# Coupled Numerical Model of Vibration-Based Harvester
## 2 FEM Numerical Model
\begin{split} k_{qqp}^{e}&=\left(\frac{1}{6\Delta \Omega^{\varepsilon}}\right)^{2}\!\!\!\int\limits_{\Omega^{\varepsilon}}\left(-b_{ i}i+c_{i}j-d_{i}k\right)\ \varepsilon^{e}\!\!\left(-b_{j}i+c_{j}j-d_{j}k\right)\\ z_{qp\varphi}^{e}&=\left(\frac{1}{6\Delta \Omega^{\varepsilon}}\right)^{2}\!\!\int\limits_{\Omega^{\varepsilon}}\left(a_{i}-b_ {i}x+c_{i}y-d_{i}z\right)\ \left(a_{j}-b_{j}x+c_{j}y-d_{j}z\right)d\Omega\\ \end{split}  \tag{72}\]  
When the hexahedral element from Figure 5 is selected, its base functions are written as  
\[ \begin{array}{l}W_{1}=\frac{1}{8}(1-\xi)(1-\eta)(1-\zeta)\\ W_{2}=\frac{1}{8}(1+\xi)(1-\eta)(1-\zeta)\\ \forall\xi,\eta,\zeta\in(-1,1)\\ W_{3}=\frac{1}{8}(1+\xi)(1+\eta)(1-\zeta)\\ W_{4}=\frac{1}{8}(1-\xi)(1+\eta)(1-\zeta)\\ W_{5}=\frac{1}{8}(1-\xi)(1-\eta)(1+\zeta)\\ W_{6}=\frac{1}{8}(1+\xi)(1-\eta)(1+\zeta)\\ W_{7}=\frac{1}{8}(1+\xi)(1+\eta)(1+\zeta)\\ W_{8}=\frac{1}{8}(1-\xi)(1+\eta)(1+\zeta)\end{array}  \tag{90}\]  
For the hexahedral element, the sizes of the coefficients from the set of Equations (57) are calculated from Equations (80)-(89) using the functions (90). Integration in the relations (80) to (89) can be solved analytically. This method is difficult and time consuming. Avoiding the use of analytically expressed integrals in most cases will not significantly reduce the accuracy of the solution of the system of Equations (57). Another easier solution is to apply the Gaussian quadrature. Its form, usage and properties are known [3]. The coefficients of the system of Equations (57) are then calculated for a numerical solution  
\[\begin{array}{l}K_{A1,x}^{e}=\sum\limits_{\xi\beta}^{ng}\sum\limits_{\xi \beta}^{ng}\sum\limits_{\xi\beta}^{\frac{\log}{\log}}H_{\xi\beta}I_{B\xi}I_{B \xi}I_{B\xi}\Bigg{[}\frac{1}{I  
Six points of integration \(n_{\text{g}}=m_{\text{g}}=l_{\text{g}}=6\) were chosen for the pentahedral element. The number corresponds to the required accuracy and speed of the quadrature calculation. The coordinates of the points are determined  
\[ \begin{array}{l}\xi_{g1}=\pm 0.577350269189626,\;H_{\text{i}}=1.0,\\ \eta_{g2}=\pm 0.577350269189626,\\ \zeta_{g3}=\pm 0.577350269189626\;.\end{array}  \tag{105}\]  
Similarly, for a hexahedral element, where \(n_{g}=m_{g}=l_{g}=8\). The coordinates of the point for calculating the quadrature according to (n102) to (n104) is expressed by the auxiliary function \(L\)  
\[ \begin{array}{l}1.\;\text{point}:\;L_{1}=0.1314458557658,\;H_{\text{i}}=1.0, \\ 2.\;\text{point}:\;L_{2}=0.4905626121623,\\ 3.\;\text{point}:\;L_{3}=L_{1},\\ 4.\;\text{point}:\;L_{4}=0.03522081090086,\\ 5.\;\text{point}:\;L_{5}=L_{4},\\ 6.\;\text{point}:\;L_{6}=L_{1},\\ 7.\;\text{point}:\;L_{7}=L_{4},\\ 8.\;\text{point}:\;L_{8}=0.09437387837656\;.\end{array}  \tag{106}\]  
The electromagnetic part of the model of the task MG can be characterized by the expression (57). The description of the mechanical part of the model by means of the FEM is known for example from theses and works [1,2] and the system [21], then the electromechanically bounded equation is written in the form  
\[(\lambda_{1}+2\lambda_{2})\text{grad\;divs}-\lambda_{2}\text{rot\;rots}+ \underbrace{\boldsymbol{J\times B}+\rho\frac{\partial^{2}\boldsymbol{s}}{ \partial\,t^{2}}+\boldsymbol{f}_{0}}_{f}=\boldsymbol{0} \tag{107}\]  
where \(\boldsymbol{f}\) is the vector of specific force, \(\lambda_{1}\), \(\lambda_{2}\) are auxiliary scalar functions. For the geometrical model, Figure 2, the external mechanical load was not considered. Such presumption will lead to \(\boldsymbol{f}_{0}=\boldsymbol{0}\). But, to respect the position of the MG in the earth's gravitational field, with the acceleration \(g\) a static force, it was considered \(\boldsymbol{f}_{0}=\frac{M_{g}}{V_{g}}\cdot g\), where \(m_{g}\) is the mass of the MG core, \(V_{g}\) is the volume of the MG core. The specific magnetic force \(\boldsymbol{f}_{m}\) is expressed as  
\[\boldsymbol{f}_{m}=\boldsymbol{J\times B} \tag{108}\]  
There are valid boundary and initial conditions for the model, for example known [21].