PSI - Issue 48

Behrooz Keshtegar et al. / Procedia Structural Integrity 48 (2023) 348–355 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

350

3

where, g ( X ) is the performance function (limit stat function) with random variables  X x 1 ,…,x n and failure region of g ( X ) ≤0; X f ( X ) is the joint probability density function of X . The reliability index is approximated by the MPP using the probabilistic model in Eq. (1). Generally, the gradient method is applied to compute the sensitivity vector of the iterative FORM formula based on a discrete nonlinear map. The HL-RF is a known iterative formula to search for the MPP, as follows:

T

( ) ( ) g U U U U U  ( ) k k g

( )

g



HL

k

( ) U

U

g

1  



(3)

k

k

T

g  

k

k

where, T n g g u g u g u / ] ,..., , / () [ / 2 1         is the gradient vector . The HL-RF is an efficient method for modestly nonlinear limit state functions but it may not converge or slowly converge for highly nonlinear reliability problems. Recently, the FORM formula has been improved using the feedback chaos control-based stability transformation method by the directional sensitivity vector. The robustness of FORM is enhanced by using modified iterative formula that may provide stable results for MPP search. The directional stability transformation method (Meng et al. 2017) is formulated based on the directional sensitivity vector to approximate the reliability index as the below iterative relation:

T

( ) U U U  k k g

( ) k

g



1 DSTM k 

U



k 

T

( ( ) ( ( ) ( ) k k k f f U C U U U C U U U U U U U           ( ) k k k ( ) ( ) T k k T g g g g 



k 



(4)

) )

k k

k k

k 



k

where k  is the steepest descent sensitivity vector, C is the involutory matrix as C I  ,  is a chaos control factor as 0 1    , k is the current iteration and ( ) k f U is the new point without control by HL-RF method as, HL k k f U U  ( ) . The gradient-based FORM may provide unstable results for some highly nonlinear performance functions and may compute an inaccurate sensitivity vector to approximate the MPP using the gradient vector. As we shall see, the non gradient FORM approaches can be used to overcome these drawbacks of FORM. 3. Refined First Order Reliability Method It is supposed that the MPP is located on the failure domain, where the reliability index is at the minimum distance from origin. Therefore, the FORM probabilistic model of Eq. (1) is refined as below: (5) in which g( ) 0  X denotes the failure domain. Unlike the ordinary FORM, this probabilistic model provides MPP on the failure region g( ) 0  X . The optimization model of Eq. (5) is presented using the penalty approach for searching the MPP using non-gradient refined FORM as below: P( ) min f U X    (6) U * =arg min (|| U || | G( X )  0)