88
4 Application of the Set-Theoretic Algorithm to CFRP’s
the next quarter. The parameters in the VIC-3D®-file for solving this problem are
shown following Fig. 4.11.
The problem shown in Fig. 4.10 models Fig. 4.8 with a single void microcrack.
This structure is embedded in an isotropic graphite-epoxy host whose conductivity
is 20,000 S/m, and is 0.3 mm high. The objective of this study is to determine the
feasibility of detecting and determining the width, W , of the microcrack when
the structure is excited by the T probe shown in Fig. 4.7 as it is scanned past the
microcrack from −1.6 mm to +1.6 mm at a frequency of 1 GHz.
To that end, we apply model-based inversion, starting with the blending functions
that define the surrogate interpolation table. These are shown in Fig. 4.11 for the
nodal values W = (0.00, 0.025, 0.050) mm. The VIC-3D®-grid used to generate
these functions was N x = 256, N y = 8, N z = 16, which was sufficient to capture
the variations in W . The test value of W was 0.030 mm.
The results of the inversion are shown in Table 4.3. Not only is the solution
virtually identical to the test value, but all 500 initial starting points in NLSE
converged to the same global minimum. The excellent quality of the inversion
testifies to the fact that the blending functions of Fig. 4.11 are highly sensitive to W .
4.2 Statistical Analysis of the Feasible Set
One of the principal efforts of research in Set-Theoretic Estimation is to determine
the number and nature of the experiments, (Z(v), E
(i)
lmj (v)), and receiver scans,
E
(n)
LMJ , that produce a good feasible set for statistical analysis. As might be expected,
this depends upon the complexity of the flaw that is to be reconstructed, and upon
the resolution desired in the reconstruction.
After deducing the feasibility set, we are then faced with the task of assigning a
single number for the conductivity of each cell; this is a problem of data analysis,
which uses robust regression, as described in Chapter 13 of [111]. The aim of the
statistical analysis of the feasible set is to fit a constant through the data, which
are produced by the algorithm described in the preceding section, for all views (or
experiments), v.
A beneficial feature of this algorithm is that the analysis of the data set for each
cell is done independently of every other cell; i.e., a decision is made on a cell-bycell basis. Since the decision to be made for each cell involves a nonlinear (robust)
estimator, the computational burden is greatly reduced when compared to using a
nonlinear estimator to solve for many cells jointly. (The bilinear conjugate-gradient
algorithm is an exception.) Furthermore, this leads us to a constrained iterative
(‘layer-stripping’) algorithm, that uses the known and accepted results for some
cells to determine the results for others at a later stage of the iteration.
4 Application of the Set-Theoretic Algorithm to CFRP’s
the next quarter. The parameters in the VIC-3D®-file for solving this problem are
shown following Fig. 4.11.
The problem shown in Fig. 4.10 models Fig. 4.8 with a single void microcrack.
This structure is embedded in an isotropic graphite-epoxy host whose conductivity
is 20,000 S/m, and is 0.3 mm high. The objective of this study is to determine the
feasibility of detecting and determining the width, W , of the microcrack when
the structure is excited by the T probe shown in Fig. 4.7 as it is scanned past the
microcrack from −1.6 mm to +1.6 mm at a frequency of 1 GHz.
To that end, we apply model-based inversion, starting with the blending functions
that define the surrogate interpolation table. These are shown in Fig. 4.11 for the
nodal values W = (0.00, 0.025, 0.050) mm. The VIC-3D®-grid used to generate
these functions was N x = 256, N y = 8, N z = 16, which was sufficient to capture
the variations in W . The test value of W was 0.030 mm.
The results of the inversion are shown in Table 4.3. Not only is the solution
virtually identical to the test value, but all 500 initial starting points in NLSE
converged to the same global minimum. The excellent quality of the inversion
testifies to the fact that the blending functions of Fig. 4.11 are highly sensitive to W .
4.2 Statistical Analysis of the Feasible Set
One of the principal efforts of research in Set-Theoretic Estimation is to determine
the number and nature of the experiments, (Z(v), E
(i)
lmj (v)), and receiver scans,
E
(n)
LMJ , that produce a good feasible set for statistical analysis. As might be expected,
this depends upon the complexity of the flaw that is to be reconstructed, and upon
the resolution desired in the reconstruction.
After deducing the feasibility set, we are then faced with the task of assigning a
single number for the conductivity of each cell; this is a problem of data analysis,
which uses robust regression, as described in Chapter 13 of [111]. The aim of the
statistical analysis of the feasible set is to fit a constant through the data, which
are produced by the algorithm described in the preceding section, for all views (or
experiments), v.
A beneficial feature of this algorithm is that the analysis of the data set for each
cell is done independently of every other cell; i.e., a decision is made on a cell-bycell basis. Since the decision to be made for each cell involves a nonlinear (robust)
estimator, the computational burden is greatly reduced when compared to using a
nonlinear estimator to solve for many cells jointly. (The bilinear conjugate-gradient
algorithm is an exception.) Furthermore, this leads us to a constrained iterative
(‘layer-stripping’) algorithm, that uses the known and accepted results for some
cells to determine the results for others at a later stage of the iteration.