In order to evaluate the proposed methodology adequacy for performing dynamic scaling using a single material topology optimization strategy, eight test cases are conducted. These test cases are also important to identify the proposed dynamic scaling methodology limitations. The first two cases, presented in
Section 3.1, consists of simply introducing a difference between the material properties of both models (full and scaled), which increases from the first to the second case. Then, to assess the impact of design space size the two different grids mentioned before are compared in
Section 3.2. Finally, the strategy of increasing the penalization factor to avoid intermediate densities is discussed in
Section 3.3 using 4 different values.
3.1. Material Influence Test Case
Before defining this test case, it is worth to recall that the full scale model is composed of two parts, an external shell (Part-2) and an internal solid part (Part-1) made of two materials (A and B), as illustrated in
Figure 2; while the scaled model consists of an internal part made of a single material and an external shell. Furthermore, the scaled model shell remains unmodified during the optimization process.
As mentioned earlier the goal of this sub-section is to evaluate the influence of varying the difference between the material properties of both models. Two case studies, with increasing difficulty, are conducted for this purpose, using the most coarse design grid (18, 24, 1). In the first case study (Case 1), the proposed dynamic scaling methodology is tested considering small differences between the material properties of the full scale model and the scaled one, as presented in
Table 1. These difference are then increased for the second case (Case 2) as can be observed also from
Table 1. In what concerns solid A the same differences between material properties as for Case 1 are maintained, however for solid B these differences are increased and consequently making it more difficult to achieve dynamic similarity. Since unitary scaling factors are defined for both density and pressure the material properties of the target and full size models are the same.
For Case 1, the highest differences in terms of elasticity modulus is observed for the shell part (); while for the density and Poisson ratio, the solid part made of material A presents the highest differences ( and , respectively). It is important to mention that although the shell thickness is 1 mm in the full size model, its reduced size counterpart is four times higher (i.e., 4 mm) to further difficult the dynamic scaling process. This last consideration is also applied in Case 2. Even though a difference of 60,506% in the elasticity modulus is considered for the shell in Case 2, its influence on the results is reduced when compared with the internal solids A and B given its small thickness value.
The optimization for Case 1 reached a reasonable good solution just after performing 16 iterations, as one can see from the convergence history in
Figure 6. After these 16 iterations, despite the solution fluctuates slightly the best result is reached at the 26th iteration with an objective function value of
. As expected, Case 2 presents a slightly slower convergence than Case 1 given the higher differences in material properties between not only target and scaled models but also between solids A and B. The lowest objective function value (
) is reached at 49th iteration. For both cases, the maximum number of iterations is set to 60.
These low objective function values indicate that a very small difference between natural frequencies is reached, which is demonstrated in
Table 2. In fact, the relative error between the target and obtained frequencies is lower than 0.1% for both case studies.
To verify if the mode shapes from both models also match, one can use the previously mentioned MAC metric. Despite not directly expressed in the objective function, the MAC is used to track the right mode shapes to compare the associated natural frequencies. This metric is illustrated for the best result of Case 1 in
Figure 7a, where the less correlated mode shape (5th mode shape) presents
of correlation concerning its peer (diagonal values), and higher correlation between each other (off-diagonal values) was
. A similar correlation is found for Case 2 as depicted in
Figure 7b.
The results obtained indeed indicate that the matching of both the natural frequencies and mode shapes is successfully accomplished even for the most coarse design grid (18, 24, 1). This proves that the developed methodology can be used for dynamic scaling with similar assumptions to those considered in this case study. For this case study, the material properties of the solid parts in the scaled model differ only slightly from the target material properties, which are equal to ones of the full scale model since unitary density and pressure scaling factors are defined. The correlation between mode shapes degrades slightly with the increase of these differences in material properties of the solid parts.
Regarding the topologically optimized internal structures, the solutions for both cases are very similar as illustrated in
Figure 8.
