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A Polynomial Chaos Expansion Trust Region Method for Robust Optimization

Published online by Cambridge University Press:  03 June 2015

Samih Zein
Samih Zein*
Affiliation:
Samtech H.Q., 8 rue des chasseurs ardennais Angleur, Belgium
*
*Corresponding author.Email:[email protected]
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Abstract

Robust optimization is an approach for the design of a mechanical structure which takes into account the uncertainties of the design variables. It requires at each iteration the evaluation of some robust measures of the objective function and the constraints. In a previous work, the authors have proposed a method which efficiently generates a design of experiments with respect to the design variable uncertainties to compute the robust measures using the polynomial chaos expansion. This paper extends the proposed method to the case of the robust optimization. The generated design of experiments is used to build a surrogate model for the robust measures over a certain trust region. This leads to a trust region optimization method which only requires one evaluation of the design of experiments per iteration (single loop method). Unlike other single loop methods which are only based on a first order approximation of robust measure of the constraints and which does not handle a robust measure for the objective function, the proposed method can handle any approximation order and any choice for the robust measures. Some numerical experiments based on finite element functions are performed to show the efficiency of the method.

Keywords

60H1565K0562K2062K05Reliability based design optimizationpolynomial chaos expansiontrust region method
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 2 , August 2013 , pp. 412 - 424
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Zein, S., Colson, B. and Glineur, F., An efficient sampling method for regression-based polynomial chaos expansion, Communications in Computational Physics, Vol. 13, pp. 1173–1188 (2013).Google Scholar
[2]Valdebenito, M. A. and Schuller, G. I., A survey on approaches for reliability-based optimization, J. Struct. Multidisc. Optim., Vol. 42, No. 5, pp. 645–663 (2010).CrossRefGoogle Scholar
[3]Beyer, H.-G. and Sendhoff, B., Robust optimization - a comprehensive survey, Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 33-34, pp. 3190–3218 (2007).CrossRefGoogle Scholar
[4]Eldred, M. S., Design under uncertainty employing stochastic expansion methods, Int. J. for Uncertainty Quantification, Vol. 1, No. 2, pp. 119–146 (2011).Google Scholar
[5]Agarwal, H. and Renaud, J. E., Reliability based design optimization using response surfaces in application to multidisciplinary systems, Engineering Optimization, Vol. 36, No. 3, pp. 291–311 (2004).Google Scholar
[6]Ju, B. H. and Lee, B. C., Reliability-based design optimization using a moment method and a kriging metamodel, Follow Engineering Optimization, Vol. 40, No. 5, pp. 421–438 (2008).Google Scholar
[7]Deb, K., Gupta, S., Daum, D., Branke, J., Mall, A. K. and Padmanabhan, D., Reliability-based optimization using evolutionary algorithms, IEEE Transactions on Evolutionary Computation, Vol. 13, No. 5, pp. 1054–1074 (2009).Google Scholar
[8]Ray, T. and Smith, W., A surrogate assisted parallel multi-objective evolutionary algorithm for robust engineering design, Engineering Optimization, Vol. 38, No. 8, pp. 997–1011 (2006).Google Scholar
[9]Eldred, M. S., Agarwal, H., Perez, V. M., Wojtkiewicz, S. F. Jr. and Renaud, J. E., Investigation of reliability method formulations in DAKOTA/UQ, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance, Vol. 3, No. 3, pp. 199–213 (2007).Google Scholar
[10]Eldred, M.S., and Bichon, B.J., Second-Order Reliability Formulations in DAKOTA/UQ, paper AIAA-2006-1828inProceedingsofthe 47thAIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference (8th AIAA Non-Deterministic Approaches Conference), Newport, Rhode Island, May 1–4 (2006).Google Scholar
[11]Bjoörk, A., Numerical Methods for Least Squares Problems, ISBN 0-89871-360-9 (1996).Google Scholar
[12]Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, Springer-Verlag (1991). ISBN 0-387-97456-3 or 0-540-97456-3.Google Scholar
[13]Cameron, R. and Martin, W., The orthogonal development of nonlinear functions in series of Fourier-Hermite functionals, The Annals of Mathematics Second Series, Vol. 48, No. 2, pp. 385–392 (1947).Google Scholar
[14]Baäck, J., Nobile, F., Tamellini, L. and Tempone, R., Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison, in Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Vol. 76, pp. 43–62 (2011).Google Scholar
[15]Xiu, D., Fast numerical methods for stochastic computations: a review, Communications in Computational Physics, Vol. 5, No. 2-4, pp. 242–272 (2009).Google Scholar
[16]Dodson, M. and Parks, G. T., Robust aerodynamic design optimization using polynomial chaos, J. of Aircraft, Vol. 46, No. 2, pp. 635–646 (2009).Google Scholar
[17]Bruyneel, M., Duysinx, P. and Fleury, C., A family of MMA approximations for structural optimization, Structural & Multidisciplinary Optimization, Vol. 24, pp. 263–276 (2002).Google Scholar
[18]Pulch, R. and Emmerich, C. van, Polynomial chaos for simulating random volatilities, Math. Comput. Simulat., Vol. 80, No. 2, pp. 245–255 (2009).Google Scholar
[19]Li, J. and Xiu, D., A generalized polynomial chaos based ensemble Kalman filter with high accuracy, J. Comput. Phys., Vol. 228, No. 15, pp. 5454–5469 (2009).Google Scholar
[20]Eldred, M. S., Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design, 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, (2009).Google Scholar