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Structural and Multidisciplinary Optimization

, Volume 57, Issue 6, pp 2109–2125 | Cite as

An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake pedal case study

  • José Romero
  • Nestor V. Queipo
RESEARCH PAPER
  • 110 Downloads

Abstract

Accounting for uncertainty reduction measures (URMs) is critical to maximize the potential benefits of probabilistic design methods such as reliability-based design optimization (RBDO) and tackle the challenges in the design and construction of lightweight, high quality and reliable products. This work formulates and solves the RBDO of a Formula SAE (FSAE) brake pedal model with two failure modes (stress-Smax and buckling-fbuck) accounting for uncertainty reduction measures (URMs) throughout the product lifecycle while establishing the URMs global relative contributions to weight savings (expected value and variability) and computational expense. Given a set of URMs such as number of coupon tests, mesh refinement and manufacturing control, the solution approach includes: i) modeling structural analysis errors, ii) construction of surrogate models for the functions of interest, e.g., mass-M, Smax, fbuck and the corresponding error functions, iii) modeling pre-design and post-design URMs, such as material property density functions from coupon tests, and manufacturing tolerances (quality control), iv) solving the RBDO problems associated with each of the entries in a DOE with replication, and v) using ANOVA to compute main effects of most significant URMs on selected performance measures, i.e., mean and standard deviation of brake pedal mass, and computational expense. Results show that in the context of the brake pedal case study: the adoption of URMs led to reductions of up to 15 and 85% of mass mean and standard deviation, respectively, design and post-design URMs were responsible for 77 and 19% of the maximum mass reduction, respectively, and it was possible to set preliminary guidelines for URMs allocation and meet a particular performance objective under alternative URMs.

Keywords

Brake pedal FSAE Design under uncertainty Reliability-based design optimization Uncertainty reduction measures Surrogate models ANOVA 

Nomenclature

ci

Design parameter i

Cov(.)

Covariance function

CT

Computational time

d

Vector of random design variables

di

Random design variable i

E

Young’s modulus

f(.)

Regression function in Kriging model

fbuck(.)

Buckling load factor function

Gj(.)

Limit state function for failure mode j

G1(.)

Limit state function of stress condition

G2(.)

Limit state function of buckling condition

I(.)

Indicator function

p

Vector of random design parameters

M(.)

Brake pedal mass function

\( \hat{\mathrm{M}}(.) \)

Random brake pedal mass function

n

Number of sample points

nrv

Number of random variables

P(.)

Probability of the statement within the braces to be true

Pfsys

Probability of system failure

PfT

Target probability of system failure

R(.)

Correlation function

Sy

Material yield strength

Smax(.)

Maximum von Mises stress function

toli

Manufacturing tolerance probability distribution of di

x

Vector of random variables, x = [d, p] or input variables vector

xi

Random variables or vector of the input variable at the ith sample point

xA

Lower [−1] or higher [1] level correponding to number of coupon tests

xB

Lower [−1] or higher [1] level correponding to the mesh refinement

xC

Lower [−1] or higher [1] level correponding to the DOE size for the construction of surrogate model

xD

Lower [−1] or higher [1] level correponding to the degree of accuracy of the selected surrogate model

xE

Lower [−1] or higher [1] level correponding to the manufacturing quality control

xURM

Set of uncertainty reduction measures level combination x URM  = {xA, xB, xC, xD, xE}

yCT

Response surface model of computational expense – total run time of RBDO

yμM

Response surface model of the mean of the brake pedal mass

yσM

Response surface model of the standard deviation of the brake pedal mass

\( \hat{\mathrm{y}}(.) \)

Surrogate prediction function

Z(.)

Random function with mean zero, and nonzero covariance

β0, βi, βiiij

Polynomial regression coefficients

εmb

Buckling load factor surrogate model error

εmM

Mass surrogate model error

εmS

Maximum von Mises stress surrogate model error

εrb

Buckling prediction error due to the mesh refinement level

εrS

Maximum von Mises stress prediction error due to the mesh refinement level

μd

Vector of mean values of random design variables

μM

Mass mean value

μp

Vector of mean values of random design parameters

ν

Poisson’s ratio

σ2

Process variance in Kriging model

σM

Mass standard deviation value

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of ZuliaMaracaiboVenezuela
  2. 2.Applied Computing InstituteUniversity of ZuliaMaracaiboVenezuela

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