## 2 Modified Ellipse Method  
When Lamb waves encounter a defect, a propagating wave will be reflected and/or partly transmitted. These damage-scattered waves will carry enough information about the damage. For instance, time of flight (ToF) and amplitude can be used for damage identification. The ToF is defined as a time duration for a specific wave packet travelling a certain distance. Generally, ToF represents, to an extent, a linear correlation with the damage parameters such as its location, size and shape. To illustrate the principle of damage localization using ToF, a sensing path \(T_{I}-T_{I}\) (\(I,J=1,2,\ldots\)N, \(I\
eq J\)) from part of a sensor network composed of N pieces of PZT transducers is schematically shown in Figure 1a. Using Cartesian coordinates, the actuator \(T_{I}\), the sensor \(T_{J}\) and the center of the damage are supposed to be situated at \((x_{I},y_{I})\), \((x_{I},y_{I})\), \((x_{m},y_{m})\), respectively. The widely known ellipse method can be used to describe their relationship as given by  
\[\frac{l_{Im}}{V_{g}}+\frac{l_{mJ}}{V_{d-s}}=t_{g}+t_{d-s}=t_{IJ}\]  
\[l_{Im}=\sqrt{\left(x_{I}-x_{m}\right)^{2}+\left(y_{I}-y_{m}\right)^{2}}\]  
And  
\[l_{mJ}=\sqrt{\left(x_{J}-x_{m}\right)^{2}+\left(y_{I}-y_{m}\right)^{2}} \tag{1}\]  
where \(V_{g}\) and \(V_{d-s}\) are the group velocities of the incident wave and the damage-scattered wave, respectively. \(t_{IJ}\) is the ToF it takes for the incident wave to travel from the actuator to the damage (\(t_{g}\)), and then to the sensor after damage scattering (\(t_{d-s}\)). \(l_{Im}\) is the distance between the actuator and the damage center, and \(l_{mJ}\) is the distance between the damage center and the sensor. By repeating the above analysis for at least two different sensing paths in the sensor network, a nonlinear equation set with two unknown parameters \((x_{m},y_{m})\) is created. For each equation described by Equation (1) in the set, the solution, which mathematically depicts an elliptical (provided \(V_{g}=V_{d-s}\)) locus, indicates all possible damage locations perceived by the sensing path \(T_{I}-T_{J}\) (Figure 1a). If the nonlinear equation set is solved, the damage locations can be presumably positioned at the intersections of most of the loci. However, the ellipse method will inevitably cause positioning errors when the group velocity changes (\(V_{g}\
eq V_{d-s}\), e.g., mode conversion) before and after interacting with the damage. Furthermore, a damage location should not be simply considered as a single point at the intersection with specific geometric parameters. Instead, it is more likely to be a damage zone. For example, a crack has a certain length, and there may be a plastic deformation zone at the tip of the crack. Thus, a modified ellipse method focusing on elliptical rings is proposed to solve the problems in accurate damage localization. Given the velocity difference \(\Delta v=V_{g}-V_{d-s}\), the nonlinear equation for damage localization could be expressed as  
\[\frac{l_{Im}}{V_{g}}+\frac{l_{Im}}{V_{g}-\Delta v}=t_{II}=\frac{l_{Im}}{V_{d-s} +\Delta v}+\frac{l_{Im}}{V_{d-s}} \tag{2}\]  
The damage locations are positioned at the intersections of most of the elliptical (when \(\Delta v=0\)) or ellipse-like (when \(\Delta v\
eq 0\)) loci. By neglecting nonzero \(\Delta v\), two different elliptical loci are derived from the left- and right-hand side of Equation (2), which are smaller and bigger than the ellipse-like locus, respectively. Moreover, no matter how small the difference between the two elliptical loci is, the ellipse-like locus indicating all the possible damage locations will situate at a region between the two elliptical loci.