To increase the robustness and functionality of piezoceramic ultrasonic sensors, e.g. for flow, material concentration or non-destructive testing, their development is often supported by computer simulations. The results of such finite-element-based simulations are dependent on correct simulation parameters, especially the material data set of the modelled piezoceramic. In recent years several well-known methods for estimation of such parameters have been developed that require knowledge of the sensitivity of a measured behaviour of the material with respect to the parameter set. One such measurable quantity is the electrical impedance of the ceramic. Previous studies for radially symmetric sensors with holohedral electrode setups have shown that the impedance shows little or no sensitivity to certain parameters and simulations reflect this behaviour making parameter estimation difficult. In this paper we have used simulations with special ring-shaped electrode geometry and non-uniform electrical excitation in order to find electrode geometries, with which the computed impedance displays a higher sensitivity to the changes in the parameter set. We find that many such electrode geometries exist in simulations and formulate an optimisation problem to find the local maxima of the sensitivities. Such configurations can be used to conduct experiments and solve the parameter estimation problem more efficiently.

Piezoelectric effect is the physical phenomenon discovered by Pierre and Jacques Curie in 1880 that is exhibited by several crystalline and synthetic ceramic materials. When voltage is applied across certain surfaces of the solid, it exhibits mechanical strain; conversely, when mechanical stresses are applied, voltage is produced between its surfaces.

Acoustic transducers are required to construct an ultrasonic measurement device of any kind. Transducers based on circular piezoelectric ceramic disks have been manufactured and used in a wide variety of applications.

Ceramics are composed of a large number of randomly oriented crystals. The
cumulative properties of each of these crystals together determine the
properties of the whole solid. This means that each batch of the ceramic
produced will have somewhat different material properties. Additionally, the
material coefficients are dependent on geometry. If a such a ceramic disk
with a specific thickness and diameter is sintered and polarised, the
resulting material coefficients are different than a disk of the same
material and of different thickness and diameter. Hence, post processing and
treatment of the material also influences the material coefficients.

Depending on the application at hand, the full knowledge of the material
data set can be crucial to the whole process: for a circular disk with full
width electrodes on the top and bottom (see Fig.

Analytical and numerical modelling of piezoelectricity and the resulting
computer simulations

Such numerical modelling requires the precise knowledge of the parameters of
the material under analysis. To this end, new methods that utilise the
mathematical concept of inverse problems to estimate the material parameters
have been designed

Previous studies

Two-electrode configuration on a circular disk

In order to improve the sensitivity of the measured impedance characteristic
to the material parameters, we designed a three-electrode configuration with
electrodes of various radii as shown in Fig.

Possible applications with non-uniform electrical excitation in piezoelectric
ceramics are interdigital transducers

In the following sections we first give a short overview of the equations governing our simulation and the excitation of the ceramic. We then define and analyse the sensitivity of the computed impedance to parameters. Finally, we formulate an optimisation problem to find a locally optimal electrode geometry that maximises the sensitivity and show that many such local optima occur.

In this paper we restrict ourselves to linear piezoelectric effect

We shall also restrict ourselves to thin circular disks.
Therefore, the use of a cylindrical coordinate system is appropriate.
We shall also assume that the configuration is rotationally
symmetric. Using the notation of

The above material equation can be extended to a full set of partial
differential equations in time and space using Newton's, Gauss' and
Faraday's Laws

Newton's law of motion

Neglecting the insignificant changes in magnetic field in this case,
Faraday's law states that the electric field is the negative gradient
of the electric potential.

Additionally, we consider a Rayleigh damping model with positive
constant

The above equations may also be transformed for harmonic analysis into
the frequency domain.
Application of a
Fourier transform changes the unknowns

The resulting time harmonic partial differential equations are given as
follows:

This transformation however contains a systematic error as

Weak formulations and finite-element methods for the solution of the
above system have been studied in

In order to simulate the behaviour of a piezoelectric transducer with
finite elements, it is excited by a delta function of the charge or a
pulse of potential. This gives the boundary conditions to solve
Eqs. (

Electrical current flow at the electrode is the time derivative of the
free charge and the impedance between any two electrodes is then
defined as the quotient of the potential difference and the
current. In the simple case of two electrodes, with a delta function
pulse of charge into one electrode with amplitude

Sensitivity of the impedance for the geometry of the ring
electrodes shown in Fig.

In our setup two different electric potentials are applied on two electrodes
and the third is grounded. An external circuit (Fig.

For each frequency of interest in the domain, the solution of
Eqs. (

For the numerical solution we use the commercial software package CAPA, which computes the impedance for a given configuration. CAPA is a simulation tool for the numerical solution of electromechanical, coupled field problems and is therefore suitable for the analysis of most mechatronic sensors and actuators such as electromagnetic loudspeakers or piezoelectric transducers.

