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Step length adjustment iterative algorithm for inverse reliability analysis

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Abstract

Performance Measure Approach (PMA) is an alternative way for evaluation of probabilistic constraints in reliability-based design optimization other than traditional Reliability Index Approach (RIA). In PMA, the probabilistic performance measure (PPM) is obtained through locating the minimum performance target point (MPTP) with the specified target reliability index in standard normal space, which is also called inverse reliability analysis. The advanced mean-value (AMV) method is well suitable for locating MPTP due to its simplicity and efficiency. However, AMV may have difficult to converge for highly nonlinear performance function. In this paper a step length adjustment (SLA) iterative algorithm, which introduces a “new” step length to control the convergence of the sequence, is proposed. This step length is new because the line search process for step length selection is not needed and it may be constant during the whole iteration process or decrease successively several times using a self-adjust strategy. It is proved that the AMV method is a special case of the SLA algorithm when the step length tends to infinity and the reason why AMV diverges is illustrated. SLA is as simple as AMV and does not need the prior knowledge of convexity or concavity of the performance function as other modified algorithms do. Numerical results of several highly nonlinear performance functions including an engineering application indicate that SLA is effective and robust.

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Abbreviations

RBDO:

Reliability-based design optimization

RIA:

Reliability index approach

PMA:

Performance measure approach

MPFP:

Most probable failure point

PPM:

Probabilistic performance measure

MPTP:

Minimum performance target point

AMV:

Advanced mean value

CMV:

Conjugate mean value method

HMV:

Hybrid mean value method

EHMV:

Enhanced hybrid mean value method

CC:

Chaos control

MCC:

Modified chaos control

SLA:

Step-length-adjustment iterative algorithm

d :

Design variable vector

x :

Original random vector

u :

Standard Gaussian vector

G j (d, x):

The j-th limit state function or performance function

P f (⋅):

Failure probability of performance measure function

P j, t :

The prescribed acceptable failure probability

d L, d U :

Lower and upper bounds of design variable

β j, t :

The j-th target reliability index

Φ(⋅):

Standard normal cumulative distribution function

g(d, u):

Performance function in standard normal u-space

G p (d):

Probabilistic performance measure

u ∗ :

Minimum performance target point

u k :

The k-th iteration point on the target reliability sphere surface

∇ U g(u k):

The gradient vector of performance function at point u k

n(uk):

Normalized steepest descent direction at point u k

λ :

Step length

c :

Adjusting coefficient for step length

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Acknowledgments

We would like to deeply thank Prof. Cheng GD and the anonymous reviewers for their useful and helpful comments and assistance.

The work was supported by the National Natural Scientific Foundation of China (51138011,51478086,51479027), which is gratefully acknowledged by the authors.

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Authors and Affiliations

  1. Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian, 116024, China

    Ping Yi & Zuo Zhu

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  1. Ping Yi
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  2. Zuo Zhu
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Correspondence to Ping Yi.

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Yi, P., Zhu, Z. Step length adjustment iterative algorithm for inverse reliability analysis. Struct Multidisc Optim 54, 999–1009 (2016). https://doi.org/10.1007/s00158-016-1464-8

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  • DOI: https://doi.org/10.1007/s00158-016-1464-8

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