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In this thesis, several numerical approaches for the development of structural health monitoring (SHM) methodologies for engineering structures are described. In particular, the first boundary element models of three-dimensional piezoelectric smart structures are introduced. Comparing to the finite element method (FEM), the boundary element method (BEM) demonstrates higher numerical stability and requires less computational resources. Also, the dual boundary integral formulation provides a natural and efficient approach for replicating the targets of SHM techniques - material discontinuities. A boundary element formulation for the ultrasonic guided wave based damage detection strategy is firstly presented. The semi-analytical finite element model of piezoelectric patches is coupled with the boundary element model of substrates via the variables of the BEM. The first systematic approach for determining the number of Laplace terms to be used for an elastodynamic boundary element analysis is also introduced. The above-mentioned formulation is then transformed to the Fourier domain for simulating the electro-mechanical impedance (EMI) based damage detection strategy. The key to attaining accurate EMI signatures is the inclusion of appropriate damping effects. In addition to the detection of the damages in substrates, a partially debonded coupling condition between substrates and piezoelectric patches is derived for modelling the diagnosis of faulty transducers. The computational efficiency of the BEM is further enhanced by the implementation of high-order spectral elements. The difficulties associated with the applications of these elements in the BEM are among the key emphases. The accelerated BEM is used to reformat the models of the two damage detection strategies. The performances of the two strategies are more deeply investigated and understood. At the end of this thesis, a technique for the characterisation of cracks in plate structures is established. By utilising a two-stage approach, the long-existed difficulty of the simultaneous localisation and sizing of arbitrary cracks can be overcome. The technique is developed mathematically using analytic models and the FEM, and is extensively assessed by numerically simulated extreme scenarios. Throughout this thesis, physical experiments are heavily relied on for validation studies. A summary of the skills and the experiences, which the author has gained on experimental testing, is reported in this thesis for further reference.