Probabilistic Structural Model Updating with Modal Flexibility Using a Modified Firefly Algorithm
Abstract
:1. Introduction
2. Model Updating Formulation
2.1. Modal Flexibility
2.2. Model Reduction
2.3. Bayesian Model Updating Based on Measured Modal Flexibility Data
3. Firefly Algorithm
3.1. Original Firefly Algorithm
- All the fireflies are unisex so they can attract each other without the influence of gender.
- The attractiveness is proportional to the brightness, and they both decrease as their distance increases.
- The brightness of a firefly is determined by the landscape of the objective function.
Algorithm 1 the firefly algorithm |
Initialize the parameters(α, β, γ, n) |
Initialize randomly a population of n fireflies |
Evaluate the fitness of the initial population at xi by objective function |
While (k < MaxGen) do |
For i =1:n |
For j = 1:n |
If Firefly j is better than i |
Firefly i moves towards j |
End if |
End for |
Evaluate the new solution |
End for |
Rank and update the best solution found so far |
Update iteration counter k; |
Update α |
End while |
3.2. Modified Nelder–Mead Firefly Algorithm
3.2.1. Modification in Parameters α and β
3.2.2. Boundary Constrain Mechanism
Algorithm 2 boundary constrain mechanism |
For i = 1:n |
For j = 1:D |
If or |
Then |
End if |
End for |
xi = xi + F*(xbest − xi) |
End for |
3.2.3. Hybrid of Nelder–Mead Algorithm and the Diverse Threshold
4. Numerical Illustrative Examples
4.1. Benchmark Work of m-NMFA
4.2. Numerical Simulation of Shear Frame
5. Experimental Illustrative Example
6. Conclusions
- In the research on the random step size formula, it is found that the smaller the random step size is at the beginning of the iteration, the faster the convergence speed of the algorithm will be. However, as the random step size gradually approaches zero at the end of the iteration, its value should be large enough to keep the algorithm with sufficient exploration ability.
- In the research on the value of diversity threshold, a fraction is used to quantify the diversity of the population to enable the NMA algorithm to start at an appropriate time. The diversity threshold is quantized into a value between (0, e−1] through exponential form. When the diversity threshold is taken to be between 10−3~10−5, the algorithm can obtain a more accurate optimal solution.
