3.1 Fragmented support volume (FSV)
Results of the ANOVA test at a 95% confidence interval are presented in Table 3. The results showed that the selected fragmented support parameters along with their interactions have significant effect on the support volume. According to the sum of squares the fragmented separation width (Fsw.) has the most significant effect on the support volume.
Table 3 ANOVA for reduced quadratic model of FSV
Source
|
Sum of Squares
|
df
|
Mean Square
|
F-value
|
p-value
|
Model
|
4.977E+06
|
8
|
6.221E+05
|
15627.09
|
< 0.0001
|
A-Th
|
5.780E+05
|
1
|
5.780E+05
|
14518.25
|
< 0.0001
|
B-Tbi
|
17848.26
|
1
|
17848.26
|
448.33
|
< 0.0001
|
C-Fsw
|
4.248E+06
|
1
|
4.248E+06
|
1.067E+05
|
< 0.0001
|
AB
|
6050.00
|
1
|
6050.00
|
151.97
|
< 0.0001
|
AC
|
87362.00
|
1
|
87362.00
|
2194.42
|
< 0.0001
|
BC
|
3528.00
|
1
|
3528.00
|
88.62
|
< 0.0001
|
B²
|
248.50
|
1
|
248.50
|
6.24
|
0.0166
|
C²
|
9363.19
|
1
|
9363.19
|
235.19
|
< 0.0001
|
Residual
|
1632.25
|
41
|
39.81
|
|
|
Lack of Fit
|
1632.25
|
6
|
272.04
|
|
|
Pure Error
|
0.0000
|
35
|
0.0000
|
|
|
Cor Total
|
4.979E+06
|
49
|
|
|
|
Figure 8 shows the effect of fragmented support parameters on the fragmented support volume. It was found that as the support tooth height, tooth base interval, and separation width increases, the support volume decreases. The decrease is significant in case of separation width as compared to other parameters. This is becasuse as the fragmented separation width increases, more material is removed from borders and hatches of the support, resulting in lesser support structures and hence the lower support volume as shown in Table 2. The low support volume was achieved with 4 mm Th, 4mm Tbi, and 2.5 mm Fsw.
Figure 9 presents the contours plots showing the interaction of fragmented support parameters on the support volume. The contour plot of tooth height and tooth base interval indicate slight interaction and that at lower levels of tooth height the support volume is higher for all levels of tooth base interval. This is because at lower tooth height the perforation at top of the support is small and in the selected range the tooth base interval does not have any significant change in the perforation at low tooth height. The contour plots indicate that the minimum fragmented support volume could be achieved with higher levels of tooth height (factor Th = 4mm), tooth base interval (Tbi = 4 mm) and fragmented separation width (Fws = 2.5 mm).
The mathematical model of the FSV was developed using the response surface methodology (RSM), given by equation 1.1.
FSV =1970.67876 - 123.89379 Th - 28.87730 Tbi - 610.36769 Fsw - 6.11111Th * Tbi + 34.83333 Th * Fsw + 7.00000 Tbi * Fsw +3.67611 Tbi² + 50.77126 Fsw² (1.1)
The developed predictive model has high R-Square of 99.95% which indicate that the parameters included explain 99.95% of variation in support volume. Therefore, it can be deduced that the developed model is adequate and the predictive model can be utilized for predicting the fragmented support volume. The developed predictive models were confirmed for their adequacy using a two-sample t-test at a 95% confidence interval. It has been found that P-value for the fitted model is greater than α (0.05). This suggests that null hypotheses H0: μ1= μ2 cannot be rejected i.e., means for actual measured responses and fitted responses are almost equal. A comparison of the actual measured results with predicted ones is shown in Figure 10.
Numerical optimization was then carried out in order to find the optimal parameters minimizing the FSV using desirability approach. The optimization results show several optimal solutions with various desirability values. Table 4 presents five solutions (out of 94), along with their respective desirability values. Figure 11 shows the selected solution by the software. Based on the practical feasibility, the selected solution is acceptable.
Table 4 Samples of optimum solutions for fragmented support volume
#
|
Th
|
Tbi
|
Fsw
|
FSV
|
Desirability
|
1
|
3.999
|
3.663
|
2.490
|
535.002
|
1.000
|
2
|
3.925
|
3.994
|
2.495
|
535.982
|
1.000
|
3
|
3.999
|
3.388
|
2.500
|
535.763
|
1.000
|
4
|
4.000
|
4.000
|
2.500
|
530.369
|
1.000
|
5
|
3.987
|
3.845
|
2.482
|
535.730
|
1.000
|
The variation of desirability function with tooth height, tooth base interval, and fragmented separation width is shown in Figure 12. It can be seen that the desirability is higher at higher level of selected parameters.
