*3.3. E*ff*ect of the Penalization Exponent*

≥ The penalization exponent *p* in Equation (2) has been discussed in previous studies, because the relaxation of the design variables yields an intermediate density called a gray-scale element, which cannot be interpreted physically in structural design problems. To clarify the physical meaning of the gray-scale element, Bendsøe et al. [42] discussed the range of *p* based on Hashin-Shtrikman (HS) bounds [43] and proved that the SIMP method is physically permissible as long as *p* is greater than a certain value (e.g., *p* ≥ 3 for two-dimensional problems with Poisson's ratio of 1/3). In previous studies of damage identification based on topology optimization, Nishizu and Neumann et al. [35,38–40] set *p* = 3, Reumers et al. [36] set *p* = 1, and Eslami et al. [37] changed *p* gradually from 3 to 1. An element

(i.e., microscopic area) will have various Young's modulus depending on the severity of the damage. Therefore, unlike structural design problems, the intermediate damage parameters are acceptable in the damage identification problem. Instead, it is essential to set a threshold to distinguish a damaged from an intact region.

Damage identification using various penalty exponents and the HS upper bound was performed. Here, the problem is defined by Equation (5) and Figure 2; the initial value of the damage parameter, *d* ini, was 0.5 in all the cases. Figure 6a depicts the estimated damage states, and Figure 6b shows a variation of Young's modulus normalized by *E*1. The inverse analysis did not converge when *p* was less than 0.5 or greater than 3. In all converged results, the maximum value of the damage parameter in *T* was smaller than its minimum in *D*\*T*, and the damage state was estimated appropriately.

**Figure 6.** Comparison of (**a**) the optimal solution of the damage state and (**b**) the values of Young's modulus as a function of damage parameters with the various penalization exponents and the Hashin-Shtrikman upper bound [43].

When *p* was greater than 1 (including the HS upper bound), the damage parameters were almost uniform in *D*\*T*. Furthermore, as *p* increased, the damage parameters in *D*\*T* approached 1, and the number of gray-scale elements decreased. This is because the gradient of Young's modulus in damage parameters close to 1 is large when *p* = 2 or more, as shown in Figure 6b. Therefore, *p* should be greater than 2 to interpret the elements as undamaged physically.

On the other hand, when *p* was 1 or less, the difference of the damage parameter in *T* and in *D*\*T* was larger than that with *p* greater than 1. Furthermore, the damage parameter in *T* became almost 0. As shown in Figure 6b, when the damage parameter is close to 0, Young's modulus changes significantly with a small variation in the damage parameter. Therefore, the damage parameter tends towards 0 in *T*, and it hardly approaches 0 in *D*\*T*. The penalization exponent of 0.5–1 enables setting a threshold to distinguish between damaged elements from intact ones. For example, when using *p* = 0.8 and the threshold *d*th = 0.1, the damaged and undamaged elements were distinguished in all initial values of the damage parameter. The above discussion indicates that the penalization exponents *p* of 0.5–1 are suitable for damage identification.
