*3.1. Numerical Models*

Figure 2a shows the analysis model. The target plate model was 100 × 100 mm, and a penetrating crack 6 mm long and 1 mm wide was introduced at the center. Four-node quadrilateral solid (plane-stress) elements were used, and the size of each element was 1 × 1 mm. There were 10,000 elements and 10,201 nodes. The crack was expressed as damaged elements with a small Young's modulus *E*0. Five period-sinusoidal wave loads multiplied by the Hanning window function were applied to all nodes on the lower surface; their amplitude and frequency were 1 N and 1 MHz. A non-reflective boundary condition [41] that removes one reflection was set on the upper, left, and right surfaces. The material was assumed to be stainless steel, and its Young's modulus, Poisson's ratio, and density were 186.6 GPa, 0.306, and 7.86 g/cm<sup>3</sup> , respectively.

Figure 3 depicts the analyzed ultrasonic wave propagation. The ultrasounds propagated from the bottom to the top and were diffracted and scattered at the crack. The waves propagating from the bottom corners were reflected waves. Figure 3b shows the distribution of the maximum amplitude of the mean stress. The maximum amplitude had a characteristic distribution near the crack (e.g., high in the lower area and low in the upper area to the actual crack). Therefore, the crack will be reproduced by focusing on the maximum amplitude distribution.

The inverse analysis model (i.e., design domain *D*) was a 10 × 10 mm domain (including 100 elements and 121 nodes), which was extracted from the center of the target model, as illustrated in Figure 2b. Topology optimization defines the same number of design variables as elements in the design domain. Therefore, the size of the inverse analysis model and the calculation cost are in a trade-off relationship. Input loads, similar to those in the forward analysis, were applied to the lower surface of the inverse analysis model. A non-reflective boundary condition [41] was set on the upper, left, and right surfaces.

**Figure 2.** Models to calculate data of ultrasonic waves: (**a**) Target model; (**b**) Inverse analysis model.

**Figure 3.** Numerical results of wave propagation on a stainless-steel plate with a crack: (**a**) Snapshots (mean stress); (**b**) Distribution of the maximum amplitude of the mean stress.
