PSI - Issue 13

Kota Kishi et al. / Procedia Structural Integrity 13 (2018) 1111–1116 Author n me / Structural Integrity Procedia 00 (2018) 000 – 000

1114 4

2 = ‖ − ‖ 2 ( ) ‖ ‖ 2 ( ) = { ∫ | − | (4) In this experiment, to examine the relationship between number of nodal degrees and relative 2 norm, the size of both meshes is decreased while the ratio ℎ G /ℎ L of the size of global mesh ( ℎ G ) to the size of local mesh ( ℎ L ) is constant. The ratio ℎ G /ℎ L is 2, 4 and 8. Fig. 4 shows displacement field obtained by this experiment. Black mesh shows global displacement field and blue mesh shows the sum of global and local displacement field. It is showed that blue mesh represents displacement accurately. Fig. 5 shows the relative 2 norm vs. number of nodal degrees. According to Fig. 5, as the number of nodal degrees of freedom increases, relative 2 norm decreases. The tendency is the same as the result of usual FEM analysis, so the result shows that displacement field is represented accurately. 2 ∫ | | 2 } 1 2

0,0005 0,0010 0,0020 0,0040 0,0080 0,0160

㻸㼛㼏㼍㼘㻌㻹㼑㼟㼔

㻳㼘㼛㼎㼍㼘㻌㻹㼑㼟㼔

Mesh size ratio = 2 Mesh size ratio = 4 Mesh size ratio = 8

Relative L2 norm

Fig. 5 Results of Relative 2 error (the ratio is fixed) 400 800 1600 3200 6400 12800 25600 Number of nodal degrees of freedom

Fig. 4 Numerical results of mesh deformation (x 5.0*10 10 )

3.3.2 Accuracy of tensile stress In order to evaluate the accuracy of tensile stress, we compare the tensile stress solved in this experiment against the exact solution. The asymptotic solution is /√2 . is stress intensity factor and is distance from the crack tip. Also, to examine effect of the global mesh size and local mesh size, firstly the local mesh size is decreased while the global mesh size is fixed, then the global mesh size is decreased while the local mesh size is fixed. Fig. 6 and Fig. 7 show the error on tensile stress vs. distance from the crack tip when the global mesh size is fixed and when the local mesh size is fixed respectively. According to Fig. 6 and Fig. 7, the local mesh size affect the accuracy very much while the global mesh size is not related to the accuracy. The result shows that the accuracy is governed by the local mesh size. In addition, the two figures show that error on tensile stress near the edge of the local area (distance from crack tip is 0.5) is large. Especially, when the ratio ℎ G /ℎ L is large, the error is also large. This shows that the ratio ℎ G /ℎ L affects the accuracy of tensile stress near the edge of the local area very much. Moreover, the value of error is less than 8 percent in the close areas from the crack tip and less than 1 percent in a little far areas from the crack tip. This result shows that mesh superposition method is applicable for static crack problems.

0,10

0,10

hG=2/23, hL=1/24 hG=2/23, hL=1/52 hG=2/23, hL=1/88

hG=2/13, hL=1/88 hG=2/35, hL=1/88 hG=2/79, hL=1/88

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Error on tensile Stress

Error on tensile Stress

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Distance from crack tip x [m]

Distance from crack tip x [m]

Fig. 6 Error of tensile stress (the global mesh size is fixed)

Fig. 7 Error of tensile stress (the local mesh size is fixed)