A Comprehensive Study on Fitness Approximation Techniques in Shape Optimization of Aerofoil Forging Preform Tools
Abstract
:1. Introduction
2. Optimization Objectives and Design Variables
2.1. Optimization Objectives
2.2. Design Variables
3. FE Simulation
4. Parameter Study
5. Surrogate Models
5.1. Response Surface Model (RSM)
5.2. Radial Basis Function (RBF) Model
5.3. Kriging Model
6. Model Estimations and Comparisons
7. The Shape Optimization of an Aerofoil Preform Tool
7.1. Particle Swarm Optimization Algorithm
7.2. Design of Fitness Function
7.3. Optimization Result
8. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Number | r | Y1 | Y2 | Y3 | Y4 | Y5 | Y6 | Y7 | Y8 | Y9 | C | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 5.0103 | 44.5009 | 44.5042 | 44.437 | 44.111 | 44.148 | 44.727 | 43.949 | 44.1495 | 44.145 | 0.663 | 0.337 | 0.0826 |
2 | 5.1448 | 44.6976 | 44.3319 | 44.884 | 45 | 44.252 | 44.238 | 44.428 | 44.4324 | 44.2924 | 0.615 | 0.442 | 0.1022 |
3 | 5.0811 | 44.6019 | 44.7148 | 44.115 | 44.078 | 44.217 | 44.124 | 44.141 | 44.3287 | 44.4186 | 0.693 | 0.419 | 0.0892 |
4 | 5.0705 | 44.4636 | 44.7052 | 44.283 | 44.061 | 44.87 | 44.564 | 44.647 | 44.1966 | 44.2871 | 0.625 | 0.365 | 0.0875 |
5 | 4.9466 | 44.5753 | 44.2458 | 44.926 | 44.654 | 44.011 | 44.499 | 44.483 | 44.1778 | 44.2134 | 0.623 | 0.388 | 0.0874 |
6 | 5.074 | 44.788 | 44.2075 | 44.674 | 44.637 | 44.441 | 44.173 | 44.537 | 43.8949 | 44.245 | 0.625 | 0.395 | 0.0853 |
7 | 5.0917 | 44.6604 | 44.6573 | 44.297 | 45.033 | 44.939 | 44.385 | 44.592 | 43.9137 | 44.2818 | 0.596 | 0.352 | 0.0891 |
8 | 4.9643 | 44.6391 | 44.7243 | 44.576 | 45.049 | 44.097 | 44.825 | 44.168 | 44.2438 | 44.2503 | 0.608 | 0.341 | 0.0849 |
9 | 5.0634 | 44.7508 | 44.265 | 44.101 | 44.802 | 44.698 | 44.597 | 44.291 | 44.1023 | 44.466 | 0.614 | 0.313 | 0.082 |
10 | 5.0846 | 44.7083 | 44.7531 | 44.227 | 44.358 | 44.544 | 44.857 | 44.25 | 44.4513 | 44.1819 | 0.628 | 0.338 | 0.085 |
11 | 5.0952 | 44.7561 | 44.3798 | 44.786 | 44.703 | 44.75 | 44.613 | 43.895 | 44.1589 | 44.1293 | 0.606 | 0.343 | 0.0867 |
12 | 5.1235 | 44.6338 | 44.3032 | 44.199 | 43.995 | 44.303 | 44.531 | 44.62 | 43.942 | 44.3713 | 0.657 | 0.354 | 0.0844 |
13 | 4.9714 | 44.6072 | 44.791 | 44.451 | 44.572 | 45.042 | 44.515 | 44.045 | 44.0835 | 44.1661 | 0.608 | 0.308 | 0.086 |
14 | 5.1129 | 44.7774 | 44.6191 | 44.395 | 44.67 | 44.956 | 43.945 | 44.127 | 44.3475 | 44.3029 | 0.638 | 0.333 | 0.0879 |
