## 2 Materials and Methods
### Acoustic Field Modeling Using FEM  
The acoustic field was modeled using the finite element method (FEM) based on the geometry shown in Figure 2. The software used for the FEM processing was COMSOL Multiphysics (COMSOL AB, Stockholm, Sweden) working in a PC of 8-core 3 GHz microprocessor and 64 RAM (Dell, Round Rock, TX, USA). Based on the cylindrical symmetry of the transducer, the problem was assumed axisymmetric. The validity of this assumption was verified with acoustic field measurements in which the profile of the radiation along the azimuthal coordinate is similar for any angle (data not shown). The mesh in the rectangular part of Figure 2 consisted of 10 square elements per wavelength, i.e., more than 530,000 elements; triangular elements were used only for the zone created with the boundaries 1 and 5 to simplify meshing. The requirements to mesh using squares are very specific, and this kind of element cannot be used in certain curved geometries; triangular elements are sometimes the only option for these complicated zones. The mesh convergence was verified by increasing the mesh resolution, which indicated an error of 0.01% at the focus amplitude between meshing using 9 and 10 elements per wavelength. Using square elements instead of triangular permitted to increase the spatial resolution with the same interpolation functions because this kind of mesh has a larger number of nodes (and therefore more degrees of freedom) than the triangular one with the same number of elements (more nodes per element area); this can be noticed with the reduction of the solution time compared to the time required for solving the problem using the same number of triangular elements. The largest solution time registered for our main model, which was barely the most demanding of computational resources, was 45 s.  
Figure 2: The finite element method (FEM) axisymmetric geometry for modeling the focused acoustic field. It was assumed the linear ultrasound propagation (i.e., rather low acoustic intensity) was in an attenuation-free homogeneous media (degassed distilled water). The ultrasound propagation is determined with FEM based on the homogeneous Helmholtz wave equation for harmonic radiation, assuming a purely harmonic source and a no-frequency dispersion. This equation can be written as  
\[\
abla^{2}p+k^{2}p=0 \tag{1}\]  
where the wavenumber is \(k=\omega/c\), \(\omega\) is the angular frequency (rad), and \(c\) is the speed of sound in water (m/s) assumed constant. Boundary 1 was set with the harmonic normal acceleration. The amplitude of this normal acceleration was adjusted for the different radiation conditions presented in this paper. For instance, in the uniform radiation model of the next section, the amplitude of the harmonic acceleration was set constant along the radius. Usually, the radiation of a boundary is expressed in terms of particle velocity [20; 26; 31]. In harmonic conditions [32], the particle acceleration \(a_{0}\) on the radiator surface can be obtained by time-deriving the constant-amplitude particle velocity \(v_{0}e^{i\omega t}\) as  
\[a_{0}=\frac{d}{dt}\Big{(}v_{0}e^{i\omega t}\Big{)}=v_{0}\omega e^{j(\omega t+ \frac{\pi}{2})}=\frac{\omega}{\rho c}p_{0}e^{j(\omega t+\frac{\pi}{2})}, \tag{2}\]  
where \(\rho\) is the medium density (kg/cm\({}^{3}\)). The last term was determined by considering \(p_{0}=\rho cv_{0}\), which is true in the field very close to the radiator surface [27]. The term \(\frac{\pi}{2}\) in Equation (2) is a time phase shift between the velocity and the acceleration, and this is not related to the spatial profile along the transducer surface [32]. Then, under harmonic simulations, using either acceleration or velocity for the radiation distributions of the plate is not relevant if adequate amplitude considerations are taken. Boundaries 2-4 were configured to match the acoustic impedance of water and to reduce the ultrasound reflections at the walls [27].