## 2 Methodology
### Convergence and Termination
#### 2.2.3 Bi-Directional Evolutionary Structural Optimization  
Bi-directional Evolutionary Structural Optimization (BESO) is a computational optimization method designed to achieve optimal structural designs by iteratively adding or removing materials based on performance criteria. Unlike traditional methods, BESO optimizes both material distribution and structural topology, making it particularly effective for lightweight design and multi-scale problems.  
Mathematical Formulation  
The optimization process in BESO aims to minimize an objective function, such as compliance or weight, while satisfying design constraints. The design domain is discretized into finite elements, where each element is assigned a density variable \(\rho_{e}\in[0,1]\), representing the presence (\(\rho_{e}=1\)) or absence (\(\rho_{e}=0\)) of material.  
1. Objective Function: The objective function for minimizing compliance is defined as \[C=\mathbf{U}^{T}\mathbf{K}\mathbf{U},\] (27) where * \(C\): Structural compliance; * \(\mathbf{U}\): Global displacement vector; * \(\mathbf{K}\): Global stiffness matrix.
2. Volume Constraint: The optimization problem must satisfy a volume constraint: \[V=\sum_{e=1}^{N_{e}}\rho_{e}V_{e}\leq V_{\text{max}},\] (28) where * \(V\): Total material volume; * \(V_{\text{max}}\): Maximum allowable material volume; * \(N_{e}\): Total number of finite elements; * \(V_{e}\): Volume of element \(e\).
3. Sensitivity Analysis: The sensitivity of the objective function with respect to the design variable \(\rho_{e}\) is calculated as \[\frac{\partial C}{\partial\rho_{e}}=-\mathbf{U}_{e}^{T}\mathbf{K}_{e}\mathbf{ U}_{e},\] (29) where \(\mathbf{U}_{e}\) and \(\mathbf{K}_{e}\) are the displacement vector and stiffness matrix of element \(e\), respectively.  
Convergence and Termination  
The algorithm iterates until a termination criterion is met, such as reaching a predefined number of iterations \(T_{\text{max}}\) or when the change in the objective function \(\Delta C\) between consecutive iterations falls below a tolerance threshold:  
\[\Delta C=\frac{|C^{(t+1)}-C^{(t)}|}{C^{(t)}}\leq\epsilon, \tag{30}\]  
where \(\epsilon\) is the convergence tolerance.  
4. Evolutionary Update Rule: The material distribution is updated iteratively using \[\rho_{e}^{(t+1)}=\begin{cases}1,&\text{if }\frac{\partial C}{\partial\rho_{e}} \leq\eta,\\ 0,&\text{if }\frac{\partial C}{\partial\rho_{e}}>\eta,\end{cases}\] (31)where \(\eta\) is a threshold determined by the volume constraint and evolutionary ratio.
5. Bi-directional Adjustment: BESO allows both the addition and removal of materials, enabling a balanced search for optimal topology: \[\rho_{e}^{(t+1)}=\rho_{e}^{(t)}+\Delta\rho,\] (32) where \(\Delta\rho\) is the material adjustment increment.  
Application in Multi-Scale Optimization  
In this study, BESO is extended to simultaneous multi-scale topology optimization, addressing both cellular and composite materials. The fitness function incorporates separate and unified volume constraints, enabling the optimization of hierarchical structures. The process is defined as  
\[f(\mathbf{\mathrm{\mathbf{x}}})=\alpha\cdot C(\mathbf{\mathrm{\mathbf{x}}})+\beta\cdot P(\mathbf{ \mathrm{\mathbf{x}}}), \tag{33}\]  
where  
* \(C(\mathbf{\mathrm{\mathbf{x}}})\): Compliance of the structure;
* \(P(\mathbf{\mathrm{\mathbf{x}}})\): Penalization term for multi-scale connectivity;
* \(\alpha,\beta\): Weighting factors balancing the objectives.  
Performance in Structural Design  
[PERSON] et al. demonstrated the effectiveness of BESO for multi-scale optimization using a simplified MATLAB (version R2022a4) framework. The method supports cellular and composite materials while handling both separate and single volume constraints, showing significant improvements in structural performance and material efficiency [20].