## 2 General Approach to Homogenization
\tag{9}\]  
Next, we will present a rigorous approach for efficient, accurate calculation of the effective parameters of a homogenized three-dimensional phononic crystal, where the inclusion and host materials in the unit cell have the densities \(\rho_{a}\) and \(\rho_{b}\), and elastic compliance constants \(S_{a}\) and \(S_{b}\), respectively. So, within this context of effective medium, to obtain \(A_{eff}(\mathbf{k})\), we will determine the matrix \(D^{-1}(\mathbf{k};\mathbf{G},\mathbf{G}^{{}^{\prime}})\) from the following equation:  
\[\sum\
olimits_{\mathbf{G}^{{}^{\prime}}}D(\mathbf{k};\mathbf{G},\mathbf{G}^{{}^{ \prime}})D^{-1}(\mathbf{k};\mathbf{G}^{{}^{\prime}},\mathbf{G}^{{}^{\prime\prime}} )=I\delta_{\mathbf{G},\mathbf{G}^{{}^{\prime}}}, \tag{10}\]  
where the unitary matrix \(I\) is sized \(9\times 9\). Applying the definition of \(D(\mathbf{k};\mathbf{G},\mathbf{G}^{{}^{\prime}})\) indicated in (8), Equation (10) can be expressed by:  
\[-\left[ \begin{array}{cc}0_{3}&K_{3\times 6}(\mathbf{k}+\mathbf{G})\\ (K_{3\times 6}(\mathbf{k}+\mathbf{G}))^{T}&0_{6}\end{array} \right]D^{-1}(\mathbf{k}; \mathbf{G},\mathbf{G}^{{}^{\prime\prime}})+\frac{\Omega}{i}\sum\
olimits_{ \mathbf{G}^{{}^{\prime}}}A(\mathbf{G}-\mathbf{G}^{{}^{\prime}})D^{-1}(\mathbf{k}; \mathbf{G}^{{}^{\prime}},\mathbf{G}^{{}^{\prime\prime}})=I\delta_{\mathbf{G}, \mathbf{G}^{{}^{\prime}}}. \tag{11}\]  
It is important to remember that the Fourier coefficients of the matrix \(A(\mathbf{r})\) are given by:  
\[A(\mathbf{G}-\mathbf{G}^{{}^{\prime}})=\left[ \begin{array}{cc}\rho_{b}I&0_{ 3\times 6}\\ 0_{6\times 3}&S_{b}I\end{array} \right]\delta_{\mathbf{G},\mathbf{G}^{{}^{ \prime}}}+\left[ \begin{array}{cc}\Delta\rho I&0_{3\times 6}\\ 0_{6\times 3}&\Delta S_{6\times 6}I\end{array} \right].F(\mathbf{G}-\mathbf{G}^{{}^{ \prime}}), \tag{12}\]  
where \(\Delta\rho=\rho_{a}-\rho_{b}\), \(\Delta S=S_{a}-S_{b}\) and \(F(\mathbf{G}-\mathbf{G}^{{}^{\prime}})\) is the form factor of the inclusion. An alternative, more simplified, expression of Equation (12) can be used by redefining the following terms:  
\[A_{b}=\left[ \begin{array}{cc}\rho_{b}I&0_{3\times 6}\\ 0_{6\times 3}&S_{b}I\end{array} \right], \tag{13}\]  
\[\Delta A(\mathbf{G}-\mathbf{G}^{{}^{\prime}})=\left[ \begin{array}{cc}\Delta \rho I&0_{3\times 6}\\ 0_{6\times 3}&\Delta S_{6\times 6}I\end{array} \right]F(\mathbf{G}-\mathbf{G}^{{}^{ \prime}}). \tag{14}\]  
By using the definition of Equation (12) in (11), and after a few manipulations, we obtain:  
\[\overline{\overline{T}}(\mathbf{G})\cdot D^{-1}(\mathbf{k};\mathbf{G},\mathbf{G}^ {{}^{\prime\prime}})+\frac{\Omega}{i}\sum\
olimits_{\mathbf{G}^{{}^{\prime}}} \Delta A(\mathbf{G}-\mathbf{G}^{{}^{\prime}})D^{-1}(\mathbf{k};\mathbf{G}^{{}^{ \prime}},\mathbf{G}^{{}^{\prime\prime}})=I\delta_{\mathbf{G},\mathbf{G}^{{}^{ \prime}}}, \tag{15}\]  
where:  
\[\overline{\overline{T}}(\mathbf{G})=-\left[ \begin{array}{cc}0_{3}&K_{3\times 6}( \mathbf{k}+\mathbf{G})\\ (K_{3\times 6}(\mathbf{k}+\mathbf{G}))^{T}&0_{6}\end{array} \right]+\frac{\Omega}{i }A_{b}=\left[ \begin{array}{cc}-\frac{\Omega}{i}\rho_{b}I&-K_{3\times 6}(\mathbf{k}+\mathbf{G})\\ -(K_{3\times 6}(\mathbf{k}+\mathbf{G}))^{T}&-\frac{\Omega}{i}S_{b}I\end{array} \right] \tag{16}\]Realizing the product of Equation (15) using the matrix \(\overset{=-1}{T}^{-1}(\mathbf{G})\), we get:  
\[D^{-1}(\mathbf{k};\mathbf{G},\mathbf{G}^{{}^{\prime\prime}})+\overset{=-1}{T}^{-1}( \mathbf{G})\frac{\Omega}{i}\sum\
olimits_{\mathbf{G}^{{}^{\prime}}}\Delta A( \mathbf{G}-\mathbf{G}^{{}^{\prime}})D^{-1}(\mathbf{k};\mathbf{G}^{{}^{\prime}}, \mathbf{G}^{{}^{\prime\prime}})=\overset{=-1}{T}^{-1}(\mathbf{G})I\delta_{ \mathbf{G},\mathbf{G}^{{}^{\prime\prime}}}, \tag{17}\]  
where the matrix \(\overset{=-1}{T}^{-1}(\mathbf{G})\), sized 9 \(\times\) 9, is defined as:  
\[\overset{=-1}{T}^{-1}(\mathbf{G})=\left[ \begin{array}{cc}T_{uu}^{-1}&T_{uv}^ {-1}\\ T_{cu}^{-1}&T_{ev}^{-1}\end{array} \right]. \tag{18}\]  
In general, Equation (18) is solved numerically.