Recently a periodic surface model was developed to assist geometric construction in computer-aided nano-design. This implicit surface model helps create super-porous nano structures parametrically and support crystal packing. In this thesis, a new approach for pathway search in phase transition simulation of crystal structures is proposed. The approach relies on the interpolation of periodic loci surface models. Respective periodic plane models are reconstructed from the positions of individual atoms at the initial and final states, and surface correspondence is found using a Simulated Annealing-like algorithm. With geometric constraints imposed based on physical and chemical properties of crystals, two surface interpolation methods are used to approximate the intermediate atom positions on the transition pathway in the full search of the minimum energy path. This hybrid approach integrates geometry information in configuration space and physics information to allow for efficient transition pathway search. The methods are demonstrated by examples of FeTi, VO2, and FePt. Additionally, two new particle swarm optimization (PSO) algorithms are developed and applied to crystal structure relaxation of the initial and final states. The PSO algorithms are integrated into the Quantum-Espresso open-source software package and tested against the default Broyden-Fletcher-Goldfarb-Shanno relaxation method.