Mathematical Modelling of Novel Ultrasonic Transducers with Fractal-Like Geometries

  • Sara Canning

Student thesis: Doctoral Thesis


Piezoelectric ultrasonic transducers have the potential to operate as both a sensor and as an actuator of ultrasonic waves. Standard designed transducers have a regular structure and therefore operate effectively over narrow bandwidths due to their single length scale. Biological transducers of ultrasound benet from a wide range of length scales giving rise to increased bandwidths. In this thesis, one-dimensional mathematical models are employed to predict the performance of novel ultrasonic transducers whose designs cover a range of length scales. In
particular, pre-fractal designs are utilised. A variety of fractal structures have been considered in this thesis. The effect of an infinitely ramified Sierpinski carpet device necessitates an adaptation on the renormalization approach so that a Green function renormalization method can be utilised. The important operating characteristics for the device are derived, and comparison of metrics between the new device alongside the standard design (Euclidean) and a previously investigated Sierpinski gasket device are performed. A model of a three-dimensional pre-fractal transducer is explored using a similar methodology to previous pre-fractal devices. The design considered is inspired by the Sierpinski tetrix fractal. The effectiveness of the design is considered through comparison with standard designs and an earlier pre-fractal device. An extension to the Green function renormalization method is applied to study the behaviour of an ultrasonic wave travelling through intricate structures that are more connected than the fractal-inspired designs. The structures considered are the Cartesian product of two Sierpinski gasket lattices and the Cartesian product of two Sierpinski carpet lattices. These structures are utilised to obtain the theoretical operating characteristics for novel devices.
Date of Award11 Aug 2017
Original languageEnglish
SupervisorPaul Roach (Supervisor) & Alan Walker (Supervisor)


  • Pizoelectric materials
  • Renormalization
  • Finite differences
  • Fractal Lattice

Cite this