15 | 4.982 | 44.687 | 44.4181 | 44.772 | 44.028 | 44.733 | 43.929 | 44.1 | 44.0552 | 44.2029 | 0.659 | 0.369 | 0.0878 |
16 | 5.0669 | 44.7721 | 44.7435 | 44.94 | 44.506 | 44.458 | 44.466 | 44.633 | 44.1684 | 44.1503 | 0.611 | 0.432 | 0.0901 |
17 | 4.936 | 44.4796 | 44.6382 | 44.269 | 44.539 | 44.355 | 43.977 | 44.373 | 44.2815 | 44.2082 | 0.657 | 0.34 | 0.0875 |
18 | 5.1023 | 44.5168 | 44.2937 | 44.129 | 44.588 | 44.715 | 44.075 | 44.579 | 44.423 | 44.366 | 0.639 | 0.357 | 0.0847 |
19 | 4.9891 | 44.5806 | 44.4085 | 44.087 | 44.407 | 44.045 | 44.841 | 44.551 | 44.3853 | 44.3345 | 0.636 | 0.348 | 0.0848 |
20 | 4.9785 | 44.4583 | 44.5329 | 44.367 | 44.95 | 44.887 | 44.759 | 44.387 | 44.2155 | 44.4134 | 0.583 | 0.323 | 0.0908 |
21 | 5.0988 | 44.5221 | 44.3989 | 44.968 | 44.16 | 44.664 | 44.645 | 44.469 | 43.9514 | 44.1871 | 0.616 | 0.407 | 0.0898 |
22 | 5.1412 | 44.7933 | 44.6956 | 44.311 | 44.736 | 43.977 | 44.417 | 44.442 | 44.1401 | 44.3502 | 0.647 | 0.36 | 0.0858 |
23 | 5.1554 | 44.4902 | 44.2171 | 44.646 | 44.901 | 44.612 | 44.483 | 44.36 | 43.9986 | 44.3871 | 0.603 | 0.367 | 0.0856 |
24 | 5.1058 | 44.5062 | 44.2267 | 44.479 | 44.341 | 43.959 | 43.912 | 44.346 | 44.1212 | 44.224 | 0.683 | 0.363 | 0.0838 |
25 | 4.9678 | 44.7455 | 44.6669 | 44.828 | 44.868 | 44.492 | 43.994 | 44.278 | 44.0646 | 44.4029 | 0.622 | 0.36 | 0.0883 |
26 | 5.0138 | 44.4849 | 44.1884 | 44.493 | 44.127 | 44.337 | 44.548 | 44.059 | 44.1872 | 44.4555 | 0.655 | 0.339 | 0.0854 |
27 | 5.1377 | 44.6923 | 44.4659 | 44.814 | 44.325 | 44.2 | 44.254 | 43.908 | 43.9892 | 44.3976 | 0.661 | 0.359 | 0.0836 |
28 | 4.9572 | 44.7402 | 44.4851 | 44.143 | 44.473 | 44.166 | 44.336 | 43.963 | 43.9797 | 44.2976 | 0.669 | 0.29 | 0.0811 |
29 | 5.0386 | 44.5275 | 44.5425 | 44.339 | 44.456 | 44.905 | 43.961 | 44.086 | 43.9326 | 44.4239 | 0.647 | 0.326 | 0.0857 |
30 | 5.0422 | 44.7668 | 44.7626 | 44.52 | 44.177 | 44.819 | 44.629 | 44.401 | 44.0458 | 44.4344 | 0.621 | 0.364 | 0.0885 |
31 | 5.166 | 44.7189 | 44.3415 | 44.423 | 44.555 | 44.836 | 44.776 | 44.743 | 44.2721 | 44.2345 | 0.593 | 0.379 | 0.0869 |
32 | 5.12 | 44.5328 | 44.5616 | 44.059 | 44.967 | 44.423 | 44.401 | 43.936 | 44.2532 | 44.2713 | 0.647 | 0.288 | 0.0821 |
33 | 5.0598 | 44.5115 | 44.2362 | 44.632 | 44.687 | 44.63 | 44.906 | 44.196 | 44.489 | 44.2187 | 0.598 | 0.356 | 0.0865 |
34 | 4.9395 | 44.7827 | 44.3894 | 44.604 | 44.851 | 44.922 | 44.694 | 44.51 | 44.3004 | 44.2661 | 0.582 | 0.343 | 0.086 |
35 | 5.0563 | 44.6764 | 44.4564 | 44.856 | 44.72 | 44.269 | 44.89 | 44.715 | 44.1118 | 44.4502 | 0.595 | 0.421 | 0.0905 |
