Multiobjective crashworthiness optimization of hollow and conical tubes for multiple load cases

Much attention of current design analysis and optimization of crashworthy structures have been largely paid to the scenarios with single load case in literature. Nevertheless the designed structures may often have to be operated in other load conditions, thus raising a critical issue of optimality. This paper aims to understand and optimize the dynamic responses and energy absorption of foam-filled conical thin-walled tubes under oblique impact loading conditions by using multiobjective optimization method. The crashworthiness criteria, namely specific energy absorption (SEA) and crushing force efficiency (CFE), are related to loading parameters and design variables by using D-optimal design of experiments (DoE) and Kriging model. To obtain the optimal Pareto solutions of hollow and foam-filled conical tubes, design optimization is first performed under different loading case (DLC) using multiobjective particle swarm optimization (MOPSO) algorithm separately. The optimal designs indicate that hollow tube has better crashing performance than the foam-filled tube under relatively high impacting velocity and great loading angle. To combine multiple load cases (MLC) for multiobjective optimization, a double weight factor technique is then adopted. It is found that the optimal foam-filled tube has better crashing performance than empty conical tube under any of overall oblique loading cases concerned. The study gains insights in deriving multiobjective optimization for multiple load cases, providing a guideline for design of energy absorber under multiple oblique loading.     

Crushing analysis and multiobjective crashworthiness optimization of tapered square tubes under oblique impact loading

In this paper, a class of axisymmetric thin-walled square (ATS) tubes with two types of geometries (straight and tapered) and two kinds of cross-sections (single-cell and multi-cell) are considered as energy absorbing components under oblique impact loading. The crash behavior of the four types of ATS tubes, namely single-cell straight (SCS), single-cell tapered (SCT), multi-cell straight (MCS) and multi-cell tapered (MCT), are first investigated by nonlinear finite element analysis through LS-DYNA. It is found that the MCT tube has the best crashworthiness performance under oblique impact regarding both specific energy absorption (SEA) and peak crushing force (PCF). Sampling designs of the MCT tube are created based on a four-level full factorial design of experiments (DoE) method. Parametric studies are performed using the DoE results to investigate the influences of the geometric parameters on the crash performance of such MCT tubes under oblique impact loading. In addition, multiobjective optimization design (MOD) of the MCT tube is performed by adopting multiobjective particle swarm optimization (MOPSO) algorithm to achieve maximum SEA capacity and minimum PCF with and without considering load angle uncertainty effect. During the MOD process, accurate surrogate models, more specifically, response surface (RS) models of SEA and PCF of the MCT tubes are established to reduce the computational cost of crash simulations by finite element method. It is found that the optimal designs of the MCT tubes are different under different load angles. It is also found that the weighting factors for different load angles are critical in the MOD of the MCT tubes with load angle uncertainty.     

Crashworthiness design of functionally graded foam-filled multi-cell thin-walled structures

Foam-filled thin-walled structure has recently gained attention due to its excellent crashworthiness. Based on the previous study, a new kind of foam-filled thin-walled structure called as functionally graded foam-filled thin-walled structure has more excellent crashworthiness than the traditional uniform foam-filled thin-walled structure. Moreover, as far as we know multi-cell thin-walled structure has more excellent crashworthiness than the traditional single-cell thin-walled structure. As an integrator of the above two kinds of excellent thin-walled structures, functionally graded foam-filled multi-cell thin-walled structure (FGFMTS) may has extremely excellent crashworthiness. Based on our study, the crashworthiness of the FGFMTSs is significantly affected by the design parameter of the graded functional parameter m. Thus, in order to obtain the optimal design parameters, the FGFMTSs with different cross sections and different wall materials are optimized using the multiobjective particle swarm optimization (MOPSO) algorithm to achieve maximum specific energy absorption (SEA) capacity and minimum peak crushing force (PCF). At the same time, the corresponding uniform foam-filled multi-cell thin-walled structures (UFMTS) which have the same weight as these FGFMTSs are also optimized in our study. In the multiobjective design optimization (MDO) process, polynomial functional metamodels of SEA and PCF of FGFMTSs are used to reduce the computational cost of crash simulations by finite element method. The MDO results show that the FGFMTS with PCF in the initial period of its crash not only has better crashworthiness than the traditional UFMTS with the same weight but also performs superior balance of crashing stability. Thus, the optimal design of the FGFMTS with PCF occurring in the initial crash is an extremely excellent energy absorber and can be used in the practical engineering.     

