Crashworthiness design of foam-filled bitubal structures with uncertainty

https://doi.org/10.1016/j.ijnonlinmec.2014.08.005Get rights and content

Highlights

  • A foam-filled bitubal structure is presented.

  • Finite element model of foam-filled structure is established and verified.

  • A multiobjective robust optimization method is adopted.

  • Comparison between deterministic and robust MOO designs is performed.

  • Crashworthiness analysis and design of foam-filled structures are conducted.

Abstract

Structural optimization has been widely used to improve the crashworthiness of foam-filled thin-walled structures. However, majority of the existing optimization studies to date have not considered uncertainties for simplication. Its associated risk is that a deterministic optimization might deteriorate its optimality and/or violate design constraints when being present in uncertain environment. In this study, a multiobjective robust design optimization (MORDO) method is adopted to explore the design of foam-filled bitubal structures. To reduce the computational burden of highly-non-linear crash analysis, adaptive Kriging models are employed in the optimization process. In this strategy, sequential sampling points are generated over the design space and Kriging models are refitted in an iterative fashion. Based on the Kriging models, the multiobjective particle swarm optimization (MOPSO) algorithm is employed to perform the optimization, integrated with Monte Carlo simulation and descriptive sampling technique. The results demonstrate that the proposed method is capable of improving the robustness of Pareto solutions within the prescribed minimum requirements of reliability. Moreover, the influence of varying the emphasis on mean and standard deviation components is also analyzed, which can provide decision-makers with insightful design information.

Introduction

Foam-filled thin-walled structures have aroused increasing interest in automotive industry recently, for their extraordinary lightweight and energy absorption capacity. Substantial research efforts have been made by various experimental [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], analytical analyses [1], [11], [13], [16] and numerical methods [3], [6], [7], [9], [12], [15], [17], [18], [19], [20]. These literatures demonstrated that foam indeed is a class of ideal materials for energy absorption because they can undergo large deformation at nearly constant load. The presence of the foam-filler materials in thin-walled structures helps improve crushing stability and collapse modes, thereby increasing the overall crashworthiness [3], [4], [5]. Although foam-filled thin-walled tubes are able to enhance the energy absorption capacity, the energy absorption was found to be highly dependent on the foam density and wall thickness. In this regard, Ahmad and Thambiratnam [17] pointed out that the mean load and energy absorption response of foam-filled conical tubes subjected to axial quasi-static loading is more influenced by the foam density and wall thickness than the semi-apical angle. Seitzberger et al. [15] revealed that filling the thin-walled circular tube with high density foam could lead to global buckling. In another study, Reyes et al. [21] found that high density aluminum foam could increase the energy absorption of thin-walled square tube considerably, but the specific energy absorption can be lowered compared with the empty tube. For this reason, the selection of tube geometry and appropriate foam density are crucial to improve the crashworthiness of such structures.

Computational optimization signifies a more effective tool by seeking for an optimal design systematically, which helps engineers to attain the best crashworthiness of foam filled thin-walled structures. Due to the complexity of non-linear finite element analysis in crashworthiness simulation for thin-walled structures, surrogate or metamodel techniques have been exhaustively adopted as an effective alternative in crashworthiness optimization of foam filled thin-walled structures. For example, Bi et al. [22] optimized foam-filled tubes with respect to tube geometry and foam density to achieve maximum specific energy absorption (SEA) while maintaining a certain level of structural rigidity measured by the mean crushing force (MCF). The optimization results showed that the SEA tended to favor slender and thicker tubes with a moderate foam density for both single- and triple-cell tubes, which was consistent with their parametric study. Song et al. [23] proposed to conduct crashworthiness optimization of foam-filled tapered thin-walled structure by using surrogate models to maximize the SEA with respect to the wall thickness, taper angle and foam density. The results demonstrated that the simultaneous use of different surrogate models could be essential for both gradient and non-gradient optimization algorithms because they may generate different outcomes in the crashworthiness design. Furthermore, in order to take into account multiple conflicting design objectives, researchers have devoted to the investigations into multiobjective optimization (MOO) for foam-filled thin-walled tubes. In this regard, Hou et al. [24] applied MOO to the design of the squared tube with aluminum foam filler, where the SEA and peak crushing force were formulated as the design criteria. The illustrative case indicated that the two objectives strongly compete with each other. Zhang et al. [25] explored the crashworthiness design for a foam-filled squared bitubal thin-walled tube. The MOO design showed that this foam-filled bitubal structure has a better crashworthiness than the foam-filled monotubal structure in practice. Yang and Qi [26] optimized the crashworthiness of empty and foam-filled thin-walled square tubes under oblique loading using multiobjective particle swarm optimization (MOPSO) algorithm. Compared to the empty tube, the optimal foam-filled tube may have better crashworthiness under pure axial loading, but the optimal empty tube has more space to enhance the crashworthiness under oblique loading. Fang et al. [27] analyzed and optimized the bending behavior of three FGF-filled structures. They found FGF-filled structures, especially axial FGF tube, help to create more competent solution, compared to traditional uniform structure.

