where 0 < r < 1; 0 < < !; =
!; ! is the maximun re-entrant angle.When is solved by piecewise linear nite element method on a quasi-uniform grid, the error is shown in the following:
4:2a 4:2b
4:2c 4:2d
Figure4.2:Error with four kinds of angle
Table 4.1 Results for four kinds of angle
! = 2 ! = 3
2 ! = 7
4 ! =
jju uhjjH1( ) 2.6377E-02 4.1243E-02 7.0759E-02 9.6003E-020:51
# of points 2205 6469 7791 8323
we can see the Fig4.2a is not a re-entrant angle , so the error of Fig4.2a didn't have peak at the origin,but the others have.We also can nd when the maximun re-entrant angle ! arise, the error also arise with the !:
The second experiment we will concern about the mesh-re nement strategy.There are three different mesh-re nement strategy we used. The rst strategy is uniform mesh
4 Numerical Results 51
with uniformly re nement,the second is singular element mesh with uniformly re nement and the last is uniform mesh with adaptive short cut-region re nement.
The rst mesh-re nement strategy is like as we re ne mesh as usual.The second strategy , we changed the uniform mesh to the singular mesh. According to our mentioned in the singular element ,here the approach is like singular element,we put the exponential grid points (x; y) in the way
(x; y) = (2 i; 0)
(x; y) = (2 icos !; 2 isin !) ; i = 1; :::; 10 (4.109) on the two rays.This may couse many grid points located near the singular point.The third strategy is using uniform mesh, but not re ne the whole domain, only re ne the region with radius r = 0:5j form the center points (0; 0) , where j represent the times of re nement.
We consider the maximun re-entrant angle ! =
0:51 case with three different mesh-re nement strategys .The times of mesh-mesh-re nement steps is 2, means that we have the original corasest domain 4h; nd domain 2hand h:
Type A Type B Type C
Figure4.3: Error of three types of mesh-re nement we caculate the H1-norm error and L2-norm in the following table
Table4.1
Type A Type B Type C
jju uhjjH1( ) 9.6003E-02 2.9259E-02 1.3185E-01 jju uhjjL2( ) 3.1643E-03 2.3718E-04 4.9345E-03
# of points 8323 15085 1373
4 Numerical Results 52
From the Table4.1, we discuss some phenomena. In the second strategy, there are many points around of center.When we re ne the mesh, the points around the center of all will be re ned.Although we reduced the error, but we also pay a high price since too many points cause the large matrix systems.In the third strategy, we only re ne the points inside the cut-region, the error outside the cut-region still not be reduced, therefore, the accuracy performance is not good. The convergence rate for type A and type B in the H1 normis therefore of order O(h( =!) ):(i:e:the theoratical convergence rate is 1.42)
Table4.2
Type A ratio A Type B ratio B Type C ratio C
4h 2.3184E-01 1.1065E-01 2.1384E-01
2h 1.4147E-01 1.5116 5.6813E-02 1.9476 1.6135E-01 1.3253
h 9.6003E-02 1.4737 2.9259E-02 1.9417 1.3185E-01 1.2237
Simultaneously, we apply the multigrid method for solving the linear systems and observed the bene ts for three dif erent mesh-re nement. For the multigrid parameter, we apply V-cycle 2th level iteration and we choose the weight-Jacobi method to relaxation, the max-imum relaxation number is 2.For the restriction operator ,we use full weighting as a restric-tion operator,and for the interpolarestric-tion operator, we consider linear interpolarestric-tion.
Type A Type B Type C
Figure4.4: Multigrid bene ts for three different mesh-re nement
From the Figure4.4 shown in above,we can observe that although the mesh size is dif erent, the slope almost not changed,meens that the converge rate of multigrid method is inden-pentent of mesh size.
4 Numerical Results 53
The third experiment we use the method with S.C.Brenner and L.-Y.Sung.The singu-lar function for is
s(r; ) = (r)r =0:51sin(
0:51 ) (4.110)
where the cut-off function (r) is de ned to be 8< The kare obtained by the extraction formula
k = 1 Z
Here,for the approximate stress intensity factors k,we choose 3-th level (l = 3) and itera-tion 5 times (m = 5)
The error is shown in the following
4h 2h h
Figure4.5: Error of three level mesh The theoretical number for the stress intensity factorys is 1.
Table4.3
jju uhjjH1( ) # of points
4h 9.9399E-01 2.3664E-01 556
2h 9.9815E-01 1.5423E-01 2128
h 9.9944E-01 1.0451E-01 8323
The fourth experiment we improve the method by including adaptive mesh-re nement techniques and adaptive cut-off function.The following table shown the stress intensity
fac-4 Numerical Results 54
torys and the error in H1 norm:
Table4.4
jju uhjjH1( ) # of points 1 9.9399E-01 2.3664E-01 556 2 1.0017E+01 2.0107E-01 913 3 9.9098E-01 1.7432E-01 1359 4 9.8648E-01 1.4946E-01 1842 5 9.8810E-01 1.3034E-01 2196 6 9.8722E-01 1.2721E-01 2747 7 9.8841E-01 1.1381E-01 3207 8 9.9204E-01 1.0953E-01 3533 9 9.9492E-01 1.0199E-01 4522 10 9.9341E-01 9.8286E-02 5494 11 9.9306E-01 9.7071E-02 6178 12 9.9560E-01 9.5075E-02 7217 13 9.9534E-01 9.5117E-02 7708 14 9.9596E-01 9.3322E-02 9533 15 9.9626E-01 9.3082E-02 10568
we can clearly compare the jju uhjjH1( ) with these two results above.Fixed points on both sides of almost equal,we can clearly see the improvement in error in H1 norm:
Table4.5
S.C.Brenner Experiment
jju uhjjH1( ) # of points jju uhjjH1( ) # of points
1 2.3664E-01 556 1 2.3664E-01 556
2 1.5423E-01 2128 4 1.4946E-01 1842
3 1.0451E-01 8323 13 9.5117E-02 7708
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