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Hybrid chimp optimization algorithm for degree reduction of ball Said–Ball curves

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Abstract

The optimal multi-degree reduction of ball Said–Ball curves is an unsolved and knotty important technique in computer aided design (CAD) and computer graphics (CG) and is potentially used in many engineering fields involving geometric modeling. In this paper, an improved chimp optimization algorithm (ICHOA, for short) is used to solve the degree reduction of BSB curves. Firstly, the multi-degree reduction of BSB curves is mathematically an optimization problem that can be efficiently dealt with by a swarm intelligence algorithm. In this regard, a novel enhanced version of CHOA called ICHOA, combined with the proportional weight, dimension learning-based hunting search and fractional order strategies, 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 ICHOA is verified by comparing it with standard CHOA, other improved CHOA and popular nature-inspired optimization algorithms on 23 classical benchmark functions, the CEC’17 test suite and five engineering optimization problems, respectively. Secondly, the optimization models of multi-degree reduction for the center curve and radius function of BSB curves are established, respectively; meanwhile, the proposed ICHOA is utilized to solve the established optimization models, and the optimal center curve and radius function with a minimum distance of the approximating BSB curves of lower degree are also obtained. Finally, experimental results illustrate the ability of the proposed ICHOA to effectively solve the optimization problems of multi-degree reduction of BSB curves in terms of precision, robustness, and convergence characteristics.

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Data availability

All data generated or analyzed during this study are included in this published article.

Code availability

The Matlab source code of the ICHOA related to this article can be found online at https://www.researchgate.net/publication/367189337_The_codes_of_ICHOA.

Abbreviations

CHOA:

Chimp optimization algorithm

ICHOA:

Improved chimp optimization algorithm

CAD:

Computer aided design

BSB:

Ball Said–Ball

CAGD:

Computer aided geometric design

t :

Number of current iterations

X prey :

Prey position

X chimp :

Chimp position vector

m :

Chaotic vector calculated based on various chaotic maps

r 1, r 2 :

Random vectors in the range of [0,1]

f :

Dynamic coefficient

H 1, H 2, H 3, H 4 :

Learning rates of chimps to attackers, obstructors, chasers and drivers

a, m, c :

Coefficient vectors

X i(t):

iTh current position of chimp

X i-CHOA(t + 1):

i-CHOATh candidate position obtained by the original CHOA

D i :

Euclidean distance between Xi(t) and Xj(t)

N :

Population number of chimps

D :

Dimension

X i - DLH(t + 1):

DLH search strategy generates another candidate

P :

Sampling period

r :

Truncation order

β :

Fractional order parameter

h :

Thickness of weld

l :

Welding joint length

t a :

Height of bar

b :

Beam thickness

T h :

Thickness of the head

T s :

Thickness of the shell

L :

Length of the pipe

R :

Inner radius of the pipe

w :

Wire diameter

d :

Coil diameter

r i :

Non-negative real number

\(\langle {\mathbf{P}}_{i} \rangle\) :

Control balls of BSB curve

\(\{{S_{i}^{n}}(t)\}_{i = 0}^{n}\) :

The Said–Ball basis functions of degree n

\(\left\lfloor x \right\rfloor\) :

Largest integer less than or equal to x

\(\left\lceil x \right\rceil\) :

Smallest integer greater than or equal to x.

\({\text{dist}}({\mathbf{C}}_{2} (t),{\mathbf{C}}_{1} (t))\) :

The Euclidean distance

\(\{ t_{j} \}_{j = 1}^{H}\) :

Sampling parameters in [0,1]

H :

Measure the distance between the two curves

\({\mathbf{C}}_{1} (t)\) :

Original center curves

\({\mathbf{C}}_{2} (t)\) :

Degree reduced center curves

R 2(t):

Radius function

\(|| \cdot ||_{\infty }\) :

Infinite norm of a continuous function

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Acknowledgements

The authors are very grateful to the Reviewers for their insightful suggestions and comments, which helped us to improve the presentation and content of the paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 51875454 and 61976176). This work is also supported by the Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China (No.2021JM320).

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Authors and Affiliations

  1. Department of Applied Mathematics, Xi’an University of Technology, Xi’an, 710054, People’s Republic of China

    Gang Hu & Wanting Dou

  2. School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, 710048, People’s Republic of China

    Gang Hu

  3. University of North Carolina at Pembroke, Pembroke, NC, 28372, USA

    Guo Wei

  4. Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan

    Muhammad Abbas

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  1. Gang Hu
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Correspondence to Gang Hu.

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Appendix 1

Appendix 1

See Fig. 33.

Fig. 33
figure 33

Convergence simulation figures of five engineering optimization problems

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Hu, G., Dou, W., Wei, G. et al. Hybrid chimp optimization algorithm for degree reduction of ball Said–Ball curves. Artif Intell Rev 56, 10465–10555 (2023). https://doi.org/10.1007/s10462-023-10416-4

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