Factor screening and multivariable crashworthiness optimization for vehicle side impact by factorial design

Abstract

This paper demonstrates the application of factor screening to multivariable crashworthiness design of the vehicle body subjected to the side impact loading. Crashworthiness, influenced unequally by disparate factors such as the structural dimensions and material parameters, represents a natural benchmark criterion to judge the passive safety quality of the automobile design. In order to single out the active factors which pose a profound influence on the crashworthiness of vehicle bodies subjected to the side impact loading, the unreplicated saturated factorial design is adopted to tackle the obstacle from the factor screening due to its huge benefits in the efficiency and accuracy. In this paper, two different kinds of vehicles are analyzed by the unreplicated saturated factorial design for multivariable crashworthiness and the optimization results enhance the crashworthiness of vehicle. This method overcomes the limitations of design variables selection which depends on experience, and solves the in-efficiency problems caused by the direct optimization design without the selection of variables. It will shorten the design cycles, decrease the development costs and will have a certain reference value for the improvement of the vehicle’s crashworthiness performance.

Appendix A

Appendix A. Appendix of functions

In case A, the approximate RS models of the 4 responses with 28 design sample points are given in (A.1)–(A.4), the RS models of U, V and A are quadratic and the RS model of M is linear.

$$\begin{array}{rll} U& = &399.1371-18.4456x_{1} -22.9388x_{2}\notag\\ && +4.157x_{3} -54.9146x_{4}-13.4024x_{1}^{2}\notag\\ &&+1.8172x_{2}^{2} +5.1301x_{3}^{2}+16.4958x_{4}^{2} \notag\\ &&+3.0099x_{1}x_{2} -0.3832x_{1} x_{3} -20.3799x_{1} x_{4}\notag\\ &&-6.2434x_{2} x_{3} +20.9541x_{2} x_{4} -19.7127x_{3} x_{4} \\\notag \end{array} $$

(A.1)

$$\begin{array}{rll} V& = &12.9801+1.1018x_{1} -1.5224x_{2}\notag\\ &&-1.4712x_{3} -3.7633x_{4} -0.0416x_{1}^{2}\notag\\ &&+0.6801x_{2}^{2} +1.3276x_{3}^{2} +1.0598x_{4}^{2} \notag\\ &&-1.0758x_{1} x_{2} -0.4423x_{1} x_{3}+0.4256x_{1} x_{4}\notag\\ &&-0.4113x_{2} x_{3} +0.33x_{2} x_{4} -0.0248x_{3} x_{4}\\\notag \end{array} $$

(A.2)

$$\begin{array}{rll} A& = &14.5324-0.6917x_{1} +2.4961x_{2}\notag\\ && +4.5982x_{3} +1.2342x_{4} -0.4358x_{1}^{2}\notag\\ &&-0.5297x_{2}^{2} -1.9054x_{3}^{2} -0.2445x_{4}^{2}\notag\\ &&-0.0846x_{1} x_{2} +0.4466x_{1} x_{3}-0.0473x_{1} x_{4} \notag\\ &&+0.0126x_{2} x_{3} +0.0621x_{2} x_{4} -0.7866x_{3} x_{4} \\\notag \end{array} $$

(A.3)

$$\begin{array}{rll} M& = &42.3885+18.9832x_{1} +1.0414x_{2}\\ &&+2.4926x_{3} +1.6616x_{4} \end{array} $$

(A.4)

In case B, the approximate RS models of the 4 responses with 41 design sample points are given in (A.5)–(A.8).

$$\begin{array}{rll} U& = &354.0743-108.1148x_{1} +98.1881x_{2} \notag \\ &&-158.5943x_{3} -108.2705x_{4}+62.4263x_{1}^{2}\notag \\ &&+18.2688x_{2}^{2} +59.05943x_{3}^{2} +54.5704x_{4}^{2} \notag \\ &&-90.9162x_{1} x_{2}+40.8179x_{1} x_{3}+59.8716x_{1} x_{4}\notag\\ &&-5.545x_{2} x_{3} -54.5496x_{2} x_{4} -18.8679x_{3} x_{4}\\\notag \end{array} $$

(A.5)

$$\begin{array}{rll} V& = &-0.4387-2.1406x_{1} +7.1207x_{2} \notag\\ &&+0.9186x_{3} -0.7663x_{4} +3.3058x_{1}^{2}\notag\\ &&-0.1958x_{2}^{2} +0.3914x_{3}^{2} +0.7315x_{4}^{2} \notag\\ &&-2.913x_{1} x_{2} -0.1118x_{1} x_{3}+0.8806x_{1} x_{4}\notag\\ &&-1.0351x_{2} x_{3}-1.1962x_{2} x_{4} +0.05x_{3} x_{4}\\\notag \end{array} $$

(A.6)

$$\begin{array}{rll} A& = &6.3973-5.5454x_{1} -2.6635x_{2}\notag\\ && +17.5526x_{3}-4.8791x_{4} +4.7710x_{1}^{2}\notag\\ &&+1.0417x_{2}^{2} -2.2446x_{3}^{2} +1.2998x_{4}^{2}\notag \\ &&-3.8751x_{1} x_{2} -3.9216x_{1} x_{3}+6.7334x_{1} x_{4} \notag \\ &&+0.1033x_{2} x_{3} +0.6676x_{2} x_{4} -6.4598x_{3} x_{4}\\\notag \end{array} $$

(A.7)

$$\begin{array}{rll} M& = &41.1069+19.9045x_{1} +2.0114x_{2}\\ &&+7.6915x_{3} +8.5555x_{4} \end{array} $$

(A.8)