## 3 Bandgap Algorithm of the PMPIC by Extended Plane Wave Expansion Method
### Transverse Motion Equation of the PMPIC
\begin{array}{ccc}\widetilde{\tau}_{11}& \widetilde{\tau}_{12}&0&-\widetilde{\tau}_{31}\\ \widetilde{\tau}_{12}&\widetilde{\tau}_{11}&0&-\widetilde{\tau}_{31}\\ 0&0&\widetilde{\tau}_{66}&0\\ \widetilde{\tau}_{31}&\widetilde{\tau}_{31}&0&\widetilde{\tau}_{33}^{E}\end{array}  \right]\left[ \begin{array}{c}\varepsilon_{x}^{p}\\ \varepsilon_{y}^{p}\\ \tau_{xy}^{p}\\ E_{3}\end{array} \right] \tag{6}\]  
where, \(\widetilde{\tau}_{11}=s_{11}^{E}/\left[\left(s_{11}^{E}\right)^{2}-\left(s_{1 2}^{E}\right)^{2}\right]\), \(\widetilde{\tau}_{12}=-s_{12}^{E}/\left[\left(s_{11}^{E}\right)^{2}-\left(s_ {12}^{E}\right)^{2}\right]\), \(\widetilde{\tau}_{66}=1/s_{66}^{E}\), \(\widetilde{\tau}_{31}=d_{31}/\left[s_{11}^{E}+s_{12}^{E}\right]\), \(\widetilde{\tau}_{33}^{E}=\frac{e_{7}^{T}}{3}-2d_{31}^{2}/\left[s_{11}^{E}+s_ {12}^{E}\right]\). The piezoelectric patch is assumed to be polarized along the direction of \(z\)-axis, and then the electric field in the \(z\)-axis can be approximated as Equation (7) [10],  
\[E_{3}=-V/h_{p} \tag{7}\]  
where \(V\) is the output voltage across the two electrodes on the piezoelectric patch, and \(h_{p}\) is the thickness of the piezoelectric patch. Based on Equation (6), the bending moments can be calculated as follows:  
\[ \begin{array}{c}H_{11}=\int_{h_{s}/2-h_{p}}^{h_{s}/2}\sigma_{x}^{p}zdz=-D_{p }\bigg{(}\frac{\partial^{2}w}{\partial x^{2}}+\mu_{p}\frac{\partial^{2}w}{ \partial y^{2}}\bigg{)}+\lambda_{p}V\\ H_{22}=\int_{h_{s}/2-h_{p}}^{h_{s}/2}\sigma_{y}^{p}zdz=-D_{p}\bigg{(}\frac{ \partial^{2}w}{\partial y^{2}}+\mu_{p}\frac{\partial^{2}w}{\partial x^{2}} \bigg{)}+\lambda_{p}V\\ H_{12}=H_{21}=\int_{h_{s}/2-h_{p}}^{h_{s}/2}\tau_{xy}^{p}zdz=-D_{p}\big{(}1-\mu_{ p}\big{)}\frac{\partial^{2}w}{\partial x\partial y}\end{array}  \tag{8}\]  
where, \(D_{p}=\widetilde{c}_{11}\Big{[}\big{(}h_{s}/2\big{)}^{3}-\big{(}h_{s}/2-h_{p} \big{)}^{3}\Big{]}/3\), \(\mu_{p}=\widetilde{c}_{12}/\widetilde{c}_{11}\), \(\lambda_{p}=\big{(}h_{s}-h_{p}\big{)}\widetilde{c}_{31}/2\). By combining Equations (2), (7), and (8), we can obtain the transverse motion equation of the piezoelectric patch as  
\[-\rho_{p}h_{p}\frac{\partial^{2}w}{\partial t^{2}}=\frac{\partial^{2}}{ \partial x^{2}}\bigg{[}D_{p}\bigg{(}\frac{\partial^{2}w}{\partial x^{2}}+\mu_{ p}\frac{\partial^{2}w}{\partial y^{2}}\bigg{)}-\lambda_{p}V\bigg{]}+\frac{ \partial^{2}}{\partial y^{2}}\bigg{[}D_{p}\bigg{(}\frac{\partial^{2}w}{ \partial y^{2}}+\mu_{p}\frac{\partial^{2}w}{\partial x^{2}}\bigg{)}-\lambda_{p }V\bigg{]}+2\frac{\partial^{2}}{\partial x\partial y}\bigg{[}D_{p}\big{(}1-\mu_{ p}\big{)}\frac{\partial^{2}w}{\partial x\partial y}\bigg{]} \tag{9}\]  
Referring to the idea in Ref. [34], an infinitesimal number \(\lambda_{s}\) multiplying the voltage is introduced into Equation (4) in order to unify the forms of Equations (4) and (9). Then, the unified motion equation can be written as  
\[-\phi_{*}\frac{\partial^{2}w}{\partial t^{2}}=\frac{\partial^{2}}{\partial x^{2} }\bigg{[}D_{*}\frac{\partial^{2}w}{\partial x^{2}}+v_{*}\frac{\partial^{2}w}{ \partial y^{2}}-\lambda_{s}V\bigg{]}+\frac{\partial^{2}}{\partial y^{2}}\bigg{[}D_ {*}\frac{\partial^{2}w}{\partial y^{2}}+v_{*}\frac{\partial^{2}w}{\partial x^{2}} -\lambda_{s}V\bigg{]}+2\frac{\partial^{2}}{\partial x\partial y}\bigg{[}\chi_{*} \frac{\partial^{2}w}{\partial x\partial y}\bigg{]} \tag{10}\]  
where the subscript (*) can be \"s\" for the thin plate or \"p\" for the piezoelectric patch, \(v_{*}=D_{*}\mu_{*}\) and \(\chi_{*}=D_{*}(1-\mu_{*})\). (iii) For the interface circuit: This electric current flowing to the interface circuit can be calculated as Equation (11) using the long-wave approximation assumption [37]. \[I(s)=D_{3}A_{e}s \tag{11}\]  
where \(s\) is the Laplace operator, and \(A_{e}\) is the surface area of the electrode or the piezoelectric patch. On the other hand, the electric current can be written as  
\[I(s)=\frac{V(s)}{Z(s)} \tag{12}\]  
where \(Z(s)\) denotes the impedance of the interface circuit.