## 4 Time Domain Spectral Finite Element Method (TDSFEM)
### Wave Propagation in 3D Elements  
3D elements, which should in fact be called solid elements, are useful in situations where complex fields of variables (displacement, deformation, electric potential, etc.) have to be modelled. In such cases, typically the three-dimensional theory of elasticity is applied. Thanks to TDSFEM modelling simplicity, it is relatively easy to derive it, especially for the analysis of elastic waves' propagation. The important disadvantage of 3D analysis is that there are more degrees of freedom, which in turn leads to more calculation time [82]. The successful application of TDSFEM to the novel formulation of a 3D spectral element for wave propagation in reinforced concrete has been described here [144]. The reinforcement has been modelled as the truss spectral element and has been embedded in the 3D solid spectral finite element. Both elements have an equal number of nodes in the reinforcement direction and have the same number of degrees of freedom per each node. It has been assumed that there exists kinematic consistency between nodal displacements of the truss element and the solid element. This simplifying assumption has been justified by a linear character of wave propagation; however, this makes it impossible to diagnose material nonlinearities such as, for instance, de-bonding between steel and concrete. The developed 3D spectral element has been applied for modelling of wave propagation in reinforced concrete subjected to mechanical degradation. Different degradation models have been considered to reflect the real behaviour of bended beams. It has been shown that the models incorporating the uniform reduction of the beam stiffness worked well at the early stage of the degradation. The model with the distributed reduction of the stiffness enables for reflecting the behaviour of wave propagation at the stage, when macro-cracks are being developed. This model supports the possibility of the use of various spatial functions to analyse different stages of degradation. The formation of physical macro-cracks of large sizes in the more advanced phase of the bending process requires appropriate representation in the numerical model in the form of rows of nodes with a significantly reduced value of the Young's modulus. The author declares that the results of the paper may be treated as pilot activities, requiring further work in this area. Another 3D solid spectral element has been derived with shape functions based on Lagrange interpolation and Gauss-Lobatto-Legendre points here [145]. This particular approach has been applied for displacement approximation suited for fundamental modes of Lamb waves as well as potential distribution in piezoelectric transducers. A reliable numerical model of a transducer improves the quality of numerical analysis of any structural health monitoring algorithm. The novelty of the proposed research is the model geometry extension from flat to curved elements for application in shell-like structures. Exemplary visualisations of waves excited by the piezoelectric transducers in curved shell structure made of aluminium alloy have been presented. Simple signal analysis of wave interaction with a crack is performed. The crack is modelled by separation of appropriate nodes between elements. An investigation of influence of the crack length on wave propagation signals is performed. Additionally, some aspects of the spectral element method implementation have been discussed. The importance of a reliable method of numerical modelling and simulations of PZT-induced Lamb wave propagation in plate-like structures has been also presented by [85]. The authors derive here a novel finite spectral plate element, which can efficiently model the three-dimensional behaviour of Lamb waves. In the formulation, linear displacement distributions in the thickness direction have been assumed for both the PZT layer and the base plate. A method to avoid the thickness locking has been proposed and used in the formulation.