- The selection of the iterative formula of the attraction parameter has great impact on the solving ability of the FA in solving the multi-dimensional optimization problems, The selection of the iterative formula of the attraction parameter has a great impact on the ability of the FA in solving multi-dimensional optimization problems, which would lead to stagnation and non-convergence if an improper formula were selected. This paper avoids such problem by selecting an appropriate formula for the coefficient of attraction.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
No. | Modified Algorithm with No NMA | T = 10−1 | T = 10−3 | T = 10−5 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | SD | Max | Min | Mean | SD | Max | Min | Mean | SD | Max | Min | Mean | SD | Max | Min | |
−2.063 | 2.68 × 10−15 | −2.063 | −2.063 | −2.063 | 2.68 × 10−15 | −2.063 | −2.063 | −2.063 | 2.68 × 10−15 | −2.063 | −2.063 | −2.063 | 2.68 × 10−15 | −2.063 | −2.063 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
−3.022 | 0.032 | −2.942 | −3.042 | −3.025 | 0.028 | −2.981 | −3.042 | −3.012 | 0.031 | −2.981 | −3.042 | −3.021 | 0.029 | −2.981 | −3.042 | |
73.192 | 28.043 | 145.588 | 19.536 | 2.90 × 10−6 | 2.12 × 10−6 | 1.30 × 10−5 | 4.38 × 10−7 | 2.61 × 10−6 | 2.25 × 10−6 | 1.48 × 10−5 | 3.05 × 10−7 | 2.71 × 10−6 | 1.99 × 10−6 | 8.48 × 10−6 | 1.95 × 10−7 | |
1.703 | 1.369 | 7.756 | 0.033 | 2.39 × 10−4 | 1.49 × 10−4 | 7.06 × 10−4 | 1.67 × 10−5 | 2.13 × 10−4 | 1.39 × 10−4 | 6.57 × 10−4 | 7.92 × 10−6 | 2.71 × 10−4 | 1.74 × 10−4 | 7.94 × 10−4 | 2.14 × 10−5 | |
4.91 × 10−3 | 6.50 × 10−3 | 0.028 | 3.77 × 10−6 | 0.032 | 0.064 | 0.460 | 1.16 × 10−13 | 0.035 | 0.034 | 0.162 | 6.91 × 10−13 | 3.89 × 10−3 | 7.00 × 10−3 | 0.044 | 1.32 × 10−13 | |
0.957 | 1.102 | 7.171 | 4.62 × 10−4 | 3.727 | 5.407 | 26.387 | 8.15 × 10−15 | 1.650 | 3.082 | 22.824 | 9.36 × 10−15 | 0.423 | 0.621 | 3.720 | 1.05 × 10−15 | |
0.155 | 1.251 | 12.226 | 2.03 × 10−5 | 26.232 | 35.036 | 100.531 | 5.94 × 10−16 | 2.743 | 9.077 | 55.762 | 6.46 × 10−14 | 0.808 | 4.779 | 45.357 | 9.50 × 10−15 | |
0.262 | 0.448 | 1.900 | 7.99 × 10−15 | 3.955 | 1.580 | 17.586 | 2.317 | 0.240 | 0.483 | 1.900 | 4.44 × 10−15 | 0.265 | 0.502 | 1.778 | 4.44 × 10−15 | |
2.90 × 10−6 | 2.60 × 10−6 | 1.69 × 10−5 | 2.55 × 10−7 | 8.93 × 10−78 | 3.94 × 10−77 | 3.64 × 10−76 | 1.74 × 10−84 | 2.20 × 10−71 | 2.20 × 10−70 | 2.20 × 10−69 | 7.04 × 10−84 | 3.68 × 10−72 | 3.68 × 10−71 | 3.68 × 10−70 | 5.99 × 10−83 | |
2.96 × 10−8 | 1.64 × 10−7 | 1.17 × 10−6 | 1.39 × 10−32 | 7.52 × 10−15 | 7.52 × 10−14 | 7.52 × 10−13 | 1.72 × 10−32 | 2.89 × 10−32 | 7.39 × 10−33 | 4.76 × 10−32 | 1.65 × 10−32 | 1.84 × 10−15 | 1.84 × 10−14 | 1.84 × 10−13 | 1.56 × 10−32 | |
