## 5 The Case Study
### The Multi-Domain Model  
Figure 7 depicts the co-simulation architecture, in which the main effort and flow variables exchanged according to the logic described in Section 2 are highlighted and reported. The leading routine discretizes the time of the simulation, managing the inter-exchange of variables among the different domains as well as estimating the parameters needed in the analysis based on the set of equations described in the previous paragraphs. In detail, at every time step, the \(AC\) resistance \(R_{AC}\), the equivalent inductance \(L\), and the stray capacitance \(C\), together with the voltage amplitude \(V\) of the generator, the voltage amplitude of the equivalent back _e.m.f._ generator \(V_{back\_e.m.f. }\), and the amplitude of the equivalent microcontroller controlling the MOSFET \(V_{gate}\) are passed to the analog electronic domain that solves the instantaneously configured circuit and outputs the voltages' drop across a shunt resistor \(\left(V_{n5}-V_{n6}\right)\). According to [PERSON]'s law, this voltage drop is used to compute the current circulating in the coil. The MBD model outputs the deflection amplitude \(w\) and the related membrane deflection velocity \(dw\). The deflection amplitude measure is passed to the magnetic domain to update the equivalent magneto-static model, which solves the 3D FVM analysis and passes back the estimated magnetic field density \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Diaphrag radius: \(r\) & 1000 & \(\left[\text{$\mu$m}\right]\) \\ Diaphrag thickness: \(h\) & 100 & \(\left[\text{$\mu$m}\right]\) \\ Diaphrag surface: \(S\) & 3.15 & \(\left[\text{$\mu$m}^{2}\right]\) \\ PDMS Young’s modulus: \(E\) & 550 & \(\left[\text{$k$Pa}\right]\) \\ PDMS Poisson’s ratio: \(v\) & 0.5 & \\ Coil current: \(i\) & 0.6 & \(\left[\text{$A$}\right]\) \\ Number of turns: \(n\) & 10 & \\ Magnetic flux density: \(B\) & 0.0416 & \(\left[\text{T}\right]\) \\ Max deflection @ 0.6 \(A\): \(w_{max}\) & 150 & \(\left[\text{$\mu$m}\right]\) \\ Equivalent coil length: \(l\) & 7384 & \(\left[\text{$\mu$m}\right]\) \\ Equivalent spring stiffness: \(k\) & 0.0078 & \(\left[\text{$\text{N}$/$mm}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured and estimated parameters. Figure 7: MEMS multi-domain co-simulation model. \(B\) to the leading routine for the computation of the actuating magnetic force \(F_{mag}\). This computed force value is passed back to the MBD system for the dynamic actuation of the device. The deflection velocity, instead, is passed to the analog electronic domain in the form of back _e.m.f._ voltage amplitude according to: \(V_{back\,e.m.f.}=k_{e.m.f. }dw\), where \(k_{e.m.f. }\) represents a numerically estimated constant: \(k_{e.m.f. }=1.02\times 10^{-12}\frac{\mathrm{V_{S}}}{\mathrm{nm}}\). Figure 8 shows an implemented numerical model of the equivalent MEMS device developed through ESRF Radia(r) package and solved through the 3D Finite Volume Method (3 DFVM). Since most of the constitutive parts do not have any ferromagnetic properties and, consequently, do not interact during the simulation, the system was modeled through a reduced number of components: the planar micro-coil, the permanent magnet, and a fictitious element depicting the PDMS diaphragm. The coil was grounded, while the fictitious disk together with the magnet were continuously updated in position to have a correspondence between the magnetic model and the MBD model. Due to the lack of iron-based materials, the part that required the most computational effort and, consequently, required a volume partition was the NdFeB magnet. The best compromise in solution convergence and computing performances was obtained with 4 \(\times\) 4 \(\times\) 4 volume subdivisions along the \(X\)-axis, \(Y\)-axis, and \(Z\)-axis, respectively. To best fit the cylindrical geometries of the components, the polyhedron finite volume shapes were adopted.