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

Table 2. Statistics for 20-D results using Optimization Algorithm (OA)

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

Table 3. Function evaluations required for same tolerance (20-D)

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

Table 5. Summary of 20-D computational efforts using approximations (RBF-OA)

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

Table 6. Material properties of the sheets

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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