36 | 5.028 | 44.7242 | 44.3607 | 44.157 | 44.983 | 44.234 | 44.271 | 44.455 | 44.3381 | 44.124 | 0.634 | 0.345 | 0.0869 |
37 | 5.1589 | 44.5966 | 44.6095 | 44.744 | 44.391 | 44.561 | 43.896 | 44.702 | 44.0929 | 44.3608 | 0.637 | 0.424 | 0.0897 |
38 | 5.1306 | 44.7295 | 44.1788 | 44.213 | 44.209 | 44.389 | 44.32 | 44.031 | 44.3758 | 44.2555 | 0.676 | 0.322 | 0.0817 |
39 | 4.9608 | 44.6285 | 44.3702 | 44.842 | 44.621 | 44.681 | 44.792 | 44.018 | 43.8854 | 44.3555 | 0.599 | 0.341 | 0.084 |
40 | 5.1518 | 44.671 | 44.6478 | 44.255 | 44.226 | 44.475 | 44.222 | 44.209 | 43.9609 | 44.1345 | 0.667 | 0.339 | 0.0813 |
41 | 5.1342 | 44.5913 | 44.4468 | 44.325 | 44.259 | 44.973 | 44.955 | 43.99 | 44.074 | 44.3239 | 0.621 | 0.297 | 0.083 |
42 | 4.9962 | 44.7349 | 44.2841 | 44.534 | 44.374 | 44.08 | 44.043 | 44.496 | 44.357 | 44.445 | 0.659 | 0.389 | 0.0868 |
43 | 5.0492 | 44.7136 | 44.5234 | 44.562 | 44.605 | 44.372 | 44.743 | 43.881 | 44.4607 | 44.4607 | 0.628 | 0.329 | 0.0871 |
44 | 5.1271 | 44.469 | 44.5138 | 44.66 | 44.242 | 44.801 | 44.108 | 43.977 | 44.3664 | 44.2292 | 0.649 | 0.359 | 0.0845 |
45 | 4.9749 | 44.6551 | 44.5712 | 44.353 | 44.44 | 44.406 | 44.922 | 44.661 | 43.9043 | 44.1766 | 0.608 | 0.359 | 0.087 |
46 | 4.9997 | 44.7614 | 44.5521 | 44.073 | 44.275 | 44.647 | 44.01 | 44.729 | 44.1306 | 44.2608 | 0.651 | 0.367 | 0.0895 |
47 | 5.1483 | 44.5859 | 44.7818 | 44.898 | 44.835 | 44.767 | 44.678 | 44.182 | 44.2249 | 44.3292 | 0.591 | 0.379 | 0.0877 |
48 | 5.0032 | 44.5647 | 44.4277 | 44.241 | 45.016 | 44.028 | 44.189 | 44.606 | 43.9703 | 44.3766 | 0.637 | 0.357 | 0.0878 |
49 | 5.0174 | 44.5487 | 44.1693 | 44.171 | 44.522 | 44.853 | 44.45 | 44.319 | 44.0175 | 44.1608 | 0.621 | 0.314 | 0.0846 |
50 | 5.0775 | 44.4955 | 44.7339 | 44.381 | 44.489 | 44.131 | 44.808 | 44.223 | 43.9232 | 44.4081 | 0.637 | 0.343 | 0.0878 |
51 | 5.0528 | 44.5381 | 44.5904 | 44.758 | 44.917 | 44.286 | 44.157 | 44.072 | 43.876 | 44.1714 | 0.629 | 0.362 | 0.0852 |
52 | 5.1625 | 44.57 | 44.4946 | 44.73 | 44.094 | 44.062 | 44.711 | 44.414 | 44.4041 | 44.345 | 0.645 | 0.395 | 0.0821 |
53 | 4.9926 | 44.6179 | 44.6861 | 44.702 | 44.012 | 43.925 | 44.206 | 44.524 | 44.008 | 44.3082 | 0.668 | 0.403 | 0.0865 |
54 | 5.0457 | 44.453 | 44.5808 | 44.8 | 44.769 | 44.114 | 44.14 | 44.237 | 44.3192 | 44.4397 | 0.637 | 0.391 | 0.0874 |
55 | 4.9855 | 44.6125 | 44.198 | 44.548 | 44.934 | 44.578 | 44.026 | 43.922 | 44.291 | 44.3134 | 0.633 | 0.288 | 0.0883 |
