Multiobjective Optimization of a Fuzzy Tracking Controller Using Differential Evolution for Nonlinear Three-Dimensional Tracking

Authors

https://doi.org/10.48313/mtei.v3i1.79

Abstract

The presented study introduces a multiobjective optimization approach that combines numerical and surrogate modeling approaches in order to enhance the crashworthiness characteristics of thin-walled S-shape energy absorbers under axial impact loading. In particular, the optimization problem includes the maximization of energy absorption and minimization of peak crushing force by changing four important geometrical variables such as cross-section width, curvature angle, curvature radius, and wall thickness. To conduct the simulation of the progressive crushing process with large deformation, local buckling, plastic folding, and self-contact phenomena, a 3D finite element model was developed using ABAQUS/Explicit. These results have been verified against reference values, with the largest difference between theoretical and experimental values in relation to the absorbed energy and peak crushing force being equal to 2.86% and 0.75%, respectively. The database consisting of 58 absorbers was developed through numerical simulation. In turn, surrogate models based on Radial Basis Function (RBF) and Group Method of Data Handling (GMDH) were built from that database. It should be noted that GMDH showed excellent performance with determination coefficients equal to 0.9882 and 0.9725 for absorbed energy and peak crushing force, respectively. The developed surrogate models were later coupled with Differential Evolution (DE) to carry out the optimization analysis in order to construct the Pareto front. Eight non-dominated solutions were achieved, highlighting the natural trade-off between the energy absorption capacity and peak crushing force. Amongst the obtained solutions, one was selected as the compromise solution, which had an energy absorption of about 33,400 J and peak crushing force of about 8,713 N. The results reveal that the proposed integration of finite element modeling, neural-network surrogate model, and DE is quite effective in performing crashworthiness optimization of S-shaped energy absorber design at a significantly lower computational cost.

Keywords:

S-shaped energy absorber, Crashworthiness, Multiobjective optimization, Radial basis function neural network, GMDH, Differential evolution, Finite element analysis, Pareto front

