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Structural and Multidisciplinary Optimization

, Volume 60, Issue 4, pp 1571–1582 | Cite as

Uncertainty-based design optimization approach based on cumulative distribution matching

  • Xingzhi HuEmail author
  • Xiaoqian Chen
  • Geoffrey T. Parks
  • Fulin Tong
Research Paper
  • 129 Downloads

Abstract

Uncertainty-based design optimization (UBDO) is becoming increasingly important with great attention to critical system reliability and robustness. However, most traditional UBDO methods only consider low-order response moments and a limited number of aleatory uncertainties, which desiderates to be resolved. In this study, an efficient UBDO approach based on cumulative distribution matching is presented to address all response moments under mixed aleatory and epistemic uncertainties. The mathematical formulation of cumulative distribution function (CDF) matching is systematically derived including the distance metric definition and numerical computation process. Uncertainty analysis within the CDF matching optimization loop is also investigated, including the identification of active subspaces and the propagation of constructed surrogate model in reduced dimensions. Extended active subspaces are furthered discussed for mixed uncertainty analysis to obtain the bounds of CDF efficiently, instead of the time-consuming full-dimensional simulation. The proposed approach is illustrated by three typical optimization examples, i.e., response function matching, standard NASA test problem, and satellite system design in aerospace engineering. The UBDO results based on the CDF matching are discussed and compared with the probability density function matching, which exhibits higher efficiency and better convergence in tackling constrained optimization design problems subject to mixed uncertainties.

Keywords

Uncertainty-based design optimization Density/distribution matching Cumulative distribution function Uncertainty analysis Reduced subspace 

Nomenclature

f

Objective response

Sd

Response CDF

T

Designer-given target CDF

L

Distance metric between target and system response

x

Overall uncertainty

d

Uncertain design variables

p

System uncertainty parameters

d

Optimal design under uncertainty

xa

Aleatory uncertainty

xe

Epistemic uncertainty

xi

ith input uncertainty

wi

ith integration weight

K

Kernel function

g

Required constraint

\( {\widehat{\mathbf{W}}}_1 \)

Estimated eigenvectors for active subspace

y

Reduced coordinate

yI

Interval reduced coordinate

Response surface surrogate model

N

Number of quadrature points

M

Number of KDE samples

PDF

Probability density function

CDF

Cumulative distribution function

CBF

Cumulative belief function

CPF

Cumulative plausibility function

Notes

Acknowledgments

The authors would like to thank Dr. Pranay Seshadri and Prof. Wen Yao for their valuable comments and suggestions for improving this paper.

Funding information

This work was supported by National Nature Science Foundation of China (Grant Nos. 11702305 and 11725211) and Pre-research Generic Technology Project (Grant No. 41406030102).

Compliance with Ethical Standards

Conflict of interests

The authors declared that they have no conflicts of interest to this work.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.China Aerodynamics Research & Development CenterMianyangChina
  2. 2.Chinese Academy of Military ScienceBeijingChina
  3. 3.University of CambridgeCambridgeUK

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