1.558 | 1.552 | 6.694 | 0.044 | 3.62 × 10−7 | 4.53 × 10−7 | 2.13 × 10−6 | 1.85 × 10−9 | 3.80 × 10−7 | 4.71 × 10−7 | 2.56 × 10−6 | 1.19 × 10−9 | 3.36 × 10−7 | 4.60 × 10−7 | 2.73 × 10−6 | 9.01 × 10−10 | |
29.276 | 7.036 | 77.985 | 22.668 | 0.997 | 1.735 | 3.987 | 4.51 × 10−13 | 0.797 | 1.603 | 3.987 | 1.81 × 10−12 | 0.678 | 1.505 | 3.987 | 5.81 × 10−12 | |
8.061 | 9.410 | 65.806 | 0.681 | 0.667 | 2.04 × 10−7 | 0.667 | 0.667 | 0.667 | 0 | 0.667 | 0.667 | 0.667 | 6.21 × 10−8 | 0.667 | 0.667 | |
118.714 | 131.326 | 674.149 | 0.563 | 3.09 × 10−7 | 3.91 × 10−7 | 1.82 × 10−6 | 5.69 × 10−10 | 3.12 × 10−7 | 4.06 × 10−7 | 2.05 × 10−6 | 1.35 × 10−10 | 2.95 × 10−7 | 4.39 × 10−7 | 2.48 × 10−6 | 2.02 × 10−10 | |
1.74 × 1026 | 1.04 × 1027 | 8.19 × 1027 | 8.19 × 1027 | 209.091 | 326.701 | 1.42 × 103 | 2.53 × 10−8 | 135.316 | 229.458 | 1.50 × 103 | 2.71 × 10−7 | 187.578 | 296.558 | 1.42 × 103 | 9.43 × 10−8 | |
2.41 × 104 | 1.58 × 104 | 8.91 × 104 | 4.35 × 103 | 6.13 × 10−4 | 4.51 × 10−4 | 2.48 × 10−3 | 7.10 × 10−5 | 7.22 × 10−4 | 6.88 × 10−4 | 5.09 × 10−3 | 1.08 × 10−4 | 6.11 × 10−4 | 3.91 × 10−4 | 1.99 × 10−3 | 1.32 × 10−4 | |
2.444 | 2.584 | 16.389 | 0.041 | 1.68 × 10−3 | 1.18 × 10−3 | 5.81 × 10−3 | 7.15 × 10−6 | 1.70 × 10−3 | 1.16 × 10−3 | 5.61 × 10−3 | 7.39 × 10−5 | 1.73 × 10−3 | 1.09 × 10−3 | 6.50 × 10−3 | 2.66 × 10−5 |
No. | FA | GA | PSO | m-NMFA | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | SD | Max | Min | Mean | SD | Max | Min | Mean | SD | Max | Min | Mean | SD | Max | Min | |
−2.0626 | 2.65 × 10−15 | −2.0626 | −2.0626 | −2.0626 | 1.01 × 10−10 | −2.0626 | −2.0626 | −2.0626 | 2.68 × 10−15 | −2.0626 | −2.0626 | −2.0626 | 2.68 × 10−15 | −2.0626 | −2.0626 | |
3.53 × 10−5 | 3.26 × 10−4 | 3.25 × 10−3 | 0.00 × 100 | 0.046 | 0.0982 | 0.475 | 2.95 × 10−14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
−3.0197 | 0.0366 | −2.9233 | −3.0425 | −3.0179 | 0.0303 | −2.9799 | −3.0425 | −2.9622 | 0.0871 | −2.6175 | −3.0425 | −3.0210 | 0.0295 | −2.9810 | −3.0425 | |
193.2893 | 48.4340 | 328.8250 | 91.7273 | 61.9113 | 71.3777 | 318.4053 | 1.5917 | 76.3396 | 77.5750 | 528.4291 | 1.1334 | 2.71 × 10−6 | 1.99 × 10−6 | 8.48 × 10−6 | 1.95 × 10−7 | |
4.7192 | 2.4243 | 12.9291 | 0.9156 | 0.1317 | 0.2206 | 1.7274 | 3.80 × 10−3 | 3.2038 | 1.9981 | 11.9646 | 0.5124 | 2.71 × 10−4 | 1.74 × 10−4 | 7.94 × 10−4 | 2.14 × 10−5 | |
0.8609 | 0.4527 | 2.8608 | 0.0800 | 4.13 × 10−3 | 9.34 × 10−3 | 0.0620 | 4.08 × 10−5 | 13.8875 | 4.8491 | 30.1011 | 4.08 × 10−5 | 3.89 × 10−3 | 7.00 × 10−3 | 0.0443 | 1.32 × 10−13 | |
