0111 11010 0111 -- 1010 0111 11010 011129 1010 0111 01111 1000 No -- 1010 0111 01111 1000 1, 13 0010 0111 01110 100030 0101 1011 01101 1101 No -- 0101 1011 01101 1101 12 0101 1011 01111 110131 1111 1111 11001 0001 No -- 1111 1111 11001 0001 -- 1111 1111 11001 000132 0101 0101 10011 0110 No -- 0101 0101 10011 0110 -- 0101 0101 10011 011033 1110 0010 11100 0100 Yes 12, 17 1110 0010 11101 0010 -- 1110 0010 11101 001034 0111 1111 10001 0010 Yes 12, 17 0111 1111 10000 0100 -- 0111 1111 10000 010035 1101 1101 10111 1011 No -- 1101 1101 10111 1011 13 1101 1101 10110 101136 0100 0111 10101 1001 No -- 0100 0111 10101 1001 -- 0100 0111 10101 100137 1001 0010 10011 0101 No -- 1001 0010 10011 0101 17 1001 0010 10011 010038 0111 0111 10110 1101 No -- 0111 0111 10110 1101 -- 0111 0111 10110 110139 1001 0110 10110 0010 Yes 4, 6 1001 0110 10110 0010 -- 1001 0110 10110 001040 0001 0110 10001 0001 Yes 4, 6 0001 0110 10001 0001 8, 13 0001 0111 10000 0001 Thus strings 12 and 27 participate in the first crossover operation. In this work, twopoint crossover [15] is adopted with the probability, Pc = 0.85 to check whether a crossover isdesired or not. To perform crossover, a random number is generated with crossoverprobability (Pc) of 0.85. If the random number is less than Pc then the crossover operation isperformed, otherwise the strings are directly placed in an intermediate population forsubsequent genetic operation. When crossover is required to be performed then crossoversites are to be decided at random by creating random numbers between (0, l-1), where lrepresents the total length of the string. For Example, when crossover is required to beperformed for the strings 3, 4 two sites of crossover are to be selected randomly. Here, therandom sites are happened to be 6, 12. Thus the portions between sites 6 and 12 of the strings3 and 4 are swapped to create the new offspring as shown in Table 4. However with therandom sites, the children strings produced may or may not have a combination of good 467
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEstrings from parent strings, depending on whether or not the crossing sites fall in theappropriate locations. If good strings are not created by crossover, they will not survive toolong because reproduction will select against those chromosomes in subsequent generation.In order to preserve some of the good chromosomes that are already present in the matingpool, all the chromosomes are not used in crossover operation. When a crossover probabilityPc is used, the expected number of strings that will be subjected to crossover is only 100Pcand the remaining percent of the population remains as they are in the current population. Thecalculations of intermediate population are shown in the Table 4. The crossover is mainlyresponsible for the creation of new strings. The third operator, mutation, is then applied on the intermediate population. Mutationis basically intended for local search around the current solution. Bit-wise mutation isperformed with a probability, Pm = 0.10. A random number is generated with Pm; if randomnumber is less than Pm then the bit is altered form 1 to 0 or 0 to 1 depending on the bit valueotherwise no action is taken. Mutation is implemented with the probability, Pm=0.10 asshown in Table 4. The procedure is repeated for all the strings in the intermediate population.This completes one iteration of the GA. The above procedure is continued until the maximumnumber of generations is completed. For better convergence of the present problem, theGenetic algorithm is run for 120 generations. GA narrows down the search space as thesearch progresses and the algorithm is converged to the objective function value of 0.2688.The convergence graph is displayed in Fig.2 and the optimal values of the control factors arelisted in Table 5. The following inference discusses the performance of proposed methodology: Fromthe experimental observations presented in the Table 1, the 10th experiment resulted for 0.486for Bead volume and Bead penetration as 2.24. After optimization using GA, it is observedfrom Table 5, that Bead volume can be decreased to 0.2688 (by 55 %) for the same Beadpenetration. Fig. 2. Convergence graph for minimization of Bead volume 468
