PCF, SEA, and Energy Absorption (EA) are metrics employed to assess structures’ crashworthiness and energy absorption. The EA is calculated as follows (Salehghaffari et al., 2010): E A = ∫ 0 s F   d δ (1) Where F is the crashing force with the function of the displacement δ , and s is the displacement before failure of the specimen.

The SEA indicates the energy the structure absorbs (EA _total ) for one unit mass (M _total ); it is expressed as: S E A = E A t o t a l M t o t a l (2)

Crush force efficiency (F _max ) is expressed as the ratio of average (F _avg ) to peak crush force (Kim, 2002): C F E = F a v g F max (3)

Figure 2 depicts the architecture of a ship fender having 100 mm and 600 mm height and length (constant value) respectively, impacted to transfersal load (Jiang et al., 2010). As Figure 3 indicates, fenders have six types such as foam-filled double and single foam fender.

The impactor speeds are 0.2 m/s for quasi-static loads and 1 m/s (approximately 2 knots) for dynamic loads. A 1200 T weight is added to simulate speed collisions that fenders typically encounter. The impactor transitions to squash the structures, and the solid wall permits better deformation characteristics. Neck berthing is a frequently encountered situation. The dynamic force on the assessed fender length is assumed uniform based on this situation. On the other hand, the impact might be unpredictable on the structure of the fender and its rough and non-uniform surface (Guo and Yu, 2011a).

Figure 3 and Figure 4 presents the cross-section and the finite element model comprising six types foam filled single and double fender structures The Finite Element (FE) ABAQUS software was used to build the fender comprising aluminium foam filling and determine the effect of a free-falling mass impacting thin walls. This research integrated the fender with a hull and it offered a stationary boundary.

The walls of the fenders were modelled using four node shell continuum elements with five integration points along the element’s thickness direction. Moreover, the foam was modelled using eight node continuum elements with a reduced integration technique combined with the hourglass control. To avoid both artificial zero energy deformation modes and volumetric locking, enhancement-based hourglass control and reduced integration were applied. Based on a mesh convergence study of shells and foam elements, a 2 mm element size was chosen. A mesh convergence was addressed to ensure a sufficient mesh density and to accurately capture the deformation process. The contact interaction between all components was the general contact algorithm used to avoid interpenetration of fender walls, which is less intense in terms of computational time. Meanwhile, the contacts between the foam and the fender walls were modelled as a finite sliding penalty based contact algorithm, with contact pairs and a hard contact. The friction coefficient value for all contact surfaces was set at 0.3.

The meta-model comprises a mathematical method to use simulations to create outputs using the gathered input data. Such a model helps assess physical properties through several datasets employed to implement distinct meta-models. The experimental design comprises setting points in an area, e.g., data collection inputs (Fang et al., 2005). Test data based on physical evaluations or computer simulations based on the finite element technique were employed to build meta-models. The design space was used to identify useful data by specifying parameter ranges that were split into several points. Every point indicates several variables having different levels. The meta-model creation process uses DOE to reduce input data. It is a statistical approach to gather data using several simulations based on different variables to understand specific characteristics (Yin et al., 2014).

Radial Basis Function (RBF) is employed for geographic data and fixes non-uniform topology boundaries. RBF was used for several situations, including objective function approximation. Fang et al. (Gao et al., 2016b) used the RBF meta-model to optimise several vehicle body requirements concerning frontal impact. It was validated using finite element simulations to estimate the general vehicle structure. Considering an equal number of reaction observations, the outcomes indicated that RBF was better than RSM because of optimal result more accuracy and stability. Additionally, impact absorption ability was determined based on the fewest applicable analytical variables to construct the RBF framework. Evolutionary algorithms are integrated with surface reactions to create single- and multiple-objective constraints. Implementing single restrictions and multiple-objective optimisation required genetic algorithms to be integrated with surface reactions. Outcomes concerning circular tubes indicated moderate eccentricity outcomes leading to high construction efficiency. Structures had uniform forward crushing reactions and desirable impact absorption. The outcomes indicated up to 7% and 20% mass reduction and vertical impact results, respectively (Fang et al., 2014b).

The multi-objective optimisation design (MOD) was performed by using sectional design parameters such as wall thickness t, height h, foam density f, yield stress y, and a 600 mm constant length tube. Furthermore, several early studies (Lanzi et al., 2004) investigated 2 crashworthiness indicators that are tweaked simultaneously to identify the lowest CFE and maximum SEA when subjected to transverse impact loadings. The GA (Genetic Algorithm) is a standard optimisation technique since it bypasses the trapping capacity when searching in local optima for an optimum (Salehghaffari et al., 2011). For grading solutions, assigning ranking fitness, and determining number problems, the NSGA (Non-dominated Sorting GA) technique, such as the NSGA versions I and II, is a more efficient and favourable approach. Figure 5 illustrates the flowchart for optimising the double fender’s crashworthiness under impact loadings. The DOE method was used in the preliminary step to specify the design space and produce sampling sites for different angle loadings. The design solutions for the preliminary D-optimal models of the design objectives were attained using FEA in the 2^nd phase. Lastly, Pareto solutions of systems under stress are obtained in the 3^rd phase. In the 3^rd phase, the NSGA II method was used to obtain Pareto solutions of systems under different loading situations.

The specimens were carried out from the casting method. The fenders molds were made using iron plate. After that, the aluminum alloy 6,063 material melts into the fender molds (shown in Figure 6).