3.2. Design Space Test Case
The objective now is to evaluate the impact of increasing the number of design variables on dynamic similarity for a more challenging test case. For this purpose the two design grids presented in
Section 2.2 are used. In the previous case studies, the material properties were fictitious and adapted to analyze differences between models. On the other hand, in the current test case existing materials are set for both full and scaled models, which increases the differences in material properties. To increase the difficult of reaching dynamic similarity, the full scale model is designed to be made of a 5000 Series Aluminum Alloy for Part-1 Solid A, Nylon 12 (PA12) for Part-1 Solid B and Nylon 610 (PA610) for the shell (Part-2). Regarding the scaled model, both solids are composed of a general Aluminum alloy and the shell is set to be made of Nylon (PA12). It is worth to note that Nylon (PA12) is considered for both full and scaled models, although for different parts (Solid B and shell for the former and latter, respectively) and considering slightly differences in the properties. Furthermore, it is also important to mention that the material selected for the skin of the scaled model presents higher mechanical properties than those of the equivalent full size counterpart. Even though the shell is expected to have a slightly higher impact on the dynamic scaling process than before, the internal structure still presents higher density and elasticity modulus. The material properties used for designing both models and design grids are shown
Table 3. Cases 3 and 4 denote coarse and fine design grids, respectively. The highest difference in
E is by far for the shell (60,506%). In regard to the
and
, the Part-1 Solid B presents the highest differences. For this internal part, the scaled model is defined with a material
denser and
stiffer than the full scaled model.
The convergence history of the objective function from both design grids is not very different, as shown in
Figure 9. None of the optimization problems converged and both cases stopped by the maximum number of iterations criterion (200). Nevertheless, reasonably good solutions are reached after 90 and 70 iterations when using coarse (18, 24, 1) and fine (54, 72, 1) design variable grids, respectively. After these iterations, small fluctuations are observed. The best solution for the Case 3 (18, 24, 1) is reached at 199th iteration with value of
, while for Case 4 the lowest objective function value (
) is obtained at the 138th iteration. If more iterations were performed, a better objective function value could have been obtained and even convergence could have been reached. Nevertheless, since the objective function values are already low (in the same magnitude as before) and given the higher computational cost, the maximum number of iterations was maintained.
The low objective function values are an indicative of a good matching between full and scaled models in terms of natural frequencies, which is confirmed in
Table 4. In relation to the previous test case, now the relative differences are higher, but still lower than
. This outcome is associated with the higher discrepancies between the material properties, i.e., further way from the target ones. As expected, the higher design freedom of the finer grid (Case 4) allowed for a lower relative errors.
Regarding the mode shapes a good correlation between models is also verified as illustrated in
Figure 10. As noticed for the natural frequencies, the mode shape correlation is also reduced for these case studies when compared with the previous ones. For Case 3 the smallest on-diagonal correlation is
for the 4th mode shape and the highest off-diagonal value is
. The correlation slightly decreased for the finest design grid used.
These results demonstrate that the proposed methodology is able to successfully design a model numerically for dynamic scaling. However, this almost perfect matching of the natural frequencies and very good agreement between mode shapes were possible without using a filter in the topology optimization process and the penalization factor was set 1. As consequence, the solution presents intermediate densities as can be seen in
Figure 11.
For Case 4, most of the intermediate densities were close to 1 or
, as one can see from
Figure 12a. When analyzing the optimized internal structure, it is worth to recall that the scaled model has a shell made of a material substantially stiffer than the target model and its thickness is 4 times higher. The solution was thus affected by the shell in such a way that the amount of material near the root is smaller compared to other wing locations. Despite the obtained solution does not present checkerboard problem, it is possible to see some “isolated” cells in
Figure 12a.
As noted before, it is possible to observe from
Figure 12a that most of the cells from Case 4 solution present values either close to 1 or
. This particular outcome is very interesting since it allows to consider a post-processing, such that these intermediate densities can be assigned to another material. The good matching of the natural frequencies and mode shapes makes this approach even more promising since it would eventually allow it to be manufactured. Moreover, if performing more iterations the results could have been improved. In this sense, in
Figure 12b is presented as a post-processing, where black and red cells correspond to the original and second material, respectively. Such intermediate material was found possible for this case after making a search in [
34], although it might be infeasible for other cases. The material in question is the Nylon 12 (PA12). From the 3D printing perspective, the challenge consists of defining materials that can be printed together. However, the topic of metal-plastic hybrid 3D printing is evolving [
35,
36].
