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

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## 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-S_{max} and buckling-f_{buck}) 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, S_{max}, f_{buck} 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

- c
_{i} Design parameter i

- Cov(.)
Covariance function

- CT
Computational time

**d**Vector of random design variables

- d
_{i} Random design variable i

- E
Young’s modulus

- f(.)
Regression function in Kriging model

- f
_{buck}(.) Buckling load factor function

- G
_{j}(.) Limit state function for failure mode j

- G
_{1}(.) Limit state function of stress condition

- G
_{2}(.) 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

- n
_{rv} Number of random variables

- P(.)
Probability of the statement within the braces to be true

- P
_{fsys} Probability of system failure

- P
_{fT} Target probability of system failure

- R(.)
Correlation function

- S
_{y} Material yield strength

- S
_{max}(.) Maximum von Mises stress function

- tol
_{i} Manufacturing tolerance probability distribution of d

_{i}**x**Vector of random variables,

*x*= [d, p] or input variables vector**x**_{i}Random variables or vector of the input variable at the i

*th*sample point- x
_{A} Lower [−1] or higher [1] level correponding to number of coupon tests

- x
_{B} Lower [−1] or higher [1] level correponding to the mesh refinement

- x
_{C} Lower [−1] or higher [1] level correponding to the DOE size for the construction of surrogate model

- x
_{D} Lower [−1] or higher [1] level correponding to the degree of accuracy of the selected surrogate model

- x
_{E} Lower [−1] or higher [1] level correponding to the manufacturing quality control

**x**_{URM}Set of uncertainty reduction measures level combination

**x**_{ URM }= {x_{A}, x_{B}, x_{C}, x_{D}, x_{E}}*y*_{CT}Response surface model of computational expense – total run time of RBDO

*y*_{μM}Response surface model of the mean of the brake pedal mass

*y*_{σM}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

- β
_{0}, β_{i}, β_{ii},β_{ij} Polynomial regression coefficients

- ε
_{mb} Buckling load factor surrogate model error

- ε
_{mM} Mass surrogate model error

- ε
_{mS} Maximum von Mises stress surrogate model error

- ε
_{rb} Buckling prediction error due to the mesh refinement level

- ε
_{rS} Maximum von Mises stress prediction error due to the mesh refinement level

**μ**_{d}Vector of mean values of random design variables

- μ
_{M} Mass mean value

**μ**_{p}Vector of mean values of random design parameters

- ν
Poisson’s ratio

- σ
^{2} Process variance in Kriging model

- σ
_{M} Mass standard deviation value

## Notes

## References

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