PSI - Issue 5

Etienne Bonnaud et al. / Procedia Structural Integrity 5 (2017) 310–317 Bonnaud / Structural Integrity Procedia 00 (2017) 0 0 – 000

5

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When welding the second bead of the other inner layer, the structure is already much stiffer and additional displacement therefore much lower. Accordingly, displacement for bead 4 welded after bead 3 (on the lower blue curve of Fig. 5) is not even +1mm. This is precisely what misleads the surrogate algorithm: the same pair (in this example 3-4) can either contribute a lot or very little to displacements, depending on the position in the sequence. A way to correct this difficulty is to store two different displacement values for certain combinations and to use the appropriate one in the optimization algorithm. Table 3 shows the new selected runs and Table 4, the new pair matrix. Not surprisingly, the selected run matrix now contains a few more runs (9 instead of 6). In the pair matrix, 4 positions now have two values: the first value corresponds to a small displacement, the second values to a large displacement (namely at closure of an inner layer). For illustration, the (3-4) layer has been highlighted in yellow and green: the displacement values stored from Run 3 and Run 9 are large whereas the displacement values stored from Run 6 is small. Figures 6a-b show a comparison between computed sequences and actual simulations for both maximal and minimal displacements. The best sequence (giving the smallest displacement) turns out to be 1-3-4-2-5-6-7-8 and the worst sequence (giving the largest displacement), 1-3-2-6-4-5-8-7. Agreement is rather good meaning that the suggested enhancement of the surrogate model works satisfactorily. Note that the minimal final value lies around 1.5mm; large displacement occurring at the beginning of a sequence are seldom recovered by subsequent displacements. This feature has been mentioned earlier, see Bonnaud et al. (2014) for example.

Table 3: Selected runs for plates (modified).

Table 4: Pair matrix for plates (modified).

1 1 1 1 1 1 1 1 1

2 4 3 3 4 2 2 4 3

5 2 4 2 3 5 3 3 4

4 6 7 4 8 3 5 2 2

3 3 2 5 2 4 6 7 8

6 7 8 7 7 6 4 6 7

8 5 5 8 6 7 8 8 6

7 8 6 6 5 8 7 5 5

1 0 0

2 3 0

3 3 1

4 3 1 0 1 1 0 0

5 0 2 1 1 0 2 1 2

6 0 1 1 1 2 0 3 1

7 0 2 1 1 1 1 0 3

8 0 2 1 1 1 2 2 0

Run 1 Run 2 Run 3 Run 4 Run 5 Run 6 Run 7 Run 8 Run 9

1 2 3 4 5 6 7 8

0 1/1

0 1/2

0 1/1 1/2

0 0 0 0

0 0 1 1

1 1 0 0

(a)

(b)

Fig. 6. Computed run rendering the largest (a) and smallest (b) vertical displacement. Comparison between estimate and simulation.

5. Pipe welding The geometry studied here consists of two long cylinders welded together, see Fig. 7. The internal radius is R = 100mm and the wall thickness t = 10mm which gives a R/t ratio of 10 and a characteristic disturbance length √ . of 32mm. The cylinders are 400mm long which is more than 10 times the disturbance length. The X-weld consists of eight 2mm thick beads, four on each side. The model is subjected to axisymmetric boundary conditions with one node at the bottom of the bottom pipe locked in the vertical direction to prevent rigid body motion. Axisymmetry is actually questionable as it can be shown that it only prevails away from the welding start/stop position, see Bonnaud et al. (2016). For the current purpose, it is nevertheless fully acceptable.