Design optimization of cross-sectional configuration of rib-reinforced thin-walled beam

In this paper, a rib-reinforced thin-walled hollow tube-like beam (named as rib-reinforced beam) is presented for potential application in vehicle bumper. Through numerical simulation of the bending behavior under impact loads, the rib-reinforced beam is compared with thin-walled hollow tube-like beams filled with and without foam materials (empty beam and foam-filled beam) in crashworthiness. The effects of the shape of the reinforced rib are investigated and the shape optimization design is performed for increasing energy absorption and reducing the initial peak force. A multi-objective crashworthiness optimization formulation including maximum energy absorption, maximum specific energy absorption and minimum initial peak force is constructed based on the ideal point method (IPM). The optimum configuration of the reinforced rib is given with a normalized cubic spline function. Numerical example results show that, compared with the empty and foam-filled beams with same weights, the optimized rib-reinforced beam has higher energy absorption performance and lower initial crash force. It is found that for the rib-reinforced beam little rumple is formed around the compressed indention, which helps to retard the collapse of the side wall and means more energy absorption.     

Multiobjective crashworthiness optimization of circular aluminum tubes

This research deals with the development of circular aluminum tubes for crashworthiness. The dynamic crash responses of aluminum tubes are determined from the finite element simulation. In order to validate the FE results some dynamic impact test on aluminum tubes are performed. Multiobjective optimization technique is adopted to solve the problem of maximization of absorbed energy and specific absorbed energy of tubes. The D-optimal design of experiments [Atkinson AC, Donev AN, Optimum experimental designs. Oxford: Oxford Science Publications; 1992] and the response surface method are applied to construct an approximated design sub-problem and the optimization process is repeated until the convergence criteria are satisfied.     

Theoretical prediction and crashworthiness optimization of multi-cell triangular tubes

The triangular tubes with multi-cell were first studied on the aspects of theoretical prediction and crashworthiness optimization design under the impact loading. The tubes׳ profiles were divided into 2-, 3-, T-shapes, 4-, and 6-panel angle elements. The Simplified Super Folding Element theory was utilized to estimate the energy dissipation of angle elements. Based on the estimation, theoretical expressions of the mean crushing force were developed for three types of tubes under dynamic loading. When taking the inertia effects into account, the dynamic enhancement coefficient was also considered. In the process of multiobjective crashworthiness optimization, Deb and Gupta method was utilized to find out the knee points from the Pareto solutions space. Finally, the theoretical prediction showed an excellent coincidence with the numerical optimal results, and also validated the efficiency of the crashworthiness optimization design method based on surrogate models.     

Crashworthiness optimization design for foam-filled multi-cell thin-walled structures

Foam-filled thin-walled structure and multi-cell thin-walled structure both have recently gained attentions for their excellent energy absorption capacity. As an integrator of the above two kinds of thin-walled structures, foam-filled multi-cell thin-walled structure (FMTS) may have extremely excellent energy absorption capacity. This paper firstly investigates the energy absorption characteristics of FMTSs by nonlinear finite element analysis through LS-DYNA. Based on the numerical results, it can be found that the FMTS with nine cells has the most excellent crashworthiness characteristics in our considered cases. Thus, the FMTSs with cell number n=9 are then optimized by adopting a multi-objective particle swarm optimization (MOPSO) algorithm to achieve maximum specific energy absorption (SEA) capacity and minimum peak crushing force (PCF). During the process of multi-objective optimization design (MOD), four kinds of commonly used metamodels, namely polynomial response surface (PRS), radial basis function (RBF), Kriging (KRG) and support vector regression (SVR) for SEA and PCF, are established to reduce the computational cost of crash simulations by the finite element method. In order to choose the best metamodel for optimization, the accuracies of these four kinds of metamodels are compared by employing the error evaluation indicators of the relative error (RE) and the root mean square error (RMSE). The optimal design of FMTSs with nine cells is an extremely excellent energy absorber and can be used in the future vehicle body.     

