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Research Papers

Robust Design Optimization in Computational Mechanics

[+] Author and Article Information
E. Capiez-Lernout1

 Université Paris-Est, Laboratoire de Mécanique (LaM), 5, Boulevard Descartes, 77454 Mame la Vallée, Cedex 2, Franceevangeline.capiez-lernout@univ-mlv.fr

C. Soize

 Université Paris-Est, Laboratoire de Mécanique (LaM), 5, Boulevard Descartes, 77454 Mame la Vallée, Cedex 2, Francechristian.soize@univ-mlv.fr

1

Corresponding author.

J. Appl. Mech 75(2), 021001 (Feb 20, 2008) (11 pages) doi:10.1115/1.2775493 History: Received January 12, 2006; Revised July 02, 2007; Published February 20, 2008

The motivation of this paper is to propose a methodology for analyzing the robust design optimization problem of complex dynamical systems excited by deterministic loads but taking into account model uncertainties and data uncertainties with an adapted nonparametric probabilistic approach, whereas only data uncertainties are generally considered in the literature by using a parametric probabilistic approach. The possible designs are represented by a numerical finite element model whose design parameters are deterministic and belong to an admissible set. The optimization problem is formulated for the stochastic system as the minimization of a cost function associated with the random response of the stochastic system including the variability of the stochastic system induced by uncertainties and the bias corresponding to the distance of the mean random response to a given target. The gradient and the Hessian of the cost function with respect to the design parameters are explicitly calculated. The complete theory and a numerical application are presented.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Finite element mesh of the dynamical system: attached spring (◼), attached lumped mass (●), attached set of three single DOF linear oscillators (▵), vibration absorbers (▴), excitation node (◆), simply supported boundary (thick line), and free boundary (thick dashed line)

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Figure 2

Reference observation of the mean master system. Graph of function ν↦20log10(w̱master(2πν)). Horizontal axis is the frequency ν in Hz.

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Figure 3

Convergence analysis: graph of function ns↦20log10(Conv(ns,N1)) for the stochastic master system with N1=300. Horizontal axis is ns.

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Figure 4

Convergence analysis: graph of function N1↦20log10(Conv(ns,N1)) for the stochastic master system with ns=300. Horizontal axis is N1.

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Figure 5

Definition of the target ν↦20log10(g(2πν)) (thick dashed line). Comparison with the reference observation ν↦20log10(w̱master(2πν)) (thin solid line) in the frequency band B1=[500,600]Hz (horizontal axis).

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Figure 8

Case 1. Comparison of the reference observation ν↦20log10(w̱master(2πν)) (thin solid line) with the confidence region (light gray region) of random response for the robust design, over the band B=[5,1200]Hz (horizontal axis) and for α=1∕2 and a probability level Pc=0.95. Horizontal axis is the frequency ν in Hz.

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Figure 12

Case 2. Low uncertainty level. Comparison of the reference solution ν↦20log10(w̱master(2πν)) (thick solid line) with the 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, and 90th quantiles of the random response ν↦20log10(WD(2πν)) (thin black lines) corresponding to the design optimization and of the random response ν↦20log10(WRD(2πν)) (thin gray lines) corresponding to the robust design optimization. Horizontal axis is the frequency ν in Hz.

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Figure 13

Case 2. Medium uncertainty level. Comparison of the reference solution ν↦20log10(w̱master(2πν)) (thick solid line) with the 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, and 90th quantiles of the random response ν↦20log10(WD(2πν)) (thin black lines) corresponding to the design optimization and of the random response ν↦20log10(WRD(2πν)) (thin gray lines) corresponding to the robust design optimization. Horizontal axis is the frequency ν in Hz.

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Figure 11

Case 1. Sensitivity analysis for the robust design optimization: Graph of function δ↦mRD(δ) for α=0.5 (thin line with ◻), for α=0.25 (thin line with ◇), and for α=0.1 (thin line with (○). Horizontal axis is δ.

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Figure 10

Case 1. Comparison of the reference solution ν↦20log10(w̱master(2πν)) (thin solid line) with the two confidence regions defined in Fig. 9.

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Figure 9

Case 1. Comparison of the reference solution ν↦20log10(w̱master(2πν)) (thin solid line) with the confidence region (dark gray region) of the random response ν↦20log10(WD(2πν)) corresponding to the design optimization and with the confidence region (light gray region) of the random response ν↦20log10(WRD(2πν)) corresponding to the robust design optimization. Horizontal axis is the frequency ν in Hz.

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Figure 7

Case 1. Comparison of the target ν↦20log10(g(2πν)) (thick dashed line) with the response of the mean model corresponding to the design optimization ν↦20log10(w̱D(2πν)) (thin dark gray line) and corresponding to the robust design optimization ν↦20log10(w̱RD(2πν)) (thin light gray line) for α=1∕2. Horizontal axis is the frequency ν in Hz.

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Figure 6

Case 1. Sensitivity of the robust design optimization with respect to the weighting factor α∊]0,0.5]: graph of α↦mRD∕mD (thick line), graph of α↦j(mRD,α) (thick dashed line), and graph of α↦‖σRD‖B1∕‖σD‖B1 (thin dashed line). Horizontal axis is α.

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