PSI - Issue 37

Alexandru Vasile et al. / Procedia Structural Integrity 37 (2022) 857–864 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

859

3

decrease the time spent on running different scenarios of the same problem. The most important output parameters we considered are the effective stiffness on X and Y directions (1) (Kaszynski et al., 2021)

1

1   NE i NE =

V

i i 

 

V

E

= =

(1)

eff

1

V



i i

V

1

i

=

3. Brute force algorithm The first algorithm we tried to use for the specified problem was a modified Greedy algorithm. To assess the accuracy of the results provided by it, we needed another type of algorithm. In computer science, brute-force search or exhaustive search is a very general problem-solving technique that consists of enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's statement. For the discussed case study, a brute force algorithm is supposed to provide all possible configurations of hard and soft materials and verify their effective stiffness. In Table 1 are given the total number of combinations (see equations (2) and (3)) and solving time estimation for a 2D domain composed from n finite elements. One can notice that for a problem made up of 4 by 4 elements with 8 cells of hard material and 8 of soft material, the solving time was around 27 minutes. Scaling that, we can approximate how long it would take for a grid made up of 6 by 6, and the time it would take for a single run makes the algorithm unfeasible for large domains. ( ) ! ! ! p n p n n n C C p n p − = = − (2) 2 n comb n = (3) where: n – number of cells that make up the domain; p – number of elements with soft material.

Table 1. Numbers of possible configuration and time estimation for domain composed of different number of finite elements.

No. of configurations with the specified (col.2) number of elements of each material type

Estimated time for determining all possible configurations[min]*

No. of configurations with no material percentage constraint

No. of elements with soft material ( p )/ hard material ( n-p )

No. of FE elements ( n )

16 16 16 20 32 36

8 / 8

65536 65536 65536

12870

27.06

4 / 12 10 / 6

1820 8008

3.83

16.84

10 / 10 16 /16 18 / 18

1.05E+06 4.29E+09 6.87E+10

1.85E+05 6.01E+08 9.08E+09

388.56

1.26E+06 1.91E+07

* Hardware configuration: Processor Intel i7-8750 H 2.2GHz, 8GB RAM; Software PyAnsys 0.61.1

Running the algorithm for the initial domain with just 16 cells, we obtained the results displayed in the chart in Fig. 2, that shows us the value of one of the objective functions (equation (4)) that we used for each possible combination for our 4 by 4 configuration with 50% hard and soft material. To keep the relation dimensionless, we divided by E O.

1

max(

, E E

)

F

=



(4)

_1

_ eff x eff y _

OBJ

_ eff x E E E −

_ eff y

1

+

O

where: E eff_x , E eff_y – effective modulus of elasticity on x and y axis; E O – 1 MPa in this paper.