Mathematical modeling of the radiated acoustic field of ultrasonic transducers

Ultrasonic inspection techniques are widely used for nondestructive evaluation of materials. Mathematical modeling provides an efficient and reliable method to assist the design of the inspection application and the analysis of its results. In this work we use a method based on the Rayleigh-Sommerfeld integral to express the pressure in a fluid generated by different ultrasonic devices like monolithic transducers and phased arrays. Analogous expressions for the displacement in a solid are used to model the case of the inspection of a solid specimen by means of immersion transducers. In order to match the pressure field in the coupling liquid with the displacements in the solid the pressure and displacement fields are expanded in terms of planar waves. Using then the stationary phase approximation to single out the relevant incidence / refraction directions and considering the corresponding transmission coefficients (for a planar interface) the computation of the field at a given point inside the solid can be reduced to an integral over the transducer surface. This integral is in turn approximated in a fast and accurate way by application of the edge elements method. Several examples are presented, involving both monolithic and phased arrays transducers, and a detailed comparison with available analytical solutions. The code developed is also applied to the computation of the delay laws aimed at concentrating the acoustic energy in a given region of the fluid or the solid for an immersion linear phased array.