Telecommunication technologies

Methods

Modeling and simulation

To model the ultrasonic waves generated using the cylindrical phased array, we used the Finite Element Method (FEM) implemented in a commercial FEM software environment (COMSOL, Inc.) using a coupled multiphysics simulation including solid mechanics analysis, electrostatics, and pressure acoustics.
Specifically, the Navier’s equation is solved for the piezoelectric structure, Gauss’ law for the electrostatics, and the Helmholtz equation for the pressure acoustics as follows:
Navier’s equation:
wavenumber k_z and circumferential wavenumber m, Q_m is the monopole source, ρ_c is the density, and c is the speed of sound. The boundary electrical potential applied to the inner and outer walls of the piezoelectric elements generate an electric field that couples with the piezoelectric strain through a compliance matrix due to the piezoelectric material property. The harmonic strain, or equivalent vibrational displacement periodically changes the density of the surrounding fluid medium and generates ultrasonic pressure waves according to the Helmholtz equation above.
Both 2D and 3D simulations were performed. In the 2D case, we simulated a cross-section of the cylindrical geometry, assuming that it is infinite along the axial direction (z-axis). The 2D simulations were used as an approximate method to design our experiments for beam steering and complex pattern generation. We verified the results of our 2D simulations for a single element phased array (i.e., a cylindrical transducer) by comparing the results against an analytical solution We achieved agreement to within 0.18% between the numerically calculated resonance mode frequencies and the analytical results. For single element transducers, we performed 3D simulations. Since we are only interested in the cylindrically-symmetric modes, we reduced the simulation domain to the radial cross-section of the structure. In all our simulations, we used a perfectly matched layer (PML) boundary condition for the outer boundaries to mimic an infinite medium and prevent any reflection and interference of waves. We set the freely vibrating boundary condition to the inner and outer walls of the piezoelectric cylinder. The inner wall is held at the ground level potential and 20 V voltage is applied to the outer wall. In this arrangement, the polarization of piezoelectric cylinder occurs in radial direction.
The material and geometrical properties used in the FEM simulations are listed in Table 1.