3.2. Fragmented support removal time (FSRT)
ANOVA test was performed in order to evaluate the effect of support parameters on the support removal time. To achieve the normality assumption, transformation (Base 10 Log) was used. It can seen from the Table 5 that most of the selected terms (A, B, C, D, E, BD, DE, B²) have a significant effect on the FSRT. The main effect plots of the fragmented support removal time are shown in Figure 13. It can be seen that the support removal time decreases with increase in the tooth height, fragmentation separation width, this is because the increase in the tooth height and fragmentation separation width increases the accessibility of the cutting tool (wire cutter). In addition, with increased fragmented separation width the supported area decreases and supports are separated into smaller blocks that are easy to remove, thereby easing the support removal. In case of process parameters, the support removal time was found to have an inverse effect with regard to the beam current and scan speed. As the beam current increases the support removal time increases significantly and as the scan speed increases support removal time decreases. This is due to the fact that at higher beam current, the input energy into the supports is higher, which in turn, increases the support strength thereby increasing the support removal time. Whereas in case of scan speed, it is opposite, higher scan speed results in less input energy and hence lower strength of the supports and lower removal times.
Table 5 ANOVA for reduced quadratic model of Log FSRT
Source
|
Sum of Squares
|
df
|
Mean Square
|
F-value
|
p-value
|
Model
|
20.25
|
8
|
2.53
|
107.62
|
< 0.0001
|
A-Th
|
0.5925
|
1
|
0.5925
|
25.19
|
< 0.0001
|
B-Tbi
|
0.2390
|
1
|
0.2390
|
10.16
|
0.0020
|
C-Fsw
|
0.4476
|
1
|
0.4476
|
19.03
|
< 0.0001
|
D-Bc
|
15.61
|
1
|
15.61
|
663.45
|
< 0.0001
|
E-Bss
|
2.34
|
1
|
2.34
|
99.54
|
< 0.0001
|
BD
|
0.1793
|
1
|
0.1793
|
7.62
|
0.0071
|
DE
|
0.3557
|
1
|
0.3557
|
15.12
|
0.0002
|
B²
|
0.4896
|
1
|
0.4896
|
20.82
|
< 0.0001
|
Residual
|
1.91
|
81
|
0.0235
|
|
|
Lack of Fit
|
1.15
|
34
|
0.0339
|
2.11
|
0.0090
|
Pure Error
|
0.7542
|
47
|
0.0160
|
|
|
Cor Total
|
22.16
|
89
|
|
|
|
The interactions of the fragmented support parameters on the FSRT is shown in Figure 14. It was found that at lower levels of tooth base interval and scan speed, FSRT increases significantly with increase in beam current. This is because at lower levels of scan speed the input energy is higher for every level of beam current resulting in stronger support structures that are difficult to remove. The FSRT was found to be minimum with low beam current and that at low beam currents variation of tooth base interval and scan speed did not had any significant effect on FSRT.
The predictive model of FSRT is is given by the equation 1.2.
Log₁₀ (FSRT) = 1.27196 - 0.063166Th + 0.389343Tbi - 0.082355Fsw + 0.122791Bc - 0.000781Bss - 0.015681Tbi * Bc + 0.000083Bc * Bss - 0.074131Tbi² (1.2)
The developed model has R –Square value of 91.40%. The high R-square value confirms the model adequacy and that all factors affecting FSRT have been considered in the predictive model. T-test of 95% confidence interval was conducted between actual and predicted data and it was found that P-value for the fitted model is is greater than α value (α= 0.05). This suggests that null hypotheses H0: μ1= μ2 cannot be rejected i.e., means for actual measured responses and fitted responses are almost equal. A comparison of the measured results with predicted ones is shown in Figure15.
To find the optimal solution minimizing the FSRT, numerical optimization was then performed using desirability approach. Five of the optimal solutions (out of 100 solutions) are listed in Table 6. Figure 16 shows the selected solution based on the software suggestions, it was found to conflict with warping deformation response. The variation of desirability function with the support parameters is shown in Figure 17. The desirability was found to be high for low beam current and higher levels of tooth base interval and beam speed.
