Key Takeaways & Executive Findings
- •• An improved BESO method with adaptive evolutionary ratio (ER) significantly accelerates topology optimization for nonlinear structures, achieving up to 37.3% faster convergence compared to fixed ER. • The method integrates Python with Abaqus for finite element analysis, enabling efficient handling of material and geometric nonlinearities. • Four distinct adaptive ER functions are proposed and validated through benchmark cases, demonstrating consistent improvements in optimization efficiency. • The approach extends the applicability of BESO to large-scale engineering structures and complex nonlinear problems, offering a robust tool for practical design.
Abstract
Current topology optimization methods for nonlinear continuum structures often suffer from low computational efficiency and limited applicability to complex nonlinear problems. To address these issues, this paper proposes an improved bi-directional evolutionary structural optimization (BESO) method tailored for maximizing stiffness in nonlinear structures. The optimization program is developed in Python and can be combined with Abaqus software to facilitate finite element analysis (FEA). To accelerate the speed of optimization, a novel adaptive evolutionary ratio (ER) strategy based on the BESO method is introduced, with four distinct adaptive ER functions proposed. The Newton-Raphson method is utilized for iteratively solving nonlinear equilibrium equations, and the sensitivity information for updating design variables is derived using the adjoint method. Additionally, this study extends topology optimization to account for both material nonlinearity and geometric nonlinearity, analyzing the effects of various nonlinearities. A series of comparative studies are conducted using benchmark cases to validate the effectiveness of the proposed method. The results show that the BESO method with adaptive ER significantly improves the optimization efficiency. Compared to the BESO method with a fixed ER, the convergence speed of the four adaptive ER BESO methods is increased by 37.3%, 26.7%, 12% and 18.7%, respectively. Given that Abaqus is a powerful FEA platform, this method has the potential to be extended to large-scale engineering structures and to address more complex optimization problems. This research proposes an improved BESO method with novel adaptive ER, which significantly accelerates the optimization process and enables its application to topology optimization of nonlinear structures.
1. Introduction
Topology optimization is a mathematical method used to optimize material distribution within a design domain based on specific load cases, constraints, and performance metrics. In Ref. [1], topology optimization has made significant advancements in addressing linear problems and has achieved considerable success in structural design. However, topology optimization presents substantial challenges due to the large number of design variables and complex constraints, which often make it more difficult to find optimal solutions compared to other optimization methods. Despite these difficulties, the bi-directional evolutionary structural optimization (BESO) method has emerged as a robust tool in both academic research and engineering applications, enabling efficient exploration of optimal structural configurations through iterative addition and removal of elements [2].
Practical engineering scenarios often involve nonlinear phenomena. To achieve more realistic designs, researchers continuously explore the extension of topology optimization to address various nonlinear aspects in structural design, such as material nonlinear, geometrically nonlinear. Xu et al. [3, 4] investigated the BESO method for material and geometric nonlinearity under stress constraints, while considering both strength and stiffness. Banh et al. [5] proposed a topology optimization method that simultaneously considers geometric nonlinearity and design-dependent pressure loads. Wang et al. [6] proposed a topology optimization method considering both geometrical and load nonlinearities. Shobeiri [7] examined the optimal topology of structures subjected to dynamic loading, considering material nonlinearity, geometric nonlinearity, and contact nonlinearity using the BESO method. Chen et al. [8] conducted research on nonlinear topology optimization for flexoelectric soft dielectrics undergoing large deformations. Da Silva et al. [9] demonstrated that the maximum output displacement method, based on nonlinear analysis, provides better solutions under large displacements while simultaneously satisfying stress and manufacturing requirements.
The primary challenge in nonlinear structural topology optimization lies in finding the optimal topology structure or, in other words, developing a robust method that can efficiently handle the complexities introduced by nonlinearities.
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Linli Tian, Wenhua Zhang (2025). Bi-Directional Evolutionary Topology Optimization with Adaptive Evolutionary Ratio for Nonlinear Structures. Chinese Journal of Mechanical Engineering. https://doi.org/10.1186/s10033-025-01276-w
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Frequently Asked Questions
What is the main contribution of this paper?
The paper proposes an improved BESO method with adaptive evolutionary ratio (ER) strategies to accelerate topology optimization for nonlinear structures, achieving significant improvements in convergence speed.
How does the adaptive ER strategy work?
The adaptive ER strategy dynamically adjusts the evolutionary ratio during optimization based on four distinct functions, leading to faster convergence compared to a fixed ER.
What types of nonlinearities are considered?
The method accounts for both material nonlinearity and geometric nonlinearity, and analyzes their effects on the optimized topology.
What software is used for finite element analysis?
The optimization program is developed in Python and can be combined with Abaqus software to facilitate finite element analysis.
What are the potential applications of this method?
The method has the potential to be extended to large-scale engineering structures and to address more complex optimization problems, making it suitable for practical engineering design.
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