PSI - Issue 44

Sourav Das et al. / Procedia Structural Integrity 44 (2023) 1680–1687 Das and Tesfamariam/ Structural Integrity Procedia 00 (2022) 000–000

1684

5

where the total number of expansion levels and number of subdomains at level N m , respectively. I represents the indicator function of subdomain , Θ Ω m n . residual up to level ( m -1) which is expressed in following form:

[0, , ] ∈ … m M are denoted by M and , ( )  m n θ is the truncated expansion of

1 m N − − ∑∑ l

( )   m θ =

( ) θ

( ) θ

( )

, Ψ ∑ a

( ) θ

( ) θ

, l n

θ ;

I

,  l n

, l n l n i i

(16)

=



, l n

Ω

Θ

0 1 n

l

= =

i

Readers may refer Marelli et al. (2021) for a complete description of stochastic spectral embedding. 4. Numerical Examples Two different numerical examples are presented to illustrate the reliability analysis using the probability density evolution method. Those are: (1) a four-branch problem in which the probability of failure is estimated from a series system; and (2) reliability-based design optimization of timber outrigger building incorporating shape memory alloy based damper.

4.1. Problem 1 As the first example, a four-branch problem is considered whose P f is given by:

  

  

7

7

+ θ θ

+ θ θ

     

     

  

  

   

2 Pr = − − + θ θ 0.1( )

2 0.1( ≥ ∪ − − −   θ θ ) 3

1 2 1 θ θ ≥ ∪ − ≥ ∪ − ≥ θ θ 3 2

P

1

2

1

2

(17)

1

2

1

2

f

2

2

2

2

where θ 1 and θ 2 are the random variables which follow the standard Gaussian distribution. According to an equivalent extreme event, the above equation is equivalent to:

∞

  ∫ =

 

Pr P Y =

7 / 2

( ) p y dy

≥

(18)

EEV

f

Y

7/ 2

EEV

 

  

7

7

+ θ θ

+ θ θ

 

   

 

{

} { , θ θ θ θ } 2 1 − − 1 2

2 ) − − + θ θ

2 3 , 0.1( + − − − −   θ θ )

max 0.1(  

3 , 

Y

=

+ −

1

2

1

2

(19)

EEV

1

2

1

2

2 2

2 2

The surrogate model needs to be trained before estimating the probability density function. For this purpose, 100 representative points of random variables are generated using the GF-discrepancy algorithm. Subsequently, Y EEV is calculated for each set of random variables using Eq. 19. With these output responses, the surrogate model is trained. Once training is completed, another set of 1500 representative points are generated, which are passed through the trained surrogate model to obtain the complete response surface.

Fig. 2. (a) Subdomains in quantile space; (b) Subdomains in physical space; (c) Degree of accuracy corresponding to number of subdomains

The stochastic spectral embedding produces a minimum residual when the entire domain of random variables is divided into 8 subdomains, which is seen in Fig. 2(b). Once the surrogate model's accuracy is determined, the virtual