56 | 5.1165 | 44.4743 | 44.5999 | 44.507 | 44.819 | 44.183 | 44.58 | 44.688 | 44.2344 | 44.1556 | 0.614 | 0.396 | 0.09 |
57 | 5.0882 | 44.6817 | 44.2745 | 44.912 | 44.308 | 45.025 | 44.368 | 44.332 | 44.3098 | 44.4292 | 0.611 | 0.388 | 0.0907 |
58 | 4.9502 | 44.6444 | 44.4372 | 44.185 | 44.144 | 44.99 | 44.434 | 44.114 | 44.4136 | 44.3397 | 0.638 | 0.3 | 0.0844 |
59 | 4.9431 | 44.5594 | 44.3128 | 44.618 | 44.292 | 44.784 | 44.303 | 44.77 | 44.0269 | 44.3923 | 0.618 | 0.378 | 0.0847 |
60 | 5.1094 | 44.6657 | 44.2554 | 44.409 | 44.753 | 43.942 | 44.939 | 44.155 | 44.0363 | 44.2398 | 0.632 | 0.33 | 0.0858 |
61 | 5.0315 | 44.6232 | 44.3511 | 44.59 | 44.045 | 44.509 | 44.287 | 44.674 | 44.4701 | 44.1398 | 0.64 | 0.452 | 0.0846 |
62 | 5.0209 | 44.5434 | 44.4755 | 44.87 | 44.884 | 45.008 | 44.059 | 44.565 | 44.2627 | 44.1977 | 0.598 | 0.374 | 0.0935 |
63 | 4.9537 | 44.554 | 44.6765 | 44.954 | 44.193 | 44.595 | 44.662 | 44.305 | 44.3947 | 44.3187 | 0.623 | 0.421 | 0.0906 |
64 | 5.0245 | 44.6498 | 44.7722 | 44.465 | 44.786 | 44.526 | 44.352 | 44.756 | 44.4796 | 44.3818 | 0.611 | 0.392 | 0.0912 |
65 | 5.0351 | 44.7029 | 44.6286 | 44.716 | 44.423 | 43.994 | 44.092 | 44.004 | 44.4418 | 44.1924 | 0.668 | 0.383 | 0.0859 |
66 | 5.0068 | 44.799 | 44.3224 | 44.688 | 43.979 | 44.32 | 44.874 | 44.264 | 44.2061 | 44.2766 | 0.651 | 0.338 | 0.0896 |
67 | 4.97 | 44.466 | 44.473 | 44.326 | 44.091 | 44.885 | 44.108 | 44.064 | 44.1 | 44.193 | 0.657 | 0.323 | 0.0841 |
Term | C | Term | C | ||||
---|---|---|---|---|---|---|---|
constant | 755.803 | −12243.136 | −3298.667 | r-Y4 | 0.010 | 0.024 | −0.005 |
Y1 | 5.681 | 127.173 | 33.392 | r-Y6 | −0.011 | 0.000 | 0.022 |
r | −12.536 | 24.776 | 13.210 | r-Y7 | −0.037 | 0.016 | 0.032 |
Y2 | −3.641 | 58.148 | 14.209 | r-Y8 | −0.046 | 0.023 | 0.005 |
Y3 | −4.003 | 16.994 | 5.745 | r-Y9 | −0.036 | 0.102 | −0.102 |
Y5 | −1.424 | 20.805 | 5.616 | Y2-Y3 | −0.004 | −0.009 | −0.024 |
Y4 | −1.796 | 6.834 | 2.890 | Y2-Y5 | −0.001 | −0.002 | 0.008 |
Y6 | −2.743 | 23.049 | 5.370 | Y2-Y4 | 0.007 | −0.097 | −0.010 |
Y7 | −2.848 | 30.270 | 8.629 | Y2-Y6 | −0.002 | −0.055 | −0.011 |
Y8 | −2.933 | 44.311 | 11.196 | Y2-Y7 | −0.009 | −0.226 | −0.028 |
Y9 | −2.502 | 220.859 | 60.002 | Y2-Y8 | −0.006 | −0.082 | −0.031 |
Y1^2 | 0.185 | −1.578 | −0.430 | Y2-Y9 | −0.003 | 0.038 | 0.014 |
r^2 | 0.142 | −6.157 | −1.473 | Y3-Y5 | 0.005 | 0.023 | −0.002 |
Y2^2 | 0.052 | −0.503 | −0.128 | Y3-Y4 | −0.006 | 0.032 | 0.002 |
Y3^2 | 0.026 | −0.208 | −0.051 | Y3-Y6 | 0.014 | −0.020 | −0.013 |
