PSI - Issue 37

A.F.F. Rodrigues et al. / Procedia Structural Integrity 37 (2022) 684–691 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

687

4

Table 2. Mechanical properties of selected specimens as reported in the references.

Density (Kg/m 3 ) 1 (GPa) 2 (GPa) 12 (GPa) 13 (GPa) 23 (GPa) 12 1978.3 31.28 27.17 6.46 - - 0.1659

Specimen

SP-1 SP-2 SP-3

649.691

7.12

3.45

1.96

- -

- -

0.28

568

8.180

4.357

0.6954

0.1216

The optimization problems solved in the paper can be defined as follows: ( 1 , 2 , 12 , 12 ) where 1 , 2 , G 12 , ν 12 are the elastic constants to be determined and Φ are objective functions. The first objective function is presented in Equation (2) (Lopes et al., 2019). The aim is to minimize the sum of the absolute difference between the circular natural frequencies obtained experimentally, ω ̃ , and the circular natural frequencies obtained computationally, ω : ∑| ̃ − | =1 (2) where are the total number of frequencies considered. One other objective function is defined according to (Soares et al., 1993): = ∑ ( ̃ 2 − × 12 ) 2 ̃ 4 =1 with = ̃ 12 12 (3) A set of lower and upper constraints, listed in Table 3, are defined for each one of the design variables. The termination criteria are the tolerance and the maximum number of iterations. The tolerance is set to 10 −6 or 10 −9 depending on the specimen and the objective function. This ensures that the optimization ends when the relative difference of successive function values reaches the specified value. The maximum number of iterations is set to one thousand for all optimization problems. This is needed so that, if the tolerance is not met, the algorithms will stop when they reach the one-thousandth iteration. (1)

Table 3. Constraints applied to each design variable.

1 (GPa) 2 (GPa) 12 (GPa) 13 (GPa) 23 (GPa) 12 50 50 20 - - 0.4

Specimen

Constraint

SP-1

Upper Lower Upper Lower Upper Lower

10 15

10 10

1 5

- - - - -

- - - - -

0.05

SP-2

0.4 0.1 0.2

4

1

0.1

SP-3

15

10

5

4

1

0.1

0.05

4. Results and discussion 4.1. Comparison of results obtained with different sets of initial search agents

The present method is based on meta-heuristics, so there is a need to validate this method. Therefore, in this section, one of the most used validation tests discussed in (Hussain et al., 2019) is presented. Thirty optimization problems are