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An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces

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Abstract

Reliability-Based Design Optimization (RBDO) algorithms, such as Reliability Index Approach (RIA) and Performance Measure Approach (PMA), have been developed to solve engineering optimization problems under design uncertainties. In some existing methods, the random design space is transformed to standard normal design space and the reliability assessment, such as reliability index from RIA or performance measure from PMA, is estimated in order to evaluate the failure probability. When the random variable is arbitrarily distributed and cannot be properly fitted to any known form of probability density function, the existing RBDO methods cannot perform reliability analysis in the original design space. This paper proposes a novel Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) to evaluate the failure probability of any arbitrarily distributed random variables in the original design space. The arbitrary distribution of the random variable is approximated by a merger of multiple Gaussian kernel functions in a single-variate coordinate that is directed toward the gradient of the constraint function. The failure probability is then estimated using the ensemble of each kernel reliability analysis. This paper further derives a linearly approximated probabilistic constraint at the design point with allowable reliability level in the original design space using the aforementioned fundamentals and techniques. Numerical examples with generated random distributions show that existing RBDO algorithms can improperly approximate the uncertainties as Gaussian distributions and provide solutions with poor assessments of reliabilities. On the other hand, the numerical results show EGTRA is capable of efficiently solving the RBDO problems with arbitrarily distributed uncertainties.

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Abbreviations

ARP:

Allowable reliability point

CCP:

Chance constrained programming

CDF:

Cumulative distribution function

EoGRA:

Ensemble of Gaussian reliability analyses

EGTRA:

Ensemble of gradient-based transformed reliability analyses

FE:

Function evaluations

GTM:

Gradient-based transformation method

GTP:

Gradient-based transformation point

HMV:

Hybrid mean value

HRA:

Hybrid reliability approach

JPDF:

Joint probability density function

KDE:

Kernel density estimation

MCS:

Monte Carlo simulations

MRIA:

Modified reliability index approach

MPFP:

Most probable failure point

MPTP:

Most probable target point

PDF:

Probability density function

PMA:

Performance measure approach

RBDO:

Reliability-based design optimization

RBMDO:

Reliability-based multidisciplinary design optimization

URF:

Unified reliability formulation

d :

Mean of random variable X; d = d j e j .

d * i :

MPFP in the original design space; d * i  = d * i,j e j .

f :

Objective function.

\( {f}_{Y_i} \) :

PDF of Y i .

F p :

p th Gaussian CDF.

g i :

i th Constraint; g i ≤0 represents safe region.

G i :

i th Probabilistic constraint; G i  ≤ 0 represents reliable region.

H :

Heaviside function

i :

Index for constraints.

j :

Index for variables.

K p :

p th Gaussian kernel function.

L :

Number of variables.

M :

Number of inequality constraints.

N :

Number of sampling points.

p :

Index for sampling points.

P :

Probability of an event.

P f, i :

i th Allowable failure probability.

P MCS,i :

i th Failure probability evaluated by MCS.

q x :

JPDF of X.

s p :

p th Sampling point; s p =s p,j e j .

u :

Design variable in the standard normal space; u = u j e j .

u G i :

GTP in γ i ; u G i  = u G i,j e j .

u # i :

MPTP in γ i ; u # i  = u # i,j e j .

u * i :

MPFP in γ i ; u * i  = u * i,j e j .

U i :

Standard normal random variable; U i=Ui,j e j .

v i :

Normalized gradient of the i th constraint; v i=vi,j e j .

x :

Original design variable; x=x j e j.

x A i :

i th ARP in the original space; x A i  = x A i,j e j .

x G i :

i th GTP in the original space; x G i  = x G i,j e j .

x * p :

MPFP for the p th kernel reliability analysis; x * p  = x * p,j e j

X :

Arbitrarily distributed random variable; X=X j e j .

y i, p :

Coordinate of the p th sampling point in Ω i .

y A i :

ARP in Ω i .

y # i :

MPTP in Ω i .

y * i :

MPFP in Ω i .

Y i :

i th Gradient-based transformed random variable.

β p :

Reliability index of the p th kernel reliability analysis.

β f,i :

i th Allowable reliability index.

Φ p :

p th Gaussian CDF.

ρ :

Shape parameter in KDE.

σ :

Standard deviation matrix; \( \boldsymbol{\upsigma} ={\displaystyle \sum_{j=1}^L{\sigma}_j{\mathbf{e}}_j{\mathbf{e}}_j} \) .

Ω i :

i th Gradient-based transformed design space.

γ i :

i th Standard normal design space.

∗:

Optimal solution.

(k):

k th Iteration.

L :

Lower bound.

U :

Upper bound.

True:

True solution.

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Acknowledgments

The supports from Ministry of Science and Technology (MOST), Taiwan (grant numbers MOST 103-2221-E-033-015 and MOST 104-2218-E-033-013) are greatly appreciated. Furthermore, the supports from Research Center for Microsystem Engineering (RCME), Center for Robotics Research (CRR) and Center for Biomedical Technology (CBT) at Chung Yuan Christian University (CYCU), Taiwan are greatly appreciated.

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  1. Intelligent Unmanned System R&D Group, Department of Mechanical Engineering, Chung Yuan Christian University, 200 Chungpei Road, Chungli, Taoyuan, Taiwan, 32023

    Po Ting Lin

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Lin, P.T. An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces. Struct Multidisc Optim 54, 1653–1670 (2016). https://doi.org/10.1007/s00158-016-1535-x

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