PSI - Issue 48

Mohamed El Amine Ben Seghier et al. / Procedia Structural Integrity 48 (2023) 356–362 Ben Seghier et al / Structural Integrity Procedia 00 (2019) 000 – 000

358

3

The constant C > 0 is the penalty parameter, which computes the trade-off between the empirical error and the model complexity (regularization factor). ε -insensitive loss function ( ( )− ) proposed by Vapnik (Vapnik 1999) is used to estimate the empirical error. Thus, the loss function is calculated using the following equation: ( ( ) − ) = {0 | ( ) − | ≤ | ( )− |− ℎ (3) where is the error tolerance. To that, the optimum parameters are obtained by formulating the constrained optimization problem as: 1 2 ‖ ‖ 2 + 1 ∑ ( − + + ) =1 { − ( ( ) + ) ≤ + + ( ( ) + ) − ≤ + − − , + ≥ 0, = 1,2, … , (4) − + are positive slack variables. These latters represent the lower and upper excess deviation, respectively. To solve this problem, the constrained optimization function in equation (5) can be transformed into dual space using Lagrange multipliers, and the solution obtained is shown below: ( ) = ∑ ( − ∗ ) ( , )+ =1 (5) where , ∗ are Lagrange multipliers, subjected to constraints 0 ≤ , ∗ ≤C, whereas the term ( , ) is called the kernel function. This latter aims to map the input space to some higher dimensional space (feature space) allowing SVR to conveniently solve non-linear regression problems. There are many kernel functions proposed in the literature, the popular ones are polynomial function, radial basis function (RBF) and Gaussian function. In this study, RBF is used as the kernel function and it is defined as shown below: ( , ) = exp(− ‖ , ‖) (6) in which is the kernel parameter. The ideal performance and the high-accuracy of SVR depend greatly on the kernel function parameter. In this current study, these parameters are given by try and errors for each problem. 3. First-order reliability method-based conjugate search direction

k U as

is

The FORM formula using Taylor’s series expansion at point

T

( ) ( )  U U

( )( U U U

) 0

g g

g



1   k

k

k

k



given as:

T

( ) ( ) g U U U U U  ( ) k k g g

( ) k

g



(7)

( ) k U

U

g

1  



k

T  

k

k

where, ( ) k g U  is the gradient vector in the standard normal space of the LSF at the point k U . Actually, the HL-RF formula using FORM can be reformulated in Eq. (8) by using the following relations:

T

k k α U U U U ( ) ( )   T k k g g

k ( )

g



(8)

HL

U

α

1  

k

k

where, k α is the negative unit normal vector at the design point k U , which is computed as: