Abstract
Fuel consumption and safety are currently key aspects in automobile design. The foamfilled thinwalled aluminium tube represents a potentially effective material for use in the automotive industry, due to its energy absorption capability and light weight. Multiobjective crashworthiness design optimization for foamfilled double cylindrical tubes is presented in this paper. The double structures are impacted by a rigid wall simulating quasistatic and dynamic loadings. The optimal parameters under consideration are the minimum peak crushing force and maximum specific energy absorption, using the nondominated sorting genetic algorithmII (NSGAII) technique. Radial basis functions (RBF) and DOptimal are adopted to determine the more complex crashworthiness functional objectives. The comparison is performed by finite element analysis of the impact crashworthiness characteristics in tubes under static and dynamic loads. Finally, the optimum crashworthiness performance of empty and foamfilled double tubes is investigated and compared to the traditional single foamfilled tube. The results indicate that the foamfilled double aluminium circular tube can be recommended for crashworthy structures.
Keywords:
Crashworthiness; cylindrical tube; optimization; axial impact; NSGAII
1 INTRODUCTION
Consumers' growing awareness of safety and fuelefficiency has led the automotive industry to increase crashworthiness and decrease the weight of vehicles. The major challenge to achieve such targets has been to identify new materials and redesign structures. Simulation methods have been adopted because physical crash testing is costly. They allow the investigation of the behaviour of thinwalled tubes and to improve the energy absorption of materials by considering various parameters, including geometry, size, crosssection, and loading conditions refer to book as Jones (1989)Jones, N., (1989). Structural impact. Cambridge University Press. and the article of Lu and Yu (2003)Lu, G., Yu, T., (2003). Energy absorption of structures and materials. Woodhead Publishing Limited.. In the case of thinwalled circular tubes under axial impact, energy absorption behaviour has been studied by several scholars such as Reid (1993)Reid, S.R., (1993). Plastic deformation mechanisms in axial compressed metal tubes used as impact energy absorbers. Int J Mech Sci 35(12): 10351052.; Alghamdi (2001)Alghamdi, A.A.A., (2001). Collapsible impact energy absorbers: an overview. ThinWalled Struct 39(2): 189213.; Alexander (1960)Alexander, J.M., (1960). An approximate analysis of the collapse of thin cylindrical shells under axial loading. Q J Mech Appl Math 13: 1015.. The dynamic instability of different tube crosssections subjected to axial impact loading was reported by Jones (1989)Jones, N., (1989). Structural impact. Cambridge University Press..
The progressive buckling, inversion, and splitting of circular tubes, has previously been discussed by Reid (1993)Reid, S.R., (1993). Plastic deformation mechanisms in axial compressed metal tubes used as impact energy absorbers. Int J Mech Sci 35(12): 10351052.. It demonstrated the deformation modes of tubular metal rods under axial compression. The energy absorption of different tube structures, such as circular and square geometries, was studied by Alghamdi (2001)Alghamdi, A.A.A., (2001). Collapsible impact energy absorbers: an overview. ThinWalled Struct 39(2): 189213.. Cellular materials, in particular foams and honeycombs that result in limited increases in volume and weight, improve the crashworthiness of thinwalled tubes (Hassen et al., 1999; 2000; Hopperstad et al., 2000Hopperstad, O.S., Langseth, M., Hanssen, A.G., (2000). Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler. Int J Impact Eng 24(5):.475507.; ^{2001}Hopperstad, O.S., Langseth, M., Hanssen, A.G., (2001). Optimum design for energy absorption of square aluminium columns with aluminium foam filler. Int J Mech Sci 43(1): 153176.; Sentosa et al., 2000; Thornton, 2005Thornton, P.H., (2005). Energy absorption by foam filled structures.SAE paper 800081.). (Seitzberger et al. 1997Seitzberger, M., Rammerstorfer, R.F., Degischer, H.P., Gradinger, R., (1997). Crushing of axially compressed steel tubes filled with aluminium foam. Acta Mech 125: 93105.; ^{2000}Seitzberger, M., Rammerstorfer, F.G., Gradinger, R., Degischer, H.P., Blaimschein, M., Walch, C., (2000). Experimental studies on the quasistatic axial crushing of steel columns filled with aluminium foam. Int. J. Solids Struct. 37(30): 41254147.); Nurick et al. (2008) studied the energy absorption capability of empty and foamfilled circular tubes using a doublecell profile. The double circular tube was investigated by Guo et al. (2010aGuo, L.W., Yu, J.L., (2010a). Bending response of sandwiched double tube structures with aluminum foam core. In: Lu, J.W.Z., Leung, A.Y.T., Iu, V,P., Mok, K.M., editors. Proceedings of the ISCM II & EPMESC XII, Hong KongMacau, 2009. AIP CP1233, Part One. Melville, New York: American Institute of Physics. p. 602607.; ^{2010b}Guo, L.W., Yu, J.L., Li, Z.B., (2010b). Experimental studies on the quasistatic bending behavior of double square columns filled with aluminum foams. Acta Mech 213: 349358.; ^{2011a}Guo, L.W., Yu, J.L., (2011a). Bending behavior of aluminum foamfilled double cylindrical tubes. Acta Mech 222: 233244.; ^{2011b}Guo, L.W., Yu, J.L., (2011b). Dynamic bending response of double cylindrical tubes filled with aluminum foam. Int J Impact Eng 38(23): 8594.) using experimental and numerical testing.
Due to faster computers and better algorithms, there is increasing interest in optimization techniques applied to structure design, particularly to enhance the configuration of foam or cellular material fillings in thinwalled columns and tubes. In addition, optimization that maximises energy absorption and minimises the weight of foam fillers (Zarei and Kröger 2008aZarei, H.R., Kröger, M., (2008a). Bending behavior of empty and foamfilled beams: Structural optimization. Int J Impact Eng 35: 521529.; ^{2008b}Zarei, H.R., Kröger, M., (2008b). Optimization of the foamfilled aluminum tubes for crush box application. Thinwall Struct 46: 214221.) and honeycombs under axial loads, were investigated by Zarei and Kröger (2008c)Zarei, H.R., Kröger, M., (2008c). Optimum honeycomb filled crash absorber design. Mater Des 29: 193204.. Multiobjective optimization was also used to maximise specific energy absorption (SEA) and minimise the Peak Crushing Force (PCF) in honeycombfilled single and double polygonal tubes (Zarei and Kröger, 2008cZarei, H.R., Kröger, M., (2008c). Optimum honeycomb filled crash absorber design. Mater Des 29: 193204.). Aluminium foamfilled single and double thinwalled square columns were optimised using multiobjective optimization (Yin et al., 2011Yin, H., Wen, G., Hou, S., Chen, K., (2011). Crushing analysis and multiobjective crashworthiness optimization of honeycombfilled single and bitubular polygonal tubes. Mater Des 32: 44494460.; Hou et al., 2009Hou, S., Li, Q., Long, S., Yang, X., Li, W., (2009). Crashworthiness design for foam filled thinwall structures. Mater Des 30: 20242032.), which showed the foamﬁlled double conﬁguration had more efficient energy absorption than the empty tube. Acar et al. (2011)Acar, E., Guler, M.A., Gerçeker, B., Cerit, M.E., Bayram, B., (2011). Multiobjective crashworthiness optimization of tapered thinwalled tubes with axisymmetric indentations. Thinwall Struct 49: 94105. identified the maximum Crush Force Efficiency (CFE) and SEA absorption of tapered circular thinwalled tubes, also using multiobjective crashworthiness optimization .Testing the crashworthiness design under oblique impact loading of foamfilled thinwalled square columns indicated better outcomes (Nariman et al., 2006NarimanZadeh, N., Darvizeh, A., Jamali, A., (2006). Pareto optimization of energy absorption of square aluminium columns using multiobjective genetic algorithms. Proc Inst Mech Eng Part B  J Eng Manuf 220: 213224.).
Different mathematical programming techniques were used to identify optimal genetic algorithm solutions (Yin et al., 2011Yin, H., Wen, G., Hou, S., Chen, K., (2011). Crushing analysis and multiobjective crashworthiness optimization of honeycombfilled single and bitubular polygonal tubes. Mater Des 32: 44494460.; Nariman et al., 2006NarimanZadeh, N., Darvizeh, A., Jamali, A., (2006). Pareto optimization of energy absorption of square aluminium columns using multiobjective genetic algorithms. Proc Inst Mech Eng Part B  J Eng Manuf 220: 213224.). A twostage multifidelity method for honeycombs, to improve the crashworthiness design of functionallygraded foam structures, was applied by (Sun et al. 2010aSun, G.Y., Li, G.Y., Stone, M., Li, Q., (2010a). Application twostage multifidelity optimization procedure for honeycombtype. Comput Mater Sci 49: 500511.; ^{2010b}Sun, G.Y., Li, G.Y., Hou, S.J., Zhou, S.W., Li, W., Li, Q., (2010b). Crashworthiness design for functionally graded foam filled thinwalled structures. Mater Sci Eng A 527: 19111919.) using multiobjective particle swarm optimization. The tests produced promising findings for the development of crashworthiness behaviour.
In each of the abovementioned studies, the double circular tube could be further explored to understand the optimal design, because it has demonstrated good energy absorption capacity (Guo et al., 2011aGuo, L.W., Yu, J.L., (2011a). Bending behavior of aluminum foamfilled double cylindrical tubes. Acta Mech 222: 233244.; Li et al., 2012Li, Z.B., Yu, J.L., Guo, L.W., (2012). Deformation and energy absorption of aluminum foamfilled tubes subjected to oblique loading. Int J of Mechanical Sciences 54: 4856.). In the current study, ways in which the performance of empty and foamfilled double circular tubes under axial impact loading can improve crashworthiness are discussed. SEA and PCF are considered as the test criteria for the design variables, such as wall thickness, wall yield stress, and the density of the foam filler. The Doptimal technique was used to determine the design sampling space, and the design variables are established using Radial Basis Functions (RBF). Finally, the nondominated sorting genetic algorithm (NSGAII) was used to optimise the SEA and PCF and to compare the optimal crashworthiness of the structures (Liao et al., 2008aLiao, X.T., Li, Q., Yang, X.J., Li, W., Zhang, W.G., (2008a). A twostage multiobjective optimization of vehicle crashworthiness under frontal impact. Int J Crashworthiness 13: 279288.; ^{2008b}Liao X.T., Li, Q., Zhang, W.G., Yang, X.J., (2008b). Multiobjective optimization for crash safety design of vehicle using stepwise regression model. Struct Multdiscipl optim 35: 561569.).
2 CRASWORTHINESS INDICATORS
In order to evaluate the crash performance of energyabsorbing structures, it is necessary to define the crashworthiness indicators. The parameters of crashes are shown in Figure 1, such as energy absorption, PFC, and SEA which can be used to evaluate the efficiency of structures. Energy absorption is calculated as:
where F is the crashing force with the function of the displacement δ, and s is the displacement before failure of the specimen. The equation for SEA indicates the absorbed energy (E_{T} ) per unit mass (E_{T} ) of a structure is:
where M_{T} is the structure's total mass. In this case, the higher value indicates the greater energy absorption efficiency of a material.
The average crush force (F_{avg} ) is the response parameter for the energy absorption capability:
where total energy is absorbed (E_{T} ) during collapse and displacement (d). The crush force efficiency is defined as the ratio of the average crush force (F_{avg} ) to the peak crush force F_{max} .
3 METHODS AND MATERIALS
3.1 Finite element models
Empty and foamfilled double structures are analyzed for the numerical solution. The length (L) of the model aluminium foamfilled double tube is 250 mm, outer and inner wall diameters are, 2b = 64 mm and b = 32 mm, respectively and outer and inner wall thickness are both, t _{0} and t_{i} = 1.8 mm (Figure 2).
Thinwalled tube under axial impact loading with dimensions and boundary loading conditions: (a) empty double tube, and (b) foamfilled double tube.
