Cambridge Ultrasonics uses finite element software to solve transient elasticity problems, heat diffusion problems, electromagnetic wave propagation problems. Studying wave propagation in detail with finite elements requires using a time step analysis with small time steps and this is computationally intensive.
The pictures to the right (at time steps: 0, 1, 2, 3 microseconds) show an example of z-displacement (vertical displacement in this case) in an axial symmetry model of a transducer (cross-section through the right-half of a transducer - the left-half is a mirror image of the right-half). This is a true transient model: the excitation voltage is turned on at time t = 0.
The model includes a piezoelectric element, which is sandwiched between the upper-most rectangle (damping material) and the larger, tapering part. The peizoelectric element is the source of motion in the model; it is being driven with a tone-burst electrical drive signal.
In the top picture, time = 0, so the colour is a uniform green - colour is used to represent z-displacement using the rainbow as a scale and in this case green corresponds to zero displacement.
In the next picture
down, time = 1 microsecond, there are regions of different colours,
particularly in the damping block at the top. Observe that there
are displacement waves in the lower piece as well; the prime objective
is to create plane waves in this component, another objective
is to control reflections from the side.
In the last picture, time = 2 microseconds, and the waves are reaching the transmitting tip.
The software allows us to create a movie of a transient analysis and view waves in much the same way as Cambridge Ultrasonics' visualization equipment allows (see Ultrasound). We can view not only z-displacement but also compressive stresses, shear stresses or a large namber of dependent variables, they can be displayed in a variety of useful ways.
It is also possible to integrate a chosen variable over a surface and calculate a mean value and this can be done at all the time steps of a transient model, creating a signal that is varying in time that can be compared with signals displayed on oscilloscopes taken from transducers. We have has two ways to verify a FE model:
One of the purposes of FE modelling is to reduce the risk in a new design of transducer before building it: if the FE model looks OK then the transducer generally performs well. However, it is useful to have a wider experience of ultrasonics when assessing FE models because it is possible to set-up a model and generate fictional results! Our experience helps us to avoid these pitfalls.
Other benefits
for modelling ultrasound
We are ble to model more than just the physics of elasticity. We can model a wide range of physical systems governed by partial differential equations (PDEs). Dynamic elasticity or elastic wave propagation, is governed by the wave equation but there are other important classes of PDEs, for example: Lapalce's equation for electrostatics and the diffusion equation for heat transfer. Some fluid flow problems can be handled.
It is also possible to model linked physical problems, such as dynamic elasticity and heat diffusion or dynamic elasticity and fluid flow.
Example - a piezoelectric disc is vibrating at high amplitude. Heat is generated in the disc at regions of greatest rate of change in strain and heat diffuses away into cooler regions. It is possible to model the temperature with time and to predict when and where the piezoelectric part should start to over-heat, depolarize and fail (most commercial piezoelectric materials are not truly piezoelectric but are ferroelectric instead and therefore have a Curie temperature above which they depolarize).
The picture along
side shows one of the results of a modal analysis of a hard piezoelectric
disc, the type used in high power transducers. In this particular
case the disc is rigidly fixed over its base (blue) and it has
a peak axial displacement (anti-node) at the centre (red), an
symmetric anti-node (blue) at a constant radial distance away
from the centre and a third anti-node at the perimeter (yellow).
