Abstract
A limited conjugate gradient method is proposed to improve the robustness and efficiency of the first-order reliability method (FORM). A new search direction vector is developed for structural reliability analysis using a limited scalar factor with Armijo’s rule and sufficient descent condition, namely limited Fletcher–Reeves (LFR) method. The conjugate gradient search direction is adaptively determined based on limited scalar factor and the instabilities of FORM formula are dynamically controlled by sufficient descent condition in the proposed LFR method. The LFR method is compared with the Hasofer–Lind and Rackwitz–Fiessler (HL–RF), stability transformation method (STM), finite-step size (FSS), chaotic conjugate stability transformation method (CCSTM), and directional stability transformation method (DSTM) using five mathematical and structural problems. Results of numerical examples illustrated that LFR is more efficient than the STM, FSS, CCSTM, and DSTM reliability methods and more robust than the HL–RF for highly non-linear performance functions.
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Abbreviations
- C :
-
Involutory matrix
- \(c\) :
-
Adjusting coefficient
- CCSTM:
-
Chaotic conjugate stability transformation method
- CGM:
-
Conjugate gradient method
- DSTM:
-
Directional stability transformation method
- FORM:
-
First-order reliability method
- FR:
-
Conjugate Fletcher–Reeves method
- FSS:
-
Finite-step size method
- HL-RF:
-
Hasofer and Lind–Rackwitz and Fiessler method
- LFR:
-
Limited conjugate Fletcher–Reeves method
- LSF:
-
Limit state function
- MPP:
-
Most probability failure point
- SORM:
-
Second-order reliability method
- STM:
-
Stability transformation method
- \(\varvec{d}_{k}^{{}}\) :
-
Conjugate search direction vector
- \(g()\) :
-
Limit state function
- \(g() \le 0\) :
-
Failure region
- \(f_{X}\) :
-
Joint probability density function
- \(P_{\text{f}}\) :
-
Failure probability
- \(\varvec{X}\), \(\varvec{U}^{\varvec{*}}\) :
-
Basic random variables
- \(\varvec{X}^{\varvec{*}}\) :
-
Most probable point (MPP) in U-space, X-space
- \(\varvec{\alpha}_{k}^{c\lambda }\) :
-
Conjugate sensitivity vector
- \(\varvec{\alpha}^{\lambda }\) :
-
Finite sensitivity vector
- \(\varvec{\alpha}_{k}^{{}}\) :
-
Sensitivity vector
- \(\beta\) :
-
Reliability index
- \(\varepsilon\) :
-
Stopping criterion
- \(\mu\), \(\sigma\) :
-
Mean standard deviation
- \(\lambda_{{}}^{C}\) :
-
Chaotic control factor
- \(\lambda\) :
-
Finite-step size
- \(\varPhi\) :
-
Standard normal cumulative distribution function
- \(\nabla g(\varvec{U}_{k}^{{}} )\) :
-
Gradient vector of the LSF at point \(\varvec{U}_{k}^{{}}\)
- \(\xi\) :
-
Step size or chaos control factor
- \(\theta_{k}\) :
-
Conjugate scalar at iteration k
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The authors are grateful to Dr. Zhen Hu for his significant comments to improve this work.
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Keshtegar, B. Limited conjugate gradient method for structural reliability analysis. Engineering with Computers 33, 621–629 (2017). https://doi.org/10.1007/s00366-016-0493-7
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DOI: https://doi.org/10.1007/s00366-016-0493-7