## 4.1. Residual stress analysis

Figure 14 shows the residual stress results of different welding sequences after welding. They are highly similar in stress maximum and distribution areas. Residual stress is mainly concentrated in the weld zone, and there are only small differences in other local areas. The peak stresses of different welding sequences are all between 240 MPa and 244 MPa. Besides, the residual stress of case 1 is the highest, which is 244.00 MPa. The case 3, which is 240.11 MPa, is the lowest among all schemes. In addition, it is obvious that the peak stress point of each scheme occurs at the edge of the air inlet, namely the arc closure point.

The average residual stress of each weld, as seen in Fig. 15, has great differences in different welding sequences. The average residual stress in the first weld, which is close to 85 MPa, is higher than all welds and in the twelfth weld is close to 72 MPa for different welding sequences. The reason for the large stress difference between the first and last welds is that in all welding sequences, the first weld is completed first and the twelfth weld is completed last. Because the two welds are neighbouring, the first weld undergoes more thermal cycles than the twelfth weld, resulting in a higher residual stress in the first weld.

Figure 15a plots the average stress of each weld under the three welding sequences including case1, case 2 and case 3. For case 1, namely skip welding with zero degree, the average stress of each weld decreases steadily, making the stress difference between adjacent welds smaller. For the skip welding of 60 degrees with a single weld (case 2), the average stress curve of the weld has a sawtooth form and the stress variation between neighboring welds is substantial. Additionally, it is discovered that the even-numbered welds completed later than the odd-numbered welds have higher residual stresses. For the symmetrical welding of single weld (case 3), the average residual stress declines linearly from the first to the sixth welds, but it abruptly jumped to 84 MPa in the seventh weld, which is comparable to the stress in the first weld, before progressively declining to 72.3 MPa at the end. It is also discovered that the stresses in the symmetrical welds are extremely near to one another. For instance, the second weld's average stress is 81.6 MPa, and the eighth weld's is 80.6 MPa; the average stresses of the third and ninth welds are 77.7 MPa and 76.8 MPa, respectively.

Figure 16b shows the average stress of each weld from case 4 to case 6. For the skip welding of 90 degrees with two adjacent welds (case 4), the mean stress curve demonstrates a quasi-periodic fluctuation with three periods, and each period consists of four weld seams. The greatest residual stress fell from 84.8MPa in the first cycle to 82.4MPa in the second cycle, and then to 79.9 MPa in the third cycle. The average stress of the eleventh weld in this process is 72.1 MPa. For the symmetrical welding of two adjacent welds (case 5), it is obvious that the weld's average stress curve also follows a quasi-periodic law. The stress curve is separated into three phases in a single period, i.e. six welds. The two neighboring continuous welded seams constitute a phase in two cycles. Taking a the first period as a example, the first phase includes the first and second welds, the second phase is consisted of the third and fourth welds. In addition, the fifth weld, which has a value of 71.5MPa, is the one with the lowest average residual stress. For the symmetrical welding of three adjacent welds (case6), the green curve reveals that the average residual stress of the weld seam performed first is greater than that of the weld finished later. For example, the first, second, and third welds all have higher average mean stress than that of the fourth, fifth, and sixth. The mean stresses of the tenth, eleventh and twelfth welds are very close, and the eleventh weld has the lowest value, which is 72.5 MPa.

The column graph depicts the stress differential between neighboring welds for various welding sequences, as illustrated in Fig. 16. These data are calculated by subtracting the stress values for the lower weld numbers by those for the larger weld numbers, and then taking the absolute value of the difference. For example, the parameter "1|2" represents the value of the second weld less the value of the first weld.The abscissa denotes the number of adjacent weld seams.

Excessive stress difference causes unequal stress distribution on the lobe mixer, which will affect the stability of the whole component. The histogram clearly illustrates that the welding sequence has a significant impact on the residual stress of nearby welds. The first set of data, weld '1|2 ', demonstrates that the residual stress differential between the first and second welds is rather considerable in the skip welding of 60 degrees with a single weld (case 2), reaching 11.38MPa, while stress differential below the symmetrical welding of three adjacent welds (case6) is the smallest, at only 1.61MPa. Other processes have a stress differential of less than 3.5 MPa. For the weld '2|3 ', it is noticeable that the stress differential in the skip welding of 90 degrees with two adjacent welds (case 4) is the greatest, reaching 10.13MPa, followed by 9.96 MPa in case 2, and the process with skip welding with zero degree (case 1)has the smallest value with 2.15 MPa. For the weld '3|4 ', case 2 has the greatest stress differential, at 10.81 MPa, followed by case 6, which is 7.88 MPa. Case 4 has the smallest stress differential, at just 0.46 MPa. For the weld '4|5 ', case 2 and case 4 welding sequences cause a stress difference with 9.48MPa and 10.03MPa, respectively. In addition, the stress difference of case 6 is the smallest, only 0.18 MPa. For the weld '5|6', only case 2 results in a stress difference of 9.62 MPa, while the others are less than 2 MPa. For the weld '6|7', except for case 1, where the stress difference is 0.64 MPa, the other welding processes cause the stress differential to exceed 8MPa, with the greatest value reaching 12.93 MPa of case 3. For the weld '7|8', the stress difference is only 0.04MPa when using the welding sequence of case 4, and case2 has the largest stress difference, which is 8.37MPa. For the weld '8|9', case 4 has the highest stress difference at 7.33 MPa, followed by case 2 at 7.00 MPa. Case 1 has the smallest value at only 0.84MPa. For the weld '9|10', case 1 and case 5 (symmetrical welding of two adjacent welds) have extremely comparable stress differences, both approximately 1MPa. Additionally, case 2 has the largest stress difference with 7.2 MPa. For the weld '10|11', the stress difference is just 0.17MPa when using the case6 welding sequence, and the maximum is 4.7MPa when using the case2 welding sequence. For the weld '11|12', the largest stress difference is 4.15MPa in Case 2, whereas the stress differences in the other five different welding sequences are all less than 1.5MPa, which in the minimum is only 0.2 MPa. For the weld '11|1', the stress difference is all around 12 MPa under six different welding sequences. This is because the average stress of the first weld is quite high, surpassing 84 MPa, but that of the final weld, i.e. the twelfth weld, is lower, both below 73.5 MPa, which makes the stress difference of these two weld seams more significant.