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME Table 5. Optimal values x1 (Pulse x2 (Pulse x3 x4 (Pulse Bead BeadVariable Duration) frequency) (Welding Energy) Penetration Width Volume (µs) (Hz) Speed) (J) (mm/min) Value 3.929 6.31 761.444 15.932 2.24 0.7 0.26885. CONCLUSIONS In the present study, Design based experiments and analysis have been carried out inorder to optimize the bead volume considering the effects of bead geometrical parameterslike: Bead penetration and Bead width in butt welding of INCONEL 600 using ND:YAGLaser beam welding setup. First Experiments were carried out by as per Central CompositeRotatable factorial design to substantially reduce the number of experiments. Then RSM isused to develop second order polynomial models between the bead geometrical parameters:Bead volume, Bead width, Depth of penetration and the chosen control variables: Pulseduration, Pulse frequency, Welding speed and the Pulse energy. Later A constrainedoptimization problem is then formulated to minimize the bead volume subject to the beadwidth and bead penetration as constraints. A binary coded Genetic algorithm was used tosolve the above said problem. The genetic algorithm was able to reach near the globallyoptimal solution, after satisfying the above constraints. The optimal values obtained by theproposed methodology could serve as a ready reckoner to conduct the welding operationswith great ease to achieve the quality and the production rate demanded by the consumers. Insummary, the proposed work enables the manufacturing engineers to select the optimalvalues depending on the production requirements and as a consequence, automation of theprocess could be done based on the optimal values.REFERENCES [1] Steen W.M., “Laser material processing,” Springer, London, 1991. [2] Dawes C., “Laser Welding,” Abington Publishing, Newyork, 1992. [3] Howard B.Cary., “Modern Welding Technology,” Prentice Hall, New Jersey, 1989. [4] Murugan N., Bhuvanasekharan G., “Effects of process parameters on the bead geometry of laser beam butt welded stainless sheets,” Int J Adv. Man Tech, 32:1125-1133, 2007. [5] Benyounis K Y., Olabi A.G., “Effect of laser welding parameters on the heat input and weld bead profile,” Journal of materials processing technology,” 164-165, 2005. [6] Yang L.J., Chandel R.S., “An analysis of curvilinear regression equations for modeling the submerged-arc welding process,” Journal of Materials Processing Technology, 37, 601–611, 1993. [7] Gunaraj V., Murugan N., “Prediction and optimization of weld bead volume for the submerged arc process Part 2,” Welding Journal, 78, 331s– 338s, 2000. [8] Tarng Y.S., Yang W.H., “Optimization of the weld-bead geometry in gas tungsten arc welding by the Taguchi method,” Int. Journal of Advanced Manufacturing Technology 14 (8), 549–554, 1998. 469
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME [9] Casalino G., “Investigation on Ti6A14V laser welding using statistical and taguchi approaches,” Int. Journal of Advanced Manufacturing technology, 2008. [10] Nagesh D S., Datta G L., “Prediction of weld bead geometry and penetration in shielded metal arc welding using artificial neural networks,” J. Material Process. Technol.123, 03–312, 2002. [11] Young whan park., “Genetic algorithms and Neural network for process modeling and parameter optimization of aluminium laser welding Automa tion,” Int. J Adv Manuf Technology 2008. [12] Mishra S., Debroy T., “Tailoring gas tungsten arc weld geometry using a genetic algorithm and a neural network trained with convective heat flowcalculations,” Materials Science and Engineering, 454-455, 477–486, 2007. [13] Saurav data., “Modeling and optimization of the features of bead geometry including percentage dilution in submerged arc welding using mixture of fresh fluz and fused slag,” International journal of advanced manufacturing technology. [14] Deb K., ““Multiobjective optimization using evolutionary algorithms”, John Wiley & Sons (ASIA) Pvt. Ltd., Singapore, 2001. [15] Deb K., “Optimization for engineering: algorithms and examples”, Prentice Hall of India, New Delhi, 2001. 470