References

  1. [1] Abramowicz, W., & Jones, N. (1984). Dynamic axial crushing of square tubes. International journal of impact engineering, 2(2), 179–208. https://doi.org/10.1016/0734-743X(84)90005-8
  2. [2] Wierzbicki, T., & Akerstrom, T. (1977). Dynamic crushing of strain rate sensitive box columns. Second international conference on vehicle structural mechanics, southfield, michigan. SAE International. https://doi.org/10.4271/770592
  3. [3] Redwood, R. C. (1964). On the buckling of thin-walled tubes under axial impact. The journal of the royal aeronautical society, 68(642), 418–419. https://doi.org/10.1017/S0368393100079633
  4. [4] Yamazaki, K., & Han, J. (1998). Maximization of the crushing energy absorption of tubes. Structural Optimization, 16(1), 37–46. https://doi.org/10.1007/BF01213998
  5. [5] Nariman-zadeh, N., Darvizeh, A., & Jamali, A. (2006). Pareto optimization of energy absorption of square aluminium columns using multiobjective genetic algorithms. Proceedings of the institution of mechanical engineers, part b: Journal of engineering manufacture, 220(2), 213–224. https://doi.org/10.1243/095440506X78156
  6. [6] Zhang, Y., Sun, G., Li, G., Luo, Z., & Li, Q. (2012). Optimization of foam-filled bitubal structures for crashworthiness criteria. Materials & Design, 38, 99–109. https://doi.org/10.1016/j.matdes.2012.01.028
  7. [7] Zarei, H., & Kröger, M. (2008). Optimum honeycomb-filled crash absorber design. Materials & Design, 29(1), 193–204. https://doi.org/10.1016/j.matdes.2006.10.013
  8. [8] Zarei, H. R., & Kröger, M. (2008). Optimization of the foam-filled aluminum tubes for crush box application. Thin-walled structures, 46(2), 214–221. https://doi.org/10.1016/j.tws.2007.07.016
  9. [9] Ohkami, Y., Shimamura, M., Takada, K., Tomizawa, H., Motomura, K., & Usuda, M. (1990). Collapse of thin-walled curved beam with closed-hat section – part 1: Study on collapse characteristics. SAE Transactions, 99, 604–615. https://doi.org/10.4271/900460
  10. [10] Liu, Y., & Day, M. L. (2008). Bending collapse of thin-walled circular tubes and computational application. Thin-walled structures, 46(4), 442–450. https://doi.org/10.1016/j.tws.2007.07.014
  11. [11] Kim, H. S., & Wierzbicki, T. (2001). Effect of the cross-sectional shape on crash behaviour of a three-dimensional space frame. International journal of vehicle design, 25(4), 295–316. https://doi.org/10.1504/IJVD.2001.005204
  12. [12] Kim, H. S., & Wierzbicki, T. (2001). Effect of the cross-sectional shape of hat-type cross-sections on crash resistance of an “S”-frame. Thin-walled structures, 39(7), 535–554. https://doi.org/10.1016/S0263-8231(01)00020-9
  13. [13] Zheng, L., & Wierzbicki, T. (2004). Quasi-static crushing of S-shaped aluminum front rail. International journal of crashworthiness, 9(2), 155–173. https://doi.org/10.1533/ijcr.2004.0284
  14. [14] Zhang, C., & Saigal, A. (2007). Crash behavior of a 3D S-shape space frame structure. Journal of Materials Processing Technology, 191(1), 256–259. https://doi.org/10.1016/j.jmatprotec.2007.03.093
  15. [15] Esmaeili-Marzdashti, S., Pirmohammad, S., & Esmaeili-Marzdashti, S. (2017). Crashworthiness analysis of S-shaped structures under axial impact loading. Latin american journal of solids and structures, 14(5), 743–764. https://doi.org/10.1590/1679-78253430
  16. [16] Soheili, E., & Feli, S. (2024). Grooves and cross-section effects on energy absorption of S-shaped tubes under axial quasi-static and impact loading. Journal of the institution of engineers (india): series c, 105(1), 215–232. https://doi.org/10.1007/s40032-023-01010-2
  17. [17] Han, J., & Yamazaki, K. (2003). Crashworthiness optimisation of S-shape square tubes. International journal of vehicle design, 31(1), 72–85. https://doi.org/10.1504/IJVD.2003.002048
  18. [18] Liu, S. T., Tong, Z. Q., Tang, Z. L., & Zhang, Z. H. (2014). Design optimization of the S-frame to improve crashworthiness. Acta Mechanica Sinica, 30(4), 589–599. https://doi.org/10.1007/s10409-014-0032-y
  19. [19] Nguyen, V. S., Wen, G., Yin, H., & Van, T. P. (2014). Optimisation design of reinforced S-shaped frame structure under axial dynamic loading. International journal of crashworthiness, 19(4), 385–393. https://doi.org/10.1080/13588265.2014.904598
  20. [20] Hosseini-Tehrani, P., & Nikahd, M. (2006). Two-material S-frame representation for improving crashworthiness and lightening. Thin-walled structures, 44(4), 407–414. https://doi.org/10.1016/j.tws.2006.04.002
  21. [21] Pyrz, M., & Krzywoblocki, M. (2019). Crashworthiness optimization of front rail structure using macro element method and evolutionary algorithm. Structural and Multidisciplinary Optimization, 60(2), 711–726. https://doi.org/10.1007/s00158-019-02233-7
  22. [22] Khakhali, A., Nariman-zadeh, N., Darvizeh, A., Masoumi, A., & Notghi, B. (2010). Reliability-based robust multiobjective crashworthiness optimisation of S-shaped box beams with parametric uncertainties. International journal of crashworthiness, 15(4), 443–456. https://doi.org/10.1080/13588261003696458