Finite element models making use of the ABAQUS were compared to outcomes of the investigation to determine if they were satisfactorily accurate for design optimisation. Moreover, simulation study was carried out using side frames with height 100 mm and a length of 50 mm to determine the value under the axial collision of fender (shown in Figure 7 and Table 2). The section view of deformations in Figure 7 compare between simulation and experiment results under axial load. It can be seen also in Table 2, the errors percentage between FEA and theoretical solutions are fairly small and it found that they have a sufficient agreement.

FEA model also compared between ABAQUS and LS DYNA results (Jiang et al., 2010). The fender impacted with velocity of 1 m/s, the weight of the 1200 T and fender length of 600 mm under the tranverse impact (in Figure 8). From Table 3 also can be found that a good agreement between both results (Jiang et al., 2010).

To find the sample points accurately, a radial basis functions meta-model was created. Moreover, within the design spheres of the 6 kinds of tubes made to undergo the axial stress, five extra random points (Hanssen et al., 2000a; Langseth et al., 2003) were generated. The responses of PCF and SEA at validation sites of the RBF and FE models were used to substantiate these models at a reduced cost. To determine the radial basis functions meta-model’s degree of approximation to the FEA results, the Relative Error (RE) (Renreng et al., 2020) is computed as follows: R E = y x − y ∼ x y x (4) where y ∼ x represents the radial basis functions models while y x represents the finite element result.

In five random sample points, the initial sample points pertain to the RBF and the FEA. Figures 9–11 show that the RE for these approximations of RBF meta-model was below 4% in model IV, V and VI. Therefore, the RBF model for the goal functions (PCF and SEA) is considered to have given satisfactory accuracy for optimisation of design. In 5 random sample points, the initial sample points pertain to the RBF and the FEA. Figures 7–9 show that the RE for these approximations of RBF meta-model was below 4%. Therefore, the RBF model for the goal functions (PCF and SEA) is considered to have given satisfactory accuracy for optimisation of design.

The deformation configurations of the fenders under quasi static and dynamic loadings are shown in Figures 12, 13. Also, the force—displacement diagrams can be seen in Figures 14, 15. From these figures, they can be seen that foam-filled double fenders (model IV—VI) have more ability to absorb energy than single fender. This phenomenon is caused the frictional interaction between the walls of the inner and outer fenders and the foam. Figure 15 shows displacement–force graphs of different cross section foam filled fenders under static and dynamic loadings. By regarding this figure, we can see that the highest PCF of 436.5 N for model IV and 762 N for model V under quasi static and dynamic loadings, respectively.

The geometry of the empty fender has weaker energy absorption compared to structures of other configurations that are exposed to transverse loadings, as displayed in Figure 16 and Figure 17. The combination structures showed higher energy absorption capacity and decreased peak crushing force because of the frictional contact between the inner/outer tubes and the foam-filler (Langseth et al., 2003; Salehghaffari et al., 2011). Consequently, these constructions have the capacity to improve the crashworthiness of shallow-walled tubes, especially those utilised in ship fenders.

Based on the crashworthiness parameters, for quasi static loading, the highest energy absorption value is the model V with SEA value of 3.62 kg/kJ and the lowest one is model I with SEA value of 2.15 kg/kJ. However for dynamic loading, the highest energy absorption value is the model V with SEA value of 7.09 kg/kJ and the lowest one is model I with SEA value of 3.79 kg/kJ.

To optimize the structure, four design variables and each varies within its domain were considered in this study. The design ranges of all the parameters are determined based on practical concerns. Considering the FE simulation runs affordable, this work selects D-Optimal design sample points (see Table 4).

Several parameters were used to produce the equations for the multi-objective optimisation of aluminium foam double tube. Multi-objectives were also employed when there was the issue of trade-offs among two or more opposing objectives. With respect to design parameters such as t, h, σy, ρ _f and objective functions, that is PCF and SEA for double circular tubes, new constraint functions and objectives were established. S E A h , t , ρ f , σ y , C F E h , t , ρ f , σ y 1.6   m m ≤ t ≤ 2.8   m m 80   m m ≤ h ≤ 95   m m 150   M P a ≤ σ y ≤ 240   M P a 110   k g / m 3 ≤ ρ f ≤ 260   k g / m 3 (5)

The Pareto fronts are figured by using MOD problems, as indicated in Eq. 5. The NSGA-II technique was used to investigate the design domain using radial basis functions meta-models. Moreover, the DOE technique was used to create an initial group of 200 design points for all MOD conditions. Pareto fronts graphs of CFE vs. SEA for model V were built using NSGA-II by determining the convergence of optimisations repeating for 20 generations. In both CFE and SEA criterion design situations, they were clearly in disagreement with one another. As shown in Figure 18, there was increase in SEA, which led to an unsuitable rise in CFE.

To define the faults between the radial basis functions models and the FEA, multi-objective optimisation was examined with respect to the design parameters of sectional fender of thickness t, height h, foam density f, and yield stress y. The NSGA-II optimisation and radial basis functions method were used to determine the Pareto fronts (shown in Figure 18) using five random sample points (Hanssen et al., 2000a). Under transversal loading conditions, all fenders were optimised. To determine the entry of foam-filler into the simulation of crashworthiness, suitable RBF models and MOD issues were used.

From Table 5, the optimal value of model V is in columns 4–7. Two objective functions such as SEA are in column 8 and PCF in column 9. It can be see that the structure designs in this table, it is found that the maximum SEA at 30.06 kJ/kg is achieved when the model V under dynamic impact. The wall thickness value is 2.49 mm which the material yield stress is 238.86 MPa.