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We developed a fully automated multiobjective optimisation framework using genetic algorithms to generate a range of optimal barrel vault scissor structures. Compared to other optimisation methods, genetic algorithms are more robust and efficient when dealing with multiobjective optimisation problems and provide a better view of the search space while reducing the chance to be stuck in a local minimum. The novelty of this work is the application and validation (using metrics) of genetic algorithms for the shape and size optimisation of scissor structures, which has not been done so far for two objectives. We tested the feasibility and capacity of the methodology by optimising a 6 m span barrel vault to weight and compactness and by obtaining optimal solutions in an efficient way using NSGA-II. This paper presents the framework and the results of the case study. The in-depth analysis of the influence of the optimisation variables on the results yields new insights which can help in making choices with regard to the design variables, the constraints, and the number of individuals and generations in order to obtain efficiently a trade-off of optimal solutions.

During the first emergency phase, there is a need not only for emergency shelters for the local population, but also for larger constructions, such as warehouses and dispensaries. The latter are generally called collective service tents (CSTs) [

Scissor units, pantographs [

The advantage of using scissor structures for emergency tents is the easy transportability, the ease and speed of erection and folding (skilled labour is thus not needed for the installation), and the high volume increase between compact and deployed state [

The structural optimisation of scissor structures is however not trivial, because of the different types of design variables. The type of units and the way they are connected have an influence on the shape (shape optimisation). A second type of variable is the number of units which does not influence the shape but the typology of the structure (typology optimisation). The variation in the number of units affects the whole structure because it must meet the geometric and kinematic constraints of scissor structures. Lastly, size optimisation is required for the third type of design variables, which are the cross-sectional dimensions.

In this research, a barrel vault scissor structure shape is chosen because it can be composed of polar and translational units and has therefore a straightforward design and high compactness in its undeployed state (Figure

A barrel vault structure is composed of arches with polar units (blue bold solid line) connected through translational units (green bold dashed line).

In this paper, a general methodology using GAs is developed to obtain a set of structural optimal solutions for barrel vault shaped scissor structures in a fully automated framework without interaction with the user. The methodology behind this framework is illustrated by applying it to a case study: the optimisation of a barrel vault structure for disaster relief. The originality of the research lies in the in-depth analysis of the results, allowing us to understand the way GAs work on scissor structures and to provide new insights into the parameters which make the optimisation of barrel vault scissor structures efficient. The importance of the number of individuals and generations in the population is shown and the evolution of the constraints and design variables is discussed. The application and validation (using metrics) of genetic algorithms for the shape and size optimisation of scissor structures for two objectives are the novel contribution of this work.

In [

Table

Comparison of current research with previous researches on optimisation of scissors structures.

Authors | Algorithm | Objective(s) | Variables | Constraints |
---|---|---|---|---|

Kaveh et al. [ | Recursive Quadratic Programming Method | Weight | Cross-sectional (CS) area | Stress & buckling |

Kaveh and Shojaee [ | Genetic algorithm | Weight | CS area | Stress & buckling |

Thrall [ | Stochastic search | Weight | Shape and CS area | Stress & buckling |

Mira et al. [ | Simulated Annealing | Weight | Shape and CS area | Stress & buckling |

This paper | Genetic algorithm | Weight & compactness | Shape and CS area | Stress, buckling & deformation |

This paragraph extensively investigates a case study of a barrel vault shaped scissor structure in order to study the algorithm parameters and to reflect on the results.

The considered barrel vault is composed of polar scissor units for the arches and translational units between the arches with equal member’s length and has an outside height of 3,45 m (Figure ^{2} (determined by the head of the Médecin Sans Frontière Operation Centre Brussels Logistic Department [

The arches of the barrel vault are made of polar units and, in-between, translational units are placed to hold the arches together. The number of arches depends on the number of unit (in this figure, there are 7 units).

For the scissor elements, unhardened 5059 aluminium (^{3}) is used.

In Figure ^{2}); however, structural interference between the membrane and the arches is not considered in this optimisation exercise. With the appropriate load combination factors [

The transverse wind load causes pressure in one zone and suction in the two other zones (a). The longitudinal wind load causes a pressure in Zone D and suction in Zones G, H, I, and E (b). Two snow loads are considered, a uniform and a drifted one (c).

