Modelling and simulation of piezoelectric micromachined ultrasonic transducers

Ultrasonography was the first and the most important civil ultrasound application appeared in the 1950s. It is a non-invasive diagnostic technique, adopted to obtain images of human tissues by means of the principle of pulse-echo, that avoids the use of X-Rays scan, a technique that could be harmful because of the exposition to radiation. Nowadays, array of several piezoelectric bulk wave resonators are employed in the commercial ultrasound imaging probes, in order to generate 3D real-time images. However, this technology is limited by the high probe manufacturing and material production cost. Modern micromachining techniques allow for a much higher precision than the reticulation techniques, adopted for the fabrication of piezoelectric bulk ceramic elements and a more easily manageable fabrication of large 1D and 2D arrays that imply high resolution and real-time imaging capabilities, in spite of the very small amount of piezoelectric material. Piezo-MEMS technology has revolutionized the actuators and sensors world, leading the micromachining and miniaturization advantages in the ultrasound systems. Piezoelectric Micromachined Ultrasonic Tranducers (PMUTs), appeared in the first decade of the XXI century, consist of layered flexural plates with a piezoelectric thin film active layer, arranged in an array configuration, to emit and detect ultrasonic pressure waves. The possibility to replace the common piezoelectric bulk wave resonators with this novel technology, extremely increases the research interests in the study and the comprehension of its capabilities. The main purposes of the thesis regard the modelling and the simulation of the Piezoelectric PMUTs working behaviour, in order to deeply investigate the acoustic performances of this device. Several multiphysics numerical models predict the behaviour of the stand-alone diaphragm and the array of transducers. The attention is focused on the effects of the residual stresses related to the fabrication process and the DC voltage bias, V$_{DC}$. They both play an important role in the determination of the starting deformed configuration and of the fundamental performing frequency, which are strongly affected by the geometric stiffness. Additionally, the non-linear dynamic behaviour, due to the involved large displacements, is affected by the internal stress resultant and by the starting deflected configuration, as well. Subsequently, the non-linear response of the system changes with the imposed DC voltage bias. As a matter of fact, the PMUT center transversal oscillation shows a soft spring behaviour at V$_{DC}$ equal to 2 V, while it becomes a hard spring response at V$_{DC}$ equal to 12 V. The effects of the protecting structure in the acoustic performance, are deeply investigated. To this purpose a 3D Finite Element (FE) model is used to study the role of the vibrating package on the pressure propagation, considering its elastic properties. Furthermore, the acoustic-structure interaction is imposed on the transducers acoustic interface and on the package surface boundaries, as well. Different numerical modelling techniques are proposed to solve PMUTs problems, reporting the main features, advantages and drawbacks of each of them. The experimental and numerical comparisons are presented, in order to validate the numerical models, together with critical aspects and discussions on the electro-mechanical-acoustic coupled response of the device. The comprehensive electro-mechanical-thermoacoustic 2D axisymmetric model of the single transducer, correctly captures the quality factor of the system. The 3D electro-mechanical-acoustic model successfully simulates the in frequency response of the device. Moreover, the proposed 3D acoustic vibrating piston-like array of transducers allow for the estimation of the in-time pressure field, by means of the common commercial FEM acoustics software. Furthermore, a novel piezoelectric-acoustic coupled FE Model Order Reduction (MOR) technique is described and successfully implemented into a Fortran custom code, in order to obtain a Reduced Order Model (ROM) of the PMUTs large array and to simulate the in-water response. To this purpose, the 11x11 PMUTs array transmitting (TX) phase and the 7x7 cluster of PMUTs receiving (RX) phase are simulated. The Reduced Order Model (ROM) custom code is characterized by extremely fast computational time with respect the hugely time-consuming standard full order FE approaches, implemented in the commercial software. Therefore, it represents a suitable tool to correctly evaluate the pressure propagation, the interference phenomena in the near-field and compute the response of the transducers in the actuation and sensing phases.