PSI - Issue 37

Anastopoulos G. Stylianos et al. / Procedia Structural Integrity 37 (2022) 485–491 Anastopoulos G Stylianos et al/ Structural Integrity Procedia 00 (2019) 000 – 000

488

4

Fig. 2 (a) Boundary inclusions positioning using Periodic Geometry algorithm; (b) created composite Geometry representation At the current work every CNT among hundreds of CNTs has its own position and orientation, through the sequential positioning using «random orientation tensor» algorithm. The direction of every CNT in space is specified according to probability of convergence to the Χ 1 , Χ 2 , Χ 3 axes (Eq. 1). Since high randomization is pursued, at the present study the below orientation tensor values are chosen: α 11 = 0.333, α 22 = 0.333, α 33 = 0.333 with all other components left as zero.

0.333 0 

0

11 12 13                =     ij 21 22 23 31 32 33 ,  

    

   

(1)

0 0.333 0 0

=

0 0.333

The created geometry of the composite using ANSA ® homogenization tool and applying the previously mentioned values for the orientation tensor is illustrated at Fig. 2b.

2.3. IPopt algorithm for volume fraction optimization

The IPOpt algorithm is used for the iterations applied to the effective modulus of elasticity of the homogenized material. The effective modulus of elasticity is assumed as volume fraction dependent and is calculated at every iteration using the multi-Step method, with the ANSA ® homogenization tool. Where x variable is the CNT volume fraction (Eq 2), objective f (x) is the difference between the E eff calculated by the homogenization within the current iteration and the target E eff (Eq. 3), g_L and g_U are the lower and upper constraints bounds (Eq. 4), and x_L and x_U are the lower and upper variable bounds (Eq. 5). At every cycle, a new volume fraction is set, and a new objective is recalculated. The points x which satisfies the constraints (inequality) lower and upper bounds (g_L and g_U Eq. 4) are considered as feasible. At every cycle the new volume fraction value must range between the x_L and x_U bounds (Eq. 5). Convergence is achieved when either the objective value is less than a limit objective value or the maximum number of iterations has been completed.

n

CNT x vf =

,

(2) (3) (4) (5)

x 

t arg et

calculated eff

min f ( x ) ,

f ( x )

-

=

E

E

eff

g _ L g( x ) g _U  

x _ L x x _U  