# A Modified Shuffled Frog Leaping Algorithm for the Topology Optimization of Electromagnet Devices
###### Abstract  
The memetic algorithms which employ population information spreading mechanism have shown great potentials in solving complex three-dimensional black-box problems. In this paper, a newly developed memetic meta-heuristic optimization method, known as shuffled frog leaping algorithm (SFLA), is modified and applied to topology optimization of electromagnetic problems. Compared to the conventional SFLA, the proposed algorithm has an extra local search step, which allows it to escape from the local optimum, and hence avoid the problem of premature convergence to continue its search for more accurate results. To validate the performance of the proposed method, it was applied to solving the topology optimization of an interior permanent magnet motor. Two other EAs, namely the conventional SFLA and local-search genetic algorithm, were applied to study the  
case of TO, due to the large degree of freedoms in the design space, the inherit complexity of the problem is much higher than those of the shape and size optimization problems, and hence the searching speed of the algorithm plays an important role affecting the optimization time. One issue with respect to the conventional EAs is the passage of genes is limited to the offsprings, because the generation of children is based on the crossover of elite parents only, and hence the searching direction is limited, resulting in a relatively poor searching efficiency. To overcome this issue, local search steps are often added to the elite population or to the best candidate [9]. In the local search step, the neighborhood of the feasible candidate is searched by adding randomly generated vectors to the candidate and with their performances investigated accordingly. Another type of meta-heuristic algorithms is the memetic algorithm, where memetic evolution is used as the searching mechanism. The entire population is divided into subgroups called memeplex, and, within each memeplex, the searching direction, which can also be regarded as the memes in the memeplex, is determined by the positions of the best solutions. Each individual in the memeplex learns from both the local best solution in the memeplex and from the global best solution in the entire population. Because the memes in the memetic algorithm can spread freely through the population, this type of algorithms is associated with a stronger searching efficiency with fewer objective function evaluations when solving problems with higher degree of freedom and larger number of control variables [10; 11; 12]. The Shuffled Frog-Leaping Algorithm (SFLA), formulated in 2006, has been successfully applied to scheduling problems, network design problems, and parameter estimation of electromagnetic devices [13; 14; 15], and it has shown great potential in handling high dimension problems. The SFLA can be viewed as an extension of the classical particle-swarm optimization (PSO) algorithm, with certain newly added features such as the usage of memeplex. In this paper, the SFLA together with the Gaussian radial basis function (RBF) modeling method is employed to optimize the two-dimensional (2D) design of the rotor in a conventional interior permanent-magnet motor (IPM). To further improve the efficiency of the SFLA and avoid the problem of finding false optima, a local search step is added to the algorithm to add randomness to the optimization process. The design space of the optimization problem is discretized into small elements that will be filled with either iron or air, and Gaussian RBFs are used to calculate the topology function of each element to create the topology. The finite element method (FEM) is employed to evaluate the objective function of the TO problem. To showcase the prominent efficiency of the SFLA in the electromagnetic TO problems, three algorithms were employed for the same IPM TO problem: the local-search GA (LS-GA), conventional SFLA, and the proposed SFLA.