Structures with supporting ribs are adopted in many fields of engineering. These ribs are attached to the main plate or shell to increase stiffness and reduce the stresses of a structure. Currently, much research in structural optimization has been devoted to size or thickness optimizations. In this study, the discrete positions of the ribs of a structure are optimized in addition to their thicknesses. The objective function, which is the total weight of a structure, is a continuous function with respect to the thickness of the ribs. However, it is a stepwise function of a dimensionless variable, which represents the set of positions of the ribs. Because of this stepwise objective function, the gradient method of optimization is not applicable. Therefore, we applied the micro genetic algorithm (MGA), which does not need derivatives of the objective function. To accelerate the rate of convergence, the stepwise objective function is interpolated to a smooth artificial objective function that does not alter the optimal solution.
With the recent construction of large-scale buildings, there has been an increase in the number of ventilation facilities for the ventilation of underground spaces with a dramatic increase in the underground sector. The entrances for ventilation shafts are installed above ground, and most of these ventilation shafts are located on sidewalks or in parks, which are easily accessible to pedestrians. Moreover, the heights of most shafts are easily accessible to passersby. Ventilation shafts consist of steel coverings and ribs. As studied by Lee et al. [
However, a systematic optimal design methodology for steel covering has not yet been developed, and in most cases, these coverings have been designed simply based on experience. This has led to several problems that are due to the large deflection or rupturing of the ribs that support the steel covering under unexpected overloads. As reported in The Law & Police News [
An accident that occurred in 2014 involving the collapse of ventilation covering [
To design this kind of structure safely and optimally, the positions and dimensions of ribs are most critical. A procedure of finite element analysis (FEA) was developed for varying locations and thickness of the ribs in order to acquire their minimum total weight while satisfying the stress constraint (and the strength constraint against maximum deflection, if necessary). Furthermore, in research by Coello and Pulido [
The positional set representing the locations of ribs is not a continuous variable; hence, it is mapped to a dimensionless position variable. A certain interval of position variables is matched to a certain set of positions. Therefore, there is no change in the locations of ribs through that interval of position variables, and the objective function of overall weight of the structure is constant. Therefore, the objective function is a stepwise function along with the position variable. In order to quickly extract a more optimal solution using this genetic algorithm, the stepwise objective function was modified to be a smooth interpolated function using a FORTRAN code. The convergence rates for the original and modified objective functions will be compared.
The conventional optimization of steel ventilation coverings is easily determined by adjusting the thickness of all of the ribs. However, it is technically difficult to optimize the locations of ribs. The overall shape of the steel ventilation covering is illustrated in Figure
Top and bottom isometric views of the steel ventilation covering.
Top view
Bottom view
Locations and shapes of basic ribs.
However, in the case of this model, the number of possible combinations becomes considerably large when three locations exist for each group; therefore, this model will be simplified for execution. The ribs within ventilation shaft are support beams that dominantly endure the load applied to the steel covering. Therefore, a small change in the location or thickness of a rib not only brings about a considerable change in the stress occurring in the steel ventilation covering but may also causes an excessive deflection or rupture of the entire structure. The objective of this study is to acquire the optimal locations and thickness of ribs that satisfy a reliable safety margin while minimizing their total weight.
The von Mises equivalent stress (and maximum displacement, if necessary) was the constraint conditions considered in this study. In particular, a stress of 50 MPa that satisfies a safety factor of 5 is considered to be the constraint condition required in this study.
The basic design of steel ventilation covering includes six ribs at the initial locations of 2, 5, 8, 11, 14, and 17, located at the center of each group, as illustrated in Figure
Model of three ribs in each group to find the best position.
As mentioned previously, the objective of this study is to determine the most appropriate locations and optimal thickness for ribs of ventilation covering in order to minimize their total weight. In order to acquire an optimal position set of discrete variables, a unique rib position variable was designated for the positional set as was done by Lee et al. [
A considerable amount of time would be required to analyze the 729 (=36) cases required to optimize the three possible locations of ribs in six groups. Therefore, the analysis and optimization were carried out using a simplified model (2nd quadrant of Figure
Simplified model considering the symmetry of the structure.
In this study, the rib position variable denotes a list of unrelated and discontinuous position sets. The rib position number associated with each position set is a value given by the user. For this example, if the quantity
Schematics of step and interpolated continuous objective functions used with the MGA.
The MGA is used in this manner for the optimization of the locations as well as the thickness of the ribs for both types of objective functions. According to PIDOtech [
Therefore, an interpolated continuous model is developed, as in Figure
There are five design variables: the thickness
PIAnO is a commercial package for executing process-integrated optimal design. It changes design variables for an objective function, executes finite element analyses, conducts a test of the objective function, checks the constraint conditions, and repeats this process as necessary to find an optimal solution. The use of this package was previously demonstrated by Lee et al. [
First, a 3D model of the steel ventilation covering and support was created using Pro-Engineer. Before the FEA began, mesh creation, boundary-condition allocation, and analysis stage setting were completed using the commercial package Hypermesh as illustrated in Figure
Boundary conditions and meshing of the steel ventilation covering (front 14th and 5th ribs).
