Structural and Multidisciplinary Optimization

, Volume 59, Issue 1, pp 117–130 | Cite as

An improved numerically-stable equivalent static loads (ESLs) algorithm based on energy-scaling ratio for stiffness topology optimization under crash loads

  • Y. C. BaiEmail author
  • H.S. Zhou
  • F. Lei
  • H.S. Lei


The standard equivalent static loads (ESLs) method for stiffness topology optimization under crash condition may lead to exaggerated equivalent loads, which is not appropriate to be incorporated into the linear static topology optimization and whereby hinder the optimization process. To overcome this disadvantage, an improved ESLs algorithm based on energy-scaling ratio is proposed to guarantee the numerical stability, especially for the first several cycles with relatively larger differences of strain energy between the original crash simulation and equivalent static analysis. At each cycle, the equivalent external static forces are calculated by multiplying the stiffness matrix and the displacement vector at the time with maximal strain energy during the crash simulation. A further adaptive energy-scaling operation for those forces are performed by a weighting factor of square root of the energy ratio to the standard equivalent static loads for the crash problem based on the judging criterion. The newly equivalent loads are incorporated into the static topology optimization, and topology results are filtered into a black-white design for the crash simulation to avoid the numerical issues due to existing of low-density elements. The process is repeated until the convergence criteria is satisfied. The effectiveness of the proposed method is demonstrated by investing two crash design problems.


Stiffness topology optimization Equivalent static loads (ESLs) Crash loads Energy-scaling Numerically-stable 



This work is supported by the Beijing Institute of Technology Research Fund Program for Young Scholars, the Science and Technology Planning Project of Beijing City (NO. Z161100001416007) and the National Key R&D Program of China (NO. 2017YFB0103801).


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Copyright information

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

Authors and Affiliations

  1. 1.National Engineering Laboratory for Electric Vehicles, School of Mechanical EngineeringBeijing Institute of TechnologyBeijingChina
  2. 2.Collaborative Innovation Center of Electric Vehicles in BeijingBeijing Institute of TechnologyBeijingChina
  3. 3.State Key Laboratory of Advanced Design and Manufacturing for Vehicle BodyCollege of Mechanical Engineering and Vehicle EngineeringChangshaChina
  4. 4.Institute of Advanced Structure TechnologyBeijing Institute of TechnologyBeijingChina

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