## Mathematical Test Functions

The test functions are Spherical, Rosenbrock, Rastrigin, Schwefel, and Ellipsoidal, and their forms are listed in Table 1. All numerical simulations are carried out using the optimization framework with the following assumptions:

• The population size of 10n was used for all the simulation.

• s, where n is the number of variables for the problem.

• For all the test cases, 10 independent runs were conducted with the number of generations being 1,000.

• The search space was between [-5.12, 5.12] for the Spherical, Ellipsoidal, Schwefel, and Rastrigin functions while a search space of[-2.048, 2.048] was selected for the Rosenbrock function in accordance to the convention in the literature.

• A radial-basis function (RBF) network was used with 5n centers and two nearest neighbor in accordance with Haykin (1999).

• The parent-centric crossover (PCX) operator was used to create a child from three parents.

• Retraining the RBF network was done after every 10 generations.

• The entire simulation process was executed using a Pentium® 4, 2.4GHz CPU processor.

Results presented in Table 2 indicate the performance of the optimization algorithm (OA) when actual evaluations have been used throughout the course of optimization. The number of actual function evaluations used by the OA model is listed in Column 3 of Table 3. To achieve the same mean level of convergence as compared to the OA model, the RBF-OA model typically uses around 50% less actual function evaluations as shown in Column 4 of Table 3. The number of approximations used by the RBF-OA model is listed in Column 5 of Table 3. The computational time required by the OA model and the RBF-OA model is presented in Table 4 and Table 5. The results of the surrogate-assisted optimization framework on these 20-dimensional highly nonlinear problems clearly show

Test Function |
Best Fitness |
Worst Fitness |
Mean Fitness |
Median Fitness |
Standard Deviation |

Spherical |
3.3851 x 10-22 |
1.0224 x 10-20 |
2.9952 x 10-21 |
1.9470 x 10-21 |
2.7881 x 10-21 |

llipsoidal |
1.5937 x 10-10 |
3.7958 x 10-7 |
4.7247 x 10-8 |
9.3724 x 10-9 |
1.1121 x 10-7 |

Schwefel |
1.3206 x 10-6 |
5.9133 x 10-5 |
2.0129 x 10-5 |
1.0319 x 10-5 |
1.987 x 10-5 |

osenbrock |
14.9216 |
19.5135 |
17.6649 |
17.5303 |
1.2013 |

Rastrigin |
10.0065 |
39.2395 |
22.0698 |
23.3051 |
7.8031 |

Test Function |
Algorithm (Actual function evaluations) |
Surrogate Assisted (Actual function evaluations) |
Surrogate Assisted (Approx. function evaluations) | |

Spherical |
2.9952x10-21 |
199200 |
110467 |
1091640 |

Ellipsoidal |
4.7247x10-8 |
199200 |
81534 |
805200 |

Schwefel |
2.0129x10-5 |
199200 |
144267 |
1426260 |

Rosenbrock |
17.6649 |
70447 |
21201 |
207900 |

Rastrigin |
22.0698 |
101650 |
28020 |
213040 |

Table 4. Summary of 20-D computational efforts using Actual Evaluations (OA)

Number of actual function evaluations 199200 Total time for Actual Evaluations 37.311s Total elapsed time (Wall clock time)_420.315s

Number of actual function evaluations |
20201 |

Number of approximate function evaluations |
198000 |

Total time for training with RBF |
77.363s |

Total time for RBF approximations |
416.851s |

Total time for actual function evaluations |
3.643s |

Total elapsed time (Wall clock time) |
887.537s |

Aluminum |
AK Steel |
HT Steel (3) | |

(1) |
(2) | ||

Young's Modulus (GPa) |
69 |
206 |
206 |

Poisson's Ratio |
0.330 |
0.300 |
0.300 |

Strength coefficient (MPa) |
570 |
567 |
671 |

Hardening exponent for yield strength |
0.347 |
0.264 |
0.219 |

Flow potential exponent in Barlat's model |
8 |
6 |
6 |

Anisotropy Coefficient |
0.710 |
1.770 |
1.730 |

Sheet Thickness (mm) |
1.00 |
1.00 |
1.00 |

that a saving of nearly 50% of actual function evaluations is possible while maintaining an acceptable accuracy. This is of great significance as it could mean cutting down expensive CFD or FEM computations while maintaining an acceptable accuracy.

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