Abstract
Polynomial chaos (PC) methods with Gauss-type quadrature formulae have been widely applied for robust design optimization. During the robust optimization, gradient-based optimization algorithms are commonly employed, where the sensitivities of the mean and variance of the output response with respect to design variables are calculated. For robust optimization with computationally expensive response functions, although the PC method can significantly reduce the computational cost, the direct application of the classical finite difference method for the analysis of the design sensitivity is impractical with a limited computational budget. Therefore, in this paper, a semi-analytical design sensitivity analysis method based on the PC method is proposed, in which the sensitivity is directly derived based on the Gauss-type quadrature formula without additional function evaluations. Comparative studies conducted on several mathematical examples and an aerodynamic robust optimization problem revealed that the proposed method can reduce the computational cost of robust optimization to a certain extent with comparable accuracy compared with the finite difference-based PC method.
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Abbreviations
- b :
-
Coefficients of polynomial chaos model
- d :
-
Dimension of design variables
- N :
-
Number of function evaluations
- m :
-
Dimension of random parameters
- E(•):
-
Operation of calculation expectation
- f(x):
-
Objective function
- g(x):
-
Constraint function
- p :
-
Order of polynomial chaos model
- K :
-
Accuracy level of sparse grid
- x :
-
Random design vector
- x :
-
Random design variable
- q :
-
Random parameter vector
- q :
-
Random parameter variable
- ω :
-
Gauss-type quadrature weight coefficient
- y(x):
-
Output response
- μ :
-
Mean value
- σ :
-
Standard deviation value
- Φ:
-
Multi-dimensional orthogonal polynomial
- φ :
-
One-dimensional orthogonal polynomial basis function
- DSA:
-
Design sensitivity analysis
- FDM:
-
Finite difference method
- FFNI:
-
Full factorial numerical integration
- MCS:
-
Monte Carlo simulation
- SGNI:
-
Sparse grid numerical integration
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Funding
The work was supported by the National Numerical Wind Tunnel Project (grant number NNW2020ZT7-B31) and Science Challenge Project (grant number TZ2018001).
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The results shown in the manuscript can be reproduced. Considering the size limit of the uploaded supplementary material, the codes for two of the mathematical examples for UP (Example 1 in Section 4.1 and Example 2 in Section 4.2) were uploaded as supplementary material. The remaining examples are very easy to implement by changing the response functions and sample points based on the codes provided to obtain the results shown in the manuscript.
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Ren, C., Xiong, F., Mo, B. et al. Design sensitivity analysis with polynomial chaos for robust optimization. Struct Multidisc Optim 63, 357–373 (2021). https://doi.org/10.1007/s00158-020-02704-2
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DOI: https://doi.org/10.1007/s00158-020-02704-2