Abstract
In the manufacturing process of free-form complex surfaces, various problems occur which degraded the performance and can be resolved by using an optimization technique. Thousands of real-world problems can be converted into optimization techniques with distinct objective functions to acquire the optimal solution. In this paper, an assembly of GHT-Bézier developable surfaces by using a nature-inspired particle swarm optimization technique combined with optimal parameters are constructed from two GHT-Bézier boundary curves to improve the efficiency of complex engineering products. For this purpose, the control points of GHT-Bézier surface are chosen as design variables. The objective function is defined as the developability degree of the ruled surface and the shape parameters are considered to be optimization variables. So, we search for the optimum shape control parameters within the value range of shape parameters by using the PSO technique, then the developable surfaces with highly accurate developability can be visualized. The modeling examples demonstrate the effectiveness of the proposed method with the fairness of the surfaces and its approximation to the original surface.
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References
Adi DIS, Shamsuddin SM, Ali A (2009) Particle swarm optimization for NURBS curve fitting. IEEE Xplore 259-253
Ammad M, Misro MY, Abbas M, Majeed A (2020) Generalized developable cubic trigonometric Bézier surfaces, Mathematics 1-17
Ammad M, Misro MY (2020) Construction of local shape adjustable surfaces using quintic trigonometric Bézier curve. Symmetry 12:1205
Aumann G (2003) A simple algorithm for designing developable Bézier surfaces. Comput Aided Geom Des 20(8):601–619
Aumann G (2004) Degree elevation and developable Bézier surfaces. Comput Aided Geom Des 21(7):661–670
BiBi S, Abbas M, Misro MY, Hu G (2019) A novel approach of hybrid trigonometric Bézier curve to the modeling of symmetric revolutionary curves and symmetric rotation surfaces. IEEE ACCESS 7(1):165779–165792
BiBi S, Abbas M, Miura KT, Misro MY (2020) Geometric modelling of novel generalized hybrid trigonometric Bézier-like curve with shape parameters and its applications. Mathematics 8:967
BiBi S, Abbas M, Misro MY, Majeed A, Nazir T (2021) Construction of generalized hybrid trigonometric Bézier surfaces with shape parameters and their applications. J Math Imaging Vis. https://doi.org/10.1007/s10851-021-01046-y
BiBi S, Misro MY, Abbas M, Majeed A, Nazir T (2021) \(G^{3}\) Shape adjustable GHT-Bézier developable surfaces and their applications. mathematics 9, 2350
Cao H, Zheng H, Hu G (2021) Generation of quasi-developable Q-Bézier strip via PSO-based shape parameters optimization. Math Meth Appl Sci 2021:1–12
Chen DR, Wang GJ (2003) Developable Bézier parametric surfaces. J Comput Aided Des Comput Graph 15(5):5705
Chu CH, Chen JT (2004) Geometric design of developable composite Bézier surfaces. Comput-Aided Design Appl 1:531–539
Cui J, Zheng Y, Ohsaki M, Luo Y (2021) Shape optimization of piecewise developable free-form grid surface using plate components. Eng Struct 245
do Carmo MP (1976) Differential geometry of curves and surfaces. 1976:5–7
Farin G (2002) Curves and Surfaces for CAGD. A Practical Guide, 5th edn. Academic Press, San Diego
Gálvez A, Cobo A, Puig-Pey J, Iglesias A (2008) Particle swarm optimization for Bézier surface reconstruction. Springer-Verlag, Berlin Heidelberg, pp 116–125
Hu G, Cao XH, Qin XQ (2017) Geometric design and continuty of developable \(\lambda\) - Bézier surfaces. Adv Eng Soft 114:235–245
Hu G, Wu J, Li H, Hu X (2020) Shape optimization of generalized developable H-Bézier surfaces using adaptive cuckoo search algorithm. Adv Eng Soft 149:102889
Hu G, Cao H, Wu J, Wei G (2020) Construction of developable surfaces using generalized C-Bézier basis with shape parameters. Comput Appl Math
Implementation of Particle Swarm Optimization (2021) Greeks for Greeks, https://www.greeksforgreeks.org/implementation-of-particle-swarm-optimization/
Liu Y, Pottmann H, Wallner J, Yang YL, Wang W (2006) Geometric modeling with conical grides and developable surfaces. ACM Trans Gr 25(3):1–16
Maqsood S, Abbas M, Miura KT, Majeed M, BiBi S, Nazir T (2021) Geometric modeling of some engineering GBT-Bézier surfaces with shape parameters and their applications. Adv Diff Equ, (1)
Misro MY, Ramli A, Ali JM (2017) Quintic trigonometric Bézier curve with two shape parameters. Sains Malaysiana 46:825–831
Naseer S, Abbas M, Emadifar H, BiBi S, Nazir T, Shah ZH (2021) A class of sextic trigonometric Bézier curve with two shape parameters. J Math 9989810:1–16
Pottmann H, Farin G (1995) Developable rational Bézier and B-spline surfaces. Comput Aided Geom Des 12(5):51331
Segiun B, Chen YC, Fried E (2021) Bridging the gap between rectifying developables and tangent developables: a family of developable surfaces assiciated with a space curve
Hu G, Junli W, Xinqiang Q (2018) A new approach in designing of local controlled developable H-Bézier surfaces. Adv Eng Soft
Hu G, Zhu XN, Wei G, Chang CT (2021) An improved marine predators algorithm for shape optimization of developable ball surfaces. Eng Appl Artif Intell 105(2021):104417
Hu G, Wu J, Wang X (2021) Constructing local controlled developable H-Bézier surfaces by interpolating characteristic curves. Comput Appl Math 40(2021):216
Wang D, Tan D, Liu L (2018) Particle swarm optimization algorithm: an overview. Soft Comput 22:387–408. https://doi.org/10.1007/s00500-016-2474-6
Weiss V, Andor L, Renner G, Varady T (2002) Advanced surface fitting techniques. Comput Aided Geomet Design 19(1):19–42
Acknowledgements
This research was supported by Ministry of Higher Education Malaysia through Fundamental Research Grant Scheme (FRGS/1/2020/STG06/USM/03/1) and School of Mathematical Sciences, Universiti Sains Malaysia. The authors are very grateful to the anonymous referees for their valuable suggestion.
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BiBi, S., Misro, M.Y. & Abbas, M. Shape optimization of GHT-Bézier developable surfaces using particle swarm optimization algorithm. Optim Eng 24, 1321–1341 (2023). https://doi.org/10.1007/s11081-022-09734-3
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DOI: https://doi.org/10.1007/s11081-022-09734-3