A Paul ion trap is a device designed to con ne charged particles in a small region of uniform, dynamic quadrupole potential. The aim of this project is to simulate numerically the distribution of charges and the potential generated by our electrodes using the method of moments (MoM), and to predict the trajectory of the con ned ions when the electrodes are subject to a harmonic potential. Starting with simpler models in 2D and 3D, we provide the optimal design parameters needed for the trap to work, depending on the number and properties of the ions and the quadrupole potential. We also provide di erent numerical mechanisms aimed at reducing the computational cost of the simulation 2020/2021