2.1 Geometry model

This paper selected a typical outline for the airship with a slenderness ratio of 4. The length of the outline is 10,000mm and it is divided into two segments, both of which are ellipses, as shown in Fig. 1. The demarcation point is at x = 4,192.35mm, which is the position with the largest radius.

The envelope material of the airship capsule is composed of a resin-based composite material, and the reinforcing cable is composed of a flexible cable. The material properties are shown in Table 1.

Table 1 Material property of membrane and cable

Figure 1. Mechanical model of the airship.

When the airship is free without cables, the mechanical model is established as shown in Fig. 1. The left endpoint is constrained by six degrees x, y, z, Rx, Ry, Rz. The right endpoint is constrained by y, z and Rx, which allows the capsule to stretch freely in x-direction. The internal pressure difference is p = 0.01MPa.

2.2 Finite element model

In ABAQUS, 2D shell element S4R is used for membrane structure and 3D truss element T3D2 is used for reinforced cables. Global size of the element is 100mm.

Considering the tangential stress between the cable and the membrane, the ‘penalty function’ type is applied in contact model. It is necessary to judge whether the slave cable elements penetrate the master membrane elements. If there is no penetrating phenomenon, no treatment is carried out. If penetrating occurs, the normal contact force is used to limit. The magnitude of the contact force is proportional to the contact stiffness and contact displacement.

Due to the large deformation of the airship structure, geometric nonlinearity cannot be neglected and should be ON in the analysis setting. The solution technique is full Newton method.

3.1 Airship without cables

The load analysis is carried out for the mechanical model of the airship and the stress results are shown in Fig. 2. It is obvious that the largest stress of the airship structure is at the position with the largest radius. Therefore, setting one cable at the largest stress position x = 4,192.35mm will obtain the best load-bearing effect.

Figure 2. Stress results of the airship without cables.

3.2 The airship with 9 cables

It is feasible to apply warp and weft cables on the airship. However, the results of load analysis show that warp cables have a smaller load-sharing effect than weft cables. Taking into account the weight increase, only the weft cables are added in the airship design. According to engineering experience, nine reinforced cables are arranged on the airship, shown as Fig. 3(a). With reference to x = 4,192.35mm, one cable is placed every 1,000mm. With the same load p = 0.01MPa, the stress results are shown in Fig. 3(b) and (c).

By observing the stress distribution of the membrane and the reinforcing cables, it is found that the stress of the ropes at both ends is small: the minimum value is 31.79MPa while the maximum value is 79.35MPa. The reinforcing cables at both ends play a lesser role. Considering weight reduction and bearing load capacity, it is more reasonable to remove the left two cables and the right one.

3.3 Airship with 6 cables

Based on the analysis of airship with nine cables, the location of the remaining six cables is shown as Fig. 4(a) and the stress results are shown as Fig. 4(b) and (c).

Table 2 Comparison of different schemes

Figure 3. Model and stress results of the airship with nine cables.

Figure 4. Model and stress results of the airship with six cables.

Comparing the stress results of the two schemes, it is shown that the maximum stress of the membrane and the cable is approximately equal after the reduction of three cables, but the minimum stress of the cable is larger. The stress range of cables changes from 31.79∼79.35MPa to 62.21∼78.75MPa, indicating that the six reinforcing cables have shared more load. At the same time, the reduction in the number of cables will greatly reduce the weight of the airship and the average areal density will be reduced from 138.99g/m² to 135.90g/m², as shown in Table 2.

4.1 Optimisation plan

In order to achieve an optimisation design for the cable distribution, this paper combines the FEM software ABAQUS and the optimisation platform named Isight, shown as Fig. 5. By using ABAQUS software, parametric modeling of the airship with cables is established by the python language script function [Reference Lai and Jiang20] and the load analysis is implemented. With the Isight platform, parameter passing and optimisation method selection are realised.

4.2 Design variables and constraint conditions

The initial solution for optimisation design is the one with six cables in the Section 3. The design variables are the location of six cables in x-direction. With the left endpoint as the coordinate origin, the variables are set as z1, z2, z3, z4, z5, z6. The initial values are listed in Table 3.