Other parameters are the yield stress of the aluminium thinwalled tubes (σ_{y}) and the density of the foam filler (ρ_{f}). The bottom of the thinwalled tube is boundary conditions and the top is impacted a constant velocity by a moving rigid wall. Two types of loading speeds usually faced in a passenger vehicle crash event are simulated, i.e. 1 m/s for quasistatic and 10 m/s for dynamic impact. The impactor (110 kg) is attached to the top free end. The circular tubes such as bumper beam can absorb the kinetic energy about 10% of the compact car's mass (1,100 kg) according to Witteman (1999)Witteman, W.J., (1999). Improved vehicle crashworthiness design by control of the energy absorption for different collisions situation. Netherlands: Eindhoven University of Technology; PhD thesis.. Simulations of the crushing behaviour of the circular tubes are created by nonlinear explicit finite element (FE) software code. The FE models for double empty and foamfilled tubes are described in Figure 3.
Finite element model of double cylindrical tubes: (a) empty tube, and (b) foamfilled tube.
The tubular tubes wall are modelled using four node shell continuum elements with five integration points along the element's thickness direction. The eight node continuum elements are used to model foam filled tube with a reduced integration technique combined with the hourglass control. Moreover, the enhancementbased hourglass control and reduced integration are applied to avoid both artificial zero energy deformation modes and volumetric locking. The element size of 2 mm is chosen based on a mesh convergence study of shells and foam elements. To ensure a sufficient mesh density and to accurately capture the deformation process, the mesh convergence is addressed. Meanwhile, the contacts between as a finite sliding penalty based contact algorithm, with contact pairs and a hard contact, the foam and the tube walls are modelled. The friction coefficient value was set at 0.3 for all contact surfaces (Guo et al., 2011aGuo, L.W., Yu, J.L., (2011a). Bending behavior of aluminum foamfilled double cylindrical tubes. Acta Mech 222: 233244.; ^{2011b}Guo, L.W., Yu, J.L., (2011b). Dynamic bending response of double cylindrical tubes filled with aluminum foam. Int J Impact Eng 38(23): 8594.).
3.2 Material properties
The thinwalled circular tubes are fabricated using aluminium alloy A6063 T6 (Guo et al., 2011aGuo, L.W., Yu, J.L., (2011a). Bending behavior of aluminum foamfilled double cylindrical tubes. Acta Mech 222: 233244.; ^{2011b}Guo, L.W., Yu, J.L., (2011b). Dynamic bending response of double cylindrical tubes filled with aluminum foam. Int J Impact Eng 38(23): 8594.) with mechanical properties of young's modulus E = 59 GPa, Poisson's ratio v = 0.3 and density ρ = 2700 kg/m^{3}. Moreover, Figure 4 showed the uniaxial tension test with different thickness (1.0, 1.6 and 2 mm).
The uniaxial tensile stressstrain curves of profile material (Guo et al., 2011aGuo, L.W., Yu, J.L., (2011a). Bending behavior of aluminum foamfilled double cylindrical tubes. Acta Mech 222: 233244.; ^{2011b}Guo, L.W., Yu, J.L., (2011b). Dynamic bending response of double cylindrical tubes filled with aluminum foam. Int J Impact Eng 38(23): 8594.).
The Von Mises's isotropic plasticity algorithm as an elasticplastic material model is used to assess the structures constitutive behaviour. To define plastic hardening in the material's constitutive model, piecewise lines are performed. It found from the true stress and the plastic strain of experimental datas. Therefore the effect of strain rate in this model is considered, the insensitivity of strain rate of aluminium alloy material (Langseth et al., 1999).
In addition, the average mechanical property values of aluminium closedcell foam filler are applied, with nominal density ρ_{f} = 0.45 g/cm^{3}, young's modulus E = 625 MPa. The engineering stressstrain curve with different foam density (0.48 and 0.43 g/cm^{3)} is shown in Figure 5.
The uniaxial compression stressstrain curves of aluminum foams (Guo et al., 2011aGuo, L.W., Yu, J.L., (2011a). Bending behavior of aluminum foamfilled double cylindrical tubes. Acta Mech 222: 233244.; ^{2011b}Guo, L.W., Yu, J.L., (2011b). Dynamic bending response of double cylindrical tubes filled with aluminum foam. Int J Impact Eng 38(23): 8594.).