The blue histogram displays the variance of average stress of the twelve welds in different welding sequences, as detailed in Fig. 17. As can be noticed, adopting the welding sequence of case 1 has the smallest variance of the weld stress, at 14.47, followed by case 6 with 22.08, while case 5 has the largest variance with 22.38. In addition, the variances of case 2, case 3, and case 4 with the welding sequence are 21.53, 21.88, and 22.04, respectively. The preceding data indicates that the welding sequence of case 1 can improve the uniformity of the weld stress distribution on the lobe mixer.

## 4.2. Deformation analysis

Figure 18 displays the deformation contour for different welding sequences after welding. It can be seen that the deformed regions are mainly concentrated near the external ducts and the entrance of inner ducts for all welding processes. Besides, the maximum deformations are close, which them are all around 0.6mm. Case 3 is the smallest with 0.54 mm, and the largest is case 1 with 0.62 mm. The maximum deformation of case 2, 4, 5 and 6 is 0.57mm, 0.58mm, 0.55 mm and 0.56 mm, respectively. Therefore, it indicates that symmetrical welding of two or three adjacent welds has little effect on the maximum deformation. But they are inferior to the single weld symmetric welding in terms of controlling deformation. It can be also concluded that symmetric welding is better than skip welding.

The average deformations for different weld seams are calculated and illustrated in Fig. 19. It can be seen the welding sequence can significantly affect the average deformation of each weld from the fluctuation range of the curves.

Figure 19a describes the average deformation of each weld for the welding sequences of case 1, case 2 and case 3. For the skip welding with zero degree (case 1), the average deformation of the weld grows initially, then reduces, and then increases from the first to the twelfth weld, which is completely different from the variation law of its stress. Additionally, the third weld has the greatest average distortion, measuring 0.31 mm, while the tenth weld has the least, measuring just 0.12 mm. It is worth noting that the first and twelfth welds have the same average deformation of 0.28 mm. For the skip welding of 60 degrees with a single weld (case 2), the mean deformation curve is noticeably jagged. The fourth weld has the largest average deformation of 0.26 mm, followed by the fifth weld of 0.25 mm, and the ninth weld with the smallest of 0.14mm. For the symmetrical welding of single weld (case 3), the dark blue curve clearly shows that the average distortion of all welds is about 0.2 mm. The maximum distortion is 0.23 mm of the six and twelfth weld seams, and the minimum is 0.18 mm of tenth and eleventh welds.

Figure 19b depicts the average deformation of each weld for the welding sequences of case 4, case 5 and case 6. For the skip welding of 90 degrees with two adjacent welds (case 4), the entire curve has a hump consisting of the third and fourth welds, and their average deformations are both 0.28 mm. Moreover, the average distortion of the remaining welds is less than 0.25 mm, with the ninth weld having the least deformation of 0.14 mm.For the symmetrical welding of two adjacent welds (case 5) and the symmetrical welding of three adjacent welds (case6), the average deformation curves’ evolution law is similar in the two welding sequences, which is comparable to the mean stress curve variation law depicted in Fig. 15(b). The curves also are smoother than in other welding sequences, and all welds had an average distortion of 0.17 to 0.24 mm.

The larger fluctuation, the greater difference of each weld is, which is not conducive to ensuring dimensional accuracy of lobe mixer. To directly reflect the degree of fluctuation of each curve, the variances of average deformation in different schemes are calculated, as shown in Fig. 20a. It is clearly that the average deformation variance is arranged from large to small in the following order: case 1 > case 4 > case 2 > case 6 > case 5 > case 3. Among these, case1 has a variance of 38.91×10− 4, which is much larger than the average deformation variance of other welding sequences. In addition, with this welding sequence, the maximum deformation of the lobe mixer reached 0.62 mm.It is therby inferred that adopting the welding sequence of a skip welding with zero degree is unsuccessful for regulating the dimensional accuracy of the complete lobe mixer. Furthermore, using the welding sequence of the symmetrical welding of single weld (case 3) has the smallest variance with 3.31×10− 4 and peak deformation at 0.54 mm, which indicates that the deformation of the lobe mixer can be effectively controlled by using this.

The lobe mixer's gas input must be coupled to other components, the distortion requirements thus are considerable. Therefore, the maximum deformation at the entrance of the lobe mixer is measured as depicted in Fig. 20b. The peak deformation at the entrance of case 1 is the largest reaching 0.536mm, followed by case 4 with 0.47mm. Under the welding sequences of cases 5 and 6, the highest distortion of the entrance is 0.45 and 0.46 mm, respectively. However, the maximum distortions, which are 0.42mm, are the same when utilizing the welding sequence of cases 2 and 3.