This paper presents the optimization result in the crashworthiness problem for maximizing absorbing energy of cylindrical tube. To simulate a complicated behavior of this kind of crash problem, a self-developed explicit finite element code is used. The response surface method based on stochastic process is used, that is especially good at modeling the non-linear, multi-modal functions that often bring about in engineering. The main characteristics of using response surface for global optimization lies in balancing the need to exploit the fitting surface for improving the approximation. It can be shown that how these approximating functions can be used to construct an efficient global optimization algorithm. Especially, with the comparison of result by classical optimization method, it can be shown that presented optimization method is independent of noise factor and existence of local minimum.
In the automotive industry, crashworthiness of a design is of special interest. Non-linear finite element analysis, such as in LS-DYNA [10], PAM CRASH [20], is applied to predict the structural responses. Conclusions from these computations can lead to significant design modifications. It is often hard to determine these design modification from the analysis results. In some cases many variations are tried before a satisfactory design is found. In design considering crashworthiness, only a few algorithms [12] are available to perform sensitivity analysis and optimization as known in linear static analysis.
Tubular structure is very important part in the point of view of crashworthiness, because this kind of structure can reduce the occupant injury in a collision. However, only several attempts [8] have been made to optimize the crashworthiness characteristics of tubular structures. With the implicit finite element analysis, many researchers make an effort to accompany sensitivity analysis for this optimization problem. Nevertheless, there are crucial difficulties for such non-linear sensitivity analysis [9] including noise factor due to contact and large deformation. A ‘noisy’ response may also result in sub-optimal solutions caused by the multitude of local minima. For the explicit finite element method, Yamazaki and Han [8] had gained successful result with general polynomial approximation. This method is very simple and general. But, there are burning questions related with computational efficiency and approximating accuracy.
The response surface method [3], [13] and [17] relies on the fact that the set of designs on which it is based on well selected. However, randomly selected designs may cause an inaccurate surface to be constructed or even prevent the ability to construct a surface at all. For the most simulation case such as sheet metal forming, car crash simulation, etc., simulation is very time-consuming. So, the overall efficiency of the design process depends heavily on the appropriate selection of a design set on experimental design (DOE) is needed. Many experimental design criteria are available but one of the most popular criterion is the D-optimality criterion [5]. In many problems, a function approximation can be constructed on the basis of function values only, and response surface methods based on polynomial functions, or several researchers have studied neural networks. With the use of polynomial interpolation, it becomes difficult to determine the order of the polynomial, and the amount and distribution of data that must be used in developing a suitable response surface. Similar problems exist in the use of neural networks. A process of trial and error is typically used in this approach. Furthermore, in the design optimization problems involving non-linear dynamic analysis, there are often ‘noisy’, the chaotic nature of their response can cause large differences in response to small design changes. In addition, the presence of high frequency noise in the acceleration response of impact or crash problems implies that the unfiltered response becomes almost meaningless.
In this paper, the response surface method based on stochastic modeling process will be adopted for optimization of crashworthiness characteristics of cylindrical tube. The simulation result that have ideal crush deformed pattern without wrinkling is shown. Furthermore, through comparison with the result by classical method, real difficulty due to noise factor could be discussed.
In order to maximize crashworthiness characteristics of cylindrical tubular structure, response surface method based on stochastic process was introduced. The explicit finite element code is used for this kind of non-linear large deformation contact problem. In the response surface procedure, approximating data are comparatively good agreement with original simulation data. In addition to composing approximating surface, general global optimization procedure has been carried out for non-linear problem. With this method, a good result could be achieved with only a few simulations and the fact that proposed method is fairly well independent of noise factor could be seen by constraint axial–force curve.
Comparing the deformed shape by optimization using classical optimization method and presented response surface method, numerical deformed shape of cylinder by response surface method has nearly ideal crush pattern without wrinkling mode.