Dehspande and Fleck (2000) developed the constitutive behaviour which was based on an isotropic uniform material of the foam model. In this work, all of model are developed by nonlinear ABAQUS/Explicit software packages. In addition, to calculate the plastic behaviour of the aluminium foam, the crushable foam and the crushable foam hardening options were used. The equation of yield criterion in this model is described as,
where,
where, σ_{e} is the effective Von Mises stress and σ_{m} denotes the mean stress. The yield strength (Shahbeyk, 2007Shahbeyk, S., Spetrinic, N., Vafai, A., (2007). Numerical modelling of dynamically loaded metal foamfilled square columns. International Journal of Impact Engineering 34: 573586.) is defined as Y. In addition, α is the parameter used to define the shape of the yield surface and the plastic coefficient function ν_{p}. The plastic Poisson's ratio for aluminium foam repoted by Ahmad and Thambiratnam (2009)Ahmad, Z., Thambiratnam, D.P., (2009). Dynamic computer simulation and energy absorption of foamfilled conical tubes under axial impact loading. Computers and Structures 87(34): 186197.; Reyes et al. (2003)Reyes, A., Hopperstad, O.S., Berstad, T., Hanssen, A.G., Langseth, M., (2003). Constitutive modeling of aluminum foam including fracture and statistical variation of density. European Journal of Mechanics A: Solids 22(6): 815835. is zero, thus the parameter α can be calculated as follows,
The strain hardening effect equation for the initial model is defined by Ahmad and Thambiratnam (2009)Ahmad, Z., Thambiratnam, D.P., (2009). Dynamic computer simulation and energy absorption of foamfilled conical tubes under axial impact loading. Computers and Structures 87(34): 186197. as,
where, σ_{p} is the plateau stress, the material constants are α_{2}, γ, ε_{D} and β, and the effective strain is defined as . The strain of densification is derived as,
According to Hou et al. (2009)Hou, S., Li, Q., Long, S., Yang, X., Li, W., (2009). Crashworthiness design for foam filled thinwall structures. Mater Des 30: 20242032.; Ahmad and Thambiratnam (2009)Ahmad, Z., Thambiratnam, D.P., (2009). Dynamic computer simulation and energy absorption of foamfilled conical tubes under axial impact loading. Computers and Structures 87(34): 186197., the density of foam and the base material is defined as ρf and respectively.
3.3 Metamodel technique using RBF and DOptimal
The Doptima as design of experiment was used to reduced sample number for constructing metamodel (Kleijnen, 2005; Song et al., 2013). In addition, Metamodel technique, i.e. Radial basis functions (RBF) represent the relationship between the individual objective functions and the design variable vector. the RBF approximation of response function f'(x) was derived from the FE simulation as the following expression:
where the design variable vector and response values at n arbitrary design (training) points, x is the vector of normalised design input variables, with x_{i} representing the normalised coordinates of the i training point, ϕ is a radial symmetric basis function, i = 1 is the unknown interpolation coefficients, and r_{i} = x  x_{i}  = . The range of r is (0, 1) with 0 <c ≤ 1, and c is problem specific to the basis function and tuning parameters.
The RBF method has a fairly good accuracy for the global approximation of highly nonlinear responses and it has been successfully used in previous crashworthiness optimizations by Rais and Singh (2003)RaisRohani, M., Singh, M.N., (2003). Comparison of global and local response surface techniques in reliabilitybased optimization of composite structures. Struct Multidisciplinary Optim 26: 333345.; (Fang et al. 2005aFang, H., RaisRohani, M., Liu, Z., Horstemeyer, M.F., (2005). A comparative study of metamodeling methods for multiobjective crashworthiness optimization. Computers and Structures 85: 21212136.; 2005b); Salehghaffari (2011)Salehghaffari, S., RaisRohani, M., Najafi, A., (2011). Analysis and optimization of externally stiffened crush tubes. ThinWalled Structures 49: 397408.. The radial basis function metamodels used the equations shown in Table 1.