These above mentioned crashworthiness optimization of foam-filled thin-walled structures are restricted on deterministic optimization, where it is assumed that all the design variables and parameters involved are certain. In real world, uncertainties caused by manufacturing and operation are unavoidable and should be taken into account in analysis and design. A design through traditional deterministic optimization methods that seek an optimum without considering uncertainties may not achieve its expected performance in practice [28], [29], [30], [31], [32], [33], [34]. On the other hand, traditional optimization with uncertainties likely leads to a large scatter, which may not only cause significant fluctuations from the desired performance, but also add to life-cycle costs, including inspection, repair and other maintenance expenses. Thus, the concept of robust design optimization (RDO), which aims to reduce the scatter of the structural performance without eliminating the source of variability, has drawn increasing attention for solving real-world problems recently.

Regarding the uncertainty existing in metal foams, Reyes et al. [35] pointed out that as a cellular material, the foam has unevenly distributed pores generated randomly, whose sizes are also varying (as shown in Fig. 1). This will definitely lead to the variation of foam density. Moradi [36] claimed that greater randomness of shape and size of voids could be inevitably introduced into metal foams during the manufacturing process compared to solid metal, resulting in more uncertainty of performance indicators. Randrianalisoa et al. [37] found that the presence of cell randomness could decrease the thermal conductivity of cellular materials. Besides the foam density, thickness and dimension of tubes always come with certain manufacturing errors. For these reasons, the nature of crashworthiness optimization for foam filled thin-walled structures is indeed nondeterministic, which involves some degree of uncertainties. Unfortunately, there are very few studies available to deal with the uncertainties in the optimization procedure for crashworthiness design of foam filled thin-walled structures to the author׳s best knowledge. Most recently, Sun et al. [38] proposed a robust optimization procedure for a foam-filled tube with hexagon cross-section based on sequential Kriging models. In their work, the statistical information was solved by using the first-order derivative approximation method. However, this method could be inaccurate for highly non-linear problems such as crashworthiness design. Furthermore, their presented sequential sampling strategy only address single objective optimization problems, while the nature of the crashworthiness design for foam-filled thin-walled structure is a multiobjective optimization problem. As multiobjective optimization provides a wide spectrum of optimal solutions, the Kriging models require to reflect true responses well around all these optimums rather than only one optimum. Therefore, it is a crucial issue that how to expand the sequential sampling strategy from single optimization problem to multiobjective optimization problem.

This paper aims to develop a multiobjective sequential robust design optimization (MOSRDO) methodology for foam-filled thin-walled structures, and to demonstrate how uncertainties and different emphases on mean and deviation components affect the Pareto optimum. The rest of the paper is organized as follows. Section 2 proposes a detailed MOSRDO procedure, which comprises Kriging metamodel technique, adaptive sequential approximation framework, Monte Carlo simulation (MCS) and MOPSO methods. Section 3 depicts a particular design of a foam-filled thin-walled bitubal cylindrical column by the presented MOSRDO method, followed by the results and discussions in Section 4 and finally Section 5 draws some conclusions.

Section snippets

Multiobjective robust design optimization (MORDO)

A deterministic multiobjective optimization problem can be formulated as follows:{minF(X)=[f1(X),f2(X),,fi(X),],(i=1,2m)s.t.gj(X)0,(j=1,2,,n)XLXXUwhere X denotes a t-dimensional vector of design variables, XL denotes the lower bound and XU the upper bound. fi(X) and gi(X) are the ith objective function and jth constraint function, respectively. Since there is no uncertainty considered, it is named deterministic optimization.

Different from the deterministic optimization, robust design

Finite element (FE) model

The structure analyzed herein is a foam-filled thin-walled cylindrical tube subjected to an axial impact loading (Fig. 3). The bitubal arrangement, consisting of outer and inner walls with foam filler in between, is adopted in the tube. The length of tube is 250 mm, the diameters of outer and inner tubes are 50 mm and 25 mm respectively. The foam-filled tube impacts onto the rigid wall at an initial velocity of v=15 m/s. To generate sufficient kinetic energy similarly to vehicle crashing, an

Effect of combined AASO–AAMO

In this paper, the optimal Latin Hypercube sampling (OLHS) is implemented to generate initial sample data. Then, the combined AASO–AAMO is used to sequentially update the Kriging models until the deterministic Pareto frontier becomes stable.

In this study, the sample size of the initial training points is 50. After 9 iterations of combined AASO–AAMO the Pareto frontier is found stable adequately, i.e. the decrease in the number of Pareto solutions through the Pareto optimality checking is lower

Concluding remarks

To ensure the robustness for accommodating uncertainties, the design of foam-filled thin-walled structure is treated as a multiobjective sequential robust design optimization (MOSRDO) problem aiming to minimize the maximum impact force (Fmax) and maximize the specific energy absorption (SEA) while maintaining a certain level of the mean crash force (Favg). To enhance the computational efficiency, Kriging models are utilized to replace costly non-linear finite element (FE) simulations. To

Acknowledgment

This work was supported by the National 973 Project of China (2011CB711205), the National Natural Science Foundation of China (11202072), the Doctoral Fund of Ministry of Education of People׳s Republic of China (20120161120005), the Hunan Provincial Science Foundation of China (13JJ4036), the Shanghai Automotive Industry Science and Technology Development Foundation (SAISTDF/12-07), and the Open Fund of Traction Power State Key Laboratory of Southwest Jiaotong University (TPL1206). The first

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