  23. [23] Yao, S., Zhou, Y., Xing, J., Xu, P., Huang, Q., & Zou, F. (2024). A crashworthiness optimization method of subway underframe structures based on the differential evolution of the weighted graph representation. Structural and Multidisciplinary Optimization, 67(4), 62. https://doi.org/10.1007/s00158-024-03780-4
  24. [24] Zhang, X., & Zhang, H. (2016). Crush resistance of square tubes with various thickness configurations. International journal of mechanical sciences, 107, 58–68. https://doi.org/10.1016/j.ijmecsci.2016.01.003
  25. [25] Marzbanrad, J., Alijanpour, M., & Kiasat, M. S. (2009). Design and analysis of an automotive bumper beam in low-speed frontal crashes. Thin-walled structures, 47(8), 902–911. https://doi.org/10.1016/j.tws.2009.02.007
  26. [26] Sun, G., Li, G., Zhou, S., Li, H., Hou, S., & Li, Q. (2011). Crashworthiness design of a vehicle using multiobjective robust optimization. Structural and Multidisciplinary Optimization, 44(1), 99–110. https://doi.org/10.1007/s00158-010-0601-z
  27. [27] Fang, J., Gao, Y., Sun, G., Xu, C., & Li, Q. (2015). Multiobjective robust design optimization of fatigue life for a truck cab. Reliability engineering & system safety, 135, 1–8. https://doi.org/10.1016/j.ress.2014.10.007
  28. [28] Acar, E., Guler, M. A., Gerçeker, B., Cerit, M. E., & Bayram, B. (2011). Multiobjective crashworthiness optimization of tapered thin-walled tubes with axisymmetric indentations. Thin-walled structures, 49(1), 94–105. https://doi.org/10.1016/j.tws.2010.08.010
  29. [29] Homsnit, T., Jongpradist, P., Kongwat, S., Jongpradist, P., & Thongchom, C. (2024). Crashworthiness design of an automotive S-rail using ANN-based optimization to enhance performance and safety. Structural and Multidisciplinary Optimization, 67(6), 93. https://doi.org/10.1007/s00158-024-03803-0
  30. [30] Gao, D., Li, Q., Liu, Z., & Chang, G. (2024). An active learning approach for multiobjective optimization and its application to the crashworthiness design of thin-walled structures. Optimization and Engineering, 25(2), 699–730. https://doi.org/10.1007/s11081-023-09821-z
  31. [31] Lian, F., Wang, D., Zhang, Z., Chen, H., & Xu, W. (2025). Multiobjective optimization of lightweight and crashworthiness of automotive front-end structures based on stacked regression surrogate. Engineering applications of artificial intelligence, 155, 111138. https://doi.org/10.1016/j.engappai.2025.111138
  32. [32] Xing, J., Xu, T., Xu, P., & Yao, S. (2025). Multi-dimensional crashworthiness performance prediction and constrained optimization of the HFC energy-absorbing structures for railway vehicles driven by deep learning frameworks. Engineering Structures, 338, 120603. https://doi.org/10.1016/j.engstruct.2025.120603
  33. [33] You, J., Zhao, Y., Jia, L., Wang, N., Sun, Z., Zou, W., … & Zhang, C. (2026). A fast crashworthiness assessment framework: Sectional force-based multi-stage physics-informed surrogate model. Advances in Engineering Software, 213, 104090. https://doi.org/10.1016/j.advengsoft.2025.104090
  34. [34] Van, H. T. (2026). Deformation behavior and energy absorption performance of thin-walled tube structures for automotive crash boxes. Thu dau mot university journal of science, 8(2), 249–263. https://doi.org/10.37550/tdmu.ejs%2F2026.02.730
  35. [35] Saber, A., Amer, A. M., Abd_Elsalam, A., Shehata, A. I., & El-Gamal, H. A. (2026). Design and optimization of a bio-inspired crash box based on seashell structure for enhanced crashworthiness. Results in Engineering, 32, 111724. https://doi.org/10.1016/j.rineng.2026.111724
  36. [36] Hosseinipour, S. J., & Daneshi, G. H. (2004). Experimental studies on thin-walled grooved tubes under axial compression. Experimental Mechanics, 44(1), 101–108. https://doi.org/10.1007/BF02427984
  37. [37] Khalkhali, A., & Samareh, M. S. (2012). Multiobjective crashworthiness optimization of the aluminum foam-filled tubes. International journal of automotive engineering, 2(3), 180–193. https://scispace.com/pdf/multi-objective-crashworthiness-optimization-of-the-aluminum-kzm63tzmcz.pdf
  38. [38] Li, Y., Lin, Z., Jiang, A., & Chen, G. (2003). Use of high-strength steel sheet for lightweight and crashworthy car body. Materials & Design, 24(3), 177–182. https://doi.org/10.1016/S0261-3069(03)00021-9
  39. [39] Hartman, E. J., Keeler, J. D., & Kowalski, J. M. (1990). Layered neural networks with Gaussian hidden units as universal approximations. Neural Computation, 2(2), 210–215. https://doi.org/10.1162/neco.1990.2.2.210
  40. [40] Xu, X., Jiang, Y., & Lee, H. P. (2017). Multiobjective optimal design of sandwich panels using a genetic algorithm. Engineering Optimization, 49(10), 1665–1684. https://doi.org/10.1080/0305215X.2016.1265304

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Published

2026-03-15

Issue

Vol. 3 No. 1 (2026): Mechanical Technology and Engineering Insights (MTEI)

Section

Articles

How to Cite

Jamali, A. ., Poorahangryan, F. ., & Hatami Pour, S. (2026). Multiobjective Optimization of a Fuzzy Tracking Controller Using Differential Evolution for Nonlinear Three-Dimensional Tracking. Mechanical Technology and Engineering Insights, 3(1), 39-55. https://doi.org/10.48313/mtei.v3i1.79

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