4.3163 | 2.2529 | 12.2211 | 0.4024 | 0.0667 | 0.0862 | 0.3263 | 4.56 × 10−5 | 3.4507 | 1.1366 | 7.0231 | 4.56 × 10−5 | 0.4225 | 0.6215 | 3.7198 | 1.05 × 10−15 | |
21.3056 | 16.1388 | 122.1799 | 1.0126 | 0.0259 | 0.0344 | 0.2279 | 2.73 × 10−4 | 7.0522 | 10.4476 | 49.7862 | 2.73 × 10−4 | 0.8075 | 4.7795 | 45.3568 | 9.50 × 10−15 | |
2.4178 | 0.9188 | 4.9678 | 0.6201 | 1.6640 | 0.5075 | 2.9591 | 9.45 × 10−2 | 4.8442 | 1.2063 | 7.6330 | 0.0945 | 0.2651 | 0.5021 | 1.7780 | 4.44 × 10−15 | |
1.94 × 10−5 | 3.23 × 10−5 | 2.21 × 10−4 | 1.14 × 10−6 | 6.81 × 10−4 | 2.05 × 10−3 | 0.0159 | 7.55 × 10−8 | 4.98 × 10−6 | 1.51 × 10−5 | 1.08 × 10−4 | 0 | 3.68 × 10−72 | 3.68 × 10−71 | 3.68 × 10−70 | 5.99 × 10−83 | |
0.2330 | 0.1597 | 0.8486 | 0.0507 | 0.0193 | 0.0219 | 0.1432 | 8.46 × 10−4 | 0.2220 | 0.1488 | 0.8418 | 0.0385 | 1.84 × 10−15 | 1.84 × 10−14 | 1.84 × 10−13 | 1.56 × 10−32 | |
15.7275 | 10.1891 | 50.2489 | 1.5753 | 0.5308 | 0.6093 | 3.2460 | 0.0150 | 7.0292 | 4.2909 | 25.7904 | 0.3848 | 3.36 × 10−7 | 4.60 × 10−7 | 2.73 × 10−6 | 9.01 × 10−10 | |
73.6154 | 26.5540 | 148.5464 | 31.3071 | 38.1670 | 40.1835 | 135.4710 | 0.2690 | 70.8902 | 50.9525 | 496.6682 | 30.3306 | 0.6777 | 1.5051 | 3.9866 | 5.81 × 10−12 | |
86.3426 | 153.5819 | 1.46 × 103 | 6.6724 | 6.7756 | 5.5898 | 33.0558 | 0.3401 | 18.2969 | 19.1600 | 125.8380 | 2.8606 | 0.6667 | 6.21 × 10−8 | 0.6667 | 0.6667 | |
1.60 × 103 | 1.35 × 103 | 7.49 × 103 | 80.0058 | 0.6165 | 0.8272 | 4.9430 | 0.0178 | 106.0857 | 99.0572 | 678.9951 | 9.3399 | 2.95 × 10−7 | 4.39 × 10−7 | 2.48 × 10−6 | 2.02 × 10−10 | |
2.04 × 1051 | 1.96 × 1052 | 1.96 × 1053 | 3.12 × 1012 | 8.54 × 1011 | 7.48 × 1012 | 7.40 × 1013 | 1.69 × 10−3 | 3.37 × 1025 | 2.51 × 1026 | 2.38 × 1027 | 1.92 × 103 | 187.5780 | 296.5578 | 1.42 × 103 | 9.43 × 10−8 | |
7.361 × 104 | 5.340 × 104 | 2.837 × 105 | 1.429 × 104 | 180.9207 | 674.6373 | 6.31 × 103 | 9.3670 | 6.02 × 103 | 3.85 × 103 | 2.44 × 104 | 961.8477 | 6.11 × 10−4 | 3.91 × 10−4 | 1.99 × 10−3 | 1.32 × 10−4 | |
19.3688 | 12.8955 | 103.0825 | 3.6407 | 2.8971 | 1.2763 | 7.2355 | 0.2871 | 6.4787 | 3.0430 | 18.5810 | 2.2303 | 1.73 × 10−3 | 1.09 × 10−3 | 6.50 × 10−3 | 2.66 × 10−5 |
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No. | Function Names | Formulations | Limits | Min | D |
---|---|---|---|---|---|
Cross-in-Tray Function | xi∈[−10,10] | −2.0626 | 2 | ||
Schaffer N2 Function | xi∈[−100,100] | 0 | 2 | ||
Hartmann 6-D Function | , where A P | xi∈(0,1) | −3.3224 | 6 | |
Zakharov Function | xi∈[−5,10] | 0 | 30 | ||