3.4 Multiobjective optimization using NSGA II
The main purpose of this study is to optimise foamfilled double aluminium tubes to maximise crashworthiness performance under axial impact loading. This is referred to as a multiobjective optimization scheme. In general, the mathematic formulation of multiobjectives can be expressed as:
Where x is the design variable vector, n is number of the objective function, f _{n} (x) is the objective function, ^{xL} = () denotes the lower bound, x_{u} = () denotes the upper bound of the design variables, s and p are the number of unequal and equal constraints, respectively. Franulovic et al. (2009)Franulovic, M., Basan, R., Prebil, I., (2009). Genetic algorithm in material model parameters identification for lowcycle fatique. Compt mater Sci 75: 505510. indicated the advantage of the genetic algorithm for instance it avoids trapping when searching for an optimum within local optima. Whereas, Deb (2001)Deb, K., (2001). Multiobjective optimization using evolutionary algorithms. John Wiley & Sons. explained the benefits of the NSGA II such as more effective and efficient in ranking solutions, assigning ranking fitness, and benchmarking number problems.
The flowchart of the optimization process for the multiobjective procedure is shown in Figure 6. It clarifies the multiobjective optimization using FE analysis, MATLAB software is used for RBF and NSGA II. However, the NSGA II based on a fast nondominated sorting principle was performanced by Hanssen et al. (2001)Hanssen, A.G., Langseth, M., Hopperstad, O.S., (2001). Optimum design for energy absorption of square aluminum columns with aluminum foam filler. Int J Mech Sci 43: 153176. to calculate the Equation 11.
4 RESULTS AND DISCUSSION
4.1 Model validation
Finite element models should be compared to theoretical solutions or experimental data in literature (Li et al., 2012Li, Z.B., Yu, J.L., Guo, L.W., (2012). Deformation and energy absorption of aluminum foamfilled tubes subjected to oblique loading. Int J of Mechanical Sciences 54: 4856.) to ensure they are sufficiently accurate for design optimization. The percentage of peak crushing force difference between the experiment test and the simulation (Figure 7) are 1.42%. The deformation patterns also between the simulation and the test shown in Figure 8 suggest the model was relatively similar (good agreement) and it could be continued for design optimization (Djamaluddin et al., 2014Djamaluddin, F., Abdullah, S., Ariffin, A.K., Nopiah, Z.M., (2014) MultiObjective Optimization of Aluminum Foam Filled Double Tubes Subjected to Oblique Impact Loading for Automobile Bumper Beam. Applied Mechanics and Materials 663: 9397; ^{2015}Djamaluddin, F., Abdullah, S., Ariffin, A.K., Nopiah, Z.M., (2015). Optimization of foamfilled double circular tubes under axial and oblique impact loading conditions. Thinwalled structures 87: 111.).
Deformation pattern of tubular tubes: (a) FE, and (b) experimental solution (Li et al., 2012Li, Z.B., Yu, J.L., Guo, L.W., (2012). Deformation and energy absorption of aluminum foamfilled tubes subjected to oblique loading. Int J of Mechanical Sciences 54: 4856.).
4.2 Finite element analysis
The configuration of the two optimal deforms of empty and foamfilled double circular are shown in Figure 9. Note that the two double circular tubes designs developed some stable folding patterns.
It can be seen that the double circular foamfilled tube has greater resilience than the empty structure in the same deformation time. This proves that the foamfilled tube has more ability to absorb energy than the empty tube, because of the frictional interaction between the walls of the inner and outer tubes and the foam filling, as reported by Zhang et al. (2012)Zhang, Y., Sun, G., Li, G., Luo, Z., Li, Q., (2012). Optimization of foamfilled bitubal structures for crashworthiness criteria. Mater Des 38: 99109..
The deformation mode represented in the crushing force versus the displacement is plotted in Figure 10. The maximum crushing force on the foamfilled tube under a dynamic loading is 74.32 kN and a quasistatic loading is 43.43 kN. Similarly, for the empty structures the maximum crushing forces under dynamic and quasistatic loads are 45.54 kN and 38.46 kN, respectively.