Quasi-static response and multi-objective crashworthiness optimization of oblong tube under lateral loading

This paper addresses the energy absorption responses and crashworthiness optimization of thin-walled oblong tubes under quasi-static lateral loading. The oblong tubes were experimentally compressed using three various forms of indenters named as the flat plate, cylindrical and a point load indenter. The oblong tubes were subjected to inclined and vertical constraints to increase the energy absorption capacity of these structures. The variation in responses due to these indenters and external constraints were demonstrated. Various indicators which describe the effectiveness of energy absorbing systems were used as a marker to compare the various systems. It was found that unconstrained oblong tube (FIU) exhibited an almost ideal response when a flat plate indenter was used. The design information for such oblong tubes as energy absorbers can be generated through performing parametric study. To this end, the response surface methodology (RSM) for the design of experiments (DOE) was employed along with finite element modeling (FEM) to explore the effects of geometrical parameters on the responses of oblong tubes and to construct models for the specific energy absorption capacity (SEA) and collapse load (F) as functions of geometrical parameters. The FE model of the oblong tube was constructed and experimentally calibrated. In addition, based on the developed models of the SEA and F, multi-objective optimization design (MOD) of the oblong tube system is carried out by adopting a desirability approach to achieve maximum SEA capacity and minimum F. It is found that the optimal design of FIU can be achieved if the tube diameter and tube width are set at their minimum limits and the maximum tube thickness is chosen.     

Optimal shape design of thin-walled tubes under high-velocity axial impact loads

In this study, the objective is to maximize the crashworthiness of thin-walled tubes under axial impact loads by shape optimization. As design variables, parameters defining the cross-sectional profile of the tube as well as parameters defining the longitudinal profile like the depths and lengths of the circumferential ribs and the taper angle are used. The methodology is applied to the design optimization of a crash-box supporting the bumper beam of a vehicle for the loading conditions in standard EuroNCAP crash tests. The crash event is simulated using explicit finite element method. While the crash-box is fully modeled, the structural response of the remaining parts during the tests is taken into account by developing a lumped-parameter model. A hybrid search algorithm combining Genetic and Nelder & Mead algorithms is developed. The results indicate significant improvement in the crashworthiness over the benchmarks designs.     

Axial crushing and optimal design of square tubes with graded thickness

Introducing thickness gradient in cross-section is a quite promising approach to increase the energy absorption efficiency and crashworthiness performance of thin-walled structures. This paper addresses the deformation mode and energy absorption of square tubes with graded thickness during axial loading. Experimental study is firstly carried out for square tubes with two types of thickness distributions and numerical analyses are then conducted to simulate the experiment. Both experimental and numerical results show that the introduction of graded thickness in cross-section can lead to up to 30–35% increase in energy absorption efficiency (specific energy absorption) without the increase of the initial peak force. In addition, structural optimization of the cross-section of a square tube with graded thickness is solved by response surface method and the optimization results validate that increasing the material in the corner regions can indeed increase the energy absorption efficiency of a square tube.     

Comparisons of honeycomb sandwich and foam-filled cylindrical columns under axial crushing loads

For the conventional thin-walled energy absorber, the energy dissipation during a collision is concentrated in relatively narrow zones. This means that a great deal of material does not participate in the plastic deformation or enter the large plastic deformation stage. To expand the plastic deformation zones and improve the energy absorption efficiency, the authors presented a new type of honeycomb sandwich circular column. This innovative energy absorber is a composite structure composed of two circular aluminum tubes filled with core shaped as a large-cell honeycomb lattice. In this paper, six different honeycomb sandwich circular columns were investigated numerically. Comparisons of the interaction effect between tubes and filler, the deformation modes and the energy absorption abilities of these columns were conducted. The results were as following. The kagome sandwich column had the best energy absorption capability, followed by the columns sandwiched with triangle, hexagon lattices. In addition, foam-filled columns with different adhesive conditions were also simulated and compared with the honeycomb sandwich columns. It was found that increasing the adhesive strength improved the energy absorption and changed the deformation mode of the foam-filled columns. Furthermore, comparison showed that the honeycomb sandwich columns had higher specific energy absorption capability than the foam-filled tubes except for the strong bonded case. The kagome sandwich column performed best in crashworthiness, followed by triangle sandwich column.     

