Applications of the finite element method to space charge beam problems in RF fields

A technique is described for obtaining space charge solutions of problems where particle beams are subjected to radiofrequency fields. Macroparticles are used to represent the beam in a finite element problem. The beam is tracked through the model over several RF cycles and a space charge solution is obtained each time step. Use of a conjugate gradient method enables the potentials for the previous time step to be used as a first approximation for each new solution. Correct trajectories are obtained when each particle throughout the final cycle has seen a space charge solution for each time step of its track. The method has been used to study space charge effects in an rf quadrupole acclerator. Other applications are foreseen.