## 3 Numerical Results and Discussion  
In this section, to check the validity of the formulas above, we will calculate the effective parameters in the quasi-static limit for three-dimensional elastic solid-solid phononic crystals with cubic lattice structures. Utilizing the fact that at the long-wavelength limit, the elastic wave that travels through the artificial material does not experience the structure of the materials that constitute it, and the phononic crystal behaves as if it were homogeneous. Hence, the numerical simulations were implemented using Equation (33). The initial procedure for calculating the effective parameters indicated in (33) begins with fixing a low frequency \(\omega\) and an initial wave vector \(\mathbf{k_{0}}\) (defined in the long-wavelength regime of the propagation of elastic waves, \(\mathbf{k_{0}}\to 0\)). In this first iteration, the effective tensors \(\overline{\rho}_{eff}(\mathbf{k_{0}},\omega)\) and \(\overline{S}_{eff}(\mathbf{k_{0}},\omega)\) are calculated for the established quasi-static wave vector. It is important to note that our numerical results are calculated by discretizing the inclusion into a small number of parts, \(n\), and according to its geometry, the value of its form factor \(F(\mathbf{G})\) can be introduced in the formulas as a function of the position of its center of gravity and filling fraction in the unit cell of the phononic crystal. This leads to the system defined in (33) for the solution of a system of linear algebraic equations of \(9n\) unknowns. Otherwise, the numerical method of the theory developed here gives the effective response of density and stiffness in terms of summations over \(\mathbf{G}\). Our results converged for \(N=(41)^{3}\) summands in the computation (\(N\) denotes the number of plane waves) over \(\mathbf{G}\), which allows us to consider a finite number of components for the displacement and stiffness fields in the expansion of plane waves in the phononic crystal. It can be seen that the term \(\mathbf{G}=0\) provides the main contribution to the Fourier expansion, to study the properties of the average field at the long-wavelength limit. It is important to mention that, in these analytical expressions, when we increase the number of plane waves, a new iteration (\(i\)) is realized; then, the effective parameters are recalculated. In this work, this iterative calculation process converged with the following condition of magnitudes for the components of the effective tensor: \(\left|\frac{A_{effi}-A_{effi-1}}{A_{effi}}\right|\leq 10^{-5}\); within this limit of convergence, 3-5 iterations were sufficient. With the expressions derived above, our formalism allows us to reduce the size of the system of equations for the calculation of the effective elastic parameters. In the following scenarios, the unit cell has a lattice parameter of \(a=0.01\) m and for the sake of convenience, we discretize the inclusion in small elements of cubic geometry with an edge ratio of \((1/11)a\); the form factor of the inclusion is calculated by using the following definition:  
\[F(\mathbf{G})=\frac{1}{V_{c}}\int_{V_{inc}}e^{-i\mathbf{G}r}dr, \tag{34}\]  
with \(V_{c}\) (\(V_{inc}\)) being the volume of the unit cell (inclusion). Consequently, we find that \(F(\mathbf{G})_{cube}=f^{\frac{sen(X)\cdot sen(Y)\cdot sen(Z)}{X\cdot Y\cdot Z}}\), where: \(f=\frac{d_{x}d_{y}d_{z}}{V_{c}}\), \(X=\frac{C_{y}d_{x}}{2}\), \(Y=\frac{G_{y}d_{y}}{2}\) and \(Z=\frac{G_{z}d_{z}}{2}\). As a first check, we have obtained the effective parameters of a three-dimensional (3D) phononic crystal formed by an inclusion of cubic geometry with an edge of \((7/11)a\), centered in a cubic lattice (see Figure 1). Unlike the previously published works, it is interesting in our results that the approach shown here allows us to model periodic structures with materials where the anisotropy in its crystalline symmetry can be different from that in the cubic form.