Both Examples 4 and 5 are to show the effectiveness of the method in dealing with multiple types of structural components. Meanwhile, with the same design parameters, optimization results from pure sizing optimization are also given in Examples 1 to 4.
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This approach has been applied in benchmark studies with high accuracy and efficiency, and also been successfully used for practical and complex engineering problems. Young's modulus is 6. By using MSC. In order to make a suitable comparison, in this study, the lower bound on the frequency constraint is replaced with The initial designs for all radii are 0. By considering structural symmetry, the other case of the optimum structure can be obtained by deleting beam 5 while retaining beam 4 with the optimum sizes unchanged, as demonstrated in the work of Cheng and Liu.
As the optimum radii are quite different from those in the aforementioned work, 43 the structural mass is quite smaller, whereas the fundamental frequency keeps unchanged. With the use of GA in the work of Cheng and Liu, 43 a population size of and a maximum generation number of were used to obtain the optimal solution. It can be seen that the radius of beam 3 reaches its lower bound, ie, 0.
By decreasing all lower bounds to 0. The optimization results in Case B does not change too much compared with the results in Case A, and the structural masses obtained in pure sizing optimizations are larger than those achieved in topology and sizing optimizations. Therefore, those solutions in pure sizing optimization can be seen as local optima. A complete independent run consists of several iterations of such execution process. After comparisons as stated previously, it can be seen that the proposed method can approach reasonable results with remarkable efficiency.
Both the length and width of this plate are 0. By considering symmetry, only a quarter of the plate is designed, and the rest parts are obtained by using variable linking. As each stiffener corresponds to a discrete variable, which is used to decide their existence, there are 5, 20, and 72 discrete variables for each case, respectively, under the symmetry consideration.
For Case 1, it can be observed that this plate appears to require only two stiffeners along two diagonal lines crossing the plate. In Case 2, besides the stiffeners along the diagonal lines, another four stiffeners are required around the center. Similar case happens in Case 3, where the required stiffeners are mainly located around the corners and the center where the force is applied. As the plate is fixed at four corners, which is similar to a clamped beam loaded by a concentrated force in the middle, the resistant bending moments in the center and corners are quite large.
Therefore, the plate is mainly strengthened with stiffeners around the corners and the center. Meanwhile, as for the middle parts between the center and corners, the resistant bending moments are relatively small so the stiffeners becomes unnecessary at those parts. As a result, the stiffeners are disconnected in Case 3. For the three cases, with the increase of the amount of stiffeners, their initial structural mass is getting larger, whereas the obtained structural mass is getting smaller, which is due to the fact that the design space gets larger when the number of stiffeners increases.
For each case, the structural mass obtained in simultaneous layout and sizing optimization of stiffeners is smaller than that achieved from pure sizing optimization, and the number of required structural analysis is comparable to sizing optimization. The optimization results are then compared with those from the references. In the work of Fatemi and Trompette, 45 a similar design problem was considered.
Even though the geometry parameters in that literature are a bit different, the comparison between them is also possible. In the aforementioned work, 45 only two stiffeners were considered and the coordinates of the ends for both stiffeners were treated as design variables.
The material distribution is also along the two diagonal lines, and the obtained structure is strengthened in the center by having more materials. After those comparisons previously, it can be said that the obtained results with the proposed method are quite reasonable. A simplified mold used for composite material component forming is designed in this example.
The length and width of the structure are 1. This design is to find the best stiffener layout for this shell structure. In the initial design, there are 58 stiffeners. Since a stiffener corresponds to a discrete variable to decide the existence, there are 17 discrete variables with the symmetry consideration. In the meantime, the shell thicknesses for each stiffener are also considered as continuous variables.
Therefore, there are also 17 continuous size variables involved. After optimization, only 16 stiffeners are retained, whereas the considered constraint is satisfied. This is very similar to the results in the aforementioned example, ie, Example 2, where the optimal layout requires stiffeners around the corners and the center.
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Thus, it has shown that this result is also reasonable. All size variables get to their lower bounds, but the structural mass is still larger than that from integrated layout and thickness optimization for stiffeners. The structure considered in Case 2 in Example 2 is used in this example for concurrent layout optimization of beam and shell components in a single structure. Therefore, besides the beam stiffeners, the base plate is also considered to be designed. Similarly, only a quarter of the plate is designed, and by considering symmetry, the rest parts are obtained by using variable linking.
For this problem, in addition to the 20 discrete variables that correspond to the existence of each stiffener, another four discrete variables, which are used to decide the existence of each region of the plate, are also involved. It is found that all regions of the plate are removed, and 14 stiffeners are retained in the optimization result.
Even though the number of required stiffeners is 14, ie, larger than 12 in Example 2, the obtained structural mass is 0. Even though the maximum displacements in both cases are around the design boundary, the structural mass obtained in pure sizing optimization is much larger than that from concurrent layout optimization of beam and shell components.
This example deals with an engineering application, which is to optimally design a microsatellite. The satellite is divided into seven sections by eight clapboards, and these plates are made of aluminum with beam stiffeners. The design problem is to optimize the plate thickness for the eight clapboards and the stiffeners layout on these plates. The initial structures are designed with empirical experiences, and the total mass of the whole satellite is The mass for the design domain, which consists of eight plates and their beam stiffeners, is around 1.
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The initial values for these vibration modes are Since each stiffener corresponds to a discrete variable and there are totally stiffeners for the eight plates, there are discrete variables involved in the problem to decide the existence of each stiffener. Moreover, the thicknesses of the eight plates are also to be optimized as continuous variables. According to the problem formulations established in Section 2 , eight continuous variables corresponding to plate thicknesses are also included.
Together with their initial values and the lower and upper bounds, the optimization results on the continuous variables are given in the Supporting Information. After the optimization, the structural mass of the design domain is 0. If represented by percentage, this mass reduction is then The first and the second order of transverse bending vibrations are Therefore, from the results earlier, it has been verified that this method is effective in dealing with problems where multiple types of design variables are involved, and this method has the capability to handle practical engineering problems so that the results can be provided to the engineers as a design reference.
Pure sizing optimization results are also given in Examples 1 to 4, and the obtained structural masses are always larger than those achieved in design cases where topology or layout variables are also considered. This is not the focus of this paper, and it will be considered in our future work. In sizing optimizations, some design variables get to their lower bounds. If the lower bounds are decreased, the size variables cannot yet reach zero or a very small value, as shown in Example 1.
That is caused by the characteristics of singular optimum and disjoint feasible domains, which are always encountered in topology design problems. On the other hand, from the numerical examples previously, it can be observed that the number of structural analysis is even comparable to pure sizing optimization.
Thence, the efficiency of this method is noticeable, especially when compared with pure GA. In addition, for topology optimization designs, it might end up with a structure that contains too many thin members. Author: Stolpe, Mathias. Author: Stidsen, Thomas K. View graph of relations. In this paper, we present a hierarchical optimization method for finding feasible true solutions to finite-element-based topology design problems. The topology design problems are initially modelled as non-convex mixed programs. Laser Photonics Rev. Sardan, O.
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