*2.2. Setup of Optimization Problem*

The design domain *D* is discretized into finite elements, and the damage parameter *d<sup>i</sup>* is assigned to element *i*:

$$d\_i = \begin{cases} 0 & \text{for } i \in \Omega\_d \\ 1 & \text{for } i \in D \backslash \Omega\_d \\ \end{cases} \qquad \text{damage} \tag{1}$$

 where Ω*<sup>d</sup>* is the damaged domain, and *<sup>D</sup>*\Ω*<sup>d</sup>* is the intact domain. The solid isotropic material with penalization (SIMP) method [31] is used to relax the damage parameter, and a continuous value from 0 to 1 represents the severity of the damage. With this setting, the damage identification problem is translated into a distribution problem of the damage parameter in the design domain *D*.

The Young's modulus of element *i* is expressed as a function of the damage parameter *d<sup>i</sup>* as:

$$E\_i(d\_i) = (E\_1 - E\_0)d\_i^p + E\_{0\prime} \tag{2}$$

where *E*<sup>1</sup> and *E*<sup>0</sup> are Young's modulus of a solid (i.e., perfectly intact) and in a void (i.e., perfectly damaged), and *p* is the penalization exponent for intermediate damage. Although *E*<sup>0</sup> is initially zero, elimination of an element (i.e., modification of the model) is cumbersome in the optimization process; therefore, a small stiffness (*E*<sup>0</sup> = 0.001 MPa) is assigned to perfectly damaged elements.

The topology optimization problem is defined as the minimization of the square error between the estimated ultrasonic feature and the target data as follows:

$$\min\_{d} f(d) = \sum\_{j=1}^{m} \left( \mathcal{U}\_{j}(d) - \mathcal{U}\_{j, \text{target}} \right)^{2} \qquad \text{subject to} \quad 0 \le d\_{i} \le 1 \quad \text{for } i = 1, \dots, n \tag{3}$$

where *Uj*(*d*) is the analytically obtained ultrasonic feature at node *j* with the present damage state *d*, *Uj*,target is the ultrasonic feature of target data at an illuminating point, and *m* and *n* are the total number of nodes and elements in the design domain *D*. The ultrasonic feature in the objective function should be selected appropriately, depending on the problem. Although a few studies [38–40] used the summation of damage parameters as a constraint condition, no constraint conditions other than Equation (3) provided better results in the preliminary investigation.