Despite the task of designing scaled model with the internal structure made of two materials is feasible, it is not the focus of this paper. Here, the strategy followed to avoid intermediate solutions is the usage of a penalization factor in the next sub-section.
3.3. Penalization Factor Test Case
The penalization factor influence on the topologically optimized structures is now the focus of study in this test case. Four different penalization values are used for the finest considered design grid (54, 72, 1): 1 (Case 5), 2 (Case 6), 3 (Case 7) and 6 (Case 8). Despite the same full size model as for Cases 3 and 4 is used in these case studies, the materials and shell thickness of the scaled model are now different. The material properties of both models are summarized in
Table 5. A Titanium alloy (Ti-6Al-4V (Grade 5)), often used for manufacturing through 3D printing [
37], is considered for the internal part (solids A and B) of the scaled model. This pose an even higher challenged than before since the difference between properties increases. Regarding the shell, it is now 1 mm thick, i.e., equal to the full size model. This choice is related to the trend of reducing material near the root observed in
Figure 12.
Given the high computational cost of these numerical optimizations, the maximum number of iterations is set to 135. A stable convergence history of the objective function can be seen in
Figure 13 for the 4 considered penalization factors. As expected, the proposed dynamic scaling methodology takes more iterations to reduce the objective function when a higher penalization factor is considered since the design freedom is reduced. The lowest obtained objective function values are:
at iteration 120 for Case 5;
at iteration 128 for Case 6;
at iteration 64 for Case 7; and
at iteration 97 for Case 8.
These relatively low objective function values indicate that the first five natural frequencies are very close to the target ones, which is confirmed in
Table 6. Even though higher objective function values and relative errors of frequencies are in general obtained by increasing the penalization factor, the relative errors are still low. In fact, the highest error (1.16%) is determined for the fifth vibration mode considering the highest penalization factor (Case 8). Thus, showing the proposed methodology effectiveness in matching the natural frequencies.
The correlation between target and scaled mode shapes is again assessed through the MAC metric, which is depicted in
Figure 14 for the considered cases. This correlation is clearly observed to degrade as the penalization factor is increased. When using the lowest value (Case 5), a very good agreement between mode shapes is found since the lowest correlation is
and the highest off-diagonal term is 16%, as illustrated in
Figure 14a. Both these values are within the range usually deemed as acceptable,
and
for correlation and off-diagonal terms, respectively, [
38]. By increasing the penalization factor to the double (Case 6,
Figure 14b), despite an off-diagonal term of
is calculated, the lowest correlation (
) is still considerably higher than the above mentioned threshold. For this reason, Case 6 is still considered a good result by the authors. However, for the Case 7 (
Figure 14c) these metrics continues degrading and the lowest correlation falls to
, which is barely above the threshold value, but the maximum off-diagonal term slightly raises to
. Given the fact that at least the correlation is still above the threshold value, this case is considered acceptable by the authors although not desirable. For Case 8 (
Figure 14d) both these thresholds are not satisfied for the first five mode shapes, thus the mode shape matching is deemed not achieved even though the frequencies present low relative errors.
The topologically optimized internal structures for the different cases are shown in
Figure 15. It is possible to identify areas with different intermediate density in all the cases. Despite increasing the penalization factor has not allowed for completely removing all intermediate densities, its amount has reduced. The identification of well-defined areas with intermediate densities makes the post-processing with different materials an attractive strategy. However, this approach might not be possible for other cases, e.g., the resulting intermediate material properties can be nonexistent. Furthermore, even if the second material exists it might not be compatible with the original one. Thus, a multi-material topology optimization can be a promising strategy for dynamic scaling.
The studies considering different penalties enabled important remarks. First, the increasing of the penalization negatively affected the quality of results in such a way that the optimizer failed in performing the matching of 5th mode shape for Case 8. The same effect was observed for the matching of the frequencies, although the worst result was which is fully accepted. Even the highest penalty factor not allowed achieving a gradient-free solution. However, observing how the penalty affected the results, a suitable solution of such optimization problems considering just gradient-free designs is really a very hard task. For instance, the matching of both natural frequencies and mode shapes with the same quality as reached with maybe impossible to be achieved through a fully gradient-free approach.