Articles techniques et présentations

One way to extract piezoelectric material data from measurements is to utilize vibrational eigenmodes of block-shaped test specimens (Rupitsch 2015). Within a simulation-aided approach this results in optimizing the values of the material data parameters to fit the experimental result, namely, the frequency-dependent resonant behavior of the electric impedance of the test sample. As this is a high-dimensional inverse problem, the question of a sensitivity analysis quickly arises. The latter provides information about the sensitivity of a model (here: resonance position) to fluctuations in the model parameters (here: piezoelectric material parameters). Knowing the most relevant parameters leads potentially to a reduced and more manageable dimension of the optimization problem. One established way of expressing sensitivity quantitatively in the form of a measure is to calculate Sobol indices (Prieur 2016). Their applicability to the non-smooth, resonant behavior of above mentioned eigenmodes upon variation of the piezoelectric material parameters is examined in this contribution.
The calculation of Sobol indices is a statistical method, which means that many randomly generated model values are required for a meaningful calculation. These are taken from a Latin hypercube sampling (LHS) of independently varied piezoelectric material parameters. For each such sampling point a frequency-domain study is performed, resulting in an electrical impedance curve rather than a single data point (see Fig. 1). From this curve the resonance frequency is extracted, by means of which the Sobol indices of the selected piezoelectric material parameters are calculated. It is shown that the Sobol index highest in magnitude, which ought to represent the material parameter with the largest sensitivity, agrees with the one known from literature to be most influential for the eigenmode properties of the considered block-shaped sample (Rupitsch 2015). All the above steps cannot – to our knowledge – be implemented as a whole in COMSOL Multiphysics®. Therefore, we choose a hybrid computational approach by means of the MATLAB® LiveLink™ feature. The LHS process, the extraction of the resonance frequencies, and the calculation of the Sobol indices is performed in MATLAB®. COMSOL Multiphysics® is used for the computation of the electrical impedance of the considered block-shaped piezoceramic test specimen. To this end, a frequency domain study is set up, combined with a parametric sweep with the LHS parameter combinations as input. As there a many thousands of samples needed, a batch sweep is employed to speed up the computation by parallel model execution.