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

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.