Quasi-static axial compression behavior of constraint hexagonal and square-packed empty and aluminum foam-filled aluminum multi-tubes

The axial crushing behavior of empty and Al close-cell foam-filled single Al tubes and Al multi-tube designs (hexagonal and square) were investigated through quasi-static compression testing. The effects of foam filling on the deformation mode and the crushing and average crushing loads of single tubes and multi-tube designs were determined. The foam filling was found to shift the deformation mode of empty single tube and empty multi-tube designs from diamond into concertina. In multi-tube designs the constraint effects and the frictional forces were found to increase the average crushing loads over those of single tubes. It was also found that foam filling induced a higher strengthening coefficient in multi-tube than single tubes. Although foam filling increased the energy absorption in single tubes and multi-tube designs, it was not effective in increasing the specific absorbed energy over that of the empty tubes. However, multi-tube designs were found to be energetically more effective than single tubes at similar foam-filler densities, proving a higher interaction effect in multi-tube designs.     

碰撞试验中泡沫填充铝管的最优化设计

对泡沫填充空心方铝管进行轴向撞击粉碎试验。此外，为获得更多有关撞击过程的信息，也对试验进行了有限元模拟分析。为找到更有效轻便的撞击减震器，并达到吸收最多能量的目的，在方矩形管的优化设计中采用了多元设计优化方法（MDO）。基于管的最佳几何尺寸考虑将具有最轻重量并且吸收能量最多作为设计目标。前期研究表明，使用高密度蜂窝材料填充会使管吸收更多能量，但重量不是最轻[Zarei HR，Kroger M．Optimum honey-comb filled crash absorber design．Mater Des 2007，29：193-204]。因此，为了解采用不同密度的泡沫填充管的撞击性能，进行了全面的研究，。采用MDO方法寻找一种优化填充管，使其吸收的能量与最优空心管吸收的能量一样多。     

Computer simulation and energy absorption of tapered thin-walled rectangular tubes

Tapered thin-walled tubes have been considered desirable energy absorbers under axial loading due to their relatively stable crush load and deformation response compared with straight tubes. This paper compares the energy absorption response of straight and tapered thin-walled rectangular tubes under quasi-static axial loading, for variations in their wall thickness, taper angle and number of tapered sides. Overall the study highlights the advantages of using tapered tubes as energy absorbers. In particular, the peak load required to crush the tubes decreases with the introduction of a taper, and as the taper angle increases. This is desirable for minimising the impact loads transmitted to the protected structure. The practical outcome of the study is design information for the use of tapered thin-walled rectangular tubes as energy absorbers in impact loading applications. Analysis has been undertaken using a finite element model, validated using existing theoretical and numerical models.     

Attempts to improve energy absorption characteristics of circular metal tubes subjected to axial loading

In this paper, experimental investigation of two new structural design solutions with the aim of improving crashworthiness characteristics of cylindrical metal tubes is performed. In the first design method, a rigid steel ring is press-fitted on top of circular aluminum tubes. When this arrangement of dissipating energy is subjected to axial compression, the rigid ring is driven into the cylindrical tube and expands its top area; then, plastic folds start shaping along the rest of the tube length as the compression of the structure continues. In the second design method, wide grooves are cut from the outer surface of steel thick-walled circular tubes. In fact, this method converts thick-walled tubes into several thin-walled tubes of shorter length, being assembled together coaxially. When this energy absorbing device is subjected to axial compression, plastic deformation occurs within the space of each wide groove, and thick portions control and stabilize collapsing of the whole structure. In the present study, several specimens of each developed design methods with various geometric parameters are prepared and compressed quasi-statistically. Also, some ordinary tubes of the same size of these specimens are compressed axially to investigate efficiency of the presented structural solutions in energy absorption applications. Experimental results show the significant efficiency of the presented design methods in improving crashworthiness characteristics and collapse modes of circular tubes under axial loading.     

