Version notes

This updated version uses larger ranges for the material parameters to account for a wider range of samples. Further, an erroneous boundary condition in the simulation model used to generate the data set in the previous version has been fixed.

Description of the dataset

This dataset contains the results of 301008 finite-element simulations of the complex, frequency-dependent electrical impedance of a piezoelectric ring with randomised material parameters. Each impedance consists of 2000 samples in the frequency domain up to 8 MHz. We assume the sample to be dielectric, thus the impedance a frequency 0 Hz is infinite. The piezoelectric ring has an outer radius of 6.35 mm an inner radius of 2.6 mm and a thickness of 1 mm. The transverse isotropic material parameters are sampled from independent uniform distributions with ranges that are intended to represent the behaviour of different piezoceramic materials. The parameters of the Rayleigh damping model (alpha_M and alpha_K) are sampled from a logarithmic distribution to account for the larger parameter range.

Files

The dataset contains the following files:

• dataset.hdf5: The main dataset file in HDF5 format. Refer to the next section on how to load the dataset.
• simulation.xml: The file describing the parameters of the finite element simulation. This file can be used with openCFS along with the mesh and material file to simulate the behaviour of the piezoelectric ceramic in the frequency domain. This simulation file was used with randomised material parameters to generate the dataset.
• material.xml: The material file used for the simulation with exemplary material parameters values.
• ring.geo: The gmsh geometry file with the axisymmetric representation of the piezoelectric ring.
• ring.msh: The mesh file generated with gmsh using the ring.geo file.

Loading the dataset

The dataset is stored as a HDF5 file, which can be opened with all libraries that support that format, e.g. in Python using the h5py library:

import h5py

# Open the dataset in read mode.
file = h5py.File("dataset.hdf5", "r")

# Impedances as a 301008 x 2000 array of complex numbers.
impedances = file["impedances"]
# Material parameter values as a 301008 x 13 array of real numbers.
parameters = file["parameters"]
# Frequency vector of the impedance with length 2000.
frequencies = file["meta"]["frequencies"]
# 13 strings with the identifiers of the material parameters.
parameter_labels = file["meta"]["parameter_labels"]

Simulating impedances

To generate a result for the electrical impedance using the supplied simulation files, download and install openCFS and call the executable with the simulation.xml, but omit the file extension, e.g.:

cfsbin.exe simulation

The path to the executable of openCFS will depend on your operating system and installation. Running the simulation will result in the creation of several files and folders. Among those files will be the result for the electric charge on one of the electrodes of the sample, which will be placed in the 'history' subfolder. We can determine the current by taking the time derivative of the charge and already know the voltage because we excited the piezoceramic with an electric potential of 1 V in the simulation. Because the simulation is conducted in the frequency regime, all we have to do is to divide voltage by current to get the frequency dependent electrical impedance. The loading of the result and calculation of the impedance is implemented in the following Python script as an example:

import numpy as np

# Load result file for electric charge
result_path = 'history/simulation-elecCharge-surfRegion-ground.hist'
data = np.loadtxt(result_path)

frequency = data[:, 0]
# Convert polar representation from file to complex numbers.
charge = data[:, 1] * np.exp(1j * 2 * np. pi / 360 * data[:, 2])

# Excitation potential is 1 V in simulation.
potential = 1
# Determine impedance by applying Z = V / I = V / (j omega Q).
impedance = potential / (1j * 2 * np.pi * frequency * charge)