A rib is placed at a certain location in each group during the optimization process (see Figure
The entire process begins by creating a model of the steel ventilation covering using a computer aided design tool. Then, the preconditioning process is executed using Hypermesh. The cmf file of Hypermesh is used to designate the location and thickness of a rib based on the stepwise or interpolated continuous objective function and is used as input file arranged by PIAnO as illustrated in Figure
Integrated optimal design procedure.
After completing the optimization process, an optimal solution satisfying all of the imposed requirements was acquired. As summarized in Table
Optimization results of each method.
Evaluation index | Design parameters | Initial state | Fixed position |
MGA |
MGA |
---|---|---|---|---|---|
Design variables | Thickness ( |
10 | 9.244 | 9.043 | 7.062 |
Thickness ( |
10 | 3.249 | 4.313 | 4.859 | |
Thickness ( |
10 | 3.249 | 4.983 | 3.496 | |
Thickness ( |
10 | 9.244 | 6.265 | 7.778 | |
Position set ( |
(5, 8, 11, 14) | (5, 8, 11, 14) | (5, 8, 12, 14) | (5, 7, 12, 14) | |
|
|||||
Objective | Weight ( |
0.195 | 0.122 | 0.119 | 0.113 |
|
|||||
Constraints | von Mises stress ( |
33.430 | 45.400 | 49.100 | 49.930 |
von Mises stress distribution in the steel ventilation covering.
Side view
Bottom view
The source of this accelerated convergence is as follows. In common genetic algorithms (GA) applied to continuous objective functions, the chromosomes of parents of good fitness are mixed to produce the offspring. Some offspring are located farther from the parents, but more are located near the parents. After many generations, offspring are more closely located near the optimal solution. For this case of a continuous objective function, the ratio of offspring with poor fitness to those with good fitness is relatively moderate. The ratio would decrease with progressed generations.
When one uses the stepwise objective function, a selected parent located near the boundary of the constant function would produce many offspring of poor fitness along with many offspring of good fitness. The ratio of offspring with poor fitness to good fitness may be relatively large. The ratio would not decrease with progressed generations, because the parents are near the fitness cliff as long as they are located near the boundary of the constant objective function.
In short, the use of the proposed interpolated smooth model removes the chance of individuals, located near the boundary of the constant objective function, being selected as parents for the next generation. With a constant objective function, individuals near the boundary have the same fitness as individuals located near the center of the division; however, with the interpolated smooth model, individuals near the boundary have lower fitness compared to individuals near the center of the division. Therefore, most parents are far from the fitness cliff, which accelerates convergence to the optimal solution. Finally, the interpolated smooth model is continuous and can be used in the gradient method of optimization from the first stage—or from a later stage where the convergence rate is very slow in most GA—to pinpoint the optimal solution in a hybrid manner.
Using a continuous function with MGA improves the accuracy for the position and thickness optimization problem of the steel ventilation covering. For verification, we also applied this method to another problem, which was studied by Lee et al. [
Schematics of the rising sector gate.
Meshes of the finite element model.
The results of the optimization are shown in Table
The optimal solution using MGA with an interpolated continuous objective function.
Evaluation index | Design parameters | Initial state | MGA |
---|---|---|---|
Design variables | Thickness ( |
20 | 12.478 |
Thickness ( |
20 | 25.825 | |
Thickness ( |
20 | 6.445 | |
Girder position set | (2, 5, 8) | (3, 4, 9) | |
|
|||
Objective | Weight ( |
94.293 | 69.192 |
|
|||
Constraints | von Mises stress |
118.993 | 119.46 |
Lateral displacement |
52.701 | 52.453 | |
1st frequency |
10.526 | 10.526 |
Finite element result of the rising sector gate.
The issues of discrete optimal design were evaluated in this study in order to minimize the weight of ribs for steel structures while satisfying various constraints. The optimization process was integrated using PIAnO, Hypermesh, Abaqus, and a FORTRAN code. The discrete optimization problem can be converted into a continuous variable problem, which resulted in a stepwise objective function problem. This stepwise objective function was modified to give an interpolated continuous function, which provides the ability for the MGA to possess fast global search characteristics.
The proposed conversion, modifying techniques, and the analysis results can be summarized as follows. The conversion technique of mapping a discrete position set into a continuous position variable was applied to design structures optimally by using the MGA with an interpolated continuous objective function. The MGA with an interpolated continuous objective function was used to effectively improve the convergence to the optimal structure design with the given constraints. It may be extended to a local gradient search—or a hybrid algorithm of the MGA followed by a local gradient search algorithm. An analysis of the ribs of ventilation covering was conducted to acquire the optimal solution, and it was revealed that the thicknesses of each rib in the optimal position set The optimization method applied to the ventilation covering yielded a 42% decrease in the total weight of the ribs while satisfying all imposed constraint conditions in comparison with the initial guess. In the previously constructed rising sector gate case, the optimization method decreased the weight by 27% with respect to the initial state.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by Kyungpook National University Research Fund, 2012. The authors are partially supported by the Brain Korea 21 Plus project of the Korean Research Foundation.