## 3 Overview of Modelling Issues for Energy Recovery Systems
The use of piezoelectric transducers is also being considered to recover the energy associated with human gait, for example  
Figure 1: Diagram of a piezoelectric energy recovery device. using transducers mounted in shoes. In [23], the use of such a solution was shown to recover between 0.03 mW and 0.35 mW depending on gait speed. In [24], the achievements in piezoelectric energy recovery of recent years were collected and compared. It was noted that the piezoelectric properties of inorganic piezoelectric materials remain superior to organic materials. However, organic materials are still being developed due to their better mechanical properties, mainly less brittleness and lack of toxicity. The piezoelectric properties of materials of natural origin, such as onion skin or membranes made from the swim bladder of fish, were also pointed out as potential applications in bioengineering. In [25], advances related to piezoelectric energy recovery were compared when dividing materials into ceramic, lead-free, polymer and composite materials. Studies on nanopiezoelectric, ferroelectric, non-ferroelectric and composite structures were also collected. Possible applications of different shapes of piezoelectric elements were also compared: cantilevered, disc-shaped, and stacked. In [26], an analysis of piezoelectric transducers that can be used to power electronics with very low power consumption based on vibration, such as some portable devices or wireless sensor networks, was carried out. Also described in [27], the feasibility of using different types of piezoelectric transducers to recover energy from a traffic-laden asphalt road was assessed. The finite element method was used for the evaluation and the feasibility of using different types of transducers was tested. It was determined that the direct application of standard transducers is limited because they are not suitable for use on asphalt roads, so a prototype transducer was developed for use under these conditions. As a result of optimisation, a PZT stack was selected as the transducer material due to its circular shape. Research related to piezoelectric energy recovery often concerns the optimisation of material or geometrical parameters of the transducer. For example, in [28], the influence of the periodically varying cross-section of an oscillating beam was investigated based on a mathematical model and results from FEM analysis in ANSYS 16.0. The results of both methods were found to be in good agreement. The performance of a beam with a periodically varying cross-section was then compared with that of a beam with a constant cross-section. It was determined that the periodically varying cross-section solution provided greater recovered energy at a wider range of vibration frequencies, the system absorbed vibration energy better and had less mass. In [29], an attempt was made to optimise a structure consisting of multiple transducers in order to increase the maximum possible power achieved (greater than 1 W). For this purpose, a computational simulation was prepared, from which it was concluded that the optimal way to combine the transducers was to use a rectifier circuit before connecting them in parallel. A physical model was then created consisting of 102 transducers connected in this way, which were loaded with the impacts of rods mounted on a rotary drive. The circuit thus constructed allowed a power recovery of 1.99 W at the optimum load, the frequency of which, however, proved to be inversely proportional to the number of transducers. In [30], an optimisation of the topology of piezoelectric transducers was performed, taking into account both the properties of piezoelectric materials and deformable elastic materials. The optimisation was carried out using the finite element method. Based on the calculations at different loads on the beam, the improvement in the energy conversion factor was determined. In bending, the improvement was 34.4 per cent and in tension 11.56 per cent.