How can we deal with moving objects on a fixed mesh?

In numerical modeling and simulation it frequently occurs that moving objects need to be considered. In these cases, dealing with such objects on a fixed mesh would be a clear advantage. Not only that the mesh could be created once and for all, but in case of applying optimization algorithms there would be no noise in the evaluation of the target function due to re-meshing. How can this be accomplished? We will explain the general idea in terms of the electromagnetic domain, and in particular for eddy current problems of moving bodies as test cases. Such problems occur in the modeling and simulation of electromechanical devices, like solenoid actuators or rotating electrical machines.