The two objectives of the optimisation are the mass of the structure (calculated as a function of the volume of all structural elements and the permanent loads such as joints and membrane) and the compactness of the structure (calculated as the volume of the undeployed barrel vault).

The following constraints are considered, each time for both the polar and the translational units and for two load combinations:

The maximum stress is 160 MPa (unhardened 5059 aluminium).

The maximum deflection is L/100. This constraint is less strict than for traditional structures and was set in agreement with our partner

The maximum horizontal displacement is H/100. This constraint is less strict than for traditional structures and was set in agreement with our partner

For the local buckling, an analytical calculation is implemented to check the member’s local buckling by taking into account the normal force and the bending moments around the two axes following Eurocode 9 [

The global buckling calculation is based on the eigenvalue buckling analysis (linear perturbation procedure) of the model using Abaqus/CAE 6.12 [

Seven design variables are defined in this research: the number units of the barrel vault structure (typology optimisation) and the height, width, and thickness of the rectangular tube cross section for both the polar and the translational units (size optimisation). The boundary values for the design variables are shown in Table

Boundary values for the seven design variables (height, width, and thickness of both the polar and the translational units).

Lower limit | Upper limit | |
---|---|---|

Number of scissor units | 5 | 10 |

Height | 40 mm | 150 mm |

Width | 40 mm | 150 mm |

Thickness | 2 mm | 10 mm |

One has

A genetic algorithm uses operators inspired from biological evolution to generate individuals, which represent in this case scissor structures with certain design parameters (e.g., 7 units with a hollow cross section of 45 × 45 × 7 mm). The quality of the individuals is then evaluated through a fitness function (e.g., a representation of the values of the objective functions). The set of solutions is then improved by applying biological operators [

For this research, the Nondominated Sorting Genetic Algorithm II (NSGA-II) is used [

Instead of opting for the Pareto optimal method, several other approaches are possible to deal with a biobjective optimisation problem. Considering a linear combination of the objectives as done by Shahvari et al. in [

In order to couple the structural analysis of the barrel vault scissor structure with multicriteria optimisation in an automated way, a framework is set up combining Matlab R2014b, Abaqus 6.12, and the programming language Python. This process is illustrated in Figure

NSGA-II generates a first population that is sent to Abaqus for static and dynamic calculations. The output is sent back to Matlab to calculate the fitness function.

The strength of this open framework is that it can be used for any parametrised structure. It is flexible as it can optimise one or multiple objectives by varying multiple continuous and discrete design variables and by taking into account many constraints.

The structural optimisation consists of many design variables and constraints; fourteen constraints are considered (for both load combinations LC1 and LC2: horizontal displacement, deflection, stress of the polar units, stress of the translational units, local buckling test of the polar units, local buckling test of translational units, and global buckling) and seven design variables are optimised (number of scissor units and the three dimensions of a box cross section for both the polar and the translational units). Analysing the behaviour of each individual constraint and design variable is cumbersome and slow and does not significantly contribute to the aim of this study. Therefore, in a first instance, a simplified structural optimisation will be run in order to analyse the global behaviour of the objective space, the constraints, and the design variables. In this simplified simulation, the cross section of the polar and translational units will be identical and only one load combination will be analysed, LC1 (dominant transversal wind load and a drifted snow distribution). For this first optimisation, 60 individuals and 20 generations are used. More explanation about the number of individuals and generations and a convergence check will be provided in Sections

This subsection handles the general behaviour of the Pareto front. After running the optimisation routine, instead of finding one single optimal solution, a front is obtained with several optimal solutions. This is the consequence of working with multiple objectives. The set of Pareto solutions in the design space is called the

The Pareto front obtained for 60 individuals and 20 generations shows the conflicting behaviour of weight and compactness. A small cross section procures a compact solution, but, to meet constraints, the cross section must be thicker, which causes heavier solutions (and vice versa).

Because of the two conflicting objectives, mass and compactness, a Pareto front is obtained (the range of the design variables of the Pareto set is given in Table ^{3}, and a weight of 573 kg). On the other hand, a bigger cross section has the opportunity to meet the criteria with a thinner cross section, but this affects the compactness of the whole structure (the lightest being the solution with a cross section of 53 × 51 × 3 mm, compactness of 0.721 m^{3}, and a weight of 433 kg).

