In production conditions (compared to laboratory installations) the prediction of the geometrical characteristics of EBW is an even more complex task due to the presence of errors, coming from the tolerances in the controlled EBW parameters, or, from other uncontrolled parameters [62]. The variations caused by these variables make it difficult to repeat weld geometry exactly under the same conditions. The quality improvement considered here is connected with finding regimes where the variation in the weld depth and width will be less sensitive to such variables [71].
In production conditions usually variations of the process conditions are usually observed. They result in increasing the variations of the performance characteristics of the produced welds. The robust engineering approach can be applied for the quality improvement related to the decrease of the variations of the obtained welds and its repeatability. The estimated regression models are used for the estimation of two new models for the performance of each quality characteristic in production conditions: a model of the mean and a model of the variance [62, 71]. These two models can be used for choosing process parameters, which satisfy both the characteristic being close to its target value and minimization of its variance.
A new method for estimation of regression coefficients takes into account both the correlation and the heteroscedasticity (the case when there are errors in the factors levels in the production stage resulting in variation of performance characteristics, which depends on the process parameters) of the performed experiments in order to improve the accuracy of the estimated regression models, as well as the models for the means and variances of the multiple responses, is proposed in [72]. This combined approach can be implemented for the sequential generation of industrial experimental designs.
The application of the proposed approach gives the possibility to use for the quality improvement using the robust engineering approach raw industrial experimental data, instead of the necessary very precise regression model estimations without errors in the factor levels, done usually in laboratory conditions.
The mean and the variance models for the two responses are estimated, applying the original new combined method. On Figure 72 contour plots of the weld depth H mean and variance at EBW of SSt in production conditions are presented. For the estimation of the models the tolerance limits given in Table 9 are used.
Figure 72 and Figure 73 present the equipotential contour lines of the mean value (solid) and the variance (dotted) for both - the weld depth and the weld width depending on the beam power and the welding velocity at focusing parameter dz=-40 mm (Figure 72) dz=-78 mm (Figure 73).
Figure 72. Contour plots of the mean ~yH
(x) (solid lines) and the variance of the weld depth H (dotted lines) depending on P and v at zo=276 mm and zp=236 mm
Figure 73. The estimated contour plots for the mean value ~y
B (solid) and the variance
2
sˆB (dotted) of the weld width B for a focusing parameter dz = -78 mm
Process Parameter Optimization and Quality Improvement at Electron Beam Welding 157
O
PTIMIZATIONUsing the multi-response surface methodology [57, 58], polynomial regression models or neural network models for the estimation of the behavior of the weld depth H and the mean weld width B (as well as the thermal efficiency or other performance characteristics) at EB welding with deep penetrating beam versus welding and material characteristics parameter optimization can be performed. A model is developed that includes the values of beam power and welding speed as well as the distances between the electron gun and both the focusing plane of the beam and the sample surface as process parameters. Computer procedures for the choice of operating conditions under some criteria for obtaining special parameters of the seam and for acquiring optimal weld parameters can be different, depending on the concrete requirements for the characteristics of the produced welds. As criteria for such optimization can be used desirability function for a property - values of the weld depth, the width or the thermal efficiency. In order to improve the quality of process (to decrease the deviation from the target value of the performance characteristics) in production conditions a model approach is applied. Two models: one describing the mean value (using the mentioned polynomial regression or other modeling method) and second calculating the variance for the weld depth and the weld width in the mass production are estimated. Utilizing these models quality improvement can be defined [62, 71] as an optimization problem of variance minimization while keeping the mean value of weld depth or/and width on the target values.
Additionally to the requirements for the geometry of the obtained welds and the process thermal efficiency, requirements for the defect-free welds are typical.
For the experimentally obtained weld cross-sections by EBW of stainless steel, the number of defects is counted. Several approaches (response surface methodology, discriminant analysis etc.) are applied for the prediction of the process parameter regions, where the probability for appearance of defects is smaller. The experimental welds are separated into two groups (classes): 1 – with defects and 2 – without defects. The type of the defects is not taken into account.
The analysis for concrete conditions shows that the most influential process parameters, which should be considered, in order to avoid the defect appearance, are electron beam power and the distance to the surface of the sample.
In the case of applying the regression analysis 94% of the observations are predicted correctly (95% - for the group 1 of observable defects in welds and 89.5% for the group 2).
The regression model for the defects is estimated as follows:
D = -0.177-0.341x1-0.113x2+0.562x4-1.188x5+0.495x1
The value of D=0.5 is accepted as a conditional limit between the regions with (D>0.5) and without (D<0.5) defects.
The estimated regression models can be used for EBW process parameter optimization fulfilling the specific performance characteristic requirements for finding individual optimum and compromise solutions.
