*3.2. Damage Identification for Numerical Results*

The maximum amplitude of the mean stress was adopted as the ultrasonic feature in the objective function. Thus, the optimization problem is defined as follows:

$$\min\_{d} f(d) = \sum\_{j=1}^{m} \left( \sigma\_{j, \text{max}}(d) - \sigma\_{j, \text{max}, \text{target}} \right)^2 \qquad \text{subject to} \quad 0 \le d\_i \le 1 \quad \text{for } i = 1, \dots, n \tag{4}$$

where σ*j*,max(*d*) is the maximum amplitude of the mean stress estimated at node *j* with the present damage state *d*, and σ*j*,max,target is that of the target data at node *j* obtained by forward analysis (Figure 3b right) in this section. A constant value of the damage parameter, *d* ini, was initially assigned to all the elements, and ten inverse analyses were performed using *d* ini from 0.1 to 1.0, with an increment of 0.1. The penalization exponent *p* of 3 was used as in previous studies [31,34–36,38]. Sequential quadratic programming in the MATLAB Optimization Toolbox (R2019a, MathWorks, Inc, Natick, MA, USA) was used for exploration and updating of the damage parameters. The gradient of the objective function was calculated by the finite-difference method.

Figure 4a depicts typical damage distributions estimated by the proposed method. The estimated damage parameters of the elements in the actual damaged domain (*T*) were smaller than that of most elements in the actual intact domain (*D*\*T*) in all of the ten cases of the initial damage parameter *d* ini distribution. This result suggests that the damage in the target can be estimated by the proposed method; however, the estimated damage states were different depending on the initial damage parameter *d* ini in the upper area of *<sup>D</sup>*\*T*. Moreover, for example, in the case of *<sup>d</sup>* ini = 0.5, the damage parameters smaller than its maximum in *T* (0.2016) were estimated in nine elements in *D*\*T*, which were erroneously determined to be damaged. Such elements were concentrated in the upper region of *D*.

**Figure 4.** The optimal solution of the damage state obtained using ultrasonic wave data (**a**) from a single direction and (**b**) from two directions.

The reason for this false identification is caused by the low sensitivity of the objective function in the area above the crack. The objective function was evaluated at various damage states depicted in Figure 5a. Damaged elements were added to the lower area of *D*\*T* in #1–#5 and to the upper area in #7–#10. Figure 5b presents the value of the objective function obtained by forward analysis in these damage states. The gradient of the objective function within #6–#10 was smaller than that within #1–#6. Moreover, the maximum amplitude in the area above the actual crack was smaller than that in the other areas, as shown in Figure 3b, because the diffracted ultrasonic waves propagated with a small amplitude above the crack as a result of the high directivity of ultrasound. Therefore, the maximum amplitude hardly changed in the area above the actual crack even if the damage state in that area was

altered. Consequently, the damage identification results depended on the relationship between the crack geometry and the direction of ultrasonic wave propagation, and it was difficult to estimate the right damage state in the area above the crack with the objective function (4).

**Figure 5.** The objective function values obtained using ultrasonic wave data in (**a**) the various damage states (**b**) from a single direction and (**c**) from two directions.

To improve the sensitivity of the objective function, the wave propagation from the top to bottom (superscript: upper) was considered in addition to that from the bottom to the top (superscript: lower); thus the optimization problem is re-defined as follows:

$$\begin{aligned} \min\_{d} f(\mathbf{d}) &= \sum\_{j=1}^{m} \left( \sigma\_{j,\text{max}}^{\text{lower}}(\mathbf{d}) - \sigma\_{j,\text{max},\text{target}}^{\text{lower}} \right)^{2} + \sum\_{j=1}^{m} \left( \sigma\_{j,\text{max}}^{\text{upper}}(\mathbf{d}) - \sigma\_{j,\text{max},\text{target}}^{\text{upper}} \right)^{2} \\ &\text{subject to} \quad 0 \le d\_{l} \le 1 \quad \text{for } i = 1, \dots, n \end{aligned} \tag{5}$$

In ultrasonic visualization experiments, ultrasonic propagation data from two directions can easily be recorded.

Figure 4b depicts the estimated damage states based on Equation (5). The damage parameter in *D*\*T* was always greater than its maximum in *T*, and thus, the actual damage *T* was successfully identified. However, the difference of the damage parameter in *T* and in *D*\*T* was small, and this issue will be discussed in the following section. The sensitivity of the objective function was high in both the upper and lower areas of the target crack, as shown in Figure 5c. Thus, when the sensitivity was enhanced in the entire design domain, the proposed method provided the damage identification results closer to the target damage state.

The above results and discussion demonstrate the feasibility of the proposed method. This study used the maximum amplitude distribution as the ultrasonic feature, and two sets of ultrasonic propagation data were required in the objective function to enhance its sensitivity. However, considering the cost of inspection, it is recommended to estimate damage using a single set of wave propagation data. To that end, the objective function should be improved and will be investigated in our future study.