Y5^2 | 0.009 | −0.270 | −0.066 | Y3-Y7 | 0.016 | 0.010 | −0.011 |
Y4^2 | 0.020 | −0.113 | −0.039 | Y3-Y8 | 0.010 | 0.070 | 0.024 |
Y6^2 | 0.022 | −0.183 | −0.045 | Y3-Y9 | 0.010 | −0.015 | −0.002 |
Y7^2 | 0.025 | −0.206 | −0.063 | Y5-Y4 | 0.002 | 0.000 | 0.004 |
Y8^2 | 0.023 | −0.572 | −0.189 | Y5-Y6 | −0.001 | 0.033 | 0.000 |
Y9^2 | 0.014 | −2.385 | −0.669 | Y5-Y7 | 0.006 | −0.053 | −0.014 |
Y1-r | −0.003 | 0.867 | 0.146 | Y5-Y8 | 0.001 | −0.047 | 0.005 |
Y1-Y2 | −0.004 | 0.147 | 0.025 | Y5-Y9 | 0.007 | 0.065 | 0.015 |
Y1-Y3 | −0.003 | −0.063 | −0.004 | Y4-Y6 | −0.009 | 0.112 | 0.009 |
Y1-Y5 | −0.005 | 0.067 | −0.006 | Y4-Y7 | 0.015 | 0.061 | 0.017 |
Y1-Y4 | −0.011 | 0.144 | 0.010 | Y4-Y8 | 0.004 | −0.124 | 0.001 |
Y1-Y6 | 0.015 | −0.010 | 0.012 | Y4-Y9 | −0.003 | −0.061 | −0.018 |
Y1-Y7 | −0.015 | −0.029 | −0.012 | Y6-Y7 | 0.001 | −0.020 | −0.007 |
Y1-Y8 | 0.006 | 0.180 | 0.106 | Y6-Y8 | −0.002 | −0.085 | −0.019 |
Y1-Y9 | 0.015 | −0.226 | −0.036 | Y6-Y9 | 0.001 | −0.110 | −0.004 |
r-Y2 | 0.004 | −0.137 | −0.054 | Y7-Y8 | 0.005 | 0.064 | −0.004 |
r-Y3 | −0.042 | 0.074 | 0.037 | Y7-Y9 | −0.001 | −0.078 | −0.014 |
r-Y5 | 0.010 | −0.128 | −0.044 | Y8-Y9 | 0.006 | 0.162 | 0.041 |
Serial Number of ω | C Value | Serial Number of ω | C Value | ||||
---|---|---|---|---|---|---|---|
1 | −0.0076 | −0.0192 | 0.0013 | 35 | −0.0070 | −0.0520 | −0.0029 |
2 | −0.0002 | −0.1173 | −0.0385 | 36 | −0.0019 | −0.0103 | 0.0003 |
3 | −0.0198 | −0.2331 | −0.0182 | 37 | 0.0066 | 0.0060 | −0.0006 |
4 | 0.0067 | 0.0434 | 0.0010 | 38 | −0.0170 | 0.0209 | 0.0046 |
5 | 0.0013 | 0.0413 | 0.0039 | 39 | 0.0116 | −0.0331 | 0.0075 |
6 | 0.0090 | −0.0444 | 0.0096 | 40 | −0.0099 | 0.0289 | 0.0080 |
7 | −0.0076 | −0.0350 | −0.0048 | 41 | −0.0095 | 0.0197 | 0.0003 |
8 | 0.0005 | −0.0009 | 0.0077 | 42 | 0.0059 | 0.0030 | 0.0050 |
9 | 0.0114 | −0.0276 | 0.0080 | 43 | 0.0094 | 0.0180 | −0.0016 |
10 | 0.0060 | 0.0113 | −0.0027 | 44 | 0.0057 | 0.0181 | 0.0023 |
11 | 0.0192 | −0.0243 | −0.0008 | 45 | 0.0117 | −0.0170 | −0.0007 |
12 | 0.0013 | 0.0246 | −0.0029 | 46 | −0.0024 | −0.0083 | −0.0129 |
13 | 0.0121 | 0.0193 | −0.0005 | 47 | 0.0044 | 0.0021 | 0.0075 |
14 | −0.0060 | 0.0298 | 0.0013 | 48 | −0.0075 | −0.0230 | −0.0039 |
15 | −0.0012 | −0.0261 | −0.0074 | 49 | 0.0123 | −0.0237 | −0.0066 |
16 | −2.5304 | −0.0167 | 0.0024 | 50 | −0.0007 | 0.0259 | −0.0089 |
17 | −0.0052 | 0.0435 | −0.0028 | 51 | 0.0075 | −0.0441 | 0.0020 |
18 | −0.0050 | 0.0077 | 0.0073 | 52 | 0.0081 | 0.0822 | 0.0198 |