Range of design variables of the Pareto set.

Range | |
---|---|

Number of scissor units | 7 |

Height cross section | 41 mm–51 mm |

Width cross section | 42 mm–69 mm |

Thickness cross section | 3 mm–6 mm |

Note that, in this simulation, the cross sections have a continuous variation in height, width, and thickness but, in reality, the cross sections will be chosen from a catalogue. This catalogue is not implemented in this simulation in order to have a better understanding of the relation between mass, cross section, and compactness. The interface is however open and the switch from continuous to discrete variables is therefore feasible.

Figure

(a) The objective space for 60 individuals and 20 generations and (b) zoom on the Pareto front. The red crosses above the Pareto front have a low number of scissor units, which is the reason why they fail.

The aim of this subsection is to determine which one(s) of the constraints are the most dominant in the structural optimisation of the barrel vault scissor structure. The evolution of the four constraints (stress, deflection, local buckling, and global buckling) was plotted using the

These figures show that global buckling is not an important constraint while the stress constraint is the most important one. The optimisation tries therefore to stay on the edge of the stress constraint.

Global buckling failure

Local buckling failure

Deformation failure

Stress failure

The first step is to analyse the global buckling behaviour of the construction. As shown in Figure

Global buckling is thus not the most important constraint; other constraints are more dominant. This is an important conclusion because it takes an important part of the computation time to make the global buckling calculation. By knowing that the global buckling calculation is not detrimental, this check can be removed from the optimisation loop (Figure

The calculation of the local buckling is an analytical calculation based on the normal force and bending moments in the most loaded beam [

The deformation failure has evolution that is quite identical to the stress evolution, with positive evolution of the deformation towards the limit but, from generation 10 on, the deformation stays quite constant and far under the limit. It could thus be stated that no individuals deform excessively. Hence, the serviceability limit state (SLS) of deformation is not the criterion on which the barrel vault is designed.

The stress constraint is the most important constraint as shown in Figure

In this subsection, the evolution of the design variables is analysed. The evolution of the number of scissor units and the three parameters of the box cross section of the scissor units (height, width, and thickness) are shown in Figure

(a) shows that the number of units is optimal at 7. The three other design variables are evolving to the bottom side showing that the simulation is reducing the cross-sectional dimensions in order to obtain better solutions in the Pareto front.

Number of scissor units

Height of the box cross section

Width of the box cross section

Thickness of the box cross section

The variation in number of scissor units is visible in Figure

In the first simulation, we opted for 20 generations. In this part, the behaviour of the optimisation routine will be analysed through some tools such as the generational distance, the hypervolume, and the rate of feasible solutions [

The GD represents how far in average the current set

While the GD is dropping with the generations, the HR is rising and the RF is fluctuating. This means that as the generations progress, the Pareto fronts are coming closer to each other and their extent is becoming bigger while procuring enough solutions that are failing.

Generational distance

Hypervolume

Rate of feasible solutions

Hypervolume for generations 3 (0.7668) and 16 (1.4978)

In biobjective problems, the hypervolume consists in calculating the area dominated by a set of solutions in the objective space and bounded by a reference point [

The rate of feasible (RF) solutions gives the percentage of individuals which satisfies all the constraints (Figure

NSGA-II is a genetic algorithm and works with populations of individuals. The first step in the optimisation process is to generate solutions which cover the whole objective space, in order to have an idea of which individuals fail and which individuals meet the constraints. As the number of generations progresses, the individuals are clustered around clouds of good solutions, which converge to the Pareto front.

It may be expected that if the number of generations increases, the Pareto front will contain more points, thus less gaps, and that it will converge to the “theoretical” actual Pareto front. Secondly, it might seem better to have more generations than more individuals, because generations are combining the genes of good solutions to obtain better children. Drawing conclusions on this matter is, however, not so straightforward.

The influence of the number of individuals on the number of needed generations was therefore studied. Taking into account the convergence criterion (general distance ≤ 0.01 for four consecutive Pareto fronts as explained in Section

This overview shows that, in general, the more the individuals, the smaller the number of generations needed until convergence but the higher the number of total iterations.