On Figure 74 is presented the result from maximization of the weld depth H. The maximum value obtained is H=43.65 mm at P=8.4 kW, v=20 cm/min, zo=176 mm and zp=146.5 mm (focus position at 29.5 mm below the sample surface). A requirement is added for lack of defects (D<0.5). The coloured zone contains all the regimes at which defects are not expected. Figure 75 shows the results from the parameter optimization for the thermal efficiency under the following constraints: H>25 mm, B<3 mm and no defects (D<0.5). The focus position in this case is on the sample surface (zo=226 mm and zp=226 mm). The maximum thermal efficiency is 0.43, obtained for maximum beam power and welding velocity of 26 cm/min.
Figure 74. Contour plot H(P,v), at zo=176 mm and zp=146.5 mm (SSt)
Figure 75. Contour plot T(P,v), at zo=226 mm and zp=226 mm (SSt)
Process Parameter Optimization and Quality Improvement at Electron Beam Welding 159
Figure 76. Pareto-optimal solutions (‗□‘) and constraints: H>25 mm and B=[13 mm] (SSt)
When optimum of more than one function at the same time is required, compromise solutions should be found, since the individual optima usually are reached at different regime conditions. Pareto-optimal solutions form a group of optimal solutions in the sense that moving away from a given Pareto-optimal point will worsen at least one of the considered performance characteristics. The choice among the Pareto optimal solutions should be made according other criteria. Figure 76 represents a set of points (calculated from 10000 randomly selected regimes within the experimental region), which fulfill the constraints: H>25 mm and B=[13 mm] and Pareto-optimal solutions (signed with ‗□‘), which maximize the depth H and minimize the width B within the acceptable region at the same time. In Table 18 are presented a few of these solutions such solutions (first three points). Each of these points is closer to one of the optimums: maximum H or minimum B.
Another approach of a compromise solution choice is the analytical technique for the optimization of a several functions, using the utility or the desirability of a property given by a certain performance characteristic function (in our case weld depth H and width B). One can specify certain desired values of the weld geometry characteristics di and they will be two-side constrained yi yi(x)yi (there yi* andyi* are acceptable values of the lower and upper deviations from the desired values). Then the individual desirability for each function is evaluated by the function:
where the values of s and t are chosen within the domain [0.1; 10] - the larger values of s and t are the desirability function is larger only for weld depths and widths that are closer to di. If all the values in the region yi yi(x)yi are almost equally acceptable, s and t are given
smaller values. A single function D is formed from all the individual desirability functions, which gives the overall assessment of the desirability of the combined responses, namely the geometric mean of the values of gi. The overall desirability function for H and B is:
D = (gH . gB)1/2.
In Table 18 are presented three of the solutions (№4-6), having the highest overall desirability value G at desirable values H=30 mm and B=2 mm (s=t=1) and acceptable regions H=[28-32 mm] and B=[1.5-2.5 mm]. In Figure 77 is shown contour plot of overall desirability function D (and maximal desirability value G) for the optimal solution №4.
The quality improvement based on process parameter optimization is the cheapest way to utilize the available equipment and materials. The estimated models applying the statistical approach can be utilized for fulfilling that task. In Table 19 the optimal process parameters for obtaining maximum (minimum) of the performance characteristics at EBW of SSt are determined.
Figure 77. The overall desirability function (zo=226 mm, zp=176 mm) (SSt)
Table 18. Optimal solutions – Pareto-optimal and desirability function P, kW v, cm/min zo, mm zp, mm H B G 1 8.21 52.86 253.88 255.90 25.92 1.19 - (Pareto) 2 7.36 73.91 201.04 234.55 42.01 2.97 - (Pareto) 3 6.35 57.03 270.30 253.03 35.28 1.93 - (Pareto) 4 6.30 29.00 226.00 176.00 30.00 2.00 0.9912 5 8.40 35.00 216.00 126.00 29.98 2.00 0.9848 6 5.88 26.00 186.00 146.00 29.97 2.00 0.9767
Process Parameter Optimization and Quality Improvement at Electron Beam Welding 161 Table 19. Optimal process parameters for maximum/minimum of
the performance characteristics at EBW of SSt P, kW V,
cm/min dZ, mm
H, mm B, mm S, mm2 T
Stainless steel
Hmax 8.106 20.0 -78.0 40.5659 2.6014 107.8733 0.3371 Bmin 4.200 80.0 -15.0 16.3507 0.8649 16.9844 0.3658 Smax 8.400 20.0 62.0 34.3465 4.3710 152.2110 0.4420
max 5.775 31.1 24.9 21.4486 3.0877 59.7869 0.5287
Figure 78. Desirability function G (2D- and 3D-view) at EBW of steel 45, dZ = 55.3 mm
On Figure 78 the desirability function is calculated for the fusion zone depth and width at EBW of St45. The required values for the weld geometry parameters are: HW=22.5 mm, BW=3.5 mm with tolerances: H in the region [2025 mm], B – [2.54.5 mm]. The maximum value of Gmax=0.9442, for P=3.4675 kW, v=1.0000 cm/s and dz=55.3 mm.