Table 6 Samples of optimum solutions for fragmented support removal time
#
|
Th
|
Tbi
|
Fsw
|
Bc
|
Bss
|
SRT
|
Desirability
|
1
|
3.468
|
3.995
|
2.499
|
1.500
|
1976.31
|
1.000
|
1
|
2
|
3.903
|
3.989
|
2.384
|
1.590
|
1996.606
|
1.000
|
1
|
3
|
3.604
|
3.977
|
2.398
|
1.503
|
1993.304
|
1.000
|
1
|
4
|
3.998
|
1.041
|
2.450
|
1.502
|
1999.891
|
1.000
|
1
|
5
|
3.844
|
1.008
|
2.457
|
1.506
|
1970.917
|
1.000
|
1
|
3.3. Warping deformation with fragmented support (WDFS)
The effects of the design and process parameters of fragmented support on the overhang warping deformation was evaluated and analyzed using the ANOVA test. Table 7 presents the ANOVA test results of the WDFS. The results show that the selected parameters along with few interaction terms have significant effect on the WDFS.
Table 7 ANOVA for Reduced cubic model of warping deformation with fragmented support
Source
|
Sum of Squares
|
df
|
Mean Square
|
F-value
|
p-value
|
Model
|
6.99
|
18
|
0.3885
|
47.82
|
< 0.0001
|
A-Th
|
0.1791
|
1
|
0.1791
|
22.05
|
< 0.0001
|
B-Tbi
|
0.1837
|
1
|
0.1837
|
22.62
|
< 0.0001
|
C-Fsw
|
0.1665
|
1
|
0.1665
|
20.49
|
< 0.0001
|
D-Bc
|
4.41
|
1
|
4.41
|
543.50
|
< 0.0001
|
E-Bss
|
0.4687
|
1
|
0.4687
|
57.70
|
< 0.0001
|
AB
|
0.0558
|
1
|
0.0558
|
6.87
|
0.0107
|
AC
|
0.0347
|
1
|
0.0347
|
4.28
|
0.0423
|
AD
|
0.0002
|
1
|
0.0002
|
0.0303
|
0.8622
|
BC
|
0.0034
|
1
|
0.0034
|
0.4181
|
0.5200
|
BD
|
0.0886
|
1
|
0.0886
|
10.90
|
0.0015
|
BE
|
0.0057
|
1
|
0.0057
|
0.7064
|
0.4035
|
CD
|
0.1269
|
1
|
0.1269
|
15.62
|
0.0002
|
CE
|
0.0047
|
1
|
0.0047
|
0.5789
|
0.4492
|
DE
|
0.3844
|
1
|
0.3844
|
47.33
|
< 0.0001
|
D²
|
0.5861
|
1
|
0.5861
|
72.15
|
< 0.0001
|
ABD
|
0.0902
|
1
|
0.0902
|
11.11
|
0.0014
|
BCD
|
0.0783
|
1
|
0.0783
|
9.64
|
0.0027
|
BCE
|
0.1206
|
1
|
0.1206
|
14.85
|
0.0003
|
Residual
|
0.5767
|
71
|
0.0081
|
|
|
Lack of Fit
|
0.5254
|
24
|
0.0219
|
20.08
|
< 0.0001
|
Pure Error
|
0.0513
|
47
|
0.0011
|
|
|
Cor Total
|
7.57
|
89
|
|
|
|
The main effect plots of the WDFS are shown in Figure 18. The WDFS increases with increase in tooth height, tooth base interval, fragemeted separation width, and scan speed. Whereas it decreases with increase in beam current. The effect is however significant in case of beam current as compared to other parameters. The increase in the tooth height, tooth base interval results in weaker contact between the support and part which inturn results in deformation. Increase in fragmented separation width increases the offset distance between the fragmented supports thereby increasing the unsupported area and resulting in increased deformation. Low beam, and high scan speed results in weaker supports due to lower input energy which in turns causes higher deformation. The contour plots showing the interactions of the fragmented support parameters on the WDFS are shown in Figure 19. The variation of deformation was found to be higher with beam current especially at high levels of fragmented separation width and tooth base interval as compared to the lower levels.The interaction of tooth height and fragmented separation width reveal insignificant variation of deformation with varying tooth height especially when the fragmented separation width is below 2mm.
The predictive model of the warping deformation achieved by the backward elimination method is is given by the equation 1.3.
