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Selective dimension reduction method (DRM) to enhance accuracy and efficiency of most probable point (MPP)–based DRM

  • Jeong Woo Park
  • Hyunkyoo Cho
  • Ikjin LeeEmail author
Research Paper
  • 16 Downloads

Abstract

To perform reliability-based design optimization (RBDO) in engineering systems, reliability analysis is required to calculate probability of failure (PF) for each performance function. Most probable point (MPP)–based dimensional reduction method (DRM) has been developed to accurately estimate PF using the Gaussian quadrature integration method. However, the existing MPP-based DRM is computationally expensive for highly nonlinear and/or high dimensional problems since it needs to increase the number of integration points in all directions to guarantee accuracy. In the proposed method, three statistical model selection methods—Akaike information criterion (AIC), AIC correction (AICc), and Bayesian information criterion (BIC)—are utilized to identify characteristic of performance functions for more efficient integration point allocation. Then, genetic algorithm and simplex optimization method are used to find the best models with the smallest AIC, AICc, and BIC values. No additional function evaluations are required for the model selection process since MPP candidate points are utilized. The best models obtained through optimization show where to allocate integration points which makes it possible not to allocate unnecessary integration points. Numerical study verifies that the proposed method can guide how to allocate integration points according to characteristic of performance functions: no integration points for almost linear performance functions, minimal additional integration points for mildly nonlinear performance functions, and more integration points for highly nonlinear functions.

Keywords

Reliability-based design optimization (RBDO) Reliability analysis Dimension reduction method (DRM) Akaike information criterion (AIC) 

Nomenclature

X

Vector of random variables; X = {X1, X2, ⋯, XN}T

x

Realization of X; x = {x1, x2, ⋯, xN}T

U

Vector of standard normal random variables; U = {U1, U2, ⋯, UN}T

u

Realization of U; u = {u1, u2, ⋯, uN}T

V

Vector of rotated standard normal random variables; V = {V1, V2, ⋯, VN}T

v

Realization of V; v = {v1, v2, ⋯, vN}T

P(•)

Probability measure

fX(x)

Joint probability density function (PDF) of random variables

G(X)

Performance function; considered failure if G(X) > 0

Φ(•)

Standard normal cumulative distribution function (CDF)

Mk

kth candidate model used in statistical model selection method

\( {L}_{M_k} \)

Likelihood estimation value in candidate model Mk

θ = {ρ, λ, σ}

Hyperparameters used in calculation of \( {L}_{M_k} \)

N

Total number of random variables

Nb

Number of variables considered in the selected interaction terms

Ndi

Number of times Vi is used in the selected interaction terms

Nu

Number of variables selected only in the axial direction

SN

Number of selected interaction terms

Notes

Funding information

This research was supported by the development of reliability validation method on computational simulation of gas turbine funded by the Korea Electric Power Corporation (No. R17XA05-06).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringKorea Advanced Institute of Science and TechnologyDaejeon 34141South Korea
  2. 2.Department of Mechanical EngineeringMokpo National UniversityMuan-gunSouth Korea

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