Besides transient analysis, which is used in this contribution, the harmonic behaviour of the piezoceramic disc can also be calculated. One disadvantage of the transient simulation method is the sole use of the Rayleigh damping model for the energy dissipation processes as mentioned in Sect. 2. This is a rather simple approximation of the damping behaviour in practice. Another disadvantage of precompiled solver packages in general is inflexibility; it is impossible to modify the computation in any way in order to be able to compute more information, like sensitivities.

In contrast to previous studies we look at the real and imaginary
parts of the complex impedance separately instead of looking at
the magnitude. The formulation in terms of magnitude and phase,
although computationally efficient, hides some geometrical
structure, which is apparent when looking at the real and imaginary
components separately. This is analogous to a polar coordinate system versus
a cartesian coordinate system. Figure

Parametrisation of the ring radii compared with Fig.

Sensitivity of the impedance for the resulting geometry of the ring
electrodes shown in Table 1 to various material parameters against frequency
w.r.t.

We used finite-difference approximations for the configuration in
Fig.

Using a slightly different geometry (see resulting geometry in
Table

Uniform weights: improvement of sensitivity of impedance w.r.t material parameters using

Binary weights: improvement of sensitivity of impedance w.r.t. material parameters using

We start by formulating a constrained minimisation problem with the electrode radii as the variables which will then be solved.

The electrode configuration is parametrised using four ring radii

Now, we wish to maximise the sensitivities

We have noticed that some parameters are very reactive towards change in
geometry and others are not. Due to unevenly distributed sensitivity of the
material parameters

The scaling is only used inside the optimiser so that all components of
the gradient have a similar scale and has no meaning outside the optimisation routine.
For comparison of the results one needs to compare
the unscaled objective function with

In the production process of these piezoceramics, a laser cuts out the
electrode rings from a metal plate lying on top of the ceramic. The laser has
a width of 0.3 mm; hence, some restrictions to the geometry of the electrodes
apply. The minimal distance of two adjacent electrode rings is at least
0.3 mm; also, all radii must be positive and smaller than the constant outer
ring radius. These constraints are all linear in the radii and can be
reformulated into a vector inequality

Hence, the resulting minimisation problem is stated as

Since the sensitivity itself is a finite-difference approximation, it
makes little sense to use an optimisation method that requires further
derivatives.
For the optimisation we used Powell's latest
derivative-free trust region optimiser LINCOA (LINearly Constrained Optimization
Algorithm)

The LINCOA method is a derivative-free optimisation algorithm for linearly constrained problems written in Fortran by M.J.D. Powell. It is based on Powell`s other derivative-free optimisation algorithms with the distinction of incorporating general linear constraints. However, a detailed description of the LINCOA software has not been published yet.

According to

Mixed weights: improvement of sensitivity of impedance w.r.t. material parameters using

In the following section we will present results of the sensitivity
optimisation. The section is divided into two parts. In the first part we
will concentrate on the influence of the weight matrix

Different starting configurations with uniform weight

Evaluation history for uniform weights, binary weights, and mixed weights. Initial values are 3.7283, 145.45821 and 7.2703; optimised values are 0.6219, 84.3588 and 1.2397, respectively.

Reference initial geometry in millimetres;

As an initial point for a first optimisation we used the electrode
configuration of a piezoceramic (see Fig.

As a reaction to this slight partial decrease in sensitivity regarding

These two resulting geometries combined demonstrate the feasibility of optimising the sensitivity with regard to all parameters, i.e. the two geometries show a combined increase in sensitivity for all parameters.

We have experimented with different mixed weights (see Table

To examine the influence of different initial electrode configurations to the optimisation process we have chosen a wide range of barely feasible configurations (see Tables 4a–c). All these are optimised using uniform weights and the resulting optimal is compared to the resulting optimal when starting from the reference initial point above with uniform weights. The resulting geometries are all further away from infeasibility than their initial configuration and locally optimal. These three cases demonstrate that the problem of identifying a single globally optimal geometry is hard, since there are many locally optimal configurations.

In contrast to

In order to systematically search for the maximal sensitivity in such a configuration we need to solve an optimisation problem with the configuration radii as variables. Due to the inflexibility of the precompiled solver we were forced to use a derivative-free optimisation algorithm. More efficient algorithms may be used if a more flexible finite-element code with the possibility of influencing the internal computations were available, so that one could compute derivatives simultaneously. However the results show clearly that there are many locally optimal configurations.

We are investigating the possibility of better sensitivity analysis by
utilising the simulation software CFS++

Apart from discrete sensitivities and adjoints, it is also possible to formulate the sensitivity and adjoint equations for the model in function spaces and discretise these along with the primal equations and solving them. Another avenue for future development is to formulate the sensitivity maximisation problem using shape and topology calculus instead of parametrised rings. This would help generalise the configuration of electrodes to ceramic geometries that are not radially symmetric.

Part of the research presented here was done under the financial grant of the Research Prize 2012 awarded by the University of Paderborn to Kshitij Kulshreshtha and Jens Rautenberg.

The authors are thankful to M. Kaltenbacher for his support and making the simulation software CFS++ available for future development. Edited by: B. Jakoby Reviewed by: two anonymous referees