*4.2. Damage Identification for Experimental Results*

Figure 9a shows the two-dimensional analysis model. The dimension of the analysis domain was 50 mm long and 50 mm wide. Four-node quadrilateral solid (plane-stress) elements were used, and the size of each element was 1 × 1 mm. There were 2500 elements and 2601 nodes. Five period-sinusoidal wave loads multiplied by the Hanning window function were applied to the center point of the lower surface (i.e., the position of the receiver); their amplitude and frequency were 1 N and 5 MHz. A non-reflective boundary condition [41] that removes one reflection was set on the upper, left, and right surfaces. Young's modulus, Poisson's ratio, and density of the aluminum plate were 69.0 GPa, 0.34, and 2.70 g/cm<sup>3</sup> .

The design domain *D* was a part of the analytical model located 40 mm from the bottom, as shown in Figure 9b. Its size was 10 × 10 mm, including 100 elements. The damage parameters only in the design domain *D* were updated, and those in the other areas remained constant at unity. The size of the element was 1 mm square, and therefore, the location and length of the crack could be identified, but the width could not be estimated. Equation (4) was used as the optimization problem setting, and the maximum amplitude distribution on the right of Figure 8b was used as the target data. The initial value of the damage parameter *d* ini was 0.5. The penalization exponent *p* = 0.8 was adopted based on the discussion in Section 3.3.

Figure 10a depicts the estimated damage state. The maximum value of the damage parameter in *T* was 0.2950, and there were five elements in *D*\*T* that had a damage parameter smaller than that value. Most of these elements were in the area above the actual crack, and this pattern was similar to Figure 4a. The sensitivity of the objective function above the damage was evaluated in Figure 10b in the same manner as in Figure 5. The curve had the same tendency as in Figure 5b, and the low sensitivity in the area above the actual crack caused the misestimation. Nonetheless, as shown in Figure 10c, when using the threshold *d*th = 0.1 for distinguishing between the damaged from the intact region, three elements out of the actual damage region *T* (four elements) were estimated as damaged, and all the other elements were estimated as intact.

**Figure 9.** Numerical models with the same configuration as the experiment: (**a**) Ultrasonic wave propagation analysis model; (**b**) Inverse analysis model.

**Figure 10.** Results of applying the proposed method to experimentally measured wave propagation: (**a**) The optimal solution of the damage state; (**b**) The objective function values in the various damage states as with Figure 5; (**c**) The estimated damage state when using the threshold *d*th = 0.1.

The feasibility of the proposed method was thus demonstrated, though these results were obtained in simple and ideal model cases. A more detailed investigation will be required for the application of the present method to real problems.