The trained neural networks can also be implemented for prediction of the considered performance characteristics over the experimental region and their individual optimization (for the H and T – maximum and for B - minimum) at EBW of stainless steels. In Table 16 are presented the optimal results, the corresponding optimal process parameter values and the values of the rest two performance characteristics predicted at the same EBW process conditions. It can be seen that the most deep welds do not coincide with the regimes with maximum thermal efficiency, the minimum width of the welds is obtained for weld depths about 25 mm, the maximum thermal efficiency is reached at regime conditions at which the focus position is 150 mm above the sample surface and the welds are comparatively wide and shallow.
In Figure 79 is presented a contour plot of the thermal efficiency, depending on the distances to the beam focus and to the sample surface (z0 and zp), at optimal values of the beam power Р = 7.14 kW and the welding velocity v = 20 cm/min, at which the maximum thermal efficiency is reached (Table 20). It can be seen that values above 0.5 (50%) are reached at focus positions considerably below the sample surface. Figure 80 shows the corresponding (the same process parameters P and v) contour plots of the weld depth and mean width. At these conditions the most deep and narrow welds are obtained for small distances to the sample surface and focus positions a below its surface. Since the optimal
solutions for each performance characteristic are different, a compromise solution must be found, fulfilling the requirements for all the characteristics at the same time.
Table 20. Optimal regimes and weld quality performance characteristics (SSt) P, kW v, cm/min zo, mm zp, mm H, mm B, mm T
Hmax 8.40 20 196 126 45.69 2.60 0.356
Bmin 8.40 74 266 126 24.69 1.00 0.266
T, max 7.14 20 176 326 12.38 5.27 0.687
Figure 79. Contour plot of the thermal efficiency, depending on the distances z0 and zp, at values of Р = 7.14 kW and v = 20 cm/min (SSt)
Figure 80. Contour plot of the weld depth (solid lines) and the weld mean width (dashed lines), depending on the distances z0 and zp, at Р = 7.14 kW and v = 20 cm/min (SSt)
Process Parameter Optimization and Quality Improvement at Electron Beam Welding 163 The optimization task in the case of quality improvement at production conditions based on robust engineering approach is defined as variance minimization, while the weld geometry parameters are kept on the required values. If we want to obtain a weld depth of H=20 mm (with 2% tolerance), the parameter regime with the lowest variance sˆ2Hmin=0.1649 in production is: Р=7.77 kW, V=13.333 mm/s, dz=-8 mm. The estimated value of the mean for the depth is~y
H= 20.2251 mm. If the target value for the width is В=2.5 mm (with 5%
tolerance), the regime with the minimum variance sˆ2Bmin=0.1649 is obtained for: Р=6.93 kW, V=3.333 mm/s, dz=-78 mm. The calculated value for the width is then~y
B=2.4887 mm. A simultaneous optimization of the weld width and depth is done for the same target values for H=20 mm and for B=2.5 mm and the regime with a minimum variance at which these values are obtained is: P=7.35 kW, v=8.333 mm/s, dz=27 mm. This is a compromise solution in favor of both the weld depth and weld width. The values of the compromise variances and the corresponding estimated values of B and H are: sˆ2BminC= 0.16519, ~y
B=2.509 mm, sˆ2HminC= 7.0979, ~y
H=19.621 mm.
C
ONCLUSIONThe results of calculations using steady state models (namely moving linear heat source) can be used for rough (initial) technology parameter choice. One can apply this model at admission of the known value of the width or the depth of the weld as well as at prognosis of the both values: the width and the depth of the weld as a pair at calculating its values on the basis of known welding and material characteristics. But such estimation has a big disadvantage due to not taking in the account the position of the beam focus relatively to the sample surface (or the beam focusing current changes and the variations of the distance gun-sample). The beam physical parameters (radial and angular distributions or the beam emittance) are not included too.
The proposed statistical approach gives more deep knowledge of the process characteristics influence on the weld geometry parameters. The region of application of created models is limited to studied material and EBW machine due to nature of the quantitative information obtained. It is appropriate for computer expert systems for EBW operator or technologist advice as well as for CNC systems and for computer optimization of results of EBW applications in the laboratories, at workshop services and mass production in the industry.
The functional elements of the developed expert system for electron beam weld characterization and parameter optimization, which gives the possibility for fulfilling various modeling and optimization tasks, are reviewed. This tool can be upgraded with new experimental data and now incorporates the accumulated knowledge for EBW of stainless steel and steel 45.
The tool integrates several options for:
Design of experiment for obtaining objective information on the influence of material and process parameters on EBW with minimum number of experiments.
Estimation of models. This permits to find acceptable regions for the EBW process; to estimate the significance parameters and to understand the interactions between the factors.
Process parameter choice at various requirements and conditions (defects, desirability function, robust engineering at industrial production processes etc.)
Multi-criteria parameter optimization - compromise Pareto-optimal regimes (for example maximum H and minimum B).
R
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