Structural and Multidisciplinary Optimization

, Volume 47, Issue 1, pp 63–76 | Cite as

Concurrent treatment of parametric uncertainty and metamodeling uncertainty in robust design

  • Siliang Zhang
  • Ping ZhuEmail author
  • Wei Chen
  • Paul Arendt
Research Paper


Robust design is an effective approach to design under uncertainty. Many works exist on mitigating the influence of parametric uncertainty associated with design or noise variables. However, simulation models are often computationally expensive and need to be replaced by metamodels created using limited samples. This introduces the so-called metamodeling uncertainty. Previous metamodel-based robust designs often treat a metamodel as the real model and ignore the influence of metamodeling uncertainty. In this study, we introduce a new uncertainty quantification method to evaluate the compound effect of both parametric uncertainty and metamodeling uncertainty. Then the new uncertainty quantification method is used for robust design. Simplified expressions of the response mean and variance is derived for a Kriging metamodel. Furthermore, the concept of robust design is extended for metamodel-based robust design accounting for both sources of uncertainty. To validate the benefits of our method, two mathematical examples without constraints are first illustrated. Results show that a robust design solution can be misleading without considering the metamodeling uncertainty. The proposed uncertainty quantification method for robust design is shown to be effective in mitigating the effect of metamodeling uncertainty, and the obtained solution is found to be more “robust” compared to the conventional approach. An automotive crashworthiness example, a highly expensive and non-linear problem, is used to illustrate the benefits of considering both sources of uncertainty in robust design with constraints. Results indicate that the proposed method can reduce the risk of constraint violation due to metamodel uncertainty and results in a “safer” robust solution.


Parametric uncertainty Metamodeling uncertainty Uncertainty quantification Kriging  Robust design 



The grant support from the National Natural Science Foundation of China (Grant No. 50875164) and US National Science Foundation (CMMI-0758557) is greatly acknowledged. This collaborative work is also made possible by the Chang Jiang Scholar Funds provided to Professor Wei Chen by China Education Ministry through Shanghai Jiaotong University.


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Siliang Zhang
    • 1
  • Ping Zhu
    • 1
    Email author
  • Wei Chen
    • 1
    • 2
  • Paul Arendt
    • 2
  1. 1.The State Key Laboratory of Mechanical System and Vibration, Shanghai Key Laboratory of Digital Autobody EngineeringShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA

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