Multimodal Particle Swarm Optimization with Phase Analysis to Solve Complex Equations of Electromagnetic Analysis

In this paper, a new meta-heuristic method of finding roots and poles of a complex function of a complex variable is presented. The algorithm combines an efficient space exploration provided by the particle swarm optimization (PSO) and the classification of root and pole occurrences based on the phase analysis of the complex function. The method initially generates two uniformly distributed populations of particles on the complex plane and extracts the function phase in a position of each particle. By collecting phase samples, the candidate regions of root and pole occurrences are selected. Then, the second population, by iteratively converging towards candidate regions, thoroughly explores an area outside candidate regions and reduces the possibility of root or pole omission. The subsequent swarms are generated locally to explore candidate regions and decrease their size. The algorithm is verified in electromagnetic benchmark that solves the equation determining surface waves on a microstrip antenna. The numerical results show that the algorithm is able to solve multimodal problems quickly even with a small initial population and a small number of generated swarms.