A model for ultrasonic transducers in a high-temperature regime with boundary dynamics : an evolutionary equations approach

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In this thesis we formulate an abstract model to describe ultrasonic transducers, taking into account a high-temperature regime as well as dynamics at the boundary. We use an abstract boundary trace theory to extend from a known thermo-piezo-electromagnetic system, and encode boundary dynamics directly within our model. Using the theory of evolutionary equations invented by Rainer Picard, we establish the well-posedness of our system. Well-posedness in this context corresponds to both Hadamard well-posedness and causal dependence on given data. Moreover, we conduct a systematic investigation into different arrangements of complicated boundary dynamics which lead to a well-posed system. Motivated by a set of known piezo-electric boundary conditions, we formulate and consider novel generalised impedance like boundary conditions. Furthermore, we formulate and analyse a specific example of these generalised boundary dynamics, which account also for the influence of heat at the boundary. The resulting example pertains to all three physical aspects of our system, and thus harnesses the full generality afforded by our system.

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