The potential of the proposed coupling process is first assessed by using singular value decomposition to assess the rank of the sensitivity matrices for 4, 8, 12, and 16 electrodes with 0, 5, 15, and 25 enhancements. Singular value decomposition is an effective way to
Figure 6.7: Rank assessment of unenhanced and enhanced sensitivity matrices by singular value decomposition. Left: plot of all scenarios considered. Right: close up of singular value indices up to 140.
visually ascertain the rank of large numerical matrices. As shown in Figure6.7, the index at which there is a sudden drop of singular value corresponds to the rank of the sensitivity matrix.
A couple of important insights can be gleaned from Figure 6.7. First, recalling that there are ∼1300 elements in theEITmesh, the singular value plots for 16 electrodes with 15 or 25 enhancements and 12 electrodes with 25 enhancements show that it is indeed possible to achieve a full rank sensitivity matrix. And, second, up to the point of being full rank, the rank seems to enhance in direct proportion to the number of enhancements. That is, the unenhanced sensitivity matrix has a rank of (L − 3)L/2 where L is the total number of electrodes. Consider the 16 electrode case. Unenhanced, the sensitivity matrix has a rank of 104. With 5 enhancements, the singular values drop just past 600 which implies the rank of the enhanced sensitivity matrix is approximately (1 + n)(L − 3)L/2 where n is the number of enhancements.
Next, the potential of this approach is demonstrated by showing how enhancements affect image reconstruction. This is shown in Figure6.8 where the distribution in Figure
6.7 is being reproduced by EIT. These results make it clear that the proposed coupling method can not only markedly enhance image quality but also significantly reduce the number of electrodes necessary to produce a quality image.
Although there are a couple of scenarios in which the sensitivity matrix is full rank, no perfect conductivity reconstructions exist in Figure6.8. This is to be expected because of the care that was taken to avoid so called inverse crimes [134]. Furthermore, the image quality seems to reach a saturation point. That is, the images stop improving better beyond a certain quality. This is also a consequence of avoiding an inverse crime. Simply put, no perfect solution is available toEITso enhancing the sensitivity matrix past the maximum rank does not produce a better image.
Lastly, a final example of the power of this approach is provided by comparing the image reconstruction with 16 electrodes and no enhancements to the image reconstruction with 16 electrodes and 25 enhancements on a much more complicated reference image as shown in Figure6.9. In this example, the reference mesh has approximately 4000 elements and the EITmesh has approximately 3000 thereby ensuring the enhanced sensitivity ma- trix is not full rank. Here, the unenhanced image captures the basic shape of the reference image, but consistent with the limitations of EIT, it is blurry and identifiable details are not discernable. The enhanced image, however, still is not perfect, but it is significantly improved to the point that the shape can be easily recognized and identified without knowl- edge of the reference image.
6.4
Summary and Conclusions
In this chapter, the influence of tailoring nanofiller alignment and directionality has been investigated to enhance delamination detection. Anisotropic conductivity values from the equivalent resistor network model developed in Chapter 4 were used to determine how alignment-induced anisotropy influences changes in boundary voltage due to damage. It was found that degree of alignment, orientation through the plate thickness, and anisotropic ratio all have a pronounced influence on sensitivity. AnisotropicEIT was then employed to locate a single and double delamination case for conductivities with φm = 45◦, 60◦,
and 75◦ at β = 0◦, 30◦, 60◦, and 90◦ and 1% filler volume fraction. Compared to random nanofiller networks with isotropic conductivity, it was found that anisotropic conductivity markedly enhancesEITsensitivity, but overly anisotropic conductivity degrades the image. This implies that incorporating microscale nanocomposite tailoring has incredible potential to bolster structural-scaleSHMand sensitivity to delaminations.
Figure 6.8: Influence of coupling enhancements on EIT images for 4, 8, 12, and 16 elec- trodes with 0, 5, 15, 25 enhancements. As expected from the singular value decomposition analysis, for a given number of electrodes, adding enhancements significantly improves the image.
Figure 6.9: Comparison of unenhanced image with an image produced with 25 enhance- ments. Both use 16 electrodes. In this more complicated example, the enhanced image very decidedly outperforms the unenhanced image.
conductivity changes was postulated and formulated. This approach, motivated by piezore- sistive nanocomposites, is predicated on the supposition that adding additional constraints to the sensitivity matrix to enhance its rank will result in a better image or, alternatively, can enable quality images to be produced with fewer electrodes. Singular value decomposition analysis showed that the coupling process does indeed enhance the rank of the sensitivity matrix, and image reconstruction examples demonstrated the potential of this approach by producing markedly superior images. Unlike other multi-physics EITenhancement tech- niques, this approach does not necessarily impose upon the advantages ofEIT. Particularly for nanocompositeSHM, this technique has considerable potential to significantly enhance image quality or reduce the implementation burden ofEIT.
CHAPTER 7
Summary of Scholarly Contributions and
Broader Impacts
This research radically expands the potential of EIT for conductivity-based health moni- toring of nanocomposites by first extending the fundamental knowledge of nanocomposite conductivity, concretely demonstrating the potential ofEIT for strain and damage identi- fication in nanocomposites, and lastly, significantly advancing state of the artEIT by ex- ploiting the unique properties of nanocomposites. The scholarly contributions and broader impacts of each chapter are summarized below.