## 3 Numerical Implementation  
### Piezoelectric Pump Finite Element Implementation  
Solid mechanics and electrostatics equations were used to calculate piezoelectric dynamics numerically in the stress-charge form. \[\{T\} =\left[c^{E}\right]\{S\}-\left[e\right]^{t}\{E\}, \tag{12}\] \[\{D\} =\left[e\right]\{S\}+\left[e^{S}\right]\{E\}, \tag{13}\]  
where \(T\) is stress vector, \(S\) is elastic strain vector, \([c^{E}]\) is elastic stiffness under constant electric field, \([e]\) is a coupling matrix, and superscript \(t\) on \([e]\) means transpose of the matrix \([e]\). \(D\) is electric flux density of piezoelectric material, \(E\) is electric field intensity, and \(\left[e^{S}\right]\) is dielectric permittivity under constant strain [34]. The values of these parameters are presented in Table 1 with other material properties. The dimensions of the pump are illustrated in the same table as well. Some of dimensions are optimized in Section 3.2, so initial values are given. For numerical simulation of the piezoelectric actuator, solid mechanics and two-way electrostatic coupling were used, and appropriate dimensions were determined through iterative design and simulations. Piezoelectric material used in the simulations were decided to be the 27 \(\times\) 0.5 mm piezoelectric disk of the STEMinc (Davenport, FL, USA) [35] company. Last but not least, the optimum design was found by conducting a three-way analysis (e.g., electrostatics, structural mechanics, and laminar flow) to the system. The requirements for a valveless design are as follows: the flow velocity needs to reach to 1 m/s but should be less than 8 m/s, the piezoelectric disk diameter should be 27 mm, the maximum piezoelectric disk displacement should be 50 \(\upmu\)m and minimum displacement should be 2.5 \(\upmu\)m, net flows should be at least 2.5 mL/h (\(\sim\)0.1 \(\upmu\)L/s), the size of the pump should fit the palm of a hand, the maximum voltage supply should be 100 Vp-p, and, finally, 0.1 A continuous, the instantaneous current (transient) value of one ampere, should be the maximum value needed for current. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material/Dimensions** & **Parameter** & \multicolumn{4}{c}{**Value**} & **Unit** \\ \hline
**PZT-Sj** & Density & \multicolumn{4}{c}{7800} & kg/m\({}^{3}\) \\  & & & 8.2 & 3.4 & 3.0 & 0 & 0 & \\ _Dinner (D\({}_{p}\))_: 25 mm & \multirow{2}{*}{Elasticity matrix \([c^{E}]\)} & 3.4 & 8.2 & 3.0 & 0 & 0 & \\ _Thickness (\(\eta_{p}\))_: 0.4 mm & & 3.0 & 3.0 & 5.9 & 0 & 0 & \\  & 0 & 0 & 0 & 2.1 & 0 & 0 & \\  & 0 & 0 & 0 & 0 & 2.1 & 0 & \\  & 0 & 0 & 0 & 0 & 0 & 2.4 & \\ \hline \hline \end{tabular}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{_Chamber Dimensions:_} & \multicolumn{4}{c}{_Chamber Dimensions:_} \\ _Inlet/Outlet Diameter (\(D_{out}\))_: 3 mm & Piezoelectric Coupling Matrix \([e]^{t}\) & 0 & 0 & -10.48 & \\ _Nozelle Small Radius (\(\eta_{p}\))_: & Piezoelectric Coupling Matrix \([e]^{t}\) & 0 & 0 & 16.58 & \\ _Nozelle Small Radius (\(\eta_{p}\))_: & 0 & 0 & 0 & 0 & \\ _50 um_ & & 14.25 & 0 & \\  & 14.25 & 0 & 0 & \\ _Nozelle Big Radius (\(R_{d}\))_: 0.25 mm & Permittivity \([e^{S}]/\varepsilon_{0}\), where \(\varepsilon_{0}=8.85*10^{-12}F/m\) & \(\left[\begin{array}{ccc}1641&0&0\\ 0&1641&0\\ 0&0&1143\end{array}\right]\) & 1 \\ _Chamber Height (\(\mu\))_: & 0.5 mm (Initial) & Damping (Isotropic Loss & 0.02 & 1 \\ _Nozelle Length (\(L\))_: 0.5 mm (Initial) & Factor) \(\eta_{p}\) & 0.02 & 1 \\  & Dielectric Loss (Isotropic Loss & & \\ _Divergence Angle (\(0\))_: \(11^{\circ}\) (Initial) & Factor) for Electrical Permittivity & 0.02 & 1 \\ \hline \hline
**Brass** & Density & 8500 & kg/m\({}^{3}\) \\ _Dinner (D\({}_{p}\))_: 27 mm & Young modulus & 90,000 & MPa \\ _Thickness(\(\eta_{p}\))_: 0.2 mm & Poisson’s ratio & 0.32 & 1 \\ \hline
**Insulin** & Density & 997 & kg/m\({}^{3}\) \\  & Kinematic viscosity & 0.0014 & kg/m\(s\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties [34; 35].