We present the logical expressions (LE) approach that allows fast computation of three-dimensional eddy current problems, including parts in motion. The approach applies time-dependent logical expressions to describe moving parts of the model on a fixed computational grid. The study is motivated by a novel nondestructive testing technique called Lorentz force eddy current testing (LET), which enables the detection of defects lying deep inside a conducting material. Depending on the definition of the frame of reference, we present two different implementations of the LE approach referred to as 1) moving magnet approach, and 2) moving defect approach. In order to demonstrate the advantages of the LE approach, we compare its results with the sliding mesh technique. The validation of the obtained results with experiments is also presented.Index Terms-Eddy current testing, finite element method (FEM), logical expressions, Lorentz force, nondestructive testing.