Alpine Function | xi∈[−10,10] | 0 | 30 | ||
Griewank Function | xi∈[−600,600] | 0 | 30 | ||
Penalized 1 Function | where | xi∈[−50,50] | 0 | 30 | |
Penalized 2 Function | xi∈[−50,50] | 0 | 30 | ||
Ackley Function | xi∈[−32.768,32.768] | 0 | 30 | ||
Sum of Different Powers Function | xi∈[−1,1] | 0 | 30 | ||
Sphere Function | xi∈[−5.12,5.12] | 0 | 30 | ||
Sum Squares Function | xi∈[−5.12,5.12] | 0 | 30 | ||
Rosenbrock Function | xi∈[−2.048,2.048] | 0 | 30 | ||
Dixon-Price Function | xi∈[−10,10] | 0 | 30 | ||
Rotated Hyper-Ellipsoid Function | xi∈[−65.536, 65.536] | 0 | 30 | ||
Perm Function 0, d, β | xi∈[−30,30] | 0 | 30 | ||
Schwefel 1.2 Function | xi∈[−100,100] | 0 | 30 | ||
Schwef 2.22 Function | xi∈[−100,100] | 0 | 30 |
No. | θa | FA | GA | PSO | m-NMFA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
θ | SD | CV | θ | SD | CV | θ | SD | CV | θ | SD | CV | ||
1 | 0 | 0.026 | 0.0012 | 0.0452 | 0.0083 | 0.0011 | 0.1374 | −0.0030 | 0.0020 | 0.6884 | 0.0081 | 0.0011 | 0.1408 |
2 | 0 | 0.007 | 0.0016 | 0.2196 | −0.0053 | 0.0015 | 0.2858 | −0.0104 | 0.0029 | 0.2772 | −0.0054 | 0.0015 | 0.2785 |
3 | 0 | 0.032 | 0.0014 | 0.0452 | −0.0044 | 0.0013 | 0.2932 | 0.0004 | 0.0042 | 9.3279 | −0.0047 | 0.0013 | 0.2729 |
4 | 0 | 0.042 | 0.0015 | 0.0350 | 0.0061 | 0.0013 | 0.2196 | 0.0414 | 0.0043 | 0.1042 | 0.0058 | 0.0013 | 0.2291 |
5 | −0.2 | −0.185 | 0.0012 | 0.0065 | −0.1932 | 0.0012 | 0.0060 | −0.2016 | 0.0024 | 0.0121 | −0.1933 | 0.0012 | 0.0060 |
6 | −0.4 | −0.404 | 0.0008 | 0.0019 | −0.4036 | 0.0008 | 0.0019 | −0.4124 | 0.0015 | 0.0036 | −0.4036 | 0.0008 | 0.0019 |
7 | −0.2 | −0.205 | 0.0012 | 0.0058 | −0.1992 | 0.0012 | 0.0061 | −0.2025 | 0.0026 | 0.0127 | −0.1993 | 0.0012 | 0.0061 |
8 | 0 | 0.028 | 0.0015 | 0.0553 | −0.0058 | 0.0014 | 0.2402 | −0.0063 | 0.0027 | 0.4338 | −0.0061 | 0.0014 | 0.2299 |
9 | 0 | 0.032 | 0.0013 | 0.0416 | −0.0030 | 0.0012 | 0.3932 | −0.0235 | 0.0030 | 0.1271 | −0.0031 | 0.0012 | 0.3779 |
10 | 0 | 0.046 | 0.0021 | 0.0452 | −0.0060 | 0.0017 | 0.2859 | −0.0019 | 0.0030 | 1.5957 | −0.0062 | 0.0017 | 0.2760 |
11 | 0 | 0.015 | 0.0015 | 0.0943 | −0.0079 | 0.0014 | 0.1722 | 0.0127 | 0.0040 | 0.3176 | −0.0080 | 0.0014 | 0.1688 |
12 | 0 | −0.001 | 0.0024 | 2.7776 | −0.0121 | 0.0022 | 0.1850 | 0.0621 | 0.0065 | 0.1043 | −0.0120 | 0.0022 | 0.1868 |
No. | θa | FA | GA | PSO | m-NMFA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
θ | SD | CV | θ | SD | CV | θ | SD | CV | θ | SD | CV | ||