19 | 0.0123 | 0.0073 | 0.0016 | 53 | 0.0045 | 0.0307 | 0.0019 |
20 | 0.0102 | −0.0011 | −0.0126 | 54 | 0.0068 | 0.0009 | 0.0089 |
21 | −0.0050 | −0.0137 | −0.0107 | 55 | 0.0005 | 0.0743 | −0.0069 |
22 | −0.0019 | 0.0562 | 0.0057 | 56 | 0.0026 | −0.0072 | −0.0055 |
23 | 0.0070 | −0.0087 | 0.0086 | 57 | 0.0026 | −0.0056 | −0.0051 |
24 | −0.0163 | 0.0592 | 0.0012 | 58 | 0.0088 | 0.0553 | 0.0046 |
25 | 0.0029 | 0.0333 | 0.0015 | 59 | −0.0065 | 0.0184 | 0.0124 |
26 | −0.0011 | −0.0148 | −0.0065 | 60 | −0.0024 | −0.0075 | −0.0075 |
27 | −0.0032 | 0.0156 | 0.0047 | 61 | 0.0124 | −0.1480 | 0.0099 |
28 | −0.0086 | 0.0195 | 0.0055 | 62 | −0.0133 | 0.0681 | −0.0041 |
29 | −0.0022 | −0.0066 | −0.0008 | 63 | −0.0182 | −0.0901 | −0.0078 |
30 | 0.0086 | −0.0095 | −0.0011 | 64 | −0.0009 | 0.0339 | 0.0023 |
31 | 0.0116 | 0.0226 | 0.0035 | 65 | 0.0033 | 0.0060 | 0.0045 |
32 | −0.0175 | 0.0379 | 0.0067 | 66 | −0.0387 | 0.0941 | −0.0156 |
33 | 0.0058 | −0.0191 | 0.0037 | 67 | −0.0058 | −0.0073 | 0.0042 |
34 | −0.0007 | 0.0164 | 0.0101 | 68 | 0.6311 | 0.3663 | 0.0885 |
Term | r | Y1 | Y2 | Y3 | Y4 | Y5 | Y6 | Y7 | Y8 | Y9 | MLEs | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
C | 0.0335 | 0.0587 | 0.0545 | 0.1046 | 0.2034 | 0.1562 | 0.1909 | 0.1241 | 0.0592 | 0.0114 | 0.1223 | 108.46 |
0.0402 | 0.0189 | 0.0897 | 0.2201 | 0.135 | 0.1583 | 0.0641 | 0.2319 | 0.0406 | 0.0492 | 0.0846 | 64.48 | |
0.0457 | 0.0828 | 0.1143 | 1.5682 | 0.2513 | 0.0039 | 0.0179 | 0.1272 | 0.0741 | 0.0095 | 0.2392 | 28.43 |
Surrogate Model | Objective Response | ERRave | ERRmax | ERRrms | R-Square |
---|---|---|---|---|---|
RSM | C | 0.001 | 0.005 | 0.002 | >0.999 |
<0.001 | <0.001 | <0.001 | 1 | ||
0.003 | 0.009 | 0.003 | >0.999 | ||
RBF | C | <0.001 | <0.001 | <0.001 | 1 |
<0.001 | <0.001 | <0.001 | 1 | ||
<0.001 | <0.001 | <0.001 | 1 | ||
Kriging | C | 0.005 | 0.014 | 0.006 | >0.999 |
0.001 | 0.004 | 0.002 | >0.999 | ||
<0.001 | <0.001 | <0.001 | 1 |
Number | r | Y1 | Y2 | Y3 | Y4 | Y5 | Y6 | Y7 | Y8 | Y9 | C | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 5.022 | 44.67 | 44.695 | 44.597 | 44.992 | 44.344 | 44.306 | 44.308 | 44.121 | 44.23 | 0.617 | 0.366 | 0.0876 |
2 | 5.13 | 44.516 | 44.559 | 44.694 | 44.954 | 44.15 | 44.803 | 43.948 | 44.434 | 44.45 | 0.618 | 0.348 | 0.0861 |
3 | 4.981 | 44.459 | 44.472 | 44.635 | 44.522 | 44.972 | 44.274 | 44.561 | 44.324 | 44.146 | 0.609 | 0.369 | 0.0897 |
4 | 4.947 | 44.652 | 44.188 | 44.709 | 44.416 | 44.133 | 44.074 | 44.044 | 44.091 | 44.127 | 0.66 | 0.344 | 0.0847 |