WDFS = -0.577325 + 0.111608 Th + 0.261658 Tbi + 0.231394 Fsw - 0.184232 Bc + 0.000733 Bss -0.040943 Th * Tbi +0.015531 Th * Fsw - 0.017962 Th * Bc - 0.072031 Tbi * Fsw -0.014017 Tbi * Bc -0.000093 Tbi * Bss + 0.006124 Fsw * Bc - 0.000159 Fsw * Bss - 0.000086 Bc * Bss + 0.036045 Bc² + 0.007417 Th * Tbi * Bc - 0.010366 Tbi * Fsw * Bc + 0.000072 Tbi * Fsw * Bss (1.3).
The developed model has R –Square value of 92.32%, which confirms the model adequacy. T-test of 95% confidence interval was performed between the actual and predicted data. It was found that the P-value for the fitted model is greater than α value (α= 0.05) which suggests that null hypotheses H0: μ1= μ2 cannot be rejected i.e., means for actual measured responses and fitted responses are almost equal. A comparison of the actual measured results with predicted ones is shown in Figure 20.
Optimization was then carried out using desirability approach in order to find the optimal parameters minimizing the WDFS. Five of the optimal solutions (out of 100 solutions) are listed in Table 8. Based on the practical feasibility the suggested solution is applicable with acceptable values for the other responses. Figure 21 shows the selected solution based on software suggestions. The variation of desirability function with the support parameters is shown in Figure 22.
Table 8 Optimum Parameters for warping deformation with fragmented support
#
|
Th
|
Tbi
|
Fsw
|
Bc
|
Bss
|
WDFS
|
Desirability
|
1
|
2.035
|
2.165
|
0.876
|
4.958
|
1340.552
|
0.000
|
1
|
2
|
2.720
|
1.234
|
0.997
|
4.797
|
1385.761
|
0.000
|
1
|
3
|
1.407
|
1.334
|
1.071
|
5.021
|
1715.529
|
0.000
|
1
|
4
|
1.326
|
1.045
|
0.586
|
4.941
|
1806.601
|
0.000
|
1
|
5
|
1.064
|
3.069
|
1.201
|
5.231
|
1281.563
|
0.000
|
1
|
3.4. Multi-response optimization of fragmented support
The results of multi-response optimization of the fragmented support are presented in bubble charts in Figure 23 and Figure 24. The 3D bubble chart is plotted using the design points against the two objective functions i.e. fragmented support volume (FSV) and fragmented support removal time (FSRT), where one of the output variables i.e. warping deformation (WDFS) is represented by the diameter of the bubbles. Design points present in the original DOE matrix are real whereas predicted ones from RSM are virtual, also the feasible design points have a gray color whereas the unfeasible design points have a yellow color (Figure 23). As the objective of the optimization problem is to minimize the fragmented support volume (FSV) and fragmented support removal time (FSRT), the feasible design points corresponding to the lower-left corner of the bubble chart (inside the red rectangular) are the candidates for the optimal solution Figure 24.
Table 9 presents the optimum solutions; The first solution is the real and corresponds to the original DOE matrix. The optimal solution consists of high levels of parameters except for the scan speed.
Table 9 Optimal solutions mimizing fragmented support volume, and support removal time
#
|
Th
(mm)
|
Tbi
(mm)
|
Fsw
(mm)
|
Bc
(mA)
|
Bss
(mm/s)
|
FSV
(mm3)
|
FSRT
(S)
|
FSDW
(mm)
|
1
|
4
|
4
|
2.5
|
6
|
1200
|
536
|
10
|
0.094000
|
2
|
3.5
|
4
|
2.5
|
6
|
1400
|
562.3
|
12.25
|
0.091536
|
3
|
4
|
3.5
|
2.5
|
6
|
1400
|
541.17
|
12.37
|
0.095600
|
Another way to analyze the design points is to use a parallel coordinate chart as shown in Figure 25. A parallel coordinate chart can show design points with all the parameters used in the study. In summary, optimum results were found with high tooth height, high tooth base interval, high fragmented separation width, high, beam current, and low beam scan speed.
The optimal fragmented support parameters were applied to the selected bearing bracket design in order to validate the optimization. The results show that the application of optimum fragmented support structures (from multi-response optimization) resulted in the decrease in support structures volume, the un-melted powder removal time, and the support removal time at the expense of around 50 µm warping deformation as shown in Figure 26. The optimum fragmented support structures resulted in the reduction of the support volume by 36%, powder removal time by 66%, and support removal time by 72%. Therefore there is a significant reduction in material consumption, and post processing time at the expense of low level of warping deformation of around one layer (50 µm).