1 | 0 | 0.017 | 0.0024 | 0.0293 | 0.0045 | 0.0020 | 0.0459 | 0.0613 | / | 1.2185 | −0.0026 | 0.0020 | 0.7461 |
2 | 0 | 0.0473 | 0.0015 | 0.0368 | 0.0383 | 0.0077 | 0.0150 | 0.1942 | / | 0.6339 | 0.0070 | 0.0044 | 0.6253 |
3 | 0 | 0.0436 | / | 0.0039 | 0.0036 | 0.0023 | 0.2470 | 0.1399 | 0.0023 | 0.0077 | 0.0199 | 0.0081 | 0.4085 |
4 | 0 | 0.0117 | / | 0.0244 | 0.0271 | / | 0.0141 | 0.1616 | 0.0097 | 0.0346 | −0.0272 | 0.0062 | 0.2278 |
5 | −0.2 | −0.0973 | 0.0121 | 0.3250 | −0.1704 | 0.0015 | 0.0026 | 0.1186 | / | 0.4796 | −0.2036 | 0.0037 | 0.0184 |
6 | −0.4 | −0.3829 | 0.0038 | 0.0348 | −0.3727 | 0.0069 | 0.0104 | −0.3700 | / | 0.0330 | −0.4072 | 0.0037 | 0.0091 |
7 | −0.2 | −0.2059 | 0.0037 | 0.0098 | −0.0327 | 0.0018 | 0.0041 | −0.0968 | / | 0.0248 | −0.1977 | 0.0047 | 0.0235 |
8 | 0 | 0.0682 | 0.0036 | 0.1014 | 0.0266 | 0.0038 | 0.0056 | 0.1807 | / | 0.0366 | −0.0156 | 0.0089 | 0.5712 |
9 | 0 | −0.0001 | 0.0022 | 0.0083 | 0.0813 | 0.0048 | 0.0841 | 0.1135 | / | 0.4384 | −0.0244 | 0.0071 | 0.2907 |
10 | 0 | 0.0468 | 0.0067 | 0.0263 | −0.0314 | 0.0037 | 0.2940 | 0.0855 | / | 0.2556 | 0.0058 | 0.0062 | 1.0808 |
11 | 0 | 0.0299 | 0.0043 | 0.0431 | 0.0151 | 0.0045 | 0.0155 | 0.1274 | 0.0018 | 0.0393 | 0.0046 | 0.0027 | 0.5872 |
12 | 0 | 0.0211 | 0.0030 | 0.0867 | 0.0336 | 0.0018 | 0.0134 | 0.1086 | / | 0.0620 | −0.0079 | 0.0030 | 0.3828 |
Modal Order | Undamaged | Damaged |
---|---|---|
1 | 2.6616 | 2.6104 |
2 | 7.8329 | 7.3901 |
3 | 12.5790 | 12.3932 |
4 | 16.6569 | 16.5127 |
5 | 19.7584 | 19.0000 |
6 | 21.4556 | 21.2004 |
n | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
0.4402 | −0.0904 | −0.1299 | −0.1259 | −0.0910 | −0.0126 | |
0.0051 | 0.0022 | 0.0033 | 0.0030 | 0.0028 | 0.0023 |
n | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
0.4923 | −0.0792 | −0.1438 | −0.2008 | −0.3099 | −0.0250 | |
0.0050 | 0.0022 | 0.0034 | 0.0032 | 0.0018 | 0.0023 |
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Feng, Z.; Wang, W.; Zhang, J. Probabilistic Structural Model Updating with Modal Flexibility Using a Modified Firefly Algorithm. Materials 2022, 15, 8630. https://doi.org/10.3390/ma15238630
Feng Z, Wang W, Zhang J. Probabilistic Structural Model Updating with Modal Flexibility Using a Modified Firefly Algorithm. Materials. 2022; 15(23):8630. https://doi.org/10.3390/ma15238630
Chicago/Turabian StyleFeng, Zhouquan, Wenzan Wang, and Jiren Zhang. 2022. "Probabilistic Structural Model Updating with Modal Flexibility Using a Modified Firefly Algorithm" Materials 15, no. 23: 8630. https://doi.org/10.3390/ma15238630