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Eddy currents induced in steel are analyzed by the boundary element method. As the periodic electromagnetic quantities in the steel are distorted due to the nonlinear magnetization property, they are expressed by a Fourier series, whose fundamental and harmonic waves are determined by solving the corresponding integral equations with the surface magnetic fields given as the boundary values. In this paper, we introduce a new combined integration scheme that enables efficient, volume-mesh-less treatment of the practical problems typically characterized by the geometrical singularities like edges and corners. Using this procedure, the internal and surface electromagnetic fields are obtained numerically one after another by applying an adaptive equation technique to obtain the boundary values.