# Microstructured Phononic Crystal Isolates from Ultrasonic Mechanical Vibrations
## 2 Design of 3D Phononic Crystals  
The design of the unit cell of the phononic crystal proposed in this paper is inspired by the work recently published by co-authors [41]. The phononic crystal proposed in [41] is indeed very promising for our target applications since it exhibits an ultra-wide bandgap, i.e., gap to mid-gap ratio of 132% and is fully compatible with the 2 PP fabrication technique we want to employ. In [41], the design was obtained as a three-dimensional generalization of the two-dimensional phononic crystals proposed and optimized in terms of the gap to mid-gap ratio in [30]. The 2D unit cell topology chosen was made by an external frame with half circles centred in the middle-point of the edges, the bidirectional evolutionary structural optimization (BESO) gave the optimal frame thickness and radius of the circles. In particular, the high mass of the circles and low frame thickness and stiffness allow to separate the global and local modes resulting in a wide band-gap. In the 3D extension proposed in [41], the symmetries and proportions of the optimal 2D geometry are preserved, ending up with a cubic structure where each face has the planar topology of the 2D counterpart. Again, the high mass of the spheres and low stiffness of the connecting beams allow for a large band-gap formation. In the present work, a slightly different geometry is proposed, with masses that are no more semi-spherical and with modified relative dimensions between the masses and the beams connecting the masses. The new unit cell is reported in Figure 1a: it consists of six masses obtained by cutting a semi-sphere through a plane parallel to the sphere section, connected through three straight beams of length \(0.44a\) and a square cross section of dimension \(0.075a\), with \(a\) being the dimension of the unit cell along the three orthogonal directions. Thanks to the cubic symmetry of the unit cell, the dispersion diagram can be computed by chosing the tetrahedron drawn in Figure 1b as the irreducible Brillouin zone (IBZ). The phononic band structure can be obtained by imposing Floquet periodic boundary conditions on the six sphere sections of the unit cell and performing a parametric sweep over the wave vector according to the Bloch theorem [42]:  
\[u_{j}(x+a,y+a,z+a)=u_{j}(x,y,z)e^{i(k_{x}a+k_{y}a+k_{z}a)} \tag{1}\]  
where \(u_{j}\) denotes the elastic displacement vector along the \(x\)-, \(y\)- and \(z\)-directions and \(k_{x}\), \(k_{y}\) and \(k_{z}\) are the components of the Bloch wave vector. In a finite element method (FEM) code, Floquet boundary conditions can be implemented, as explained, e.g., in [43]. To compute the dispersion diagram, the 3D geometry of the unit cell is then discretized through 30,000 quadratic tetrahedral elements to guarantee at least four elements in the beams thickness and a consequently good accuracy of the results. The numerical phononic band structure is finally computed by means of the solid mechanics module of COMSOL Multiphysics v5.6, and it is shown in Figure 1b in a non-dimensional form. The non-dimensional frequency \(f_{nd}\) is defined as the product of the dimensional frequency \(f\) with the ratio \(a/v\), with \(v~{}=~{}\sqrt{E/\rho}\) being the sound velocity of the material (\(E\) and \(\rho\) are the Young's modulus and the mass density of the bulk material). The first full 3D bandgap is characterized by a gap to mid-gap ratio of 134.87%, with non-dimensional frequency limits equal to 0.113 and 0.581. According to the Bragg-based bandgap formation mechanism, as the cell dimension \(a\) decreases, the actual, dimensional, frequency of the band-gap increases. An increase in the Young's modulus \(E\) of the bulk material also results in a shift of the bandgap towards higher frequencies. The ultra-wide bandgap exhibited by the proposed phononic crystal is justified by the modal shapes of the bandgap opening and closing modes.