# A Modified Shuffled Frog Leaping Algorithm for the Topology Optimization of Electromagnet Devices
## 4 Problem Formulation and Numerical Experiments  
In this paper, the proposed SFLA is applied to the topology optimization of the iron part in the rotor of a benchmark IPM. To reduce the time for objective value evaluation, the 2D model of the 1/4 cross-section of the prototype machine with periodic boundary condition is used for FEM evaluation, and the stack length is set to be 120 mm. All other parameters in the model are set to be constants. Table 1 lists all the fixed parameters. The design space of the problem is the 1/16 th portion of the rotor which is then replicated with two mirror processes to re-construct the quarter model. The usage of smaller design space helps to improve the resolution of the obtained topology with the same number of elements, as well as ensuring a symmetrical rotor structure. The entire design domain is divided into 990 elements, 22 in a row and 45 in a column, with each element occupying 0.5\({}^{\circ}\) in width and 1 mm in length and is filled with either iron or air. Numerical experiments were conducted to search for the best number of RBFs and Gaussian variance for the RBF modeling process. To obtain a smooth topology with relatively small \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Stator outer radius & 110 & mm \\ Stator inner radius & 75 & mm \\ Air gap length & 0.45 & mm \\ Rotor outer radius & 74.55 & mm \\ Rotor inner radius & 52.5 & mm \\ Permanent magnet length & 20.4 & mm \\ Permanent magnet width & 5.5 & mm \\ Phase current amplitude & 9.69 & A \\ Coil number of turns & 30 & \\ Motor speed & 2800 & rpm \\ Number of slots/poles & 48/8 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: IPM fixed parameter values. number of design variables, \((4\times 9)=36\) RBFs are used to cover the entire design domain. Each RBF is set to be fiveelements from its neighbor RBF, in both radial direction and along the arcs, and the variance of the Gaussian function is set to 1. The control parameters, which are the weight of each RBF, are set to be bounded within [0, 1]. Figure 2 depicts the initial design domain and the 36 RBFs are used to construct the topology. The red circles in the design domain indicate the RBFs and their effective area, where inside the effective area the values of RBFs are positive, and outside the effective area their values are assumed to be zero. Hence, each RBF controls only the topology of a small region in the design domain, and a large variety of topology can be generated using the RBF method. The aim of the topology optimization is to maximize the average torque and minimize the torque ripple, and hence the objective function of the TO problem is defined as the weighted sum of the normalized average torque and the torque ripple of the IPM evaluated using Ansys Maxwell, evaluated by the virtual work method employed in the software:  
\[ \begin{split}\max f(X)=c_{1}\frac{T_{\text{avg}}(X)}{T_{\text{ torque}}}-c_{2}\frac{T_{\text{max}}(X)-T_{\text{min}}(X)}{T_{\text{avg}}(X)},X=[X_{1},X_{2}, \ldots X_{36}]\\ \text{s.t.}\ Area_{\text{irron}}\leq Area_{\text{design}}\end{split}  \tag{9}\]  
where \(c_{1}\) and \(c_{2}\) are two predefined weight coefficients and equal to 1 in this paper, \(Area_{\text{irron}}\) is the area of iron part in the IPM of the design, and \(Area_{\text{design}}\) is the area of the design space, which has a fixed size in the optimization problem, and hence can serve as the volume constraint of the TO problem. Moreover, a transient analysis in Ansys Maxwell is used to compute the average torque and the torque ripple. Consequently, the length of the time step has a significant impact on the computational results. To address this issue, one first conducts a numerical experiment to decide the time step before the optimization, i.e., for a fixed topology, one incrementally reduces the length of time step until the errors of the computed average torque and the torque ripple between two successive time step reductions are smaller than a given tolerance.