# Two-Dimensional Topology Optimization of Headtube in Electric Scooter Considering Multiple Loads
## 2 Design Optimization
### Topology Optimization: Single-Load Conditions  
Methodology  
Topology optimization iteratively redistributes materials in the design domain to determine the optimal material arrangement. The control mechanism is a self-enhancing system in which more loads attract a larger mass. The update scheme in topology optimization adds material to parts with relatively high strain energy and removes material from parts with relatively low strain energy.  
Single-load conditions were applied to the nodes of the structure as concentrated forces, combining each of the four directions at two load points. The design variable was the density of each element. For topology optimization, the total strain energy under each of the 16 load conditions was selected as the objective function to be minimized, and the perimeter constraint volume fraction was implemented. Topology optimization [37] is formulated as follows:  
\[\begin{array}{c}\min\limits_{\rho}\qquad f(\rho)=\sum\limits_{i=1}^{N}{(\rho _{i})^{3}\mathbf{u}_{i}^{7}\mathbf{k}_{0}\mathbf{u}_{i}}\\ \text{subject to}\quad g(\rho)=\frac{\sum\limits_{i=1}^{N}{\rho_{i}}}{V_{0}}= \text{VF}\\ \qquad 0.001\leq\rho_{i}\leq 1\end{array} \tag{2}\]  
where \(\rho_{i}\) is the density of each element as a design variable, \(N\) is the total number of elements, \(\mathbf{u}_{i}\) is the displacement vector, \(\mathbf{k}_{0}\) is the stiffness matrix of the elements, \(V_{0}\) is the volume of the total design area, and \(VF\) is the volume fraction. The volume fraction (0.2) and filter size (4 pixels) were applied as hyperparameters. To reduce numerical instabilities in topology optimization, the sensitivity was filtered [38; 39] as follows:  
\[\frac{\partial f}{\partial\rho_{i}}=\frac{1}{\rho_{i}\sum\limits_{j=1}^{N}{r_ {\min}-dist(i,j)}}\sum\limits_{j=1}^{N}{(r_{\min}-dist(i,j))\rho_{j}\frac{ \partial f}{\partial\rho_{j}}} \tag{3}\]  
where \(dist(i,j)\) represents the distance between the centers of the \(i\)-th and \(j\)-th elements and \(j\) denotes an element satisfying the condition of \(dist(i,j)\leq r_{\min}\), where \(r_{\min}\) is the filter size. The design completion criteria are defined by the maximum allowable tolerance of the design variables between iterations. Topology optimization was performed using the Optimality Criteria Method [40], which can be formulated as \[\rho_{i}^{new}=\left\{\begin{array}{l}\max(\rho_{\min},\rho_{i}-m)\\ \text{ if }\rho_{i}B_{i}^{\eta}\leq\max(\rho_{\min},\rho_{i}-m)\\ \rho_{i}B_{i}^{\eta}\\ \text{ if }\max(\rho_{\min},\rho_{i}-m)<\rho_{i}B_{i}^{\eta}<\min(1,\rho_{i}+m)\\ \min(1,\rho_{i}+m)\\ \text{ if }\min(1,\rho_{i}+m)\leq\rho_{i}B_{i}^{\eta}\end{array}\right. \tag{4}\]  
where \(m\) is the positive move limit, and \(\eta\) is the numerical damping coefficient, set to 0.5.  
The optimality condition for updating the design variable is derived from the method introduced by [PERSON] [37] and later refined by [PERSON] [34] for topology optimization, as follows: The design variable \(B_{i}\), which is determined from the optimality condition, is given by Equation (5):  
\[B_{i}=\frac{p(\rho_{i})^{p-1}\mathbf{u}_{i}^{T}k_{0}\mathbf{u}_{i}}{\lambda\frac{ \partial V}{\partial\rho_{i}}} \tag{5}\]  
where \(\lambda\) is the Lagrangian multiplier, determined using the bi-sectioning algorithm, and the denominator represents the sensitivity of the objective function. Topology optimization was initially performed under 16 single-load conditions to evaluate the structural response of the headtube under various force applications. However, as electric scooters experience complex and varying loads during operation, it was necessary to extend the optimization framework to consider multiple loading conditions. To achieve this, representative load cases were selected based on the observed structural behavior patterns, ensuring that the multiple-load optimization accounted for realistic force interactions while maintaining computational efficiency.