**2. Materials and Methods**

In this section, the main details regarding the numerical modelling tools and choices that were made are given. Firstly, the numerical methods used to estimate the uncertainty associated to blade damage are introduced. The hypotheses regarding the input uncertainties are then explained. Finally, in the following subsections the numerical tools used in the required deterministic model evaluations are detailed.

## *2.1. Stochastic Approach*

The stochastic approach exploited in the present work falls into the class of the arbitrary polynomial chaos (aPC) as implemented by Oladyshkin and Novak [13]. This technique falls into the field of the study of aleatory uncertainty, which only accounts for deviations of boundary condition and geometrical parameters. The present approach does not include the contribution of the limits of the numerical approach adopted. The deviation or the effect of such limitation have been considered as negligible. CFD has been validated and run according to best practices, including grid independence study. This approach has the advantage of providing stochastic results (or PDFs) without the need to change the algorithm of the numerical tools employed in the simulations. These kinds of approaches are generally known as non-invasive methods as reviewed by Iaccarino [14] and more detailed in Carnevale [15] and Ahfield [16]. The PDF of a specific quantity of interest is extracted by reproducing a surface response obtained by a certain number of simulations (or deterministic realization). The boundary conditions for these simulations are set to reproduce the PDF representing the aleatory parameter. The process of selecting appropriate boundary conditions is known as sampling. The sampling process is usually obtained by means of selecting the boundary condition using the Monte Carlo method filtered by the proper PDF. The approach as described implies a large number of simulations and it is not reliable for application where CFD solvers are used for each single deterministic prediction. This would require a high computational cost to complete the simulation campaign.

A strategy to overcome this limitation consists of a clever choice of the boundary conditions resulting in a limited number of simulations. The convolution of this boundary conditions is representative of a specific PDF. This approach is known in literature as the probabilistic collocation point (PCM). The PCM are obtained as quadrature points of a linear system built on the basis consisting in a set of polynomials (polynomial chaos, PC). The choice of these polynomials corresponds to make a strong assumption on how the response surface is determined. The surface response will be as the weighted functions corresponding to a specific PDF. Mathematic foundations can be found in Tatang et al. [17]. This particular approach has been successfully applied to CFD simulations in Carnevale et al. [15,18] and Salvadori et al. [19]. The particular approach proposed allows weaker hypothesis to be considered on the PDF of the aleatory parameter. The aPC only demands the existence of a finite number of moments and does not require the complete knowledge or even the existence of a probability density function. This approach has also been employed in Ahlfield et al. [16], where the stochastic behavior physical parameters are characterized by discontinuity and Gibbs phenomena. The aPC extends chaos expansion techniques by employing a global polynomial basis.

Let's consider a generic aleatory variable ξ propagating on a specific output of interest *Y* = *f*(ξ), where *f* is a general unknown stochastic model (or PDF); it can be expressed as a *d*-order expansion:

$$Y(\xi) \approx \sum\_{i=1}^{d} c\_i P^{(i)}(\xi) \tag{1}$$

According to the general theory of PCM the characteristic statistical quantities of *Y*(ξ) can be evaluated by the coefficient *ci*, and the momentum and variance are expressed as follows:

$$
\mu\_Y = c\_{1\prime} \ \sigma\_Y^2 = \sum\_{i=1}^d c\_i \tag{2}
$$

The peculiarity of the aPC approach is related to the strategy adopted to determine the orthonormal basis of polynomial *P*(*i*). These polynomials have been determined by the moment-based approach detailed in Oladyshkin et al. [13]. Once the aPC, which represents an orthonormal basis, has been identified, the collocation points are obtained by means of a quadrature procedure. Given an aleatory variable ξ ± σ associate with a PDF *f*(ξ), the more general expression of its quadrature is

$$\int\_{-\sigma}^{\sigma} \chi(\xi) f(\xi) d\xi = \sum\_{k=0}^{d} \omega(\xi\_k) P(\xi\_k) + R\_M(Y) \tag{3}$$

In the previous equation, the left-hand side is the stochastic representation of the aleatory variable ξ associated with the PDF *f*(ξ). The right-hand side is its expansion on the basis *P*(ξ*k*), where the ω(ξ*k*) is the weighting term (in this context we can consider ω(ξ*k*) = 1), *RM*(*Y*) is the remainder approaching zero as *d*-order of the expansion increases and the collocation points ξ*<sup>k</sup>* are such that the formula σ <sup>−</sup><sup>σ</sup> *<sup>Y</sup>*(ξ)*f*(ξ)*d*<sup>ξ</sup> <sup>−</sup> *d <sup>k</sup>*=<sup>0</sup> ω(ξ*k*)*P*(ξ*k*) = 0 is satisfied for the moment μ(ξ) and the μ(ξ) ± σ.