Topology optimization method of lattice structures based on a genetic algorithm

A two-stage topology optimization method of lattice structures based on a genetic algorithm is proposed. The first stage is the form-finding analysis of lattice structures, and the optimal initial shape was achieved with the numerical inverse hanging method. The second stage is the topology optimization of single-layer lattice structures, which can be realized by changing the mesh size and the tube configurations to minimize the total weight of steel tubes subject to the design requirements. The mesh configuration optimization is realized through the adjustment of the nodal horizontal co-ordinates and the removal of tubes with lower stress. The maximum displacement of the structure, the maximum stress of the circular steel tubes, and the nonlinear buckling load are the state variables, and a genetic algorithm (GA) is the optimization algorithm. Different stress-limiting values used to delete the tubes were discussed. The numerical examples show that the two-stage topology optimization method for lattice structures proposed in this paper is correct and efficient. Furthermore, the forms of the optimized structure are rich, and the structure is lightweight and efficient.     

Optimization of the foam-filled aluminum tubes for crush box application

Axial impact crush tests on empty and foam-filled square aluminum tubes have been performed. Furthermore, in order to find more details about the crush processes, finite element simulations of the experiments have been done. In terms of finding more efficient and lighter crush absorber and achieving maximum energy absorption, multidesign optimization (MDO) technique has been applied for optimizing the square rectangular tubes. Based on practical requirements the optimum tube geometry, which absorbs maximum energy and has a minimum weight, has been determined. Results of previous work indicated that using high density honeycomb for filling the tubes will results more energy absorption but the weight efficiency has been lost [Zarei HR, Kröger M. Optimum honeycomb filled crash absorber design. Mater Des 2007;29:193–204]. Therefore, a comprehensive study has been performed in order to find out the crush behavior of tube filled with foam with different densities. The MDO procedure has been implemented to find an optimum filled tube that absorbed the same energy as an optimum empty tube can absorb.     

基于连续体结构拓扑优化的树状结构拓扑创构

针对概念设计阶段树状结构拓扑创构问题,提出了连续体结构拓扑优化方法。应用连续体结构拓扑优化法,建立体积约束下结构总应变能极小化（即结构总体刚度极大化）的拓扑优化模型,依据单约束优化问题的鞍点条件建立了优化迭代格式。研究了体积比、屋面结构刚度、屋面结构几何形式、设计空间高度等对树状结构拓扑形态的影响,并给出了具体应用算例。结果表明：随着材料用量的增加,树状结构拓扑随之复杂;随着屋面刚度的增加,树状结构分枝趋向于减少;不同的屋面结构几何形式会导致不同的树状结构拓扑形态;满足一定高度后,树状结构分枝形态不会受高度影响,仅是增加主树杆高度。     

Experimental and numerical crashworthiness investigation of empty and foam-filled end-capped conical tubes

Foam-filled thin-wall structures exhibit significant advantages in light weight and high energy absorption. They have been widely applied in automotive, aerospace, transportation and defense industries. Quasi-static tests were done to investigate the crash behavior of the empty and polyurethane foam-filled end-capped conical tubes. Non-linear dynamic finite element analyses were carried out to simulate the quasi-static tests. The predicted numerical crushing force and fold pattern were found to be in good agreement with the experimental results. The energy absorption capacities of the filled tubes were compared with the empty end-capped conical tubes. The results showed that the energy absorption capability of foam-filled tube is somewhat higher than that of the combined effect of the empty tube and the foam alone. Finally, the crash performance of the empty and foam filled conical and cylindrical tubes were compared. Results from this study can assist aerospace industry to design sounding rocket carrier payload based on foam-filled conical tubes.     

轴压下金属圆管耗能特性的改善

为提高金属圆管的抗冲击性能,分别通过两种设计方法进行试验研究。第一种方法将刚性钢环压入铝质圆管端部。轴压下,首先,刚性环扩大圆管端部;接着,沿管长方向形成塑性褶层。第二种方法在厚壁钢管外表面开槽。这个方法实际是将厚壁管分解成几个同轴相连的薄壁短管。受压时,每个钢槽内将出现塑性变形,结构的倒塌取决于厚壁部分。采用两种方法进行了不同尺寸试件的受压试验,并与同尺寸的普通管轴压下的耗能试验对比。结果表明,本设计方法和提出的圆管轴压倒塌模型均很有效。