BOUNDARY
CONDITIONS
The boundary conditions needed for the problem are: electrical
conditions on the piezoelectric ceramics, definition of the
polarization direction of the ceramics, and the mechanical
symmetry conditions.
Potential
Data -> Conditions -> Lines -> Potential 0.0 (or
Ground in ATILA 5.2.4 or higher version) -> Assign the side
surfaces of the PZT -> Forced Potential 1.0 -> Assign
the center surface
To apply the electrical boundary conditions on the ceramics,
select the Lines tab, and then Electric Potential in
the drop-down list. First, apply the Forced condition
on the hot electrodes. The Amplitude and Phase of
the harmonic signal are 1V and 0º, respectively.
By using a 1V excitation, the results we will obtain (displacement,
stress, and pressure) will be per volt. Note
that the results computed by ATILA are proportional to the
magnitude of the
input signal because a linear formulation is used. This means
that
an input of 100V generates displacements, for instance, 100
times larger than with 1V.
Next, assign a Ground (reference) potential on the
cold electrode.
Line
Contraints
The axisymmetrical nature of the model requires that the
displacement of nodes on the axis of axisymmetry (the X axis)
be constrained
to that axis (nodes on the axis cannot move away from the axis).
To assign this condition select Line-Constraints in the
drop-down list. Select the Clamped option for the Y component
of the constraint.
Apply this condition to the lines of the solid parts that are
on the X axis.
Polarization
The piezoelectric ceramics are poled along
their thickness direction such that the transduction occurs
in the d33 mode. Because the poling of piezoelectric elements
such as the rings used in Tonpilz
transducers can be assumed to be uniform, we will use a so-called Cartesian polarization
for each one of the transducing elements. To define Cartesian polarizations, Local
Axes are used. The direction
of the polarization is given by the z axis of the local axes.
First, select the Surface tab of the
conditions window; then Polarization from
the drop-down list. Select Local-Axes | Define to create new local
axes. Enter P1 for the name of the local axis.
Select the 3 Points XZ definition mode,
for instance. Here, it is convenient to use geometry points to
define the center and directions of the local axes. Remember
that it is also possible
to enter point coordinates in the command line.
To use the mouse cursor to select the points, click once on the
lower left corner of the PZT region. GiD should prompt you to
use the existing point. Click Join. Repeat for the top left and
lower right points to define the local axes.
Continue in a similar way for the second polarization, P2.
Once this is done, make sure Geometry Polarization is
set to Cartesian. This will impose a uniform polarization
in the selected surfaces. Also make sure that Local-Axes is
set to P1. Then assign the condition to one of the piezoelectric
surfaces.
Select P2 and repeat for the second piezoelectric surface.
Finally, you may verify that all conditions have been applied
by selecting Draw | All Conditions | Exclude Local Axes from
the conditions window.
Radiation
Boundary
The only additional boundary condition that
is needed is the definition of the radiating boundary. From
the Boundary Conditions window, select the Lines tab,
then the Radiating Boundary condition. Applying this
condition to the arc will cause the pressure flux to be absorbed
(and not reflected), thus allowing the acoustic energy to
radiate in an infinite medium. Further, the elements of that
boundary will perform the computation of the far-field pressure.
Rules to position the radiating boundary are complex and
vary depending on the problem.
Finally, you may verify that all conditions
have been applied by selecting Draw | All Conditions |
Exclude Local Axes from the conditions window.
The Zigzag Arrow indicates the location of the radiating boundary.