Crushing force vs. Displacement of the structures for the cases of: (a) dynamic impact, and (b) quasistatic impact.
In addition, the values under quasistatic load conditions are lower than the values under dynamic load conditions, showing the effects of impacts differed according to the characteristics of the structures. The energy absorption and specific energy absorption capability of both tubes showed the foamfilled double tube is more effective than the empty tube under dynamic and quasistatic load, as shown in Figure 11.
Structures under quasistatic and dynamic impacts: (a) energy absorption, and (b) specific energy absorption.
In addition, when the velocity is analyzed, it was found that the EA and SEA of the foamfilled double tube increases approximately 49.43% when compared to the empty tube, at a rigid wall impact speed of 10 m/s. The EA of the empty double tube increases more than 46.43% under dynamic impact when compared to the quasistatic impact.
These results describe significantly different values between the EA and SEA of foamfilled and empty tubes, due to the frictional interaction between the aluminium wall and the foam filling. Moreover, the EA and SEA values also increase when the velocity is increased. To reduce the required sample number for constructing the radial basis function metamodels, in this case the crashworthiness performance is formulated using the Doptimal technique.
However, the constructed radial basis function metamodels must be validated, because they directly affect the optimization results. The RBF metamodel should be accurate at the sample points around in which it is constructed. To validate the models, five extra random points are generated in the design domains of the four types of tubes, which are impacted at 1 m/s and 10 m/s. Both the FE model and the RBF metamodel are used to predict the SEA and PCF responses at these validation points. To measure the approximation degree of the RBF metamodel against the FE analysis, the relative error (RE) is evaluated as:
where (x) is the radial basis function model and y(x) is the finite element result. The model results for the FE and RBF of the double empty and foamfilled tubes under axial and oblique impact angles, are obtained from the equation 12. The error between the FE analysis and the RBF model at the five random sample points are summarised in Table 2. The RBF metamodel approximation is less than 3.5%. However, in the design optimization the RBF model provided sufficient accuracy.
For multiobjective optimization, there are more than one objective and tradeoffs between two or more conflicting objectives. The design variables are the level of SEA and PCF defined in Equation 6, enabling a comparison to be made of the calculated crashworthiness of empty and foamfilled structures.
New objectives and constraint functions for double circular tubes with respect to design variables such as b, t, σ_{y} and ρ_{f} were constructed from Equations 1214 and the SEA and PCF. MOD problems can be calculated to obtain the Pareto fronts, as shown in Equations 1314. Based on radial basis function metamodels, the NSGAII algorithm is adopted to investigate the design space and create an initial 200 design point population for all cases of MOD. By considering the convergence of optimizations iterating for 30 generations, the PCF versus SEA Pareto front graphs are generated using NSGAII for the tube structures. It can be clearly seen that they conflict with each other in all SEA and PCF design case criteria. Each of the double tubes shows an increasing SEA which leads to undesirable PCF levels, as seen in Figures 1213.
Pareto fronts of SEA vs. PCF or empty and foamfilled double circular tubes under quasistatic impact loading.
Pareto fronts of SEA vs. PCF for empty and foamfilled double circular tubes under dynamic impact loading.
Case one: Empty double circular tube
This tube is represented by the inner diameter wall (b_{i} ), outer diameter wall (b _{0}) which 2b_{i} = b _{0} = b, tube wall thickness (t), and tube wall material (σ_{y}). The multiobjective design problem is shown as follows,
Case two: Foamfilled double circular tube
The foamfilled double tube with ρ_{f} as the density of the foam filler is calculated using multiobjective optimization:
4.3 Comparison of empty and foamfilled double circular tubes
The multiobjective optimizations shown in Eqs. 1415 are used to compare the crashworthiness of the structures. The foamfilled double circular tube has the same design variable dimensions and boundary and loading conditions. The comparison of deformation patterns in the different structures can be seen Figure 11. The energy absorption was plotted against the deformation length for different design concepts. In summary, the empty geometry had lower energy absorption under axial loads. Based on Figs. 1415, it is shown that the foamfilled configuration has better energy absorption than the empty geometry. The combination structures have lower peak crushing force and more energy absorption capacity due to the frictional interaction between the foamfiller and the inner and outer tubes. Thus, this type of structure can improve the crashworthiness performance in thinwalled tubes, especially associated with vehicle design.
