## 3 Bandgap Algorithm of the PMPIC by Extended Plane Wave Expansion Method
### Bandgap Structures of the PMPIC by an Extended Plane Wave Expansion Method  
For the complex 2D wave motion equation as Equation (10), it is difficult to obtainits analytical solution. Several numerical methods have been proposed to investigate 2D phononic crystals, such as the finite difference time domain method, finite element method, and PWE method. Due to its convenience, the PWE method is one of the most widely used methods to calculate bandgap structures of a 2D phononic crystal. As for piezoelectric coupling effects, an extended PWE method is presented here. Due to a periodic structure, material properties (\(\phi_{*}\), \(D_{*}\), \(v_{*}\), \(\lambda_{*}\) and \(\chi_{*}\)) of the PMPIC are periodic functions of the spatial coordinates (**r**). In Figure 2, it is assumed that \(\Theta_{p}\) denotes the set of small elements filled with piezoelectric patch and the center element  
Figure 3: The structure of a self-powered SCEC with a resonant inductor. belongs to \(\Theta_{p}\). Then, material properties of the center element can be represented as Equation (15) according to its Fourier series expansion  
\[g_{0}(\mathbf{r})=\sum_{G}g_{0}(\mathbf{G})\exp(j\mathbf{G}\mathbf{r}) \tag{15}\]  
where \(g_{0}(\mathbf{G})\) is the Fourier coefficient, \(g_{0}\) can be any of \(\phi_{*}\), \(D_{*}\), \(\
u_{*}\), \(\lambda_{*}\) or \(\chi_{*}\), \(\mathbf{G}=n_{1}\mathbf{b}_{1}+n_{2}\mathbf{b}_{2}\) is the reciprocal lattice vector and \(\mathbf{b}_{1},\mathbf{b}_{2}\) are the \(x\)-axis and \(y\)-axis lattice vectors in the reciprocal space, respectively, and \(\mathbf{r}=n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}\) is the primitive space vector. Here, \(\mathbf{b}_{1}=2\pi(1,0)/a,\mathbf{b}_{2}=2\pi(0,1)/a\), and \(a\) is the lattice constant of the unit cell. Furthermore, \(g_{0}(\mathbf{G})\) can be defined as  
\[g_{0}(\mathbf{G})=\frac{1}{S}\iint_{S}g_{0}(\mathbf{r})\exp(-j\mathbf{G} \mathbf{r})d^{2}\mathbf{r} \tag{16}\]  
where \(S\) is the area of the unit cell. Furthermore, the integration of Equation (16) can be calculated as  
\[g_{0}(\mathbf{G})=\left\{ \begin{array}{l}g_{p}f+g_{s}(1-f),\mathbf{G}=0\\ (g_{p}-g_{s})P(\mathbf{G}),\mathbf{G}\
eq 0\end{array} \right. \tag{17}\]  
where \(g_{p}\) and \(g_{s}\) denote material properties (\(\phi_{*}\), \(D_{*}\), \(\
u_{*}\), \(\lambda_{*}\) or \(\chi_{*}\)) of the piezoelectric patch and host structure, respectively. \(f=1/M^{2}\) is the filling ratio of each small element in the unit cell. \(P(\mathbf{G})\) is called a structural function, which is defined as  
\[P(\mathbf{G})=\frac{1}{S}\int_{S_{p}}e^{-j\mathbf{G}\mathbf{r}}d^{2}\mathbf{r} \tag{18}\]  
where \(S_{p}\) is the area of the small element. In this paper, the piezoelectric patch in any small element is square, so we will have  
\[P(\mathbf{G})=f\text{sinc}\bigg{(}\frac{\mathbf{G}_{x}a}{2M}\bigg{)}\text{sinc }\bigg{(}\frac{\mathbf{G}_{y}a}{2M}\bigg{)} \tag{19}\]  
where \(\mathbf{G}_{x}\) and \(\mathbf{G}_{y}\) are x and y components of the reciprocal vector \(\mathbf{G}\), and sinc(\(\bullet\)) is the cardinal sine function. According to the basic properties of Fourier transform, any other small element belonging to \(\Theta_{p}\) will have \(g_{\mathbf{r}}(\mathbf{G})=g_{0}(\mathbf{G})\exp(j\mathbf{G}\mathbf{r})\). Then, Fourier transform of material constants of the piezoelectric patch in the unit cell can be expressed as  
\[g(\mathbf{G})=\sum_{\mathbf{r}}g_{\mathbf{r}}(\mathbf{G})\delta(\mathbf{r})= g_{0}(\mathbf{G})\sum_{\mathbf{r}}\exp(j\mathbf{G}\mathbf{r})\delta(\mathbf{r}) \tag{20}\]  
where \(\mathbf{r}=[\mathbf{r}_{1},\dots,\mathbf{r}_{N}]^{T}\), \(\mathbf{r}_{i}\) denotes the center position of the ith small element and \(N=M^{2}\). When the ith small element belongs to \(\Theta_{p}\), \(\delta(\mathbf{r}_{i})\)\(=1\), otherwise \(\delta(\mathbf{r}_{i})\)\(=0\). Thus \(\delta(\mathbf{r})\) can be looked at as a distribution function of piezoelectric material in the unit cell, which is just the optimization target of this paper.