3D Thin Plate Piezoelectric Resonator
This model is the 3D compliment of the 2D thin plate piezoelectric resonator presented earlier. Admittedly, this it is an overkill to run this problem in 3D since it really is only a 1D problem, but much can be learned from setting up and analyzing this simple case.
For the 3D case, a square region of .1x.1 mm is created in 2D space and extruded in the Z direction the thickness of the water and piezoelectric materials. Note that it must be in the Z direction so that the anisotropic material constants of the PZT-5H ceramic line up correctly. A scale factor relating the true current to the calculated current is also required, in this case it is calculated to be 83820.0. The sketch indicates how to extrude the 2D geometry. The geometry follows the sketch:
Absorbing boundary conditions are placed on the front and back of the fluid volume to represent an infinite fluid domain. The length of the fluid is roughly one wavelength at 2 Mhz with 16 finite element layers, which amounts to 8 layers at the maximum frequency of 4 Mhz. The preferred finite element/wavelength criteria mentioned in the other examples is relaxed due to the simplicity of the solution. Planes of symmetry exist for the ceramic slab in which Ux = 0 exist at X = 0 and X = W planes. Additional ceramic symmetry planes at which Uy = 0 exist at Y = 0 and Y = W. These conditions force a thickness mode in the ceramic consistent with an infinitely large thin plate. The finite element model looks like:
Running this model over the 2 to 4 Mhz solution range yields the following mechanical impedance plot (the 3D and 2D solutions are compared with theory):
Again, the only difference between the 2D and 3D FEWaves solutions and theory is the addition of a mass loss term present in the finite element material parameter for PZT-5H.