Pareto fronts of SEA vs. PCF of single and double foamfilled tubes under quasistatic impact loading.
Pareto fronts of SEA vs. PCF of single and double foamfilled tubes under dynamic impact loading.
Table 3 lists the optimal configuration of empty and foamfilled tubes (columns 47) and the ideal values for two objective functions: SEA (column 8) and PCF (column 9). First, for each of the structure designs in this table, the maximum SEA at 18.54 kJ/kg is achieved when the empty tube is under axial impact. The preferable wall thickness value is 2.643 mm which the material yield stress is 200.73 MPa.
This shows a contradiction between the two objective crashworthiness functions, and the MOD is applied in such situations. Second, the optimal diameter of each double circular tube for maximum SEA generally varies under each type of impact condition. For example, the optimal sectional diameter of the foamfilled tube under axial impact to achieve maximum SEA is 85.61 mm. Third, the minimum PCF is close to 146.32 kg/m^{3} for both impact angles in the foamfilled tubes. The optimal SEA value on impact of the double tube, was found at the highest foam density of 219.65 kg/m^{3}. However, the optimal foam density for SEA maximisation is lower under pure quasistatic impact than dynamic impact (Hou et al., 2009Hou, S., Li, Q., Long, S., Yang, X., Li, W., (2009). Crashworthiness design for foam filled thinwall structures. Mater Des 30: 20242032.).
The multiobjective optimization was explored with respect to the design variables of wall diameter (b); outer thickness of wall (t _{0}); inner thickness of wall (t _{i}); yield stress of the tubes (σ_{y}), and density of the foam filler (ρ_{f}).
4.4 Comparison of single and double foamfilled circular tubes
The traditional foamfilled single tube was chosen and compare to the double cylindrical tube, where Eq. 15 defines the multiobjective optimization.
Case three: Foamfilled double circular tube
The single circular tube had a diameter of 64 mm, was 250 mm in length, and had the same design variables, boundary, and loading conditions described in Figure 2. New objective and constraint functions were constructed by the RBF, with respect to design variables b, t, σ_{y}, ρ_{f} for the foamfilled single circular tube.
The Pareto fronts for quasistatic and dynamic impact loads on foamfilled single and double circular tubes, using RBF metamodels and NSGA II optimization, are plotted in Figs. 1415. Again, it can be seen that the Pareto front of the single tubular structure is worse than the double circular tube. This means from a safety view point the double tube absorbs more energy under the same level of peak impact force. In these conditions the double structure appears to be a potentially better crashworthiness structure.
5 CONCLUSIONS
The crashworthiness design for thinwalled aluminium foamfilled circular tubes was explored in this paper. Criteria such as the specific energy absorption and peak crushing force were calculated under axial impact loadings at different impact velocities. The multiobjective problems based on the radial basis function were constructed using finite element analysis for the design variables. The maximum SEA and minimum PCF under quasistatic loading is 16.46 kJ/kg and 53.91 kN, respectively. However, under dynamic impact loading, the maximum SEA is 19.54 kJ/kg and the minimum PCF is 46.54 kN. It was also found that an increase in the impact speed on the tube led to an increase in the specific energy absorption and peak crushing force.
NSGAII was used in the multiobjective optimization of the SEA and PCF values in circular double structures. The optimization process examined both empty and foamfilled double and single tubes. Under pure axial impact, the results showed that the crashworthiness performance of the foamfilled tube improved compared to the empty tube, and the performance of the double structure improved compared to the single tube. This demonstrates that double cylindrical tubes have good potential energy absorbing attributes in crashworthiness structure applications, which can protect vehicle occupants involved in accidents or collisions.
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Publication Dates

Publication in this collection
June 2015
History

Received
17 Oct 2014 
Reviewed
28 Nov 2014 
Accepted
01 Dec 2014