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Figure1. Design parameters. The optimization parameters regard a population of 40 individuals (20 times the design parameters) and 150 generations (or as many generations as it takes for a con- vergence criterion to be reached), with 20% of mutation parameters and 50% crossover probability [12]. As mentioned before, the objective was to minimize the force to detach the toilet paper regarding specific design constrains, i.e., the angle, α , which ranged from 0 ◦ to55 ◦ , and the blank distance, d , between the cuts, which ranged from 0.1 to 1.0 mm. The GA created an angle, α , and a blank distance, d , population at random based on the angle range of interest. These parameters needed to be qualified according to how they may be more able than others to achieve the design objective. When this was carried out by using the finite element (FE) model, population crossing could produce a new generation, which was again qualified by the FE model, and this process was repeated until the best generation was found, as shown by the flowchart in Figure 2. After each crossing, the algorithm made an elitism pre-definition, comparing the new generation with the previous one, and selecting the best members to compose the next generation to be crossed. For the genetic algorithm, the mutation probability is 1% and the crossover probability is 100%. Regarding the optimization flowchart presented in Figure 2, four routines were devel- oped separately: i. a Python script to modify the FE model regarding the GA design parameters; ii. a Python script to perform the FE results analyses (post-processing); iii. a Fortran subroutine for the material model (more details in the section below); iv. aMATLAB ® script to control the FE analysis and GA. The optimization process was controlled using the MATLAB ® GA algorithm. The analysis started when MATLAB ® GA generated the first generation of design parameters. Then, a Python script was called to modify the FE model regarding the design parameters. After that, the MATLAB ® ran the FE analysis with the material model. Die to the fact that explicit FE analyses can take a long time and the GA algorithm demands a considerable number of analyses, it was necessary to obtain the maximum force value and terminate the current analysis. This was performed by the MATLAB ® code and a Python script that accesses the ABAQUS TM results several times until it detected a reduction of 20% in terms of the maximum force.
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