Table 3 Design variables and constraint conditions

Figure 5. Optimisation plan of combining Isight with ABAQUS.

Figure 6. Process of the NSGA-II optimisation.

According to the analysis of Section 3, z3 should be fixed at the maximum radius position. The reinforcement cables should be distributed uniformly on the airship and cannot be very close. Therefore, it is reasonable to set the range of the other five variables as shown in Table 3.

4.3 Objective functions

One important factor of the airship is lightweight and the technical requirement in this paper is that the average areal density cannot exceed 140g/m². The other factor is bearing capacity, and we choose the maximum stress of the airship membrane structure under 10,000Pa pressure as the second objective function.

So the objective functions are listed as follows:

$$objective\;functions:\left\{ \begin{array}{l}{f_1}:\;\min \left( {average\;areal\;density} \right)\\{f_2}:\;\min \;\left( {max\;stress\;of\;membrane} \right)\end{array} \right.$$

5.1 Optimisation analysis

The optimisation of cable distribution belongs to multi-objective optimisation problem. It is difficult to obtain the best solution, which makes all objectives optimum because the objectives are in conflict with each other. Therefore, the Pareto optimal solution set is the suitable choice. There are many optimisation algorithm, such as EA (Evolution Algorithms), SA (Simulated Annealing), ACO (Ant Colony Optimisation), PSO (Particles Swarm Optimisation), G (Genetic Algorithm) and so on. Compared with other algorithms, GA has some advantages in convergence, solution diversity and avoiding the non-inferiority local optimal. New genetic algorithm named NSGA-II further introduces the elite strategy and retains the Pareto solutions of every parent generation directly to the next generation. This strategy improves the search efficiency and avoids the non-inferiority local optimum [Reference Deb, Agrawal and Pratap21,Reference Zhang, Liao and Chai22].

Table 4 Parameters of NSGA-II algorithm

Figure 7. Historical process of NSGA-II iteration.

The flow chart of NSGA-II algorithm is shown in Fig. 6. In the optimisation process, the new generation parameters are transmitted to the airship model, and the results of two objective functions are returned to the algorithm process before the number of generation. The parameters are listed in Table 4.

5.2 Optimisation results

5.2.3 Pareto solution set

An important feature of the NSGA-II algorithm is that the generation of the Pareto solution set is not just one solution. In engineering applications, the best optimal results may not be achieved easily. In the Pareto solution set, decision makers can choose the suitable one according to their preferences or engineering needs. The partial ones of the Pareto solution set are shown in Table 5.

Table 5 Part of Pareto solution set

Table 6 Optimal solution of NSGA-II

Figure 8. Correlation analysis of variables.

It can be seen from the Table 5 that the range of average areal density is small and is from 135.833g/m² to 135.857g/m². The technical requirement of the project is less than 140g/m², so there are enough margins for the cable connection kn, welded part of membrane and other factors. When the stress factor is more important, we can choose the solution with the least stress. When the mass factor is more important, we can choose the solution with smallest average areal density. Considering the sum of two objective functions, the result of the 234th step is the optimal solution that meets our requirements.

5.2.4 The optimal solution

This paper compares two designs of the arrangement with six cables and the optimal solution, as shown in Table 6. As we can see, the optimisation of the NSGA-II produces very good results. Besides, the finite element analysis of the optimal solution is conducted and the stress cloud diagram is shown in Fig. 9.

The results are listed in Table 7. It shows that the number and position optimisations of the cables have brought the obvious improvement on the stress distribution and weight reduction. The max stress of the membrane and the areal density are reduced obviously. Combining Fig. 9 with Fig. 3, the max stress is also reduced from 78.75MPa to 76.49MPa. The whole airship strength performance has been improved. The cable optimisation design is very important for the airship structure design, and the weight reduction and stress reduction could provide more design space for the fabric weaving of the membrane.

Table 7 Results of different designs

Figure 9. Stress results of the optimal scheme.