Structural and Multidisciplinary Optimization

, Volume 58, Issue 4, pp 1823–1843 | Cite as

Multi-objective lightweight and crashworthiness optimization for the side structure of an automobile body

  • Feng Xiong
  • Dengfeng Wang
  • Shuming Chen
  • Qiang Gao
  • Shudong Tian


This paper demonstrates a conjoint method integrating the proposed Hybrid Contribution Analysis (HCA) method, the Artificial Neutral Network (ANN) meta-model, the modified Non-dominated Soring Genetic Algorithm II (MNSGAII) and the Ideal Point Method (IPM), used for multi-objective lightweight and crashworthiness optimization of the side structure of an automobile body. First of all, the static-dynamic stiffness models of the automobile body and the vehicle side crashworthiness model are separately established and validated against corresponding actual experiments. Next, the initially selected parts for optimization are screened using the proposed HCA method to determine the final parts for optimization, thicknesses of which are taken as design variables. After that, design of experiment (DoE) coupled with ANN-based meta-models are utilized to approximate the output performance indicators of the automobile body, based on which the modified NSGA-II (MNSGAII) with ε-elimination technique is then employed to solve the multi-objective optimization process, considering the total mass and the torsional stiffness of the automobile body, the maximum intrusion deformation of the measuring point P1 on the inner panel of B-pillar and the measuring point D1 on the inner panel of front door as four optimization objectives. Finally, the IPM method identifies the optimal trade-off solution from the obtained Pareto set, and a comprehensive comparison between the optimized design and the baseline design further confirms the validity of the proposed conjoint method. Specially, the four-objective Pareto set approximately embodies that of each pair of separately run two-objective optimization, thus providing more optimization schemes for designers.


Artificial neutral network Hybrid contribution analysis Ideal point method Modified NSGA-II Lightweight and crashworthiness optimization 


P(x1, x2, ⋯xk)

Any performance indicator of automobile body;


Main effect of thickness variables xi;

Rij(xi, xj)

Cross effect of any two variables xi and xj;


The constant term;


The error;

\( {\widehat{\varphi}}_i \)

The coefficient of main effect of design variable xi;


Direct contribution value of design variable xi;


Relative contribution value of design variable xi


Hybrid contribution value of design variable xi.


Number of static-dynamic stiffness indicators;


Number of crashworthiness indicators;


DCV of variable xi to total mass of automobile body;


DCVs of variable xi to static-dynamic stiffness indicators;


DCVs of variable xi to crashworthiness indicators

ω1, ω2, …, ωp

Weight coefficients of RCV of static-dynamic stiffness indicators;

ξ1, ξ2, ⋯, ξq

Weight coefficients of RCV of crashworthiness indicators;

α, β and γ

Weight coefficients;


R square;


Maximum relative error;


Average relative error;


Root mean square value of relative error;


Number of checking points;

\( {f}_i^Z(e) \)

Normalized data of fi(e);

\( {\overline{f}}_i \)

Mean value of fi;


Standard deviation of fi;


The non-dominated set;


Number of objective functions;


Euclidean distance of solution e;


Total mass of automobile body;


First-order torsional frequency;


First-order bending frequency;


Torsional stiffness;


Bending stiffness;


Twist angle;


The loaded torque;


Vector of thickness variables


Total mass of the automobile body


Vector of performance indicators


ith performance indicator of automobile body

\( {T}_i^L \)

Lower bound of thickness variables Ti;

\( {T}_i^U \)

Upper bound of thickness variables Ti;


Area of the ith component;


Number of independent components;


Number of employed materials;


Materials properties of the ith material;


Density of material;


Elastic modulus;


Yield strength;

\( {f}_M\left(\mathbf{x}\right) \)

Function of total mass of automobile body;

\( {f}_T\left(\mathbf{x}\right) \)

Function of torsional stiffness of automobile body;


Function of maximum intrusion deformation of P1;

\( {D}_1\left(\mathbf{x}\right) \)

Function of maximum intrusion deformation of D1;


Total number of the static-dynamic stiffness indicators as constraints


Total number of the crashworthiness indicators as constraints


Lower constraint limits of x;


Upper constraint limits of x;

\( {H}_j^{\ast } \)

Five-star reference thresholds of the jth indicator;


Elimination threshold of modified NSGA-II;



This research work is supported by the national key research and development project (2016YFB0101601) and State Scholarship Fund of China Scholarship Council ([2016]3100). The authors would like to express their appreciation for the above fund supports.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Automotive Simulation and Control, College of Automobile EngineeringJilin UniversityChangchunChina
  2. 2.Department of Mechanical Engineering, College of EngineeringThe University of MichiganAnn ArborUSA
  3. 3.School of Mechanical EngineeringNanjing University of Science and TechnologyNanjingChina
  4. 4.FAW Car Co., LtdChangchunChina

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