In this research, a new zeroth-order (non-gradient based) topology optimization methodology for compliance minimization was developed. It is called the Element Exchange Method (EEM). The principal operation in this method is to convert the less effective solid elements into void elements and the more effective void elements into solid elements while maintaining the overall volume fraction constant. The methodology can be integrated with existing FEA codes to determine the stiffness or other structural characteristics of each candidate design during the optimization process. This thesis provides details of the EEM algorithm, the element exchange strategy, checkerboard control, and the convergence criteria. The results for several two- and three-dimensional benchmark problems are presented with comparisons to those found using other stochastic and gradient-based approaches. Although EEM is not as efficient as some gradient-based methods, it is found to be significantly more efficient than many other non-gradient methods reported in the literature such as GA and PSO.