2.6 Application to Aircraft Structures
51
Fig. 2.25 Results for the fourth (top) layer of Fig. 2.24, when cells in only the fourth layer are
constrained
When we force the appropriate cells of only the fourth (top) layer to be zero,
we get the results shown in Fig. 2.25 for the fourth layer. There is a significant
improvement compared to Fig. 2.24.
If, however, we constrain the appropriate cells in all of the layers, then we get the
dramatic improvement shown in Fig. 2.26.
Even though the results are excellent, there remains an interesting anomaly
in them. Note the twelve circled cells; their conductivities are too low. This is
a manifestation of the failure of the ‘two-cell hypothesis,’ because two of the
four nearest neighbor cells have perfect zero anomalous conductivities. Hence, the
hypothesis must fail in 50% of the cases for these cells. Nevertheless, this anomaly
is easily recognized, and easily cured. We simply redo the two-cell hypothesis
using the two nearest neighbors that have nonzero conductivities, and recalculate
the conductivities using the robust estimator. When we do this we get excellent
agreement with the true value for these cells.
Problem No. 4: Another Layer-Stripping Example Using Multifrequencies
This problem concerns a buried flaw in an aluminum slab. The anomalous region
is again modeled as a two-layer system, as in Fig. 2.17, but the top layer is simply
host material (anomalous conductivity of zero for each cell). The bottom layer is
identical to the bottom layer of Fig. 2.15.
Because we do not know a priori that the top layer is host material, we run the
problem first at 85 kHz, in order to reconstruct the top layer independently of the
bottom, as in Problem No. 2. This is done by forcing the bottom cells to have zero
anomalous conductivity, as in Fig. 2.27.
The reconstruction of the top layer produces excellent zeros for all of the cells,
as we would hope. This is not a violation of the heuristic rule that was defined in
51
Fig. 2.25 Results for the fourth (top) layer of Fig. 2.24, when cells in only the fourth layer are
constrained
When we force the appropriate cells of only the fourth (top) layer to be zero,
we get the results shown in Fig. 2.25 for the fourth layer. There is a significant
improvement compared to Fig. 2.24.
If, however, we constrain the appropriate cells in all of the layers, then we get the
dramatic improvement shown in Fig. 2.26.
Even though the results are excellent, there remains an interesting anomaly
in them. Note the twelve circled cells; their conductivities are too low. This is
a manifestation of the failure of the ‘two-cell hypothesis,’ because two of the
four nearest neighbor cells have perfect zero anomalous conductivities. Hence, the
hypothesis must fail in 50% of the cases for these cells. Nevertheless, this anomaly
is easily recognized, and easily cured. We simply redo the two-cell hypothesis
using the two nearest neighbors that have nonzero conductivities, and recalculate
the conductivities using the robust estimator. When we do this we get excellent
agreement with the true value for these cells.
Problem No. 4: Another Layer-Stripping Example Using Multifrequencies
This problem concerns a buried flaw in an aluminum slab. The anomalous region
is again modeled as a two-layer system, as in Fig. 2.17, but the top layer is simply
host material (anomalous conductivity of zero for each cell). The bottom layer is
identical to the bottom layer of Fig. 2.15.
Because we do not know a priori that the top layer is host material, we run the
problem first at 85 kHz, in order to reconstruct the top layer independently of the
bottom, as in Problem No. 2. This is done by forcing the bottom cells to have zero
anomalous conductivity, as in Fig. 2.27.
The reconstruction of the top layer produces excellent zeros for all of the cells,
as we would hope. This is not a violation of the heuristic rule that was defined in