# ind. | # gen. | # iterations | HR |
---|---|---|---|

20 | 21 | 420 | 93 |

40 | 16 | 640 | 78 |

60 | 12 | 360 | 287 |

80 | 13 | 1040 | 139 |

100 | 10 | 1000 | 171 |

120 | 13 | 1560 | 88 |

140 | 9 | 1260 | 441 |

160 | 9 | 1440 | 167 |

In general, it can be seen that the more the individuals, the smaller the number of needed generations until convergence but the higher the total number of iterations. It is therefore more time-consuming to opt for more individuals as the total number of calculated solutions will be higher (1440 for 160 individuals versus 420 for 20 individuals). Secondly, there is no correlation between the number of individuals, generations, or iterations and the hypervolume. For example, the hypervolume of 140 individuals is much higher than the one of 160 individuals. In order to understand this phenomenon, the Pareto fronts of the different simulations have been plotted in Figure

The Pareto fronts of the simulations with different numbers of individuals per generation show that the high value of the hypervolume of simulations 140-6 and 60-12 is related to the extent of the Pareto front and not to the quality of the latter.

In general, it can be said that the more the individuals, the smaller the gaps between Pareto solutions. The Pareto fronts are therefore denser, contain more optimal solutions, and have solutions that are closer to the theoretical Pareto front (e.g., note the difference between case 80-13 and case 20-21). It takes however more time to calculate the optimal solutions. Thus, the following important design conclusion can be formulated: if the aim is to have a couple of Pareto solutions inside a wide objective space in a quick way, the designer should opt for a small number of individuals with a minimum (in this case) of 60 individuals. Below 60 individuals, the Pareto front has too few solutions. On the other hand, if the aim is to have a dense and accurate Pareto front and if enough calculation time is provided, then it is best to opt for a higher number of individuals (e.g., 160).

In Section

This Pareto front obtained with 160 individuals per generation reached convergence after 23 generations. It shows the design variables of the most extreme optima and of the solution that will be used for further research.

This research paper has presented a general, open, and flexible methodology using Abaqus and Matlab to obtain a set of optimal barrel vault scissor structure solutions in a fully automated framework without interference from the user. For the structural optimisation interface, several Python scripts are linked with Matlab (which contains the optimisation algorithm) and Abaqus (for the structural analysis). The fact that genetic algorithms are used gives, besides a wide search in the objective space, also the possibility of optimising other shapes of scissor structures, using discrete or continuous design variables. The objectives and the constraints can also be changed and the same methodology can be used even if the aim is the optimisation to one objective.

This paper shows the feasibility and capacity of this methodology by optimising a 6 m span barrel vault to weight and compactness and by obtaining a Pareto set of optimal solutions in an efficient way. The in-depth analysis of the influence of the optimisation variables on the results yields new insight which can help in making judicious choices with regard to the design variables, the constraints, and the number of individuals and generations in order to obtain efficiently a trade-off of optimal solutions.

As shown by the analysis of the design variables evolution with the indicators generational distance, hypervolume, and rate of feasible solutions, each solution of the Pareto set has an optimal number of scissor units of 7.

By mapping the constraints violations in function of the number of generations, the influence of the constraints was analysed; the conclusion is that stress is more dominant than global and local buckling and deformation. Because global buckling is not preponderant, the global buckling analysis was removed from the optimisation loop and only verified a posteriori for the optimal scissor structures. This means a significant gain in computation time (30%).

Studying the influence of the number of individuals and needed generations to obtain convergence (using the generation distance metric), it was found that the main clusters are mostly formed during the first generations, and, from then on, the new individuals are clustered into those clouds of good solutions. The more the individuals per generation, the smaller the number of needed generations until convergence but the higher the number of iterations, causing a higher computational time. Yet, a denser and smoother Pareto front is then obtained. In conclusion, the following design rule was formulated: if time is scarce and the aim is to have some Pareto solutions inside a wide objective space, the designer should opt for a small number of individuals per generation (minimum of 60 individuals). However, if the aim is to have a dense and accurate Pareto front, a higher number of individuals per generation are necessary. The final Pareto front of the full structural optimisation has been found with 160 individuals and 23 generations.

The authors declare that there are no competing interests regarding the publication of this paper.

The authors would like to thank the Red Cross and Médecins Sans Frontières for providing them with useful information about their expertise in the field of sheltering.