## 2 Numerical Analysis
### Determination of the Relieved Displacements
However, a more versatile approach can be obtained by replacing the summation with an integral of any continuous stress distribution. Thus, for each depth position there is a function \(f_{Us}(z)\), here called the _influence function_, which resembles the unitary weight \(\overline{U}_{si}\) for an infinitesimal depth increment \(\mathrm{d}z\) (see Figure 3b). Examples of the concept of the influence functions (sometimes called _weight functions_, such as in fracture mechanics) can be found in [42; 43] in the context of the HDM with strain pages. In principle, given the integral definition of such functions, the value at a certain depth \(z_{0}\) can be found as the integral with a Dirac delta function which is singular at \(z_{0}\), as shown in Figure 3c, in which the units are recalled. The finite element (FE) technique offers a direct way to apply a Dirac delta function, as the load can be concentrated in single points (which are the nodes), while the values of these functions can be alternatively inferred from their cumulative results, as proposed by [PERSON] et al. [42]. However, this requires the introduction of a model, such as a polynomial function, for each of them. In the present application, the use of a polynomial function would be complicated compared to the classical HDM with strain gage rosette due to the number of IFs being high. In fact, each point of the surface mesh grid starting from the hole rim up to a certain distance requires its own IFs in order to cover a large enough region around the hole. Figure 2: Details of the axisymmetric harmonic FE model: **(a)** identification of the modelled plane and definition of the hole radius, **(b)** overall view of the FE model and dimensions, **(c)** number of nodes and elements, and element sizes, **(d)** mesh transition method. The plane stress components at the surface of the specimen or the part to be investigated are \(\sigma_{x},\sigma_{y},\tau_{xy}\). It is well known that for the HDM the use of alternative stress components is preferential, according to the following:  
\[ \begin{split}& P=\frac{\sigma_{y}+\sigma_{x}}{2}\\ & Q=\frac{\sigma_{y}-\sigma_{x}}{2}\\ & T=\tau_{xy}\end{split}  \tag{1}\]  
Here, the \(P\) load is the equibiaxial stress, while \(Q\) and \(T\) are the shear stresses at \(45^{\circ}\) and \(0^{\circ}\), respectively. Therefore, the \(Q\) and \(T\) loads are the same type of loading; for this reason, the FE simulation only requires \(P\) and one of the other two. Here, the \(Q\) load is reported in the initial Figure 1 and implemented in the following. The \(P\) load only generates a radial displacement, whereas the \(Q\) load implies both normal and shear tractions and in turn generates two components of displacements (the radial and the tangential ones; see Figure 4a and 4b, respectively). The use of point loadings according to the Dirac delta function can be easily performed for just the nodes along the depth; the displacements are recorded in terms of amplitude to be multiplied by either \(\cos(0\times\theta)=1\) for the \(P\) load or \(\cos(2\theta)\) or \(\sin(2\theta)\) for the radial and the tangential components related to the \(Q\) load. We used a resolution of 0.01 mm at the hole region for both the radial distribution of the IFs and the depth variable, as shown in Figure 4.  
In principle, the IFs are properly defined (i.e., not singular) for any point of the surface around the hole except for the rim point. When the Dirac delta function is applied at that point, the local displacement is singular. This problem is avoided by simply discarding the point at the very boundary of the hole. Thus, the first point, where the IFs are defined, is at \(r=1.01\). At this point, when \(z=0\) the Dirac delta function is close to but not exactly at that point; nevertheless, when the delta function is too close to the point under consideration, even when it is not superimposed, the solution is not expected to be extremely accurate, as the finite element introduces a local perturbation of the solution.