An enhanced manta ray foraging optimization algorithm for shape optimization of complex CCG-Ball curves

Gang Hu*, Min Li, Xiaofeng Wang, Guo Wei, Ching Ter Chang

*Corresponding author for this work

Research output: Contribution to journalJournal Article peer-review

55 Scopus citations


The shape optimization of complex curves is a crucial and intractable technique in computer aided geometric design and widely used in many product design and manufacturing fields involving complex curve modeling. In this paper, an enhanced manta ray foraging optimization (MRFO) algorithm is used to optimize the shape of complex composite cubic generalized Ball (CCG-Ball, for short) curves. Firstly, to solve the problems of shape optimization for Ball curves, we construct a class of new cubic generalized Ball basis, and then present the CCG-Ball curves with multiple shape parameters based on the constructed basis functions. The shapes of the curves can be modified and optimized easily by using the shape parameters. Secondly, the shape optimization of CCG-Ball curves is mathematically an optimization problem that can be efficiently dealt with by swarm intelligence algorithm. In this regard, an enhanced MRFO called WMQIMRFO algorithm, combined with control parameter adjustment, wavelet mutation and quadratic interpolation strategy, is developed to enhance its capability of jumping out of the local minima and improve the calculation accuracy of the native algorithm. Furthermore, the superiority of the WMQIMRFO algorithm is verified by comparing with standard MRFO, other improved MRFO and popular nature-inspired optimization algorithms on the well-known CEC’14 and CEC’17 test suite as well as four engineering optimization problems, respectively. Finally, by minimizing the bending energy of the CCG-Ball curves as the evaluation standard, the shape optimization models of the curves with 1th-order and 2th-order geometric continuity are established, respectively. The WMQIMRFO algorithm is utilized to solve the established models, and the CCG-Ball curves with minimum energy are obtained. Some representative numerical examples illustrate the ability of the proposed WMQIMRFO algorithm in effectively solving the shape optimization problems of complex CCG-Ball curves in terms of precision, robustness, and convergence characteristics.

Original languageEnglish
Article number108071
JournalKnowledge-Based Systems
StatePublished - 15 03 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2021 Elsevier B.V.


  • Cubic generalized ball curves
  • Energy minimization
  • Manta rays foraging optimization
  • Morlet wavelet mutation
  • Quadratic interpolation strategy
  • Shape optimization


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