# Magnetic Force Enhancement Using Air-Gap Magnetic Field Manipulation by Optimized Coil Currents
## 3 Optimization Strategy  
This section explains the optimization strategy for finding the optimal coil currents for maximizing the magnetic force. Figure 3 shows the two-dimensional finite element model of a benchmark actuator used in this work. The magnetostatic analysis is performed using [PERSON]'s equation with vector potential \(\mathbf{A}\) formulation. \[\
abla\times\left(\frac{1}{\mu_{r}\mu_{0}}\
abla\times\mathbf{A}\right)=\mathbf{ J}\quad\left(\mathbf{B}=\
abla\times\mathbf{A}\right) \tag{2}\]  
where \(\mathbf{B}\) is magnetic flux density, \(\mu_{r}\) is relative permeability, and \(\mathbf{J}\) is a current density. In the two-dimensional analysis, only z-directional vector potential \(A_{z}\) is a state variable. The yoke and plunger are made of silicon steel with a relative permeability of 1200, and the coil winding is located around the yoke pole. A current in the winding is also set to 10 A with 100 number of turns. The Dirichlet boundary condition is applied at the outer boundary of the model. For the optimization, \(x\) directional (i.e., tangential direction to the air-gap line) magnetic force \(F_{x}^{m}\) at 21 equally spaced plunger locations (\(m=0,\,\ldots,\,20\)) is calculated. In the benchmark model in Figure 3, the design domain is located at the tip of the yoke near the air-gap. As with topology optimization [27], the design domain is divided into \(N\) small areas, and the design variable \(\beta_{i}(i=1,\ldots,N)\) is assigned at \(i\)th design area. Here, the physical meaning of the design variable \(\beta_{i}\in[-\beta_{max},\beta_{max}]\) is set as the number of coil turns at \(i\)th design area, and the sign of \(\beta_{i}\) represented the direction of coil currents. If the number of coil turns is zero (i.e., \(\beta_{i}=0\)), the \(i\)th design area represented the iron material instead of a coil. It means that the design variable \(\beta_{i}\) determined whether the \(i\)th design area is iron or coil material. For this, not only the coil current density \(\mathbf{J}_{a}^{i}\) but also relative magnetic permeability \(\mu_{r}^{i}\) of \(i\)th design area are determined by the design variable \(\beta_{i}\). In order to apply the gradient-based optimization algorithm, \(\mathbf{J}_{a}^{i}\) and \(\mu_{r}^{i}\) are interpolated using the following functions:  
\[\mathbf{J}_{a}^{i}=\frac{\mathbf{I}_{0}\beta_{i}}{S_{i}} \tag{3}\]  
Figure 3: Finite element model of a benchmark actuator with boundary condition and design domain. Figure 2: Conceptual comparison of (**a**) evenly distributed normal-direction magnetic field, (**b**) evenly distributed and 45 degrees inclined magnetic field, (**c**) focused and inclined magnetic field. \[\mu_{r}^{i}=\mu_{coll}+(\mu_{iron}-\mu_{coil})e^{-\eta\beta_{i}^{2}} \tag{4}\]  
where \(\mathbf{I}_{0}\) is the coil current, \(S_{i}\) is the volume of \(i\)th design area, \(\mu_{coll}\) and \(\mu_{iron}\) are, respectively, the relative permeability of coil and iron material, and \(p\) is the parameter for the relaxed interpolation function. Figure 4a,b, respectively, show the plot of the proposed interpolation functions (3) and (4) with respect to the design variable \(\beta_{i}\). The coil current density \(\mathbf{J}_{a}^{i}\) in (3) is simply proportional to the design variable \(\beta_{i}\). The relative permeability \(\mu_{r}^{i}\) in (4) became the iron permeability \(\mu_{iron}\) when the design variable \(\beta_{i}\) is zero (i.e., zero coil turns), while it became the coil permeability \(\mu_{coil}\) when the design variable \(\beta_{i}\) is not zero. The exponential function in (4) is introduced to relax a discontinuous function for discrete material states (i.e., iron or coil) into the continuous function. Here, the continuous function is required to apply a gradient-based optimization algorithm; refer to interpolation functions in topology optimization to control material states in [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19].