In summary, drilling simulation, a set of physic-based models run through time or depth steps to mirror events in the drilling rig, is the backbone of all field testing of technologies or procedures. If a model has been validated using drilling simulation, the risk of wasted field trial is lowered significantly. This is why the formulation of models that make up drilling simulation is key and this is what this thesis has focused on. 20 functions were used to simulate the processes described in this research. Finite element formulation of space models linked with time-based models have been developed for the 2-node system in X (axial loading and axial torsion), Y (transverse bending of Z), and Z (transverse bending of Y) directions. Laplace transform was used to solve the time based partial differential equation paving way for the development of velocity, acceleration, force, and torque equations. Drill ahead modeling using build and walk relation to resultant forces was validated. Stick slip mitigation using the optimized RPM objective function was used to optimize the mechanical efficiency of drilling. Particle swarm optimization was the process used for optimization where each solution is considered a particle in search of the global minimum. An expression of the optimized RPM was developed and simulated with field data. Confined compressive strength of the field data was compared with the CCS obtained from the simulation but there was no perfect match yet. Further runs of the simulation would show more lessons as to how to improve the results. It can be concluded that the MSE minimization process should rather be called MSE optimization process as the decision to raise or lower MSE should be based on the data supplied to the particle swarm optimizer since the objective function is built with constraints to lower drill string vibrations. When tested with field data, the objective function and optimizer built in this research was found to increase MSE but lower the downhole stick slip index by 28 percent. The downhole stick slip index was below 0.5.