following equation 14 where j 0 denotes the current density in the coil of the electromagnet when the applied current to the electromagnet is large nonlinear magnetic analysis becomes necessary in this case the magnetic potential a may be obtained by an iteration method such as the newton raphson method and the k 1 th approximate value a i k 1 for location i is given by the following equations the k th u03b4a i k is obtained by the following equation 16 and the updating a i by equation 15 is repeated until a i converges where k nonlinear and j nonlinear are related to the nonlinear property the sensitivity analysis for this case can be obtained by an equation similar to equation 11 in the second embodiment the design variables u03c1 can vary from 0 to 1 if u03c1 is 0 actually very close to 0 then the element is interpreted as an air element on the contrary if u03c1 approaches 1 it represents the bias yoke in the design domain 16 as marked in fig4 a the optimized shape of the yoke can be obtained as illustrated in fig4 c by disposing the magnetic core of the electromagnet perpendicular to the beam 12 assuming that permeability is linear in fig4 b is the direction where the current flows out and is the direction where current flows in in addition the optimized.