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A Geometric Whale Optimization Algorithm with Triangular Flight for Numerical Optimization and Engineering Design
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  • Published: 12 February 2026

A Geometric Whale Optimization Algorithm with Triangular Flight for Numerical Optimization and Engineering Design

  • Junhao Wei1,2,
  • Ran Zhang1,
  • Yanzhao Gu1,
  • Wenxuan Zhu1,
  • Yanxiao Li1,
  • Zikun Li4,
  • Shuai Wu1,3,
  • Zhiwen Wang5,
  • Ngai Cheong1,
  • Sio-Kei Im6,
  • Yapeng Wang1 &
  • …
  • Xu Yang1 

Scientific Reports , Article number:  (2026) Cite this article

We are providing an unedited version of this manuscript to give early access to its findings. Before final publication, the manuscript will undergo further editing. Please note there may be errors present which affect the content, and all legal disclaimers apply.

Subjects

  • Engineering
  • Mathematics and computing

Abstract

Engineering design optimization problems are often characterized by high dimensionality, complex constraints, and multimodal search landscapes, which pose significant challenges to conventional metaheuristic algorithms. Although the Whale Optimization Algorithm (WOA) has demonstrated competitive performance, it still suffers from premature convergence, limited population diversity, and an imbalanced exploration-exploitation mechanism in complex optimization scenarios. To overcome these limitations, this paper proposes a Geometric Whale Optimization Algorithm (ESTGWOA), in which multiple geometric strategies are systematically embedded into the canonical WOA framework to enhance population initialization, search guidance, and position update behaviors. By incorporating geometric-based mechanisms, ESTGWOA effectively improves search space coverage and strengthens the coordination between global exploration and local exploitation. Comprehensive experiments on 23 benchmark functions demonstrate that ESTGWOA, with an overall effectiveness of 97.10%, outperformed other algorithms on benchmark functions with different dimensions of 30, 50 and 100 dimensions. And the simulations on a series of constrained engineering design problems demonstrate that ESTGWOA consistently outperforms selected state-of-the-art metaheuristic algorithms. Quantitative results show that ESTGWOA achieves superior average fitness values and lower standard deviations in most cases, with statistical significance verified by Wilcoxon rank-sum and Friedman tests. Furthermore, qualitative analyses of search history, population diversity, and convergence behavior confirm the robustness and stability of the proposed approach. These results indicate that ESTGWOA is a reliable and effective optimization algorithm for complex continuous engineering design problems.

Data availability

To support the experimental study in this paper, we used the Standard Benchmark Functions. The relevant data has been uploaded to Figshare, and the link for the specific modeling of Standard Benchmark Functions is: https://doi.org/10.6084/m9.figshare.28440863 , only for reference and further analysis by the readers. The specific modeling of engineering optimization challenges in this study has been uploaded to Figshare, and the link is: https://figshare.com/articles/thesis/engineering_m/28673777?file=53256305, only for reference and further analysis by the readers.

Code availability

To support the experimental study in this paper, we used the Standard Benchmark Functions. The relevant data has been uploaded to Figshare, and the link for the specific modeling of Standard Benchmark Functions is: https://doi.org/10.6084/m9.figshare.28440863, only for reference and further analysis by the readers67. The specific modeling of engineering optimization challenges in this study has been uploaded to Figshare, and the link is: https://doi.org/10.6084/m9.figshare.28673777.v1, only for reference and further analysis by the readers.

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Acknowledgements

The supports provided by Macao Polytechnic University (MPU Grant no: RP/FCA-03/2022; RP/FCA-04/2022; RP/FCA-06/2022; RP/FCA-01/2025) and MacaoScience and Technology Development Fund (FDCT Grant no: 0044/2023/ITP2; FDCT-MOST (0018/2025/AMJ) enabled us to conduct data collection, analysis,and interpretation, as well as cover expenses related to research materials and participant recruitment. MPU and FDCT investment in our work (MPU Submission Code: fca.def5.c31c.0) have significantly contributed to the quality and impact of our research findings.

Funding

This work is supported by the grant from FDCT-MOST (0018/2025/AMJ) and Macao Polytechnic University (RP/FCA-01/2025).

Author information

Authors and Affiliations

  1. Faculty of Applied Sciences, Macao Polytechnic University, Macao, 999078, China

    Junhao Wei, Ran Zhang, Yanzhao Gu, Wenxuan Zhu, Yanxiao Li, Shuai Wu, Ngai Cheong, Yapeng Wang & Xu Yang

  2. Pazhou Lab (Huangpu), Guangzhou, 510555, China

    Junhao Wei

  3. School of Artificial Intelligence, Dongguan City University, Dongguan, 523419, China

    Shuai Wu

  4. School of Economics and Management, South China Normal University, Guangzhou, 510006, China

    Zikun Li

  5. School of Nursing, Peking University, Beijing, 100191, China

    Zhiwen Wang

  6. Macao Polytechnic University, Macao, 999078, China

    Sio-Kei Im

Authors
  1. Junhao Wei
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  2. Ran Zhang
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Contributions

J.W. conceived and designed the study; J.W. developed the methodology and implemented the software; J.W. performed the validation; R.Z., S.W., Z.L. and W.Z. conducted the formal analysis; J.W. carried out the investigation; Z.W., N.C., S.-K.I., Y.W, and X.Y. provided the resources; W.Z., S.W., Z.L., J.W., Y.L., Y.G., and R.Z. performed the data curation; J.W. prepared the original draft; Z.W., N.C., S.-K.I., Y.W, and X.Y. reviewed and edited the manuscript; W.Z., S.W., Z.L., J.W., Y.L., Y.G., and R.Z. contributed to the visualization; Z.W., N.C., S.-K.I., Y.W, and X.Y. supervised the research and managed the project; N.C, Y.W and X.Y. acquired the funding. All authors reviewed the manuscript.

Corresponding author

Correspondence to Xu Yang.

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Wei, J., Zhang, R., Gu, Y. et al. A Geometric Whale Optimization Algorithm with Triangular Flight for Numerical Optimization and Engineering Design. Sci Rep (2026). https://doi.org/10.1038/s41598-026-37387-0

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  • Received: 14 November 2025

  • Accepted: 21 January 2026

  • Published: 12 February 2026

  • DOI: https://doi.org/10.1038/s41598-026-37387-0

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Keywords

  • Whale Optimization Algorithm
  • Triangular flight
  • Numerical optimization
  • Engineering design
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