G. Marques et al.: Adaptive meshing in 3D multi-static problem with variable sources
193
Table 1. Meshes characteristics: number of nodes and ele- been pointed out. We have seen that a refined mesh for
ments, the average and the ripple of the error.
a given time t is not necessarily optimized for any time
t0. We have proposed a method which consists in averag-
M0
715
M1
1 987
M2
1 894 ꢁ 82
M3
1 911
M4
3 455
ing the numerical error on the whole time domain. The
mesh refinement is then carried out from these averaged
error distributions. This method leads to a refined mesh
which respects the symmetry of the problem. The error
oscillations versus time are then smoothed.
Nodes
Elements
Average
Magnitude 0.64% 1.49%
2 830
9 776 9 209 ꢁ 468 9 328 16 503
9.41% 6.79%
5.41%
0.37%
5.77% 6.72%
0.33% 0.19%
This alternative gives similar results, but with smaller
computation times than the method which consists in re-
fining an initial mesh at each instant.
At least, the proposed error estimator may be tested
on more complex devices with variable sources such as
electrical rotating machines (with thin airgap).
10.00
9.50
9.00
8.50
8.00
7.50
7.00
6.50
6.00
5.50
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1 980. For the mesh M3, at each time step, a different map
error of M0 is calculated. Then, all these errors maps are
averaged to finally compute the error map used to build
M3.
The three methods enable to improve the average er-
ror. Among these, M1 is the less accurate and leads to
important error ripples. M2 and M3 give equivalent re-
sults, but M2 requires a mesh adaptation for each time
step. Thus, the method 2 is more time consuming than
the method 3.
At least, a comparison with a non optimized mesh M4
having 3 455 nodes is done. Results, compared in Figure 15
show the effectiveness of the proposed method. With ap-
proximatively a half number of nodes, a weaker error on
the whole time domain is obtained.
5 Conclusion
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In this paper, the problem of adaptive meshing in the
case of time dependent problem in 3D magnetostatics has
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