5 | 5.079 | 44.535 | 44.187 | 44.281 | 44.549 | 44.971 | 44.669 | 44.707 | 43.879 | 44.344 | 0.594 | 0.358 | 0.0884 |
6 | 5.106 | 44.513 | 44.364 | 44.61 | 44.534 | 44.507 | 43.989 | 44.177 | 44.375 | 44.464 | 0.648 | 0.365 | 0.0862 |
7 | 5.117 | 44.493 | 44.649 | 44.382 | 44.314 | 44.983 | 44.427 | 44.054 | 44.339 | 44.369 | 0.633 | 0.329 | 0.0862 |
8 | 5.147 | 44.592 | 44.702 | 44.658 | 44.393 | 44.652 | 44.463 | 43.927 | 44.334 | 44.236 | 0.636 | 0.35 | 0.0845 |
9 | 5.000 | 44.538 | 44.285 | 44.105 | 45.02 | 44.163 | 44.538 | 43.954 | 44.083 | 44.296 | 0.64 | 0.281 | 0.0824 |
10 | 5.145 | 44.527 | 44.528 | 44.664 | 44.417 | 44.237 | 43.924 | 44.198 | 44.212 | 44.276 | 0.664 | 0.381 | 0.0845 |
Surrogate Model | Objective Response | ERRave | ERRmax | ERRrms | R-Square |
---|---|---|---|---|---|
RSM | C | 0.039 | 0.075 | 0.044 | 0.979 |
0.549 | 1.184 | 0.668 | 0 | ||
2.009 | 4.384 | 2.441 | 0 | ||
RBF | C | 0.018 | 0.039 | 0.021 | 0.995 |
0.065 | 0.104 | 0.071 | 0.928 | ||
0.123 | 0.444 | 0.173 | 0.611 | ||
Kriging | C | 0.058 | 0.123 | 0.071 | 0.945 |
0.071 | 0.119 | 0.082 | 0.905 | ||
0.161 | 0.561 | 0.226 | 0.341 |
Parameter Name | Value | Description |
---|---|---|
Maximum iterations | 100 | Maximum number of iterations during optimization |
Number of particles | 100 | The size of the population |
Inertia | 0.9 | The inertia of each particle in the swarm |
Global increment | 0.9 | The maximum increment to the particle position based on the global best position |
Particle increment | 0.9 | The maximum increment to the particle position based on the particle best position |
Maximum velocity | 0.1 | The maximum value of the particle velocity |
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Shao, Y.; Yan, L.; Guo, P.; Yang, H.; Shi, F.; Feng, D. A Comprehensive Study on Fitness Approximation Techniques in Shape Optimization of Aerofoil Forging Preform Tools. Metals 2019, 9, 617. https://doi.org/10.3390/met9060617
Shao Y, Yan L, Guo P, Yang H, Shi F, Feng D. A Comprehensive Study on Fitness Approximation Techniques in Shape Optimization of Aerofoil Forging Preform Tools. Metals. 2019; 9(6):617. https://doi.org/10.3390/met9060617
Chicago/Turabian StyleShao, Yong, Lin Yan, Pingyi Guo, Hongyu Yang, Fengjian Shi, and Di Feng. 2019. "A Comprehensive Study on Fitness Approximation Techniques in Shape Optimization of Aerofoil Forging Preform Tools" Metals 9, no. 6: 617. https://doi.org/10.3390/met9060617