A Modified Mean Value of Performance Measure Approach for Reliability-Based Design Optimization

The advanced mean value (AMV) is generally implemented to evaluate the probabilistic constraints of reliability-based design optimization (RBDO) problems based on performance measure approach (PMA). The PMA-based AMV is efficient method but yields unstable results for highly nonlinear probabilistic constraints. In this paper, a modified mean value (MMV) method is proposed to improve the efficiency and robustness of inverse reliability method to evaluate the reliable level in RBDO-based PMA. The modified PMA using MMV is adaptively evaluated using a modified search direction based on the two previous performance values. The modified search direction is determined using an adaptive step size, which is simply computed based on a power function and adaptive factor between 0.95 and 1. The robustness and efficiency of proposed MMV are compared with several reliability methods-based PMA including the AMV, hybrid mean value (HMV), enriched HMV (HMV\(^{+}\)) and modified chaos control (MCC) through four mathematical and structural RBDO problems with nonlinear probabilistic constraints. The results illustrated that the proposed MMV is as robust as the MCC and HMV\(^{+}\) but is computationally more efficient. In addition, the MMV is more robust than the HMV and AMV for RBDO problems with highly probabilistic constraints.

AMV:

Advanced mean value

\(A_g \) :

Adaptive factor

\({\varvec{d}}\) :

Design variables

\({\varvec{d}}^{L}\) :

Lower bound of the design vector

\({\varvec{d}}^{U}\) :

Lower bound of the design vector

DLA:

Double loop approaches

EHMV, HMV\(^{+}\) :

Enhanced hybrid mean value

f :

Objective or cost function

FORM:

First-order second-moment method

\(f_X (x)\) :

Joint probability density function

k :

Number of iterations

MCC:

Modified chaos control

MPFP:

Most probable failure point

MPTP:

Minimum performance target point

\({{\varvec{n}}}(u_k^\mathrm{AMV} )\) :

Normalized steepest descent search direction

\({{\varvec{n}}}(u_k^\mathrm{MMV} )\) :

Normalized modified descent search direction

p :

Number of performance functions

\(P_\mathrm{f} \) :

Acceptable failure probability

PMA:

Performance measure approach

RBDO:

Reliability-based design optimization

RIA:

Reliability index approach

SLA:

Single loop approaches

SORA:

Sequential optimization and reliability assessment approach

\({\varvec{X}}\) :

Random variables in X-space

\({\varvec{U}}\) :

Independent standard normal random variable

\({\varvec{U}}^{*}\) :

The most probable failure point

\(\tilde{{{\varvec{U}}}}_{k+1}^\mathrm{MMV} \) :

Modified mean value search direction

\(\beta \) :

Reliability index

\(\beta _t^j \) :

Target reliability index of the jth probabilistic constraint (\(g_j\))

\(\delta \) :

Modified factor

\(\Phi \) :

Standard normal cumulative distribution function

\({\varvec{\mu }}_{\varvec{x}}^L \) :

Lower mean of the random design vector

\({\varvec{\mu }}_{\varvec{x}}^U \) :

Upper mean of the random design vector

Authors and Affiliations

1. Department of Civil Engineering, University of Zabol, P.B. 9861335-856, Zabol, Iran

   Behrooz Keshtegar

Corresponding author

Correspondence to Behrooz Keshtegar.

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