Noise level | ||||
0.010 | 0.015 | 0.039 | 0.063 | |
0.031 | 0.033 | 0.051 | 0.07 | |
0.032 | 0.043 | 0.066 | 0.117 |
We introduce a new algorithm to solve the problem of detecting unknown cavities immersed in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the cavities are inclusions and the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse inclusion Cauchy-Stokes problem is ill-posed for both the cavities and missing data reconstructions, and designing stable and efficient algorithms is not straightforward. We reformulate the problem as a three-player Nash game. Thanks to an identifiability result derived for the Cauchy-Stokes inclusion problem, it is enough to set up two Stokes boundary value problems, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the inclusion detection. We used a level-set approach to get rid of the tricky control dependence of functional spaces, and we provided the third player with the level-set function as strategy, with a cost functional of Kohn-Vogelius type. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for three different test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy problems.
Citation: |
Figure 1.
An example of the geometric configuration of the problem : the whole domain including cavities is denoted by
Figure 5.
Test case A. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over
Figure 6.
Test case A. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over
Figure 7.
Test case A. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over
Figure 8.
Test case B. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over
Figure 9.
Test case B. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over
Figure 10.
Test case C. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over
Figure 11.
Test case C. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over
Figure 3.
Test case A. (Left) sensitivity of the reconstruction w.r.t. the mesh size (on abscissae : the number of F.E. nodes on the boundary
Table 1.
Test-case A.
Noise level | ||||
0.010 | 0.015 | 0.039 | 0.063 | |
0.031 | 0.033 | 0.051 | 0.07 | |
0.032 | 0.043 | 0.066 | 0.117 |
Table 2.
Test-case C.
Noise level | ||||
0.042 | 0.044 | 0.046 | 0.08 | |
0.095 | 0.1 | 0.13 | 0.16 | |
0.099 | 0.11 | 0.13 | 0.15 |
Table 3. Relative errors on the reconstructed missing data and inclusion shape for the Stokes problem (with noise free measurements), compared for a classical Nash algorithm and Algorithm 2: (left) test-case A (right) test-case C
Case A | Classical algorithm | Algorithm 2 |
0.058 | 0.033 | |
0.106 | 0.032 | |
0.358 | 0.140 | |
Case C | Classical algorithm | Algorithm 2 |
|
0.067 | 0.058 |
|
0.208 | 